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0 and λ ∈ (λ1 , λV ) such that for a.a. z ∈ Z and all |x| ≤ δ, we have λ1 |x|p ≤ pF (z, x) ≤ λ|x|p , x where F (z, x) = 0 f (z, r)dr (the potential function corresponding to f (z, x)); (v)
lim
|x|→∞
pF (z,x) |x|p
= λ1 and lim
most all z ∈ Z.
|x|→∞
pF (z, x) − λ1 |x|p ) = −∞ uniformly for al-
REMARK 5.1.20 Hypothesis H(f )6 (iv) implies that problem (5.1) is resonant near zero at the first eigenvalue λ1 > 0 from the right. On the other hand by virtue of hypothesis H(f )6 (v), problem (5.1) is resonant near infinity from the left. Therefore we are dealing with a problem which is resonant both near zero and near infinity and we cross λ1 as we move from 0 to infinity. The Euler functional ϕ : W01,p (Z) −→ R is given by
1 F z, x(z) dz. ϕ(x) = Dxpp − p Z
1,p 1 We know that ϕ ∈ C W0 (Z) . PROPOSITION 5.1.21 If hypotheses H(f )6 hold, then ϕ is weakly coercive; that is, ϕ(x) −→ +∞ as x −→ +∞.
5.1 Variational Method
373
PROOF: We proceed by contradiction. So suppose that ϕ is not weakly coercive. Then we can find a sequence {xn }n≥1 ⊆ W01,p (Z) and M > 0 such that xn −→ ∞
and
ϕ(xn ) ≤
for all n ≥ 1.
(5.52)
Set yn = xn /xn , n ≥ 1. We may assume that w
yn −→ y in W01,p (Z)
and
yn −→ y in Lp (Z).
From (5.52) we have 1 Dyn pp − p
Z
F z, xn (z) M dz ≤ xn p xn p
for all n ≥ 1.
(5.53)
From hypotheses H(f )6 (iii), (v), and the mean value theorem, we have |F (z, x)| ≤ α1 (z) + c1 |x|p for a.a. z ∈ Z, with α1 ∈ L∞ (Z)+ , c1 > 0,
F z, xn (z) α1 (z) ⇒ ≤ + c1 |yn (z)|p a.e. on Z for all n ≥ 1. (5.54) xn p xn p Because of (5.54) and the Dunford–Pettis theorem, we may assume that
F ·, xn (·) w −→ h in L1 (Z) as n → ∞. xn p
(5.55)
For every ε > 0 and every n ≥ 1, we consider the set
F z, xn (z) 1 1 Cε,n = z ∈ Z : xn (z) = 0, (λ1 − ε) ≤ ≤ (λ1 + ε) . p p |xn (z)| p Due to hypothesis H(f )6 (v), we have χCε,n (z) −→ 1 Note that
a.e. on {y = 0}.
(1 − χCε,n ) F ·, xn (·) 1 −→ 0 as n → ∞, p L ({y=0}) xn
F ·, xn (·) w −→ h in L1 ({y = 0}) (see (5.55)). ⇒ χCε,n (·) xn p
From the definition of the set Cε,n , we have
F z, xn (z) F z, xn (z) 1 = χ (z) |yn (z)|p (λ1 − ε)|yn (z)|p ≤ χCε,n (z) C ε,n p xn p |xn (z)|p 1 ≤ (λ1 + ε)|yn (z)|p a.e. on Z. p Taking weak limits in L1 ({y = 0}) and using Mazur’s lemma, we obtain 1 1 (λ1 − ε)|yn (z)|p ≤ h(z) ≤ (λ1 + ε)|yn (z)|p p p Because ε > 0 was arbitrary, it follows that
a.e. on {y = 0}.
374
5 Boundary Value Problems–Hamiltonian Systems h(z) =
λ1 |y(z)|p p
a.e. on {y = 0}.
(5.56)
On the other hand it is clear from (5.54) that h(z) = 0
a.e. on {y = 0}.
(5.57)
From (5.56) and (5.57) it follows that h(z) =
λ1 |y(z)|p p
a.e. on Z.
(5.58)
We return to (5.53) and pass to the limit as n → ∞. Using (5.55) and (5.58), we obtain Dypp ≤ λ1 ypp , ⇒ y = 0 or y = ±u1
(see Remark 4.3.43).
If y = 0, then we have yn −→ 0 in W01,p (Z), a contradiction to the fact that yn = 1, n ≥ 1. If y = ±u1 , then |xn (z)| −→ +∞ for a.a. z ∈ Z (see Theorem 4.3.47). By virtue of hypothesis H(f )6 (v), we have
pF z, xn (z) − λ1 |xn (z)|p −→ −∞ a.e. on Z as n → ∞. (5.59) We have
λ1
1 λ1 F z, xn (z) − Dxn pp − |xn (z)|p dz − xn pp p p p Z
λ1
≥− F z, xn (z) − |xn (z)|p dz (see Remark 4.3.43). p Z
ϕ(xn ) =
Using Fatou’s lemma and (5.59), we obtain ϕ(xn ) −→ +∞
as n → ∞,
which contradicts (5.52). This proves the weak coercivity of ϕ.
COROLLARY 5.1.22 If hypotheses H(f )6 hold, then ϕ satisfies the P S-condition and it is bounded below. PROOF: Let {xn }n≥1 ⊆ W01,p (Z) be a P S-sequence, namely {ϕ(xn )}n≥1 ⊆ R is
bounded and ϕ (xn ) −→ 0 in W −1,p (Z) as n → ∞ p1 + p1 = 1 . From Proposition 5.1.21 we know that ϕ is weakly coercive. So it follows that {xn }n≥1 ⊆ W01,p (Z) is bounded. Then arguing as in the last part of Proposition 5.1.2, we obtain xn −→ x in W01,p (Z). This proves that ϕ satisfies the P S-condition. Next suppose that {xn }n≥1 ⊆ W01,p (Z) satisfies ϕ(xn ) −→ −∞
as n → ∞.
(5.60)
Proposition 5.1.21 implies that {xn }n≥1 ⊆ W01,p (Z) is bounded and so we may w assume that xn −→ x in W01,p (Z). Exploiting the compact embedding of W01,p (Z)
5.1 Variational Method
375
into Lp (Z), we can easily see that ϕ is weakly lower semicontinuous on W01,p (Z). So we have −∞ < ϕ(x) ≤ lim inf ϕ(xn ), n→∞
which contradicts (5.60). This proves that ϕ is bounded below.
Now we can have a multiplicity result for problem (5.1). THEOREM 5.1.23 If hypotheses H(f )6 hold, then problem (5.1) has at least two nontrivial solutions x0 , x1 ∈ C01 (Z). PROOF: Recall the direct sum decomposition W01,p (Z) = Ru1 ⊕ V with V = u ∈ W01,p (Z) : Z u(z)u1 (z)p−1 dz . We know that u1 ∈ C01 (Z) (see Proposition 4.3.39). So we can find t0 > 0 such that |tu1 (z)| ≤ δ for all z ∈ Z and all |t| ≤ t0 . Then hypothesis H(f )6 (iv) implies that
λ1 |t|p u1 (z)p ≤ p F z, tu1 (z) ≤ λ|t|p u1 (z)p
a.e. on Z, with λ1 < λ < λV . (5.61)
So for |t| ≤ t0 , we have
|t|p F z, tu1 (z) dz Du1 pp − p Z |t|p |t|p p ≤ Du1 p − λ1 u1 pp (see (5.61)) p p =0 (recall that Du1 pp = λ1 u1 pp , see Remark 4.3.43).
ϕ(tu1 ) =
(5.62) Note that for a.a. z ∈ Z and all x ∈ R, we have F (z, x) ≤
λ p x + c2 |x|r p
with c2 > 0 and p < r < p∗ .
(5.63)
So if u ∈ V , we have
1 F z, u(z) dz Dupp − p Z 1 λ p ≥ Dup − upp − c2 urr (see (5.63)), p p 1 λ ≥ for some c3 > 0 Dupp − c3 Durr 1− p λV
ϕ(u) =
(5.64)
(see (5.51) and recall r < p∗ ). Recall that λ < λV and r > p. So we can find > 0 small such that ϕ(u) ≥ 0
for all u ∈ V, u ≤
(see (5.64)).
(5.65)
376
5 Boundary Value Problems–Hamiltonian Systems
On the other hand from (5.62), if > 0 is small, we have ϕ(u) ≤ 0
for all u ∈ Ru1 , with u ≤ .
(5.66)
If inf ϕ = 0, then because of (5.66) all the elements of Ru1 ∩ B (0) \ {0} are critical points of ϕ and so we have a continuum of nontrivial solutions for problem (5.1), which by virtue of Theorems 4.3.34 and 4.3.35 belong in C01 (Z). If inf ϕ < 0, then because of (5.65), (5.66), and Corollary 5.1.22, we can apply Theorem 4.1.32 and produce two nontrivial critical points x0 , x1 ∈ W01,p (Z) of ϕ. These are solutions of (5.1) and as above the nonlinear regularity theory implies that x0 , x1 ∈ C01 (Z). We prove a multiplicity theorem for semilinear problems (i.e. p = 2). In this multiplicity theorem, we combine the variational method with the method of monotone iterations. This way we are well placed to consider the method of upper-lower solutions, examined in the next section. The problem under consideration is the following. ⎧ ⎫ 2 ⎨ − x(z) = x(z) + h(z) a.e. on Z, ⎬ . ⎩ ⎭ x ∂Z = 0, h ∈ L∞ (Z)
(5.67)
First using the monotone iteration method, we produce a nontrivial solution for problem (5.67). PROPOSITION 5.1.24 If h ∈ L∞ (Z), h = 0 and h(z) ≤ 0 a.e. on Z, then problem (5.67) has a solution u ∈ −int C01 (Z)+ . PROOF: Let x0 ∈ H01 (Z) be the unique solution of − x0 (z) = h(z) a.e. on Z. From Theorems 4.3.34 and 4.3.35 we have that x0 ∈ C01 (Z). Moreover, Theorem 4.3.37 implies that x0 ∈ −int C01 (Z)+ . We set M = 2 min x0 (z) > 0. z∈Z
Given y ∈ H01 (Z), we consider the problem ⎧ ⎫ 2 ⎨ − u(z) + M u(z) = M y(z) + y(z) + h(z) a.e. on Z, ⎬ ⎩
y ∂Z = 0
⎭
.
(5.68)
Problem (5.68) has a unique solution in C01 (Z) denoted by G(y). So we define a map G : H01 (Z) −→ H01 (Z). Evidently a fixed point of G solves problem (5.67). Claim: If y ∈ H01 (Z), x0 ≤ y ≤ 0, then x0 ≤ G(y) ≤ 0. We have M y + y 2 = y(M + y) ≤ 0. So from (5.68) and Theorem 4.3.37 it follows that u = G(y) ≤ 0. Then
5.1 Variational Method
377
− (u − x0 ) + M (u − x0 ) = M y + y 2 + h − h − M x0 = M (y − x0 ) + y 2 ≥ 0, ⇒ u ≥ x0
(see Theorem 4.3.37).
In a similar fashion, we show that G is increasing; that is, x0 ≤ y1 ≤ y2 ≤ 0 ⇒ G(y1 ) ≤ G(y2 ).
(5.69)
Now let u0 = 0 and define un+1 = G(un ), n ≥ 0. By virtue of the claim and (5.69), we have x0 ≤ · · · ≤ un ≤ un−1 ≤ · · · ≤ u0 = 0,
n ≥ 1.
(5.70)
We have |un | ≤ |x0 |
for all n ≥ 1,
⇒ {un }n≥1 ⊆ L2 (Z)
is bounded.
(5.71)
Moreover, by definition we have − un+1 + M un+1 = M un + u2n + h,
⇒ un+1 2 + M un+1 22 = M un un+1 dz + u2n un+1 dz + hun+1 dz Z
Z
Z
≤ (M un 2 + h2 )un+1 ≤ M1 un+1 for some M1 > 0, all n ≥ 1 (see (5.71)), ⇒ un n≥1 ⊆ H01 (Z) is bounded. From this and (5.70), it follows that for the whole sequence {un }n≥1 , we have
w un −→ u in H01 (Z) with u(z)=inf un (z). If A∈L H01 (Z), H −1 (Z) , is defined by n≥1
A(v), y= Z
(Dv, Dy)RN dz
for all v, y ∈ H01 (Z),
then A is maximal monotone and Aun+1 + M un+1 = M un + u2n + h,
(5.72)
⇒ lim Aun+1 , un+1 − u = 0, ⇒ un+1 −→ u in H01 (Z). So if we pass to the limit as n → ∞ in (5.72), we obtain Au = u2 + h, ⇒ u solves problem (5.67) and u ∈ C01 (Z). Finally, it is clear that u =0 and from Theorem 4.3.37 we have u∈−int C01 (Z). If h ≥ 0, h = 0, it can happen that problem (5.67) has no solution.
378
5 Boundary Value Problems–Hamiltonian Systems
EXAMPLE 5.1.25 Let u1 ∈ C01 (Z) be the principal eigenfunction of − , H01 (Z) . We know that u1 (z) > 0 for all z ∈ Z. Let h(z) = tu1 (z) and t > 0 to be specified in the sequel. Suppose that we could find u ∈ H01 (Z) a solution of (5.67) with h as above. We have
uu1 dz= (Du, Du1 )RN dz = u2 u1 dz + hu1 dz λ1 Z Z Z
Z = u2 u1 dz + t (because u1 2 = 1). Z
(5.73) Note that
√ √ (u u1 ) u1 dz Z 1/2 1/2 ≤ u1 dz u2 u1 dz .
uu1 dz = Z
Z
Z
Using the estimate in (5.73), we obtain
1/2 1/2 u2 u1 dz + t ≤ λ1 c u2 u1 dz with c = u1 dz = u1 1 . Z
Z
Z
But this is possible only if 4t ≤ λ21 c2 . So if t > h = tu1 has no solution.
λ1 c 2
2
, then problem (5.67) with
Now using the variational method we produce a second solution for problem (5.67). So we have the following multiplicity theorem. THEOREM 5.1.26 If h ∈ L∞ (Z), h = 0, and h(z) ≤ 0 a.e. on Z, then problem (5.67) has at least two nontrivial solutions u, v ∈ C01 (Z). PROOF: The first solution u ∈ C01 (Z) was obtained in Proposition 5.1.24. Next using the mountain pass theorem (see Theorem 4.1.24), we produce a second nontrivial solution for problem (5.67). Let y = u + εw, w ∈ H01 (Z), w = 1, ε > 0. We have
ε2 u2 wdz ϕ(u + εw) − ϕ(u) = ε (Du, Dw)RN dz + Dw22 − ε 2 Z Z
ε3 −ε uw2 dz − w3 dz − ε hwdz 3 Z Z Z
ε2 ε3 = uw2 dz − w3 dz, (5.74) Dw22 − ε 2 3 Z Z the second equality following from the fact that u ∈ C01 (Z) solves (5.67). Because u ∈ −int C01 (Z)+ (see Proposition 5.1.24), we have Z uw2 dz ≤ 0. Also Z w2 dz ≤ c1 for some c1 > 0 and all w = 1. Therefore ϕ(u + εw) − ϕ(u) ≥ ε2
1 2
−
c1 ε 3
(see (5.74)).
(5.75)
5.1 Variational Method
379
Choosing 0 < ε < 34 c1 , from (5.75) we have ϕ(y) − ϕ(u) ≥
ε2 4
for all y ∈ H01 (Z) with y − u = ε.
(5.76)
Next let y = tu1 . Then
t2 t3 hu1 dz, λ1 − u1 33 − t 2 3 Z ⇒ ϕ(y) −→ −∞ as t −→ +∞. ϕ(y) =
Therefore we can find y ∈ H01 (Z) such that ϕ(y) < ϕ(u)
and
y − u > ε.
(5.77)
Claim: ϕ satisfies the P S-condition. Suppose {xn }n≥1 is a P S-sequence; that is,
and
|ϕ(xn )| ≤ c2
for some c2 > 0 and all n ≥ 1
(5.78)
ϕ (xn ) −→ 0
in H −1 (Z).
(5.79)
Because of (5.79), we have
ϕ (xn ), xn = Dxn 22 − x3n dz − hxn dz ≤ εn xn with εn ↓ 0, Z
Z 1 2 ⇒ 3ϕ(xn ) − Dxn 2 + 2 hxn dz ≤ εn xn (see (5.78)). 2 Z
From Poincar´e’s inequality and (5.78), it follows that {xn }n≥1 ⊆ H01 (Z) is bounded. So we may assume that w
xn −→ x in H01 (Z)
xn −→ x in L2 (Z).
Then as before, due to the maximal monotonicity of A ∈ L H01 (Z), H −1 (Z) , we conclude that xn −→ x in H01 (Z) and so ϕ satisfies the P S-condition. Because of (5.76), (5.77), and the claim, we can apply the mountain pass theorem (see Theorem 4.1.24) and obtain v ∈ H01 (Z) such that ϕ(v) ≥ ϕ(u) +
and
ε2 , i.e. v = w, v = 0 4
and ϕ (v) = 0. So v solves problem (5.67) and also v ∈ C01 (Z) (regularity theory).
We conclude this section with a remarkable theorem, which provides a bridge between variational methods and methods based on the nonlinear strong maximum principle. The result is a valuable tool in the study of nonlinear boundary value problems involving the p-Laplacian. For a proof of it we refer to Gasi´ nski–Papageorgiou [258, p. 655]. So let f : Z × R −→ R be a Carath´eodory function such that |f (z, x)| ≤ α(z) + c|x|r−1
for a.a. z ∈ Z and all x ∈ R,
380
5 Boundary Value Problems–Hamiltonian Systems
with α ∈ L∞ (Z)+ , c > 0 and 1 ≤ r < p∗ . We introduce the C 1 -functional ϕ : W01,p (Z) −→ R defined by
1 F z, x(z) dz for all x ∈ W01,p (Z), ϕ(x) = Dxpp − p Z x where F (z, x) = 0 f (z, r)dr. THEOREM 5.1.27 If ϕ ∈ C 1 (W01,p (Z)) is as above and x0 ∈ W01,p (Z) is a local C01 (Z)-minimizer of ϕ; that is, there exists > 0 such that ϕ(x0 ) ≤ ϕ(x0 + y)
for all yC 1 (Z) ≤ , 0
then x0 ∈C01 (Z) and it is also a local W01,p (Z)-minimizer of ϕ; that is, there exists 0 > 0 such that ϕ(x0 ) ≤ ϕ(x0 + y) for all y ≤ 0 .
5.2 Method of Upper–Lower Solutions In this section we present the method of upper and lower solutions in the study of nonlinear boundary value problems. With this method, via truncation and penalization techniques, we produce solutions that are located in the order interval determined by an ordered pair of upper and lower solutions, which serve as upper and lower bounds, respectively. Under appropriate monotonicity conditions on the data, this method leads to monotone iterative processes that are amenable to numerical treatment. For a given problem one may try several different methods in order to produce the necessary ordered pair of upper and lower solutions. There is no specific methodology and in general one should try simple functions (such as constants, linear, quadratic, exponential, eigenfunctions of simpler operators, solutions of simpler auxiliary equations, etc.). Here we illustrate this method by analyzing two nonlinear boundary value problems. The first deals with elliptic differential equations, whereas the second is a boundary value problem for ordinary differential equations with general boundary conditions which unify the Dirichlet, Neumann, and Sturm–Liouville problems. We start with a nonlinear eigenvalue problem driven by the p-Laplacian. So let Z ⊆ RN be a bounded domain with a C 2 -boundary ∂Z. We consider the following problem,
⎧ ⎫ p−2 Dx(z) = λ|x(z)|p−2 x(z) −f z, x(z) a.e. on Z, ⎬ ⎨ −div Dx(z) . (5.80) ⎩ ⎭ x ∂Z = 0, λ ∈ R, 1 < p < ∞ We produce a multiplicity result for problem (5.80) under the hypothesis that the right-hand side nonlinearity f (z, x) exhibits a p-superlinear growth both near 0 and ±∞. For this purpose our hypotheses on f (z, x) are the following. H(f )1 : f : Z × R −→ R is a function such that f (z, 0) = 0 a.e. on Z and
5.2 Method of Upper–Lower Solutions
381
(i) For all x ∈ R, z −→ f (z, x) is measurable. (ii) For almost all z ∈ Z, x −→ f (z, x) is continuous and f (z, x)x ≥ 0. (iii) For almost all z ∈ Z and all x ∈ R, we have |f (z, x)| ≤ α(z) + c|x|r−1 with α ∈ L∞ (Z)+ , c > 0 Np if N > p ∗ N −p . 1 ≤ r
0, we can find ϑµ ∈ L∞ (Z)+ , ϑµ = 0, such that
and
f (z, x) > µ|x|p−2 x − ϑµ
for a.a. z ∈ Z and all x ≥ 0
(5.81)
f (z, x) < µ|x|p−2 x + ϑµ
for a.a. z ∈ Z and all x ≤ 0.
(5.82)
We use (5.81) and (5.82) to produce an ordered pair of upper and lower solutions for problem (5.80). Let us start by recalling the definitions of upper and lower solutions for (5.80). 1,p DEFINITION 5.2.2 (a) A function x ∈ W (Z) is an upper solution for problem (5.80), if x ∂Z ≥ 0 and
Dxp−2 (Dx, Du)RN dz ≥ λ |x|p−2 xudz − f (z, x)udz Z
Z
Z
for all u ∈ u ≥ 0. (b) A function x ∈ W (Z) is a lower solution for problem (5.80), if x∂Z ≤ 0 and
Dxp−2 (Dx, Du)RN dz ≤ λ |x|p−2 xudz − f (z, x)udz W01,p (Z), 1,p
Z
for all u ∈
W01,p (Z),
Z
Z
u ≥ 0.
Motivated by inequality (5.81), we consider the following auxiliary boundary value problem.
⎧ ⎫ p−2 Dx(z) = (λ − µ)|x(z)|p−2 x(z) +ϑµ (z) a.e. on Z, ⎬ ⎨ −div Dx(z) . (5.83) ⎩ ⎭ x ∂Z = 0 PROPOSITION 5.2.3 If λ > λ1 and we choose µ > λ − λ1 , then problem (5.83) has a solution x ∈ int C01 (Z)+ .
382
5 Boundary Value Problems–Hamiltonian Systems
PROOF: In what follows by ·, · we denote the duality brackets for the pair
1,p W0 (Z), W −1,p (Z) , (1/p) + (1/p ) = 1. Let A : W01,p (Z) −→ W −1,p (Z) be the nonlinear operator defined by
A(x), y = Dxp−2 (Dx, Dy)RN dz for all x, y ∈ W01,p (Z). Z
We know that A is maximal monotone. Also let K : W01,p (Z) −→ Lp (Z) be defined by K(x)(·) = (λ − µ)|x(·)|p−2 x(·). Clearly K is completely continuous (recall that W01,p (Z) is embedded compactly into Lp (Z)). It follows then that A−K : W01,p (Z) −→ W −1,p (Z) is pseudomonotone. Moreover, we have A(x) − K(x), x = Dxpp − (λ − µ)xpp .
(5.84)
If λ ≤ µ, then it is clear from (5.84) that A − K is coercive. If λ > µ, then by hypothesis λ − µ = λ1 − ε with 0 < ε < λ1 and so A(x) − K(x), x ≥ Dxpp − ⇒ A − K is coercive.
λ1 − ε ε Dxpp = Dxpp , λ1 λ1
So in both cases A − K is coercive. Then by virtue of Theorem 3.2.60 we can find x ∈ W01,p (Z) such that A(x) − K(x) = ϑµ , ⇒ x ∈ W01,p (Z) is a solution of problem (5.83) and belongs in C01 (Z)+ . Because ϑµ = 0, it follows that x = 0. Moreover, using as a test function x − ∈ W01,p (Z), we obtain Dx − pp ≤ (λ − µ)x − pp
(recall that ϑµ ≥ 0).
If λ ≤ µ, then Dx − p = 0 and so we have x = 0. If λ − µ > 0, then λ − µ = λ1 − ε and so Dx − pp ≤ (λ1 − ε)x − pp , which contradicts Theorem 4.3.37, unless x ≥ 0. Therefore we have established that x ≥ 0, x = 0. Also from (5.83), we have
div Dx(z)p−2 Dx(z) ≤ |λ − µ| |x(z)|p−2 x(z) a.e. on Z and this by virtue of Theorem 4.3.37 implies that x ∈ int C01 (Z)+ .
COROLLARY 5.2.4 If λ > λ1 and we have µ > λ − λ1 , then the solution x ∈ int C01 (Z)+ of problem (5.83) produced in Proposition 5.2.3 is an upper solution for problem (5.80). Hypothesis H(f )1 (iv) implies that given ε > 0, we can find δ = δ(ε) > 0 such that and
f (z, x) < ε|x|p−2 x
for a.a. z ∈ Z and all x ∈ [0, δ]
(5.85)
f (z, x) > ε|x|p−2 x
for a.a. z ∈ Z and all x ∈ [−δ, 0].
(5.86)
The next lemma helps us generate a lower solution for problem (5.80).
5.2 Method of Upper–Lower Solutions
383
LEMMA 5.2.5 If X is an ordered Banach space with order cone K such that int K = ∅ and x0 ∈ int K, then given y ∈ X, we can find δ > 0 such that βx0 −y ∈ int K. PROOF: Because x0 ∈ int K, we can find δ > 0 such that B δ (x0 ) = x ∈ X : x − x0 ≤ δ ⊆ int K. Let y ∈ X, y = 0 (if y = 0, then the lemma is trivially true for all β > 0). We have y x0 ± δ ∈ B δ (x0 ) ⊆ int K, y y ⇒ x0 − y ∈ int K. δ So if β = β(y) = y/δ, then we have βx0 − y ∈ int K.
Now let u1 ∈ int C01 (Z)+ be the principal eigenfunction of − p , W01,p (Z) . Using Proposition 5.2.3 and Lemma 5.2.5, we can find ξ > 0 small such that ξu1 (z) ∈ [0, δ]
for all z ∈ Z and ξu1 ≤ x.
We set x = ξu1 . Evidently x ∈ int
C01 (Z)+
(5.87)
(of course x depends on ε > 0).
PROPOSITION 5.2.6 If λ > λ1 , then x ∈ int C01 (Z)+ defined above is a lower solution for problem (5.80). PROOF: Let ε > 0 be such that λ = λ1 + ε. Then we have
−div Dx(z)p−2 Dx(z) = λ1 |x(z)|p−2 x(z) = (λ − ε)|x(z)|p−2 x(z)
< λ|x(z)|p−2 x(z) − f z, x(z)
a.e. on Z
(see (5.85) and (5.87)), ⇒ x ∈ int C01 (Z)+ is a lower solution of (5.80)
(see Definition 5.2.2(b)).
Therefore we have produced an ordered pair {x, x} of lower and upper solutions. We introduce the truncation map τ+ : R −→ R defined by 0 if x ≤ 0 . τ+ (x) = x if x ≥ 0 We set
x
f+ (z, x) = f z, τ+ (x) , F+ (z, x) = f+ (z, r)dr
and
x F (z, x) = f (z, r)dr.
0
0
W01,p (Z) −→ R
We also consider the Euler functionals ϕ+ , ϕ :
1 λ ϕ+ (x) = Dxpp − xpp + F+ z, x(z) dz p p
Z
1 λ p p and ϕ(x) = Dxp − xp + F z, x(z) dz p p Z
defined by
for all x ∈ W01,p (Z).
384
5 Boundary Value Problems–Hamiltonian Systems
We have ϕ+ , ϕ ∈ C 1 W01,p (Z) and the critical points of ϕ are solutions of (5.80). Also we introduce the order interval E+ =[ x, x ]={x ∈ W01,p (Z) : x(z) ≤ x(z) ≤ x(z)
a.e. on Z}.
In the proof of the next proposition, we need the following nonlinear strong comparison principle, whose proof is postponed until Section 5.5 (see Theorem 5.5.11). THEOREM 5.2.7 If x, y ∈ C01 (Z), x = 0, f, g ∈ L∞ (Z), f ≥ 0,
a.e. on Z −div Dx(z)p−2 Dx(z) = f (z)
p−2 and −div Dy(z) Dy(z) = g(z) a.e. on Z f (z) ≥ g(z)
a.e. on Z and the set
C = {z ∈ Z : f (z) = g(z)} has an empty interior, then x(z) > y(z) for all z ∈ Z and (∂x/∂n)(z) λ1 , then we can find x0 ∈ E+ which is a local minimizer of ϕ. PROOF: Because of (5.81), we have F+ (z, x) ≥
µ p |x| − ϑµ (z)|x| p
for a.a. z ∈ Z and all x ∈ R+ .
(5.88)
Choose µ > 0 such that λ − µ < λ1 . Then
1 λ ϕ+ (x) = Dxpp − xpp + F+ z, x(z) dz p p Z 1 λ−µ p p ≥ Dxp − xp − c1 Dxp for some c1 > 0 (see (5.88)) p p 1
λ − µ = Dxpp − c1 Dxp , 1− p λ1 ⇒ ϕ+ is coercive on W01,p (Z)+
(because λ − µ < λ1 ).
Because of compact embedding of W01,p (Z) into Lp (Z), we can easily see that ϕ+ is weakly lower semicontinuous. So we can find x0 ∈ E+ such that ϕ+ (x0 ) = inf ϕ+ .
(5.89)
E+
Now let y ∈ E+ and set η(t)=ϕ+ ty + (1 − t)x0 for all t ∈ [0, 1]. Then η(0) ≤ η(t)
for all t ∈ [0, 1]
⇒ 0 ≤ η (0 ), +
⇒ 0 ≤ A(x0 ), y − x0 −λ
(see (5.10)),
|x0 |p−2 x0 (y − x0 )dz +
Z
for all y ∈ E+ . Given v ∈ W01,p (Z) and ε > 0, we define
f+ z, x0 (z) (y − x0 )dz
Z
(5.90)
5.2 Method of Upper–Lower Solutions ⎧ ⎪ if z ∈ {x0 + εv ≤ x} ⎨ x(z) y(z) = x0 (z) + εv(z) if z ∈ {x < x0 + εv < x} . ⎪ ⎩ x(z) if z ∈ {x ≤ x0 + εv} Evidently y ∈ E+ . We return to (5.90) and use this y ∈ E+ . We obtain
0≤
Dx0 p−2 (Dx0 , Dv)RN dz − λε
{x<x0 +εv<x}
f+ (z, x0 )vdz +
+ε
{x<x0 +εv<x}
+
− x0 )dz
− Dx0 )RN dz − λ
|x0 |p−2 x0 (x {x0 +εv≥x}
− x0 )dz
− x0 )dz =
Dx0 p−2 (Dx0 , Dv)RN dz − λε |x0 |p−2 x0 vdz Z Z
+ε f+ (z, x0 )vdz − Dxp−2 Dx, D(x0 + εv − x) RN dz
=ε
f+ (z, x0 )(x {x0 +εv≤x}
Dx0 p−2 (Dx0 , Dx {x0 +εv≥x} f+ (z, x0 )(x {x0 +εv≥x}
− Dv)RN dz
{x0 +εv≤x}
+
Dx0 p−2 (Dx0 , Dx {x0 +εv≤x}
|x0 |p−2 x0 (x − x0 )dz +
−λ
|x0 |p−2 x0 vdz Z
{x0 +εv≥x}
Z
|x|p−2 x(x0 + εv − x)dz −
+λ
{x0 +εv≥x}
+ εv − x)dz
Dxp−2 Dx, D(x − x0 − εv) RN dz
+
{x0 +εv≤x}
|x|p−2 x(x − x0 − εv)dz +
−λ
{x0 +εv≤x}
−
f+ (z, x)(x0 {x0 +εv≥x}
f+ (z, x)(x {x0 +εv≤x}
− x0 − εv)dz
f+ (z, x0 ) − f+ (z, x) (x0 + εv − x)dz
{x0 +εv≤x}
−
f+ (z, x) − f+ (z, x0 ) (x − x0 − εv)dz
{x0 +εv≥x}
(Dxp−2 Dx − Dx0 p−2 Dx0 , Dx0 − Dx)RN dz
+
{x0 +εv≥x}
(|x|p−2 x − |x0 |p−2 x0 )(x0 − x)dz
−λ
{x0 +εv≥x}
−
(Dxp−2 Dx − Dx0 p−2 Dx0 , Dx − Dx0 )RN dz
{x0 +εv≤x}
(|x|p−2 x − |x0 |p−2 x0 )(x − x0 )dz
+λ
+ε
{x0 +εv≤x}
(Dxp−2 Dx − Dx0 p−2 Dx0 , Dv)RN dz
{x0 +εv≥x}
385
386
5 Boundary Value Problems–Hamiltonian Systems
+ ε (Dxp−2 Dx − Dx0 p−2 Dx0 , Dv)RN dz {x0 +εv≤x}
− λε
(|x|p−2 x − |x0 |p−2 x0 )vdz
{x0 +εv≥x}
− λε
(|x|p−2 x − |x0 |p−2 x0 )vdz.
(5.91)
{x0 +εv≤x}
If u = (x0 +εv −x)+ ∈W01,p (Z)+ , then because x is an upper solution for problem (5.80), we have
− Dxp−2 Dx, D(x0 + εv − x) RN dz + λ |x|p−2 x(x0 + εv − x)dz {x0 +εv≥x}
{x0 +εv≥x}
−
f+ (z, x)(x0 {x0 +εv≥x}
+ εv − x)dz ≤ 0.
(5.92)
If u = (x − x0 − εv)+ ∈W01,p (Z)+ , then because x is a lower solution for problem (5.80), we have
Dxp−2 Dx, D(x − x0 − εv) RN dz − λ |x|p−2 x(x − x0 − εv)dz {x0 +εv≤x}
{x0 +εv≤x}
+
f+ (z, x)(x {x0 +εv≤x}
− x0 − εv)dz ≤ 0.
(5.93)
Recall that the map ψp : RN −→RN defined by yp−2 y if y = 0 ψp (y)= , 0 if y = 0 is a strictly monotone homeomorphism. Therefore
(Dxp−2 Dx − Dx0 p−2 Dx0 , Dx0 − Dx)RN dz ≤ 0
(5.94)
{x0 +εv≥x}
and
(Dxp−2 Dx − Dx0 p−2 Dx0 , Dx − Dx0 )RN dz ≤ 0.
(5.95)
{x0 +εv≤x}
Because x0 ≤ x and x − x0 ≤ εv on {x0 + εv ≥ x}, we have
−λ (|x|p−2 x − |x0 |p−2 x0 )(x0 − x)dz ≤ λε (|x|p−2 x − |x0 |p−2 x0 )vdz. {x0 +εv≥x}
Similarly because x ≤ x0 and x − x0 ≥ εv on {x0 + εv ≤ x}, we have
p−2 x − |x0 |p−2 x0 (x − x0 )dz ≤ λε (|x|p−2 x − |x0 |p−2 x0 )vdz. |x| λ {x0 +εv≤x}
(5.96)
{x0 +εv≤x}
(5.97)
{x0 +εv≤x}
Because x, x ∈ C01 (Z)+ , x ≤ x0 and using hypothesis H(f )1 (iii), we have
− f+ (z, x0 ) − f+ (z, x) (x0 + εv − x)dz ≤ c2 ε (−v)dz for some c2 > 0. (5.98) {x0 +εv≤x}
Similarly we have
{x0 +εv≤x<x0 }
5.2 Method of Upper–Lower Solutions 387
− f+ (z, x) − f+ (z, x0 ) (x − x0 − εv)dz ≤ c3 ε vdz for some c3 > 0. (5.99) {x0 +εv≥x}
{x0 +εv≥x>x0 }
We return to (5.91) and use (5.92)−→(5.99). This way we have
Dx0 p−2 (Dx0 , Dv)RN dz − λε x0 p−2 x0 vdz + ε f+ (z, x0 )vdz 0≤ε Z Z Z
+ c2 ε (−v)dz + c3 ε vdz. {x0 +εv≤x<x0 }
{x0 +εv≥x>x0 }
Divide with ε > 0 and then let ε ↓ 0. If by | · |N we denote the Lebesgue measure in RN , then |{x0 + εv ≤ x < x0 }|N −→ 0
and
|{x0 + εv ≥ x > x0 }|N −→ 0
as ε ↓ 0.
Therefore in the limit as ε ↓ 0, we obtain
Dx0 p−2 (Dx0 , Dv)RN dz − λ x0 p−2 x0 vdz + f+ (z, x0 )vdz. 0≤ Z
Z
Z
(5.100) Because x0 ∈W01,p (Z) was arbitrary, from (5.99) we infer that
⎧ ⎫ p−2 Dx0 (z) = λ|x0 (z)|p−2 x0 (z)−f z, x0 (z) a.e. on Z, ⎬ ⎨ −div Dx0 (z) ⎩
x∂Z = 0
⎭
,
(5.101)
(note that f+ z, x0 (z) = f z, x0 (z) ). From (5.101) we have that x0 ∈ C01 (Z). Moreover, from (5.81) and Proposition 5.2.6, we have x − x0 ∈ int C01 (Z)+ . Similarly because of (5.85) and Proposition 5.2.6, we have x0 − x ∈ int C01 (Z)+ . From the definition of f+ , it follows that x0 is a C01 (Z)-local minimizer of ϕ. Invoking Theorem 5.1.27, we infer that x0 is a W01,p (Z)-local minimizer of ϕ. From the above proof we have that x0 ∈ C01 (Z)+ is a solution of problem (5.80). We repeat a similar argument on the negative semiaxis of R. In this case motivated by (5.82), we consider the following auxiliary problem.
⎧ ⎫ p−2 Dx(z) = (λ − µ)|x(z)|p−2 x(z) − ϑµ (z) a.e. on Z, ⎬ ⎨ −div Dx(z) . (5.102) ⎩ ⎭ x ∂Z = 0 Following the reasoning of Proposition 5.2.3, we solve problem (5.102) and obtain a solution v ∈ −int C01 (Z)+ , which we can show is a lower solution for problem (5.80). Moreover, taking ξ > 0 as in the definition of x, we introduce v = ξ(−u1 ) which is an upper solution for problem (5.80). Using the ordered pair {v, v}, we consider the corresponding eodory truncation τ− (z, x) and then set f− (z, x) =
Carath´ x f z, τ− (z, x) , F− (z, x) = 0 f− (z, r)dr, and
388
5 Boundary Value Problems–Hamiltonian Systems
1 λ F− z, x(z) for all x ∈ W01,p (Z). ϕ− (x) = Dxpp − xpp + p p Z
We work as above, this time on the order interval E− = [ v, v ] and obtain the following. PROPOSITION 5.2.9 If hypotheses H(f )1 hold and λ > λ1 , then there exists v0 ∈ E− which is a local minimizer of ϕ. Again v0 ∈ −int C01 (Z)+ is a solution of problem (5.80). Therefore we have produced two nontrivial solutions x0 , v0 ∈ C01 (Z) of problem (5.80) which have constant sign. Next we show that if we restrict further the parameter λ > 0, namely we require that λ > λ2 , then we have a third nontrivial solution of problem (5.80), distinct from the other two. THEOREM 5.2.10 If hypotheses H(f )1 hold and λ > λ2 , then problem (5.80) has at least three distinct nontrivial solutions x0 , v0 , y0 ∈ C01 (Z). PROOF: We have produced two solutions x0 , v0 ∈ C01 (Z) that have constant sign and are also local minimizers of ϕ. We assume that these are the only nontrivial critical points of ϕλ , or otherwise we are done. Also clearly without any loss of generality, we can say that both x0 , v0 are strict local minimizers of ϕλ . We choose δ > 0 small such ϕ(v0 ) < inf ϕ(x) : x ∈ ∂Bδ (v0 ) and ϕ(x0 ) < inf ϕ(x) : x ∈ ∂Bδ (x0 ) . Let ϕ(v0 ) ≤ ϕ(x0 ), V =∂Bδ (x0 ), W0 ={v0 , x0 }, and W =[v0 , x0 ]={x ∈ W01,p (Z) : v0 (z) ≤ x(z) ≤ x0 (z) a.e. on Z}. The pair {W0 , W } links with V in W01,p (Z) via the identity map. Using hypothesis H(f )1 (v) it is easy to verify that ϕ is coercive from which it follows that ϕ satisfies the P S-condition. So by virtue of Theorem 4.1.28, we can find y0 ∈ W01,p (Z) such that ϕ (y0 ) = 0
and
(i.e., y0 is a critical point of ϕ)
ϕ(v0 ), ϕ(x0 ) < ϕ(y0 ) = inf max ϕ γ(t) ,
(5.103) (5.104)
γ∈Γ t∈[−1,1]
where Γ= γ ∈C [−1, 1], W01,p (Z) :γ(−1)=v0 , γ(1)=x 0 . Our goal is to produce a path γ ∈ Γ such that ϕγ < 0. Then y0 = 0 and of Lp (Z)
course it is distinct from v0 , x0 . To this end let S = W01,p (Z) ∩ ∂B1 where Lp (Z) Lp (Z) 1,p p 1 B1 = x ∈ L (Z) : xp ≤ 1 and Sc = W0 (Z) ∩ C0 (Z) ∩ ∂B1 . Evidently Sc is dense in S. From Theorem 4.1.28, we know that we can find γ0 ∈ Γ0 = γ0 ∈ C([−1, 1], S) : γ(−1)=−u1 , γ0 (1)=u1 such that γ0 ([−1, 1]) ⊆ Sc and max Dupp : u ∈ γ0 ([−1, 1]) ≤ λ2 + δ0 . (5.105) We choose δ0 > 0 such that λ2 + δ0 < λ. Also because of hypothesis H(f )1 (iv), given ε > 0 we can find δ=δ(ε) > 0 such that F (z, x) ≤
ε p |x| p
for a.a. z ∈ Z and all |x| ≤ δ.
(5.106)
Recall that γ0 ([−1, 1]) ⊆ Sc and −v0 , x0 ∈ int C01 (Z)+ . So we can choose ε > 0 small so that
5.2 Method of Upper–Lower Solutions |εu(z)| ≤ δ
389
for all z ∈ Z, all u ∈ γ0 ([−1, 1]),
λ2 + δ0 + ε < λ
and
εu ∈ W = [v0 , x0 ]
for all u ∈ γ0 ([−1, 1]). (5.107)
Therefore, if u ∈ γ0 ([−1, 1]), we have ϕ(εu) =
εp λεp Dupp − upp + p p
F z, εu(z) dz
Z
εp λεp εp+1 ≤ (λ2 + δ0 ) − + p p p and recall that up = 1), εp = (λ2 + δ0 + ε − λ) < 0 p
(see (5.105) and (5.106)
(see (5.107)).
Hence the path εγ0 joins −εu1 and εu1 and we have ϕεγ < 0.
(5.108)
0
Next we produce another continuous path γ+ : [0, 1] −→ W01,p (Z) which joins εu1 and x0 and along which ϕ is negative. For this purpose, we consider the truncation map x if x ≥ 0 τ+ (x) = . 0 if x < 0
x We set f+ (z, x)=f z, τ+ (x) , F+ (z, x)= 0 f+ (z, r)dr, and ϕ+ (x) =
1 λ Dxpp − x+ pp + p p
F+ z, x(z) dz
Z
for all x ∈ W01,p (Z).
Because of hypothesis H(f )1 (v), ϕ+ is coercive and so it satisfies the P Scondition. Also {0, x0 } are the only critical points of ϕ+ and ϕ+ (x0 ) = inf ϕ+ . 1,p
W0
(Z)
We set b+ = ϕ+ (εu1 )
and
a+ = ϕ+ (x0 ) = inf ϕ+ . 1,p
W0
(Z)
Invoking the second deformation theorem (see Theorem 4.6.1), we can find h ∈
b b C [0, 1] × ϕ++ , ϕ++ such that
and
a
h(t, x) = x
for all t ∈ [0, 1] and all x ∈ ϕ++ ,
h(0, x) = x
for all ϕ++
h(1, x) = x0
for all ϕ++ .
b
b
Then consider the continuous path γ+ (t) = h(t, εu1 ) for all t ∈ [0, 1], which joins εu1 and x0 . We have
ϕ γ+ (t) ≤ ϕ+ γ+ (t) ≤ b+ = ϕ+ (εu1 ) < 0 (5.109) ⇒ ϕγ < 0. +
390
5 Boundary Value Problems–Hamiltonian Systems
In a similar fashion, we produce a continuous path γ− : [0, 1] −→ W01,p (Z) joining −εu1 and v0 such that ϕγ < 0. (5.110) −
Concatenating the paths εγ0 , γ+ , and γ− we generate a path γ ∈ Γ such that ϕγ < 0 (see (5.108) through (5.110)) ⇒ ϕ(y0 ) < 0 = ϕ(0) (i.e. y0 = 0). Finally, nonlinear regularity theory implies that y0 ∈ C01 (Z).
As a second example of the method of upper–lower solutions, we consider the following nonlinear boundary value problem. ⎧ ⎫
⎨ − |x (t)|p−2 x (t) = f t, x(t), x (t) + ϑ t, x(t) a.e. on T = [0, b], ⎬ . (5.111)
⎩ ⎭ x (0) ∈ ξ1 x(0) , −x (b) ∈ ξ2 x(b) In this problem ξ1 and ξ2 are maximal monotone graphs in R2 . First let us specify the notions of upper and lower solutions for problem (5.111). DEFINITION 5.2.11 (a) A function ϕ ∈ C 1 (T ) such that |ϕ |p−2 ϕ ∈
W 1,p (0, b) , (1/p) + (1/p ) = 1 is said to be an upper solution of problem (5.111), if ⎫ ⎧
a.e. on T, ⎬ ⎨ − |ϕ (t)|p−2 ϕ (t) ≥ f t, ϕ(t), ϕ (t) + ϑ ϕ(t) .
⎭ ⎩ ϕ (0) ∈ ξ1 ϕ(0) − R+ , −ϕ (b) ∈ ξ2 ϕ(b) − R+
(b) A function ψ ∈ C 1 (T ) such that |ψ |p−2 ψ ∈ W 1,p (0, b) , (1/p) + (1/p ) = 1 is said to be a lower solution of problem (5.111), if ⎫ ⎧
⎨ − |ψ (t)|p−2 ψ (t) ≤ f t, ψ(t), ψ (t) + ϑ ψ(t) a.e. on T, ⎬ .
⎭ ⎩ ψ (0) ∈ ξ1 ψ(0) + R+ , −ψ (b) ∈ ξ2 ψ(b) + R+ We make the following hypothesis. H0 : There exist a lower solution ψ ∈ C 1 (T ) and an upper solution ϕ ∈ C 1 (T ) such that ψ(t) ≤ ϕ(t) for all t ∈ T . The hypotheses on the data of problem (5.111) are the following. H(f )2 : f : T × R × R −→ R is a function such that (i) For all x, y ∈ R, t −→ f (t, x, y) is measurable. (ii) For almost all t ∈ T, (x, y) −→ f (t, x, y) is continuous.
5.2 Method of Upper–Lower Solutions
391
(iii) For almost all t ∈ T , all x ∈ [ψ(t), ϕ(t)] and all y ∈ R, we have
|f (t, x, y)| ≤ η(|y|p−1 ) α(t) + c|y| with α ∈ L1 (T )+ , c > 0 and η : R+ −→R+ \{0} a Borel measurable nondecreasing function such that
∞ ds > α1 + c(max ϕ − min ψ) T T λp−1 η(s) b + sup |ϑ(z)| : |z| ≤ max{ϕ∞ , ψ∞ } η(λ) with λ = (1/b) max |ψ(0) − ϕ(b)|, |ψ(b) − ϕ(0)| .
(iv) For every r > 0, we can find γr ∈ Lp (T ) such that for almost all x, y ∈ R with |x|, |y| ≤ r we have |f (t, x, y)| ≤ γr (t). REMARK 5.2.12 Hypothesis H(f )2 (iii) is known as a Bernstein–Nagumo– Wintner growth condition and produces a uniform a priori bound for the derivatives of the solutions of problem (5.111). It is clear that if for almost all t ∈ T , all x ∈ [ ψ(t), ϕ(t) ], and all y ∈ R we have |f (t, x, y)| ≤ α(t) + c|y|p with α ∈ L1 (T )+ , c > 0, then hypothesis H(f )2 (iii) is satisfied. This is the growth condition initially used by Bernstein in his existence theory for second-order boundary value problems.
H(ξ): ξ1 , ξ1 : R −→ 2R are maximal monotone maps such that 0 ∈ ξ1 (0) and 0 ∈ ξ2 (0). REMARK 5.2.13 We know that there exist functions k1 , k2 ∈ Γ0 (R) such that ξ1 = ∂k1 and ξ2 = ∂k2 . H(ϑ): ϑ : R −→ R is a function that maps bounded sets to bounded sets and there exists M > 0 such that x −→ ϑ(x) + M x is increasing. REMARK 5.2.14 We emphasize that ϑ need not be continuous. We start with a lemma that produces a uniform bound for the derivatives of the solutions of problem (5.111). As we already mentioned, hypothesis H(f )2 (iii) is the crucial tool for this. LEMMA 5.2.15 If hypothesis H(f )2 (iii) holds and x ∈ C 1 (T ) satisfies
a.e. on T − |x (t)|p−2 x (t) = f t, x(t), x (t) + ϑ x(t) and
ψ(t) ≤ x(t) ≤ ϕ(t)
for all t ∈ T,
then there exists M1 > 0 (depending only on ψ, ϕ, η, α, c) such that |x (t)| ≤ M1
for all t ∈ T.
392
5 Boundary Value Problems–Hamiltonian Systems
PROOF: Set µ = ( 1/η(λ) sup |ϑ(z)| : |z| ≤ max{ϕ∞ , ψ∞ } < +∞ (see Hypothesis H(ϑ)). By virtue of hypothesis H(f )2 (iii), we can find M1 > λ such that
p−1
M1
λp−1
ds > α1 + c(max ϕ − min ψ) + µb. T T η(s)
We claim that |x (t)| ≤ M1 for all t ∈ T . Suppose that this is not the case. Then we can find t ∈ T such that |x (t )| > M1 . By the mean value theorem, we can find t0 ∈ (0, b) such that x(b) − x(0)=x (t0 )b. Without any loss of generality, we assume that t0 ≤ t (the analysis is similar if t0 > t ). Note that ψ(b) − ϕ(0) ≤ x(b) − x(0) ≤ ϕ(b) − ψ(0). So we have 1 1 |x(b) − x(0)| ≤ max |ψ(0) − ϕ(b)|, |ψ(b) − ϕ(0)| , b b ⇒ |x (t0 )| ≤ λ > M1 and |x (t )| < M1 (i.e., t0 < t ). |x (t0 )| =
Because x ∈ C 1 (T ), by the intermediate value theorem we can find t1 , t2 ∈ [t0 , t ] with t1 < t2 such that |x (t1 )| = λ and |x (t2 )| = M1 . By hypothesis we have
− |x (t)|p−2 x (t) = f t, x(t), x (t) + ϑ x(t) a.e. on T,
⇒ |x (t)|p−1 ≤ | |x (t)|p−2 x (t) | ≤ |f t, x(t), x (t) | + |ϑ x(t) |
≤ η |x (t)|p−1 α(t) + c|x (t)| + |ϑ x(t) | a.e. on T,
p−1 |ϑ x(t) | |x (t)| ≤ α(t) + c|x (t)| + ⇒ a.e. on [t1 , t2 ] η(λ) η |x (t)|p−1
t2 |x (t)|p−1
dt ≤ α1 + c(max ϕ − min ψ) + µb ⇒ p−1 T T t1 η |x (t)|
M p−1 1 ds ≤ α1 + c(max ϕ − min ψ) + µb, ⇒ T T η(s) p−1 λ
which contradicts the choice of M1 > 0.
The upper–lower solutions method employs truncation and penalization techniques. So we introduce a truncation map u : T×R×R−→R2 and a penalty function β : T ×R−→R defined by ⎧
⎪ ψ(t), ψ (t) if x < ψ(t) ⎪ ⎪
⎪ ⎪ ⎪ if x > ϕ(t) ⎨ ϕ(t), ϕ (t) u(t, x, y)= (x, M0 ) (5.112) if ψ(t) ≤ x ≤ ϕ(t), y > M0 ⎪ ⎪ ⎪(x, −M0 ) if ψ(t) ≤ x ≤ ϕ(t), y < −M ⎪ 0 ⎪ ⎪ ⎩ (x, y) if otherwise, where M0 > max{M1 , ϕ ∞ , ψ ∞ } and ⎧ p−2 ⎪ ψ(t) − |x|p−2 x ⎨ |ψ(t)| β(t, x)= 0 ⎪ ⎩ |ϕ(t)|p−2 ϕ(t) − |x|p−2 x
if x < ψ(t) if ψ(t) ≤ x ≤ ϕ(t) if ϕ(t) < x.
(5.113)
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393
We set f1 (t, x, y)=f t1 , u(t, x, y) . Note that for almost all x ∈ [ψ(t), ϕ(t)] and all |y| ≤ M , we have f1 (t, x, y) = f (t, x, y). Moreover, for almost all t ∈ T and all x, y∈R, we have |f1 (t, x, y)| ≤ αr (t) with r =max{M0 , ϕ∞ , ψ∞ }. For every x ∈
W 1,p (0, b) , let N1 (x)(·)=f1 ·, x(·), x (·) and β(x)(·)=β ·, x(·) (i.e., the Nemitsky operators corresponding
to f1 and β, respectively). Let G(x) = N1 (x) + β(x) for every x ∈ W 1,p (0, b) . The next proposition is an immediate consequence of the continuity of the truncation map (x, y) −→ u(t, x, y) (see (5.112)).
PROPOSITION 5.2.16 If hypotheses H(f )2 hold, then G : W 1,p (0, b) −→ Lp (T ), (1/p) + (1/p ) = 1 is continuous. Next we introduce the set
D = x ∈ C 1 (T ) : |x |p−2 x ∈ W 1,p (0, b) , x (0) ∈ ξ1 x(0)
and − x (b) ∈ ξ2 x(b)
and then define the nonlinear operator U : D ⊆ Lp (T ) −→ Lp (T ) by
for all x ∈ D. U (x)(·) = − |x (·)p−2 x (·)|
PROPOSITION 5.2.17 If hypotheses H(ξ) hold, then U : D ⊆ Lp (T )−→Lp (T ) is maximal monotone.
PROOF: Given h ∈ Lp (T ), we consider the following nonlinear boundary value problem, * ) = h(t) a.e. on T, − |x (t)|p−2 x (t) + |x(t)|p−2 x(t)
. (5.114) x (0) ∈ ξ1 x(0) , −x (b) ∈ ξ2 x(b) We show that problem (5.114) has a unique solution x ∈ C 1 (T ). To this end, given v, w ∈ R, we consider the following two-point boundary value problem. * ) − |x (t)|p−2 x (t) + |x(t)|p−2 x(t) = h(t) a.e. on T, . (5.115) x(0) = v, x(b) = w
Let us set γ(t) = 1 − (t/b) v + (t/b)w. Then γ(0) = v and γ(b) = w. We consider the function y(t) = x(t) − γ(t) and rewrite problem (5.115) in terms of this function )
* − |(y + γ) (t)|p−2 (y + γ) (t) +|(y + γ)(t)|p−2 (y + γ)(t) = h(t) a.e. on T, . y(0) = 0, y(b) = 0 (5.116) This is a homogeneous Dirichlet problem for y. To solve (5.116), we argue as follows.
Let V1 : W01,p (0, b) −→W −1,p (0, b) , be the nonlinear operator defined by
V1 (u), z0 =
b
|(u + γ) (t)|p−2 (u + γ) (t)z (t)dt
0
b
|(u + γ)(t)|p−2 (u + γ)(t)z(t)dt
+ 0
for all u, z∈W01,p (0, b) .
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5 Boundary Value Problems–Hamiltonian Systems
It is easy to check that V1 is strictly monotone, demicontinuous, hence maximal monotone too. Moreover, we have V1 (u), u0 ≥ u + γp − c3 u + γp−1
for some c3 > 0,
⇒ V1 is coercive.
So from Corollary 3.2.28, we infer that there exists y ∈ W01,p (0, b) such that V1 (y) = h and due to the strict monotonicity of V1 , this solution is unique. From the equation V1 (y) = h it follows easily that y ∈ C 1 (T ) and it solves problem (5.116). Therefore x = y + γ ∈ C 1 (T ) and it is the unique solution of problem (5.115). We can define the solution map s : R × R −→ C 1 (T ) which to each pair (v, w) assigns the unique solution of problem (5.115). Then let : R × R −→ R × R be defined by
(v, w) = s(v, w) (0), −s(v, w) (b) . (5.117) Using integration by parts, we check easily that is monotone. Also let αn −→ α and βn −→ β in R. Set t t αn + βn xn = s(αn , βn ), x = s(α, β), γn (t) = 1 − b b t t and γ(t) = 1 − α+ β for all n ≥ 1. b b
Then directly from (5.115) we obtain that {xn }n≥1 ⊆ W 1,p (0, b) is bounded.
It follows that |xn |p−2 xn n≥1 ⊆Lp (T ) and |xn |p−2 xn n≥1 ⊆W 1,p (0, b) are both bounded. So we may assume that
w w xn −→ u in W 1,p (0, b) , |xn |p−2 xn −→ w in Lp (T ),
w and |xn |p−2 xn −→ v in W 1,p (0, b) .
Due to the compact embeddings of W 1,p (0, b) and W 1,p (0, b) into C(T ), we have xn −→ u in C(T ) and |xn |p−2 xn −→ v in C(T ). The map σ : C(T )−→C(T ) defined by
y(·) = ϑp y(·) , σ(y)(·) = ϑ−1 p where for any 1 < r < ∞,
ϑr (y) =
|y|r−2 y 0
if y = 0 , if y = 0
is continuous and maps bounded sets to bounded sets. Therefore xn −→ σ(v) in C(T ) ⇒ u = σ(v) (i.e., v = |u |p−2 u ). Therefore in the limit as n → ∞, we obtain ) * − |u (t)|p−2 u (t) + |u(t)|p−2 u(t) = h(t) a.e. on T, . u(0) = α, u(b) = β ⇒ x = s(α, β) (i.e., s : R × R−→C 1 (T ) is continuous).
5.2 Method of Upper–Lower Solutions
395
From the continuity of s it follows at once the continuity of (see (5.117)). Finally it is straightforward to check that is coercive. So is maximal monotone (being monotone, continuous) and coercive. Evidently so is k = + ξ : R × R −→ 2R×R . So we can find (α, β) ∈ R × R such that (0, 0) ∈ k(α, β). Then x0 = s(α, β) is the unique solution of the auxiliary problem (5.114). If K : Lp (T )−→Lp (T ) is the maximal monotone operator defined by K(x)(·) = |x(·)|p−2 x(·), then from the previous argument we have that
R(U + K) = Lp (T ), (i.e., U + K is surjective).
(5.118)
We claim that this surjectivity implies the maximal monotonicity of U . For this purpose suppose that for some y ∈ Lp (T ) and some v ∈ Lp (T ), we have U (x) − v, x − yp ≥ 0
for all x ∈ D.
(5.119)
Here by ·, ·p we denote the duality brackets for the pair Lp (T ), Lp (T ) . Because of (5.118), we can find x1 ∈ D such that U (x1 ) + K(x1 ) = v + K(y). We use this in (5.119) with x = x1 . So we obtain U (x1 ) − U (x1 ) − K(x1 ) + K(y), x1 − yp ≥ 0, ⇒ K(y) − K(x1 ), x1 − yp ≥ 0.
(5.120)
Because K is strictly monotone, from (5.120) we conclude that y = x1 ∈ D and v = U (x1 ). Therefore U is maximal monotone.
The operator U + K : D ⊆ Lp (T )−→ Lp (T ) is a maximal monotone and strictly monotone operator that is coercive (indeed note that U (x) + K(x), xp ≥ xpp , because U (x), xp ≥ 0 and K(x), xp = xpp ). Therefore U + K is surjective. This
means that the operator L : (U + K)−1 :Lp (T )−→D⊆W 1,p (0, b) is well-defined, single-valued, and maximal monotone (from Lp (T ) into Lp (T )).
PROPOSITION 5.2.18 If hypothesis H(ξ) holds, then L : Lp (T ) −→ D ⊆
1,p W (0, b) is completely continuous. w
PROOF: Suppose that vn −→ v in Lp (T ). We need to show that L(vn ) −→ L(v) in Lp (T ). We set xn = L(vn ) for all n ≥ 1. Then xn ∈ D and U (xn ) + K(xn ) = vn , ⇒ U (xn ), xn p + K(xn ), xn p = vn , xn p .
(5.121)
Because of the integration by parts formula, we have U (xn ), xn p = −|xn (b)|p−2 xn (b)xn (b) + |xn (0)|p−2 xn (0)xn (0) + xn pp . (5.122)
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5 Boundary Value Problems–Hamiltonian Systems
Because xn ∈ D, we have xn (0) ∈ ξ1 xn (0) and −xn (b) ∈ ξ2 xn (b) for all n ≥ 1. Then recalling that (0, 0) ∈ Gr ξi , i = 1, 2, we have xn (0)xn (0) ≥ 0 ⇒ ⇒
− xn (b)xn (b) ≥ 0,
and
−|xn (b)|p−2 xn (b)xn (b) ≥ U (xn ), xn p ≥ xn pp
0
and
|xn (0)|p−2 xn (0)xn (0) ≥ 0
(see (5.122)).
(5.123)
Using (5.123) in (5.121), we obtain xn pp + xn pp = xn p ≤ vn p xn p ⇒ xn p−1 ≤ c3
for some c3 > 0 and all n ≥ 1, (0, b) is bounded. ⇒ {xn }n≥1 ⊆ W
w Therefore we may assume that xn −→ x in W 1,p (0, b) and xn −→ x in C(T ). Also directly from the equation U (xn ) + K(xn ) = vn for all n ≥ 1, it follows that p−2
|xn | xn n≥1 ⊆W 1,p (0, b) is bounded. So by passing to a suitable subsequence if necessary, we may assume that
w (5.124) |xn |p−2 xn −→ h in W 1,p (0, b) and |xn |p−2 xn −→ h in C(T ). 1,p
If ϑp : R −→ R is the homeomorphism |y|p−2 y ϑp (y) = 0
if y = 0 if y = 0
(see also the proof of Proposition 5.2.17), then
ϑ−1 |xn (t)|p−2 xn (t) −→ ϑ−1 h(t) for all t ∈ T, p p −1
⇒ xn (t) −→ ϑp h(t) for all t ∈ T, −1
p ⇒ xn −→ ϑp h(·) in L (T ) (by the dominated convergence theorem). But recall that xn −→ x in Lp (T ). So we infer that
x = ϑ−1 h(·) , p
⇒ h = ϑp |x (·)|p−2 x (·) , w
⇒ |xn |p−2 xn −→ |x |p−2 x ⇒ xn −→ x ⇒ xn −→ x
in C(T )
(see (5.124)),
in C(T ),
in W 1,p (0, b) .
This proves the complete continuity of the operator L.
We introduce the order interval
E=[ψ, ϕ]={x ∈ W 1,p (0, b) : ψ(t) ≤ x(t) ≤ ϕ(t) for all t ∈ T }.
Also let τ : W 1,p (0, b) −→W 1,p (0, b) , be the truncation operator defined by ⎧ ⎪ if x(t) < ψ(t) ⎨ ψ(t) τ (x)(t) = x(t) if ψ(t) ≤ x(t) ≤ ϕ(t) . ⎪ ⎩ ϕ(t) if ϕ(t) ≤ x(t)
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397
Clearly τ is continuous and bounded (i.e., maps bounded sets to bounded ones). Given w ∈ E, we consider the following auxiliary boundary value problem: ⎧
⎫
⎨ − |x (t)|p−2 x (t) =f1 t, x(t), x (t) + β t, x(t) −M τ (x)(t) + ϑ w(t) ⎬ . + M w(t) a.e. on T,
⎩ ⎭ x (0) ∈ ξ1 ψ(0) , −x (b) ∈ ξ2 ψ(b) (5.125) PROPOSITION 5.2.19 If hypotheses H0 , H(f )2 , H(ξ), and H(ϑ) hold, then problem (5.125) has a solution x ∈ C 1 (T ) ∩ E.
PROOF: Let G1 : W 1,p (0, b) −→Lp (T ) be the nonlinear operator defined by G1 (x) = G(x) + K(x) − M τ (x) + ϑ(w) + M w
for all x ∈ W 1,p (0, b) .
From Proposition 5.2.16 and the continuity of
the operators K and τ , we infer that G1 is continuous. For every x ∈ W 1,p (0, b) , we have p−1 G1 (x)p ≤ γr p + b1/p max ϕp−1 + M b1/p max ϕ∞ , ψ∞ ∞ , ψ∞ + ϑ(w)p + M wp = M2 , M2 > 0. Recall that r = max M0 , ϕ∞ , ψ∞ . Set C = {g ∈ Lp (T ) : g p ≤ M2 }. Clearly G1 maps bounded sets to bounded ones and LG1 W 1,p (0, b) ⊆ L(E)
1,p which is relatively compact in (0, b) (see Proposition 5.2.18). Therefore we
W 1,p can find x ∈ D ⊆ W (0, b) such that x = LG1 (x), ⇒ U (x) + K(x) = G(x) + K(x) − M τ (x) + ϑ(w) + M w, ⇒ U (x) = G(x) − M τ (x) + ϑ(w) + M w, ⇒ x ∈ D ⊆ C 1 (T ) solves problem (5.125)). It remains to show that x ∈ E. Because ψ ∈ C 1 (T ) is a lower solution for problem (5.111), we have ) *
− |ψ (t)|p−2 ψ (t) ≤ f t, ψ(t), ψ (t) + ϑ ψ(t) a.e. on T, . ψ (0) ∈ ξ1 ψ(0) + R+ , −x (b) ∈ ξ2 ψ(b) + R+
(5.126)
Subtracting (5.126) from (5.125), we obtain
|ψ (t)|p−2 ψ (t) − |x (t)|p−2 x (t) ≥ f t, x(t), x (t) + β t, x(t) − M τ (x)(t)
+ ϑ w(t) + M w(t) − f t, ψ(t), ψ (t)
− ϑ ψ(t) a.e. on T. (5.127)
We multiply (5.127) with (ψ − x)+ ∈ W 1,p (0, b) and then integrate on [0, b] the resulting inequality. We obtain
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5 Boundary Value Problems–Hamiltonian Systems
|ψ (t)|p−2 ψ (t) − |x (t)|p−2 x (t) (ψ − x)+ (t)dt
b
0
b
b
f1 t, x(t), x (t) −f t, ψ(t), ψ (t) (ψ−x)+ (t)dt + β t, x(t) (ψ−x)+ (t)dt
≥ 0
0 b
ϑ w(t) +M w(t) − ϑ ψ(t) − M τ (x)(t) (ψ − x)+ (t)dt.
+ 0
(5.128) First we estimate the left-hand side in inequality (5.128). Performing integration by parts, we obtain
b
|ψ (t)|p−2 ψ (t) − |x (t)|p−2 x (t) (ψ − x)+ (t)dt 0 = |ψ (b)|p−2 ψ (b) − |x (b)|p−2 x (b) (ψ − x)+ (b) − |ψ (0)|p−2 ψ (0) − |x (0)|p−2 x (0) (ψ − x)+ (0)
b
(5.129) − |ψ (t)|p−2 ψ (t)−|x (t)|p−2 x (t) (ψ − x)+ (t)dt. 0
Recall that
+
(ψ − x)
(t) =
(ψ − x)(t) 0
if ψ(t) > x(t) . if ψ(t) ≤ x(t)
(5.130)
Also from the boundary conditions in (5.125) and (5.126), we have
−x (b) ∈ ξ2 x(b) and − ψ (b) ∈ ξ2 ψ(b) + eb with eb ≥ 0. If ψ(b) ≥ x(b), then of ξ2 (see hypothesis H(ξ)), we have the monotonicity
from ψ (b) ≤ x (b) ⇒ ϑp ψ (b) ≤ ϑp x (b) ⇒ |ψ (b)|p−2 ψ (b) ≤ |x (b)|p−2 x (b). So it follows that |ψ (b)|p−2 ψ (b) − |x (b)|p−2 x (b) (ψ − x)+ (b) ≤ 0. (5.131)
In a similar fashion using the boundary conditions x (0) ∈ ξ1 x(0) and ψ (0) ∈ ξ1 ψ(0) + e0 with e0 ≥ 0, we obtain that |ψ (0)|p−2 ψ (0) − |x (0)|p−2 x (0) (ψ − x)+ (0) ≥ 0. (5.132) Also we have
=
|ψ (t)|p−2 ψ (t)−|x (t)|p−2 x (t) (ψ − x)+ (t)dt
b
0
|ψ (t)|p−2 ψ (t) − |x (t)|p−2 x (t) (ψ − x )(t)dt ≥ 0.
(5.133)
{ψ>x}
Using (5.131) through (5.133) in (5.129), we see that
b
|ψ (t)|p−2 ψ (t) − |x (t)|p−2 x (t) (ψ − x)+ (t)dt ≤ 0. 0
(5.134)
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399
Next we estimate the right-hand side of (5.128). We have
f1 t, x(t), x (t) −f t, ψ(t), ψ (t) = f t, ψ(t), ψ (t) −f t, ψ(t), ψ (t) = 0
b
⇒
a.e. on {ψ ≥ x},
f1 t, x(t), x (t) − f t, ψ(t), ψ (t) (ψ − x)+ (t)dt = 0.
(5.135)
0
Also from the definition of the penalty function β (see (5.113)), if |{ψ > x}| > 0 (by | · | we denote the Lebesgue measure on R), then
b
β t, x(t) (ψ − x)+ (t)dt
0
= |ψ(t)|p−2 ψ(t)−|x(t)|p−2 x(t) (ψ − x)(t)dt > 0. (5.136) {ψ>x}
Finally by virtue of hypothesis H(ϑ) and since w ∈ E, we see that
b
ϑ w(t) + M w(t) − ϑ ψ(t) − M τ (x)(t) (ψ − x)+ (t)dt ≥ 0.
(5.137)
0
Using (5.135) through (5.137), we have
b
b
f t, x(t), x (t) −f t, ψ(t), ψ (t) (ψ−x)+ (t)dt+ β t, x(t) (ψ−x)+ (t)dt 0
0
b
+ ϑ w(t) +M w(t) − ϑ ψ(t) −M τ (x)(t) (ψ−x)+ (t)dt > 0,
(5.138)
0
provided |{ψ > x}| > 0. Returning to (5.128) and using (5.134) and (5.138), we reach a contradiction when |{ψ > x}| > 0. So it follows that ψ(t) ≤ x(t) for all t ∈ T . In a similar fashion we show that x(t) ≤ ϕ(t) for all t ∈ T ; that is, x ∈ E. We use the solvability of the auxiliary problem (5.125) in order to produce a solution for the original problem (5.111). To do this, we need a fixed point theorem for multifunctions in an ordered Banach space. More on the fixed point theory for multifunctions can be found in Section 6.5. THEOREM 5.2.20 If X is a separable, reflexive, ordered Banach space, E ⊆ X is a nonempty and weakly closed set, and S :E −→ 2E \{∅} is a multifunction with weakly closed values, S(E) is bounded and (i) The set M ={x ∈ E : x ≤ y for some y ∈ S(x)} is nonempty. (ii) If x1 ≤ y1 , y1 ∈ S(x1 ) and x1 ≤ x2 , then we can find y2 ∈ S(x2 ) such that y 1 ≤ y2 , then S has a fixed point; that is, there exists x ∈ E such that x ∈ S(x). We apply this theorem, using the
following data. The separable, reflexive, ordered Banach space X = W 1,p (0, b) , E = [ψ, φ] and S:E−→2E \ {∅} is the solution multifunction for the auxiliary problem (5.125); that is, for every w ∈ E, S(w) is the set of solutions for problem (5.125). From Proposition 5.2.19, we know that S(w) = ∅ and S(w) ⊆ E.
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5 Boundary Value Problems–Hamiltonian Systems
THEOREM 5.2.21 If hypotheses H(f )2 , H(ξ), and H(ϑ) hold, then problem (5.111) has a solution x ∈ C 1 (T ).
PROOF: Let X = W 1,p (0, b) , E = [ψ, φ] ⊆ W 1,p (0, b) and S : E −→ 2E \{∅} the solution multifunction for problem (5.125). From Proposition 5.2.19 we know that for every w ∈ E, S(w) = ∅ and
S(w) ⊆ E. Moreover, it is routine to check that S(w) is weakly closed in W 1,p (0, b) . In addition, from the proof of Proposition 5.2.19 we know that S(E) ⊆ W 1,p (0, b) is bounded. Therefore it remains to verify conditions (i) and (ii) of Theorem 5.2.20. Note that if w = ψ, then by Proposition 5.2.19, S(ψ) = ∅ and S(ψ) ⊆ E. So we have satisfied condition (i) of Theorem 5.2.20. Next we verify condition (ii) of Theorem 5.2.20. So let w1 , w2 ∈ E, w1 ≤ w2 , and x1 ∈S(w1 ) with w1 ≤x1 . Because x1 ∈ S(w1 )⊆E, we have β t, x1 (t) = 0 for all t ∈ T (see (5.113)) and τ (x1 ) = x1 . Also note that if pM0 :R−→R is the truncation function defined by ⎧ ⎪ if y < −M0 ⎨ −M0 pM0 (y) = y if − M0 ≤ y ≤ M0 , ⎪ ⎩ M0 if M0 < y then from the definition of the function u (see (5.112)), we have
u t, x1 (t), x1 (t) = τ (x1 )(t), pM0 τ (x1 ) (t) = x1 (t), pM0 x1 (t) and so it follows that
f1 t, x1 (t), x1 (t) = f t, x1 (t), x1 (t)
for all t ∈ T.
Therefore the auxiliary problem (5.125) becomes ⎫ ⎧
− |x1 (t)|p−2 x1 (t) = f t, x1 (t), x1 (t) + ϑ w1 (t) + M w1 (t) − M x1 (t) ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ a.e. on T, . ⎪ ⎪ ⎪ ⎪
⎭ ⎩ x1 (0) ∈ ξ1 x1 (0) , −x1 (b) ∈ ξ2 x1 (b) (5.139) Because w1 ≤ w2 , by virtue of hypothesis H(ϑ), we have
ϑ w1 (t) + M w1 (t) ≤ ϑ w2 (t) + M w2 (t) for all t ∈ T. (5.140) Using (5.141) in (5.140), it follows that x1 ∈C 1 (T ) is a lower solution for the problem ⎫ ⎧
− |x (t)|p−2 x (t) = f t, x(t), x (t) + ϑ w2 (t) + M w2 (t) ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ −M x(t) a.e. on T, (5.141) ⎪ ⎪ ⎪ ⎪
⎭ ⎩ x (0) ∈ ξ1 x(0) , −x (b) ∈ ξ2 x(b) (see Definition 5.2.11(b)). Also because ϕ ∈ C 1 (T ) is an upper solution for problem (5.111), we have ⎧ ⎫
⎨ − |ϕ (t)|p−2 ϕ (t) ≥ f t, ϕ(t), ϕ (t) + ϑ ϕ(t) a.e. on T, ⎬ . (5.142)
⎩ ⎭ ϕ (0) ∈ ξ1 ϕ(0) − R+ , −ϕ (b) ∈ ξ2 ϕ(b) − R+
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401
But f t, ϕ(t), ϕ (t) + ϑ ϕ(t) + M ϕ(t) ≥ f t, ϕ(t), ϕ (t) + ϑ w2 (t) + M w2 (t) a.e. on T (see hypothesis H(ϑ) and recall that w2 ∈ E). Using this inequality in (5.142), we obtain ⎧ ⎫
− |ϕ (t)|p−2 ϕ (t) ≥ f t, ϕ(t), ϕ (t) + ϑ w2 (t) + M w2 (t) ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ −M ϕ(t) a.e. on T, . ⎪ ⎪ ⎪ ⎪
⎩ ⎭ ϕ (0) ∈ ξ1 ϕ(0) − R+ , −ϕ (b) ∈ ξ2 ϕ(b) − R+ This means that ϕ ∈ C 1 (T ) is an upper solution for problem (5.142) (see Definition 5.2.11(a)). Then for problem (5.142), using truncation and penalization techniques based on the ordered upper–lower solution pair {ϕ, x1 }, as in Proposition 5.2.19 we obtain a solution x2 ∈ E1 = [x1 , ϕ] for problem (5.142). Evidently x2 ∈ S(w2 ) and x1 ≤ x2 . So we have satisfied condition (ii) of Theorem 5.2.20. Therefore we can apply that theorem and obtain x ∈ E such that x ∈ S(x). Clearly x ∈ C 1 (T ) solves problem (5.111). Next we establish the existence of a greatest and of a smallest solution in the order interval E.These solutions are called extremal solutions. So let C1 = x ∈ C 1 (T ) : x be a solution of (5.111) and x ∈ E . On L∞ (T ) we consider the partial order structure induced by the order cone L∞ (T )+ = {x ∈ L∞ (T ) : x(t) ≥ 0 a.e. on T }. So x ≤ y in L∞ (T ) if and only if x(t) ≤ y(t) a.e. on T . From Theorem 5.2.21, we know that under hypotheses H0 , H(f )2 , H(ξ), and H(ϑ), the set C1 is nonempty. Recall that a set C in a partially ordered set is a chain (or totally ordered subset), if for every x, y ∈ C, either x ≤ y or y ≤ x. PROPOSITION 5.2.22 If hypotheses H0 , H(f )2 , H(ξ), and H(ϑ) hold, then every chain C in C1 has an upper bound. PROOF: Because C ⊆ L∞ (T ) is bounded and L∞ (T ) is a complete lattice, we can find {xn }n≥1 ⊆ C such that sup xn = sup C ∈ L∞ (T ). Moreover, because n≥1
of the lattice structure of L∞ (T ), we can assume that the sequence {xn }n≥1 is increasing. Invoking the monotone convergence theorem, we have that xn −→ x in Lp (T ). Because of Lemma 5.2.15 and H(f )2 (iv) and H(ϑ), we have |xn (t)|p−2 xn (t) ≤ γ(t) a.e. on T, for all n ≥ 1 with γ ∈ Lp (T ),
⇒ |xn |p−2 xn n≥1 ⊆ W 1,p (0, b) is bounded. Also
x n ∞ , xn ∞ ≤ r=max{M1 , ϕ∞ , ψ∞ } for all n ≥ 1, hence {xn }n≥1 ⊆ W (0, b) is bounded. Therefore we may assume that 1,p
|xn |p−2 xn −→ v w
in W 1,p (0, b)
and
w
xn −→ u
in W 1,p (0, b) .
Evidently u = x and as in proof of Proposition 5.2.18, we can check that v = |x |p−2 x . Note that xn −→ x in C(T ) and |xn |p−2 xn −→ |x |p−2 x in C(T ). So we have xn (t) −→ x(t) and xn (t) −→ x (t) for all t ∈ T . From the dominated convergence theorem, we have
f ·, xn (·), xn (·) −→ f ·, x(·), x (·) in Lp (T ).
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5 Boundary Value Problems–Hamiltonian Systems
Also from the monotone convergence theorem we have
ϑ(xn ) + M xn −→ ϑ(x) + M x in Lp (T ).
Because M xn −→M x in C(T ), it follows that ϑ(xn )−→ ϑ(x) in Lp (T ). Therefore in the limit as n → ∞, we have
− |x (t)|p−2 x (t) = f t, x(t), x (t) + ϑ x(t) a.e. on T. Moreover, exploiting the fact that Gr ξ1 , Gr ξ2 are closed in R × R, we also have
x (0) ∈ ξ1 x(0) and − x (b) ∈ ξ2 x(b) . Therefore x ∈ C 1 (T ) is a solution of problem (5.111) and in addition x ∈ E. Hence x ∈ C1 and clearly is an upper bound of C. Recall that if (C0 , ≤) is a partially ordered set, we say that C0 is directed, if for each pair u1 , u2 ∈ C0 , we can find u3 ∈ C0 such that u1 ≤ u3 and u2 ≤ u3 . PROPOSITION 5.2.23 If hypotheses
H0 , H(f )2 , H(ξ), and H(ϑ) hold, then C1 ⊆ W 1,p (0, b) is directed, when W 1,p (0, b) is endowed with the pointwise order (order induced by C(T )). + PROOF: Let
x1 , x 2 ∈C1 and set x3 =max{x1 , x2 }. We have x3 =(x1 −x2 ) +x2 and so x3 ∈ W 1,p (0, b) . We introduce the following truncation and penalty functions. ⎧
⎪ x3 (t), x3 (t) if x < x3 (t) ⎪ ⎪
⎪ ⎪ ⎪ (t) if x > ϕ(t) ϕ(t), ϕ ⎨ u0 (t, x, y)= (x, M1 ) if x3 (t) ≤ x ≤ ϕ(t), y > M1 ⎪ ⎪ ⎪ (x, −M1 ) if x3 (t) ≤ x ≤ ϕ(t), y < −M1 ⎪ ⎪ ⎪ ⎩ (x, y) if otherwise
and
⎧ p−2 ⎪ ϕ(t) − |x|p−2 x ⎨ |ϕ(t)| β0 (t, x)= 0 ⎪ ⎩ |x3 (t)|p−2 x3 (t) − |x|p−2 x
if x > ϕ(t) if x3 (t) ≤ x ≤ ϕ(t) . if x < x3 (t).
Then we introduce the following modification of the nonlinearity f ,
f0 (t, x, y) = f t, u0 (t, x, y) . Note that for almost all t ∈ T , all x ∈ [x3 (t), ϕ(t)], and all |y| ≤ M1 we have f0 (t, x, y) = f (t, x, y). Moreover, for almost all t ∈ T and all x, y ∈ R, |f0 (t, x, y)| ≤ γr (t)
with γr ∈ Lp (T ),
r = max{M1 , ϕ∞ , ψ∞ }.
Also we introduce the truncation operator τ0 : W 1,p (0, b) −→W 1,p (0, b) defined by
5.2 Method of Upper–Lower Solutions 403 ⎧ ⎪ if x(t) < x3 (t) ⎨x3 (t) τ (x)(t)= x(t) if x3 (t) ≤ x(t) ≤ ϕ(t) . ⎪ ⎩ ϕ(t) if ϕ(t) < x(t)
1,p (0, b) : x3 (t) ≤ x(t) ≤ ϕ(t) for all t ∈ T . Given Let E0 = [x3 , ϕ] = x ∈ W w ∈ E0 , we consider the following auxiliary problem ⎧ ⎫
− |x (t)|p−2 x (t) = f0 t, x(t), x (t) + β0 t, x(t) − M τ0 (x)(t) ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ +ϑ w(t) + M w(t) a.e. on T, . (5.143) ⎪ ⎪ ⎪ ⎪
⎩ ⎭ x (0) ∈ ξ1 x(0) , −x (b) ∈ ξ2 x(b) Following the reasoning of Proposition 5.2.19, we can show that problem (5.143) 1 has a solution x ∈ C (T ) ∩ E0 . So we can define the solution multifunction S:E0 −→ 1,p
(0,b) \{∅} and then as in Theorem 5.2.21, via the use of Theorem 5.2.20 we 2W can produce x ∈ C 1 (T ) ∩ E0 such that it solves problem (5.111). Therefore x ∈ C1 and clearly x1 ≤ x, x2 ≤ x which proves that the set C1 is directed.
Now we use this proposition to establish the existence of extremal solutions for problem (5.111) in the order interval E = [ψ, ϕ]. THEOREM 5.2.24 If hypotheses H0 , H(f )2 , H(ξ), and H(ϑ) hold, then problem (5.111) has extremal solutions in the order interval E = [ψ, ϕ]. PROOF: By virtue of Proposition 5.2.22 and Zorn’s we can find x∗ ∈C1 a
lemma, maximal element for the pointwise ordering on W 1,p (0, b) . If x ∈ C1 , then because of Proposition 5.2.23, we can find y∈C1 such that x ≤ y, x∗ ≤ y. Because x∗ ∈ C1 is maximal, we must have x∗ = y. Then x ≤ x∗ and because x∈C1 was arbitrary, it follows that x∗ ∈ C1 is the greatest element of C1 . If on W 1,p (0, b) we use the partial order ≤1 defined by x ≤1 y
if and only if y(t) ≤ x(t)
a.e. on T,
then by the same argument we produce x∗ ∈ C1 , which is the smallest element of C1 . Hence {x∗ , x∗ } are the extremal solutions of (5.111) in E = [ψ, ϕ]. The framework of problem (5.111) is general and it incorporates as special cases standard boundary value problems. EXAMPLE 5.2.25 (a) Let I1 , I2 ⊆ R be nonempty closed intervals containing zero. For i = 1, 2, we set 0 if x ∈ Ii (the indicator function of Ii ). iIi (x)= +∞ if otherwise Set ξi = ∂iIi , i = 1, 2 (the subdifferential in the sense of convex analysis. Then problem (5.111) becomes ⎧ ⎫
− |x (t)|p−2 x (t) = f t, x(t), x (t) + ϑ x(t) a.e. on T, ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ x(0) ∈ I1 , x(b) ∈ I2 . (5.144) ⎪ ⎪ x (0)x(0) = sup[vx (0) : v ∈ I1 ] ⎪ ⎪ ⎩ ⎭ −x (b)x(b) = sup[−wx (b) : w ∈ I2 ]
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5 Boundary Value Problems–Hamiltonian Systems
(b) If I1 = I2 = {0}, then ξi (x) = R for all x ∈ R and for i = 1, 2. Hence problem (5.144) is the classical Dirichlet problem. (c) If I1 = I2 = R, then ξi (x) = {0} for all x ∈ R and for i = 1, 2. Hence problem (5.144) is the classical Neumann problem. (d) If ξ1 (x) = (1/β)x and ξ2 (x) = (1/γ)x, β, γ > 0, then problem (5.111) becomes the following Sturm–Liouville problem. ) *
− |x (t)|p−2 x (t) = f t, x(t), x (t) + ϑ x(t) a.e. on T, . x(0) − βx (0) = 0, x(b) + γx (b) = 0 (e) If u1 , u2 : R −→ R are two contractions and ξ1 = u1 − id, ξ2 = u2 − id (both maximal monotone functions), then problem (5.111) becomes * )
− |x (t)|p−2 x (t) = f t, x(t), x (t) + ϑ x(t) a.e. on T,
. x(0) + x (0) = u1 x(0) , x(b) − x (b) = u2 x(b) REMARK 5.2.26 The boundary conditions of problem (5.111) do not include the periodic ones. However, a careful analysis of the methods of the proof, reveals that they are also valid for the periodic problem. So Theorems 5.2.21 and 5.2.24 are also true for the periodic problem ) *
− |x (t)|p−2 x (t) = f t, x(t), x (t) + ϑ x(t) a.e. on T, . (5.145) x(0) = x(b), x (0) = x (b)
5.3 Degree-Theoretic Methods In this section we use degree theoretic methods to study second-order nonlinear boundary value problems. We consider two examples. The first is a nonlinear elliptic equation driven by the p-Laplacian with Dirichlet boundary conditions. Using degree-theoretic methods based on the degree map for operators of monotone type (see Section 3.3), we establish the existence of multiple nontrivial solutions. In the second example, we deal with a second-order scalar equation driven by the ordinary p-Laplacian with periodic boundary conditions and using the Leray–Schauder degree, we produce solutions under conditions of nonuniform nonresonance and of Landesman–Lazer type for the nonlinearity. We start with the elliptic Dirichlet boundary value problem. So let Z ⊆ RN be a bounded domain with a C 2 -boundary ∂Z. The problem under consideration is the following.
⎧ ⎫ p−2 Dx(z) = f z, x(z) a.e. on Z, ⎬ ⎨ −div Dx(z) . (5.146) ⎩ ⎭ x ∂Z = 0, 1 < p < ∞ The hypotheses on the nonlinearity f (z, x) are the following. H(f )1 : f : Z ×R −→ R is a function such that f (z, 0) = 0 a.e. on Z, f (z, x) ≥ 0 a.e. on Z for all x ≥ 0 and
5.3 Degree-Theoretic Methods
405
(i) For all x ∈ R, z −→ f (z, x) is measurable. (ii) For almost all z ∈ Z, x −→ f (z, x) is continuous. (iii) For almost all z ∈ Z and all x ∈ R, we have |f (z, x)| ≤ α(z) + c|x|p−1
with α ∈ L∞ (Z)+ , c>0.
(iv) There exists ϑ ∈ L∞ (Z)+ such that ϑ(z) ≤ λ1 a.e. on Z with strict inequality on a set of positive measure and f (z, x) ≤ ϑ(z) xp−1
lim sup x→+∞
uniformly for a.a. z ∈ Z,
and there exist η1 , η2 ∈ L∞ (Z)+ such that η1 (z) ≥ λ1 a.e. on Z with strict inequality on a set of positive measure and η1 (z) ≤ lim inf
x→−∞
f (z, x) f (z, x) ≤ lim sup p−2 ≤ η2 (z) |x|p−2 x x x→−∞ |x|
uniformly for a.a. z ∈ Z. (v) There exist η, η ∈ L∞ (Z)+ such that η(z) ≥ λ1 a.e. on Z with strict inequality on a set of positive measure and η(z) ≤ lim inf x→0+
and lim
x−→0−
f (z, x) f (z, x) ≤ lim sup p−1 ≤ η(z) xp−1 x + x→0
f (z,x) |x|p−2 x
=0
uniformly for a.a. z ∈ Z
uniformly for a.a. z ∈ Z.
REMARK 5.3.1 Hypotheses H(f )1 (iv) and (v) are nonresonance conditions at +∞ and 0+ , respectively, with respect to λ1 > 0 the principal eigenvalue of
− p , W01,p (Z) . In both conditions we allow partial interaction with λ1 > 0 (nonuniform nonresonance). Note that when p = 2, these conditions incorporate in the example the so-called asymptotically linear problems. Let τ+ : R −→ R+ be the positive truncation function defined by 0 if x ≤ 0 . τ+ (x) = x if x ≥ 0
x We set f+ (z, x) = f z, τ+ (x) and F+ (z, x) = 0 f+ (z, r)dr. Then we produce the energy functional ϕ+ : W01,p (Z) −→ R defined by
1 ϕ+ (x) = Dxpp − F+ z, x(z) dz for all x ∈ W01,p (Z). p Z
We know that ϕ+ ∈ C 1 W01,p (Z) . Also ϕ(x) = (1/p)Dxpp − Z F z, x(z) dz for all x∈W01,p (Z). Again ϕ∈C 1 (W01,p (Z)). PROPOSITION 5.3.2 If H(f )1 hold, then there exists x0 ∈ int C01 (Z) a local minimizer of ϕ.
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5 Boundary Value Problems–Hamiltonian Systems
PROOF: By virtue of hypothesis H(f )1 (iv), given ε > 0, we can find M =M (ε) > 0 such that
f+ (z, x) ≤ ϑ(z) + ε xp−1 for a.a. z ∈ Z and all x ≥ M. (5.147) On the other hand from hypothesis H(f )1 (iii) and because f+ (z, x)=0 for a.a. z ∈ Z and all x ≤ 0, we have f+ (z, x) ≤ αε (z)
for a.a. z ∈ Z all x ≤ M, with αε ∈ L∞ (Z)+ .
Combining (5.147) and (5.148), we obtain
f+ (z, x) ≤ ϑ(z) + ε |x|p−1 + αε (z) 1
⇒ F+ (z, x) ≤ ϑ(z) + ε |x|p + αε (z)|x| p
(5.148)
for a.a. z ∈ Z, all x ∈ R, for a.a. z ∈ Z, all x ∈ R. (5.149)
Then for all x ∈ W01,p (Z) we have
1 F+ z, x(z) dz ϕ+ (x) = Dxpp − p Z
1 1 ε p ≥ Dxp − ϑ(z)|x(z)|p − xpp − c1 Dxp p p Z p for some c1 > 0, (see (5.149)) ε 1 ξ− Dxpp − c1 Dxp ≥ p λ1
(5.150)
(see Lemma 5.1.3 and Theorem 4.3.47). We choose ε < λ1 ξ. Then from (5.150) it follows that ϕ+ is coercive. Exploiting the compact embedding of W01,p (Z) into Lp (Z), we can see that ϕ+ is weakly lower semicontinuous. So we can find x0 ∈ W01,p (Z) such that ϕ+ (x0 ) = inf ϕ+ (x) : x ∈ W01,p (Z) . Using hypothesis H(f )1 (v), we see that ϕ+ (tu1 ) < 0 for t > 0 small. Hence m = ϕ+ (x0 ) < 0 = ϕ+ (0) and so x0 = 0. As before from nonlinear regularity theory and the nonlinear maximum principle (see Theorems 4.3.35 and 4.3.37), we obtain that x0 ∈ int C01 (Z). Then by virtue of Theorem 5.1.27, it follows that x0 is a local minimizer of ϕ. In what follows by N we denote the Nemitsky operator corresponding to the Carath´eodory function f (z, x); that is,
N (x)(·) = f ·, x(·) for all x ∈ Lp (Z).
We know that N :Lp (Z)−→Lp (Z), (1/p)+(1/p ) = 1 is continuous and bounded (see hypotheses H(f )1 (i), (ii), and (iii)). Also let A : W01,p (Z) −→ W −1,p (Z) = 1,p ∗ W0 (Z) be the nonlinear operator defined by
Dxp−2 (Dx, Dy)RN dz for all x, y ∈ W01,p (Z). A(x), y = Z
Here by ·, · we denote the duality brackets for the pair W01,p (Z), W −1,p (Z) . We know that A is of type (S)+ (see Proposition 4.3.41). Due to the compact
5.3 Degree-Theoretic Methods
407
embedding of W01,p (Z) into Lp (Z), we can see that N is completely continuous. So it follows at once that the operator x −→ (A − N )(x) is of (S)+ –type and we can consider the degree in the sense of Definition 3.3.63. PROPOSITION 5.3.3 If hypotheses H(f )1 hold, then there exists R0 > 0 such that for all R ≥ R0 we have d(S)+ (A − N, BR , 0) = 0.
PROOF: Let K− : W01,p (Z) −→ W −1,p (Z) be defined by
p−1 K− (x) = x− (·)
for all x ∈ W01,p (Z).
Fix h ∈ L∞ (Z)+ , h(z) ≥ λ1 a.e. on Z with strict inequality on a set of positive measure. We consider the (S)+ -homotopy h1 : [0, 1] × W01,p (Z) −→ W −1,p (Z) defined by h1 (t, x) = A(x) − tN (x) + (1 − t)hK− (x). Claim: There exists R0 > 0 such that for all t ∈ [0, 1], all x ∈ ∂BR (0) and all R ≥ R0 , 0 = h1 (t, x). Suppose that the claim is not true. Then we can find {tn }n≥1 ⊆ [0, 1] and xn ⊆ W01,p (Z) such that tn −→ t
in
[0, 1], xn −→ ∞
and A(xn ) = tn N (xn ) − (1 − tn )hK− (xn )
for all n ≥ 1.
Acting with the test function x+ n and using hypothesis H(f )(iv), we obtain that + {xn }n≥1 ⊆ W01,p (Z) is bounded. Then we must have x− n −→ ∞. We set yn = − (x− n /xn ) and we may assume that w
yn −→ y and
in W01,p (Z),
|yn (z)| ≤ k(z) a.e. on Z
yn −→ y in Lp (Z), yn (z) −→ y(z) a.e. on Z for all n ≥ 1, with k ∈ Lp (Z)+ .
We have N (xn ) 1 A(x+ − (1 − tn )hynp−1 . n ) − A(yn ) = tn p−1 p−1 x− x− n n Acting with yn − y and passing to the limit as n → ∞, we obtain lim A(yn ), yn − y = 0. Because A is of type (S)+ , it follows that yn −→ y in W01,p (Z) and so y = 1. Also arguing in the proof of Proposition 5.1.2, we see that N (xn ) w −→ h xp−1
in Lp (Z),
with h = −gy p−1 where g ∈ L∞ (Z)+ , η1 (z) ≤ g(z) ≤ η2 (z) a.e on Z. Hence A(y) = gy p−1
(5.151)
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5 Boundary Value Problems–Hamiltonian Systems
with g = tg + (1 − t)h. From (5.151) we have
⎧ ⎫ p−2 Dy(z) = g(z)|y(z)|p−2 y(z) a.e. on Z, ⎬ ⎨ −div Dy(z) . ⎩ ⎭ y ∂Z = 0
(5.152)
Note that 1 = λ1 (λ1 ) > λ1 (g) and so from (5.152) it follows that y must change sign, a contradiction. This proves the claim. So we have d(S)+ (A − N, BR , 0) = d(S)+ (A + hK− , BR , 0)
for all R ≥ R0 .
(5.153)
But from Godoy–Gossez–Paczka [268], we have that d(S)+ (A + hK− , BR , 0) = 0
for all R > 0.
(5.154)
Using (5.154) in (5.153), we conclude that d(S)+ (A − N, BR , 0) = 0
for all R ≥ R0 .
Next we prove an analogous result for small balls. PROPOSITION 5.3.4 If hypotheses H(f )1 hold, then there exists 0 > 0 such that d(S)+ (A − N+ , B , 0) = 0 for all 0 < ≤ 0 .
PROOF: In what follows K+: Lp (Z) −→ Lp (Z) is a continuous monotone (hence maximal monotone too) map defined by
p−1 . K+ (x)(·) = x+ (·)
Let h3 : [0, 1] × W01,p (Z) −→ W −1,p (Z) be the homotopy defined by h3 (t, x) = A(x) − (1 − t)ηK+ (x) − tN (x), where η ∈ L∞ (Z)+ is as in hypothesis H(f )1 (v). Because of the complete continuity of the maps K+ , N: W01,p (Z) −→ W −1,p (Z), we can easily verify that h3 (t, x) is a homotopy of type (S)+ . Claim: There exists 0 > 0 such that for all t ∈ [0, 1], all x ∈ ∂B (0) and all 0 0. From (5.156) we infer that
N (xn ) p/(p−1) p/(p−1) dz ≤ c1 ypp xn p−1 Z N (x ) n ⇒ ⊆ Lp (Z) is bounded. xn p−1 n≥1
(5.156)
for all n ≥ 1, (5.157)
By passing to a subsequence if necessary, we may assume that N (xn ) w −→ h0 xn p−1
in Lp (Z) as n → ∞, with h0 ∈ Lp (Z).
(5.158)
As before, using hypothesis H(f )1 (v), we can check that η(z)y + (z) ≤ h0 (z) ≤ η(z)y + (z) ⇒ h0 (z) = g(z)y + (z)p−1
a.e. on Z
a.e. on Z,
with g∈L∞ (Z)+ such that η(z) ≤ g(z) ≤ η(z) a.e. on Z. Acting on (5.155) with the test function yn − y ∈ W01,p (Z), we have
N (xn ) A(yn ), yn − y = (1 − tn )η(yn+ )p−1 + tn (yn − y)(z)dz −→ 0, xn p−1 Z as n → ∞, (see (5.156)). Because A is of type (S)+ , it follows that yn −→ y
in W01,p (Z).
(5.159)
Passing to the limit as n → ∞ in (5.155) and using (5.158) and (5.159), we obtain A(y) = (1 − t)ηK+ (y) + tgK+ (y).
Recall that +
Dy (z)=
Dy(z) 0
(5.160)
a.e. on {y > 0} . a.e. on {y ≤ 0}
So we can write that A(y + ) = σK+ (y),
(5.161)
∞
where σ ∈ L (Z)+ is given by σ = (1 − t)η + tg. From (5.161) we have
⎧ ⎫ p−2 Dy(z) = σ(z)|y(z)|p−2 y(z) a.e. on Z, ⎬ ⎨ −div Dy(z) . ⎩ ⎭ y ∂Z = 0
(5.162)
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5 Boundary Value Problems–Hamiltonian Systems
Once again, the monotonicity of the principal eigenvalue on the weight function, implies that λ1 (σ) ≤ λ1 (η) < λ1 (λ1 ) = 1. 1,p + Hence because we see
of (5.162), that y ∈ W0 (Z) cannot be the principal 1,p eigenfunction of − p , W0 (Z), σ and so it must change sign, which is a contradiction, unless y + = 0. Therefore y ≤ 0 and so K+ (y) = 0, from which we have
A(y) = 0
(see (5.161)),
⇒ y = 0, a contradiction to the fact that y = 1, see (5.160). This proves the claim. Then the homotopy invariance property implies that d(S)+ (A − N, B , 0) = d(S)+ (A − ηK+ , B , 0)
for all 0 < ≤ 0 .
(5.163)
As in the proof of Proposition 5.3.3, we have d(S)+ (A − ηK+ , B , 0) = 0.
(5.164)
From (5.163) and (5.164), we conclude that d(S)+ (A − N+ , B , 0) = 0
for all 0 < ≤ 0 .
Now we can prove the existence of multiple nontrivial solutions of constant sign for problem (5.146). THEOREM 5.3.5 If hypotheses H(f )1 hold, then problem (5.146) has at least two distinct nontrivial solutions x0 ∈ int C01 (Z)+
and
x1 ∈ C01 (Z).
PROOF: Let x0 ∈ W01,p (Z) be the local minimizer of ϕ obtained in Proposition 5.3.2. We have ϕ (x0 ) = 0, ⇒ A(x0 ) = N (x0 ). Assume that x0 is an isolated critical point of ϕ+ (otherwise we have infinitely many critical points of ϕ+ , hence infinitely many nontrivial positive solutions of (5.146)). So we can find r0 > 0 such that ϕ(x0 ) < ϕ(y) and ϕ (y) = 0 for all y ∈ B r0 (x0 ) \ {x0 }, (5.165) where B r0 (x0 ) = y ∈ W01,p (Z) : y − x0 ≤ r0 . We show that for all r ∈ (0, r0 ), the following property holds inf ϕ(y) : y ∈ B r0 (x0 ) \ Br (x0 ) > ϕ(x0 ). (5.166) We argue by contradiction. So suppose that there exists r > 0 and a sequence {xn }n≥1 ⊆ B r0 (x0 ) \ Br (x0 ) such that
5.3 Degree-Theoretic Methods ϕ(xn ) ↓ ϕ(x0 )
411
as n → ∞.
Evidently, we may assume that w
xn −→ x
and
xn −→ x
in Lp (Z) as n → ∞.
(5.167)
Note that ϕ is weakly lower semicontinuous. So ϕ(x) ≤ lim inf ϕ(xn ) = ϕ(x0 ). n→∞
Because x ∈ B r0 (x0 ), it follows that ϕ(x)=ϕ(x0 ) and by virtue of (5.165), we have x = x0 . From the mean value theorem, we have x + x n 0 ∗ xn − x0 ϕ(x0 ) − ϕ = wn , , (5.168) 2 2 x0 where wn∗ =ϕ λn xn + (1 − λn ) xn + , λn ∈ [0, 1]. We know that 2 xn + x0 (5.169) + u∗n , wn∗ =A λn xn + (1 − λn ) 2 x0 , n ≥ 1. Using (5.169) in (5.168) and then passing to the limit where u∗n =N xn + 2 as n → ∞, we obtain xn + x0 , xn − x0 ≤ 0. lim sup A λn xn + (1 − λn ) 2 n→∞ We have
xn + x0 xn + x0 , λn xn + (1 − λn ) − x0 lim sup A λn xn + (1 − λn ) 2 2 n→∞ λ + 1 xn + x0 n lim sup lim sup A λn xn + (1 − λn ) ≤ 0. , xn − x0 2 2 n→∞ n→∞
Because A is of type (S)+ , it follows that λn xn + (1 − λn )
xn + x0 −→ x0 2
in W01,p (Z).
(5.170)
But note that λn xn + (1 − λn ) xn + x0 − x0 = (1 + λn ) xn − x0 ≥ r . 2 2 2
(5.171)
Comparing (5.170) and (5.171), we reach a contradiction. So (5.166) is valid. Let µ=inf ϕ(x) : x ∈ Br0 (x0 ) \ Br0 /2 (x0 ) − ϕ(x0 ). By (5.171) we have µ > 0. Also we introduce the set V = x ∈ B r0 (x0 ) : ϕ(x) − ϕ(x0 ) < µ . 2
Because x0 ∈ V , the set V is nonempty. It is clear that V is also bounded open. Fix r ∈ (0, r0 /2) with B r (x0 ) ⊆ V . Choose a number λ ∈ R such that (see (5.171)). 0 < λ < inf ϕ(x) : x ∈ Br0 (x0 ) \ Br (x0 ) − ϕ(x0 )
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5 Boundary Value Problems–Hamiltonian Systems
Note that λ < µ because r < r0 /2. Obviously V ⊆Br0 (x0 ) and V = x ∈ Br0 (x0 ) : ϕ(x) − ϕ(x0 ) < µ . From the definition of λ > 0, we see that x ∈ Br0 (x0 ) : ϕ(x) − ϕ(x0 ) ≤ λ ⊆ Br (x0 ) ⊆ V. Moreover, from (5.170) we see that ϕ (x) = 0
for all x ∈ Br0 (x0 ) with λ ≤ ϕ(x) − ϕ(x0 ) ≤ µ.
Therefore we can apply Corollary 3.3.70 and obtain d(S)+ (A − N, V, 0) = 1
(recall ϕ = A − N ).
(5.172)
From the excision property of the degree map, we have d(S)+ (A − N, V, 0) = d(S)+ (A − N, Br0 (x0 ), 0) ⇒ d(S)+ (A − N, Br0 (x0 ), 0) = 1
(see (5.172)).
(5.173)
Now we fix R0 > 0 in Proposition 5.3.3 sufficiently large and 0 > 0 in Proposition 5.3.4 sufficiently small in order to have x0 ∈ BR0 \ B r0 . Then we can find r > 0 such that Br (x0 )⊆BR0 and Br (x0 ) ∩ B0 =∅.
Let R > R0 and ∈ (0, 0 ). We claim that there exists x1 ∈ B R\ Br (x0 )∪B with
/ (A − N ) B R\ Br (x0 )∪B A(x1 ) = N (x1 ). Indeed, if this is not the case, then 0 ∈ and so invoking the additivity property of the degree map, we obtain d(S)+ (A−N, BR , 0) = d(S)+ (A−N, Br (x0 ), 0) + d(S)+ (A−N, B , 0), ⇒ 1 = 1 + (−1)
(see (5.172) and Propositions 5.3.3 and 5.3.4),
which is a contradiction. So we can find x1 ∈ W01,p (Z) such that x1 = x0 , x1 = 0
and
A(x1 ) = N (x1 ).
Therefore x1 > 0 is a nontrivial solution of problem (5.146) and from nonlinear regularity x1 ∈ C01 (Z)+ . The second boundary value problem that we study in this section is the following nonlinear second-order periodic differential equation. ) *
− |x (t)|p−2 x (t) = f t, x(t) + h(t) a.e. on T, . (5.174) x(0) = x(b), x (0) = x (b), h ∈ L1 (T ), 1 < p < ∞ We use degree-theoretic methods to prove two existence theorems for problem (5.174). In the first we assume conditions of nonuniform nonresonance between two successive eigenvalues of the negative scalar ordinary p-Laplacian with periodic boundary conditions. In the second existence theorem, at infinity the asymptotic f (z,x) are replaced by nonuniform nonresonance conditions for the “slopes” |x| p−2 x x=0
certain Landesman–Lazer conditions.
5.3 Degree-Theoretic Methods From Section 4.3, we know that the nonlinear eigenvalue problem ) * − |x (t)|p−2 x (t) = λ|x(t)|p−2 x(t) a.e. on T, , x(0) = x(b), x (0) = x (b), λ ∈ R, 1 < p < ∞
413
(5.175)
has a sequence of eigenvalues λ2n =
2nπ p p
b
where
x=0
1
πp =
2π(p − 1) p . p sin πp
These are all the eigenvalues of (5.175). Also if we consider the nonlinear weighted eigenvalue problem ) *
− |x (t)|p−2 x (t) = λ + m(t) |x(t)|p−2 x(t) a.e. on T , (5.176) 1 x(0) = x(b), x (0) = x (b), λ ∈ R, m ∈ L (T ), 1 < p < ∞, we [624]) that (5.176) has a double sequence of eigenvalues know(see Zhang λ 2n (m) n≥1 and λ2n (m) n≥0 such that −∞ < λ0 (m) < λ 2 (m) ≤ λ2 (m) < · · · < λ 2n (m) ≤ λ2n (m) < · · · −→ +∞ as n → ∞. If p = 2 (linear case), the two sequences and λ 2n (m) λ 2n (m) n≥1 and λ2n (m) n≥0 are all the eigenvalues of (5.176). If p = 2 (nonlinear case), we do not know if this is true. For the first existence result under the nonuniform nonresonance conditions in an arbitrary spectral interval, our hypotheses on the nonlinearity f (t, x), are the following. H(f )2 : f : T × R −→ R is a function such that (i) For all x ∈ R, t −→ f (t, x) is measurable. (ii) For almost all t ∈ T, x −→ f (t, x) is continuous. (iii) For every r > 0, there exists αr ∈L1 (T )+ such that for almost all t ∈ T and all x ∈ R with |x| ≤ r, we have |f (t, x)| ≤ αr (t). (iv) There exist functions ϑ1 , ϑ2 ∈ L∞ (T )+ such that for some n ≥ 0, we have λ2n ≤ ϑ1 (t) ≤ ϑ2 (t) ≤ λ2n+2
a.e. on T
with the first and third inequalities strict on sets (not necessarily the same) of positive measure and ϑ1 (t) ≤ lim inf |x|→∞
f (t, x) f (t, x) ≤ lim sup p−2 ≤ ϑ2 (t) |x|p−2 x |x| x |x|→∞
uniformly for almost all t ∈ T .
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5 Boundary Value Problems–Hamiltonian Systems
REMARK 5.3.6 Hypothesis H(f )2 (iv) is the nonuniform nonresonance condition in the spectral interval [λ2n , λ2n+2 ]. We start with a simple observation about the eigenvalues of problem (5.176). PROPOSITION 5.3.7 If ϑ1 , ϑ2 ∈ L∞ (T )+ are as in hypothesis H(f )2 (iv) and m∈L1 (T )+ satisfies ϑ1 (t) ≤ m(t) ≤ ϑ2 (t)
a.e. on T,
then all the eigenvalues of problem (5.176) are nonzero and do not have zero as a limit point. PROOF: By virtue of the monotonicity of the eigenvalues λ 2n (m) n≥1 and λ2n (m) n≥0 on the weight function m ∈ L1 (T )+ , we have λ2n (m) ≤ λ2n (ϑ1 ) < λ2n (λ2n ) = 0
and
0 = λ 2n (λ2n ) < λ 2n (ϑ2 ) ≤ λ 2n (m).
(5.177) (5.178)
Also we know that if λ ∈ R is an eigenvalue of (5.176), then !
λ 2k (m), λ2k (m) ∪ − ∞, λ0 (m) λ∈
(5.179)
k≥1
(see Zhang [624]). Combining (5.179) with (5.177) and (5.178), we see that λ = 0. So if by σ(p) we denote the spectrum of the nonlinear weighted eigenvalue problem (5.176), we have that 0 ∈ / σ(p). Suppose that we can find {λn }n≥1 ⊆ σ(p) such that λn −→ 0. We can find 1 un ∈ Cper (T ) = x ∈ C 1 (T ) : x(0) = x(b), x (0) = x (b) such that un = 0 and
− |un (t)|p−2 un (t) = λ + m(t) |un (t)|p−2 un (t)
a.e. on T.
(5.180)
Because of the (p − 1)-homogeneity of problem (5.180), we may assume that 1,p for all n ≥ 1 ( · denotes the norm of the Sobolev space Wper un =1 (0, b) = x∈
W 1,p (0, b) : x(0)=x(b) . So by passing to a suitable subsequence if necessary, we may assume that w 1,p
un −→ u in Wper (0, b) and un −→ u in C(T ) as n → ∞.
∗ 1,p 1,p (0, b) , Wper (0, b) By ·, · we denote the duality brackets for the pair Wper
∗ 1,p 1,p and we let A:Wper (0, b) −→Wper (0, b) be the nonlinear operator defined by
A(x), y=
b
|x (t)|p−2 x (t)y (t)dt
1,p
for all x, y ∈ Wper (0, b) .
0
We know that A is monotone hence it is maximal monotone.
demicontinuous, 1,p (0, b) −→Lp (T ), (1/p) + (1/p ) = 1 we denote the operator Also by K :Wper K(x)(·) = |x(·)|p−2 x(·).
5.3 Degree-Theoretic Methods
415
Evidently K is bounded continuous. Then in terms of A and K, we can equivalently rewrite (5.180) as the following abstract operator equation A(un )=(λn + m)K(un ), n ≥ 1,
b
⇒ A(un ), un − u= λn + m(t) |un (t)|p−2 un (t)(un − u)dt,
n ≥ 1.
0
(5.181) Note that
λn + m(t) |un (t)|p−2 un (t)(un − u)(t)dt −→ 0.
b
0
So we obtain lim A(un ), un − u = 0.
(5.182)
n→∞
But A being maximal monotone, it is generalized pseudomonotone and so from (5.182) it follows that A(un ), un −→ A(u), u , ⇒ un p −→ u p . Because un −→ u in Lp (T ) and Lp (T ) being uniformly convex it has the Kadec– 1,p Klee property, it follows that un −→ u in Lp (T ) and so un −→ u in Wper (0, b) . Hence u = 1 and so u = 0. Passing to the limit as n → ∞ in (5.181), we obtain w
A(u) = mK(u). From (5.183) it follows that ) − |u (t)|p−2 u (t) = m(t)|u(t)|p−2 u(t) u(0) = u(b), u (0) = u (b)
(5.183)
a.e. on T,
* .
(5.184)
Because u = 0, from (5.184) it follows that 0 ∈ σ(p), which contradicts the first part of the proof. From the above proposition we have that 0 ∈ / σ(p) and we can find ε0 ∈ (0, 1) such that (−ε0 , ε0 ) ∩ σ(p) = ∅. We fix ε ∈ (0, ε0 ) and we consider the following periodic problem. ) * − |x (t)|p−2 x (t) + ε|x(t)|p−2 x(t) = h(t) a.e. on T, . (5.185) x(0) = x(b), x (0) = x (b), h ∈ L1 (T ), 1 < p < ∞ PROPOSITION 5.3.8 For every h∈L1 (T ) problem (5.185) has a unique solution 1 1,p Sε (h) ∈ Cper (T ) and the solution map Sε : L1 (T ) −→ Wper (0, b) is completely
w 1 1,p (0, b) . continuous; that is, if hn −→ h in L (T ), then Sε (hn )−→Sε (h) in Wper
∗ 1,p 1,p (0, b) −→Wper (0, b) be as in the proof of Proposition PROOF: Let A, K :Wper
∗ 1,p 5.3.7 (recall that Lp (T ) is embedded compactly in Wper (0, b) ). We consider the
∗ 1,p 1,p (0, b) −→Wper (0, b) defined by nonlinear operator Lε :Wper
416
5 Boundary Value Problems–Hamiltonian Systems Lε (x) = A(x) + εK(x)
1,p
for all x ∈ Wper (0, b) .
hence it is maximal monoClearly Lε is strictly monotone and demicontinuous, 1,p tone. Moreover, for every x∈Wper (0, b) , we have Lε (x), x = x pp + εxp ⇒ Lε is coercive.
1,p So Lε is surjective and we find x ∈ Wper (0, b) such that Lε (x) = h.
(5.186)
Due to the strict monotonicity of Lε , the solution x ∈ (0, b) of (5.191) is unique and we denote it by S (h). Thus we have defined the solution map Sε : ε 1,p L1 (T ) −→ Wper (0, b) , which to each forcing term h ∈ L1 (T ) assigns the unique solution of problem (5.185). We show that Sε is completely continuous. To this end w 1 let hn −→ h in L1 (T ) and set xn = Sε (hn ) ∈ Cper (T ), n ≥ 1. We have ) * − |xn (t)|p−2 xn (t) + ε|xn (t)|p−2 xn (t) = hn (t) a.e. on T, , xn (0) = xn (b), xn (0) = xn (b) 1,p Wper
(5.187) ⇒ A(xn ) + εK(xn ) = hn , n ≥ 1.
∗ 1,p 1,p 1 Taking duality brackets in Wper (T ), we (0, b) , Wper (0, b) with xn ∈ Cper obtain
b hn (t)xn (t)dt ≤ c1 xn , εxn p ≤ xn pp + εxn pp = 0
for some c1 > 0, all n ≥ 1,
1,p
(0, b) ⇒ {xn }n≥1 ⊆ Wper
is bounded.
So we may assume that w 1,p
xn −→ x in Wper (0, b)
and
xn −→ x in C(T ).
1,p As before, acting on (5.187) with the test function xn − x ∈ Wper (0, b) and p using the Kadec–Klee property of the space L (T ), we can show that xn −→ x in 1,p Wper (0, b) . Passing to the limit as n → ∞ in (5.187), we obtain A(x) + εK(x) = h,
⇒ − |x (t)|p−2 x (t) + ε|x(t)|p−2 x(t) = h(t)
a.e. on T,
x(0) = x(b), x (0) = x (b), ⇒
x = Sε (h).
From Urysohn’s criterion for the convergence
of sequences, we conclude for the 1,p original sequence xn = Sε (hn ) n≥1 ⊆ Wper (0, b) , that 1,p
xn −→ x = Sε (h) in Wper (0, b)
⇒ Sε is indeed completely continuous.
5.3 Degree-Theoretic Methods
417
THEOREM 5.3.9 If hypotheses H(f )2 hold, then for every h ∈ L1 (T ) problem 1 (5.174) has a solution x ∈ Cper (T ).
1,p PROOF: Let Nf : Wper (0, b) −→ L1 (T ) be the Nemitsky operator corresponding to the function f (t, x); that is,
Nf (x)(·) = f ·, x(·) .
1,p Because of the compact embedding of Wper (0, b) into C(T ), we see that Nf is compact. Let g ∈ L1 (T ) such that ϑ ϑ 2 (t) a.e.
1 (t) ≤ g(t) ≤1,p on T and consider the 1,p compact homotopy h : [0, 1] × Wper (0, b) −→ Wper (0, b) defined by
h(β, x) = Sε εK(x) + βNf (x) + βh + (1 − β)gK(x) . Claim: exists R0 > 0 such that h(β, x) = x for all β ∈ [0, 1] and all x ∈
There 1,p Wper (0, b) with x = R and all R ≥ R0 . We argue indirectly. Suppose that
the claim is not true. Then we can find 1,p (0, b) such that {βn }n≥1 ⊆ [0, 1] and {xn }n≥1 ⊆ Wper βn −→ β
in [0, 1],
xn −→ ∞
and
for all n ≥ 1.
xn = h(βn , xn )
We have A(xn ) = βn Nf (xn ) + βn h + (1 − βn )gK(xn ) Set yn = xn /xn , n ≥ 1. We may assume that w 1,p
yn −→ y in Wper (0, b) and yn −→ y
for all n ≥ 1.
(5.188)
in C(T ) as n → ∞.
Using hypothesis H(f )2 (iii) and (iv), we can check that N (x ) n f ⊆ L1 (T ) is uniformly integrable. xn p−1 n≥1 So by the Dunford–Pettis theorem, we may assume (at least for a subsequence), that Nf (xn ) w −→ w in L1 (T ). (5.189) xn p−1 As in previous proofs, we can check that w(t) = g0 (t)|y(t)|p−2 y(t)
a.e. on T,
(5.190)
with g0 ∈ L1 (T ) such that ϑ1 (t) ≤ g0 (t) ≤ ϑ2 (t) a.e. on T . Recall that A(xn ) = βn Nf (xn ) + βn h + (1 − βn )gK(xn ), Dividing with xn
p−1
A(yn ) = βn
n ≥ 1.
, we obtain
Nf (xn ) h + βn + (1 − βn )gK(yn ), xn p−1 xn p−1
n ≥ 1.
(5.191)
1,p (0, b) and using the As before, acting with the test function yn − y ∈ Wper
1,p Kadec–Klee property, we can conclude that yn −→ y in Wper (0, b) . So if we pass to the limit as n → ∞ in (5.191), we obtain
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5 Boundary Value Problems–Hamiltonian Systems
A(y) = βg0 + (1 − β)g K(y).
Let g = βg0 + (1 − β)g. Then g ∈ L1 (T )+ and ϑ1 (t) ≤ g(t) ≤ ϑ2 (t) a.e. on T . Also ) * − |y (t)|p−2 y (t) = g(t)|y(t)|p−2 y(t) a.e. on T, . (5.192) y(0) = y(b), y (0) = y (b) Note that y = 1 and so y = 0. Hence from (5.192), we infer that 0 ∈ σ(p) which contradicts Proposition 5.3.8. This proves the validity of the claim. Then from the homotopy invariance of the Leray–Schauder degree (see Section 3.3), we have
d I − Sε ◦ (ε + g)K, BR , 0 = d I − Sε ◦ (εK + Nf + h), BR , 0
for all R ≥ R0 . (5.193)
From the choice of ε > 0 and Proposition 5.3.8, we see that x = Sε ◦ (ε + g)K(x)
for all x = R ≥ R0 .
Moreover, it is clear that the map x −→ Sε ◦ (ε + g)K(x) is odd. So invoking Borsuk’s theorem (see Theorem 3.3.41), we have
d I − Sε ◦ (ε + g)K, BR , 0 = 0 for R ≥ R0 ,
⇒ d I − Sε ◦ (εK + Nf + h), BR , 0 = 0 for R ≥ R0
(see (5.193)).
Therefore by the solution property of the Leray–Schauder degree, we can find x ∈ BR such that x = Sε ◦ (εK + Nf + h)(x), ⇒
A(x) = Nf (x) + h,
⇒ − |x (t)|p−2 x (t) = f t, x(t) + h(t)
a.e. on T,
x(0) = x(b), x (0) = x (b), ⇒
1 (T ) is a solution of problem (5.174). x ∈ Cper
In the second existence theorem for problem (5.174), we employ a Landesman– Lazer type condition instead of the nonuniform nonresonance condition in the spectral interval [λ2n , λ2n+2 ]. More precisely, our hypotheses on the nonlinearity f (t, x) are the following. H(f )3 : f : T × R −→ R is a function such that (i) For all x ∈ R, t −→ f (t, x) is measurable. (ii) For almost all t ∈ T, x −→ f (t, x) is continuous. (iii) For every r > 0, there exists αr ∈ L1 (T )+ such that for almost all t ∈ T and all x ∈ R with |x| 0 large enough so that h(β, x) = x for all β∈[0, 1] and all x∈Wper (0, b) with x=R. We proceed by contradiction. So suppose that we can find {βn }n≥1 ⊆[0, 1] and 1,p {xn }n≥1 ⊆Wper (0, b) such that
βn −→ β in [0, 1], xn −→ ∞ and xn = Sε ◦ βn (εK + Nf + h) (xn ),
1,p
for all n ≥ 1. We set yn = xn /xn , n ≥ 1. We may assume that w 1,p
yn −→ y in Wper (0, b) and yn −→ y in C(T ) as n → ∞. For every n ≥ 1, we have A(xn ) + εK(xn ) = εβn K(xn ) + βn Nf (xn ) + βn h,
n ≥ 1.
Dividing by xn p−1 , we obtain A(yn ) + εK(yn ) = εβn K(yn ) + βn
Nf (xn ) h + βn , xn p−1 xn p−1
From hypotheses H(f )3 (iii) and (iv), we have that Nf (xn ) −→ 0 xn p−1
in L1 (T ).
Also we have
and
h −→ 0 in L1 (T ) xn p−1
b yn (t)(yn − y)(t)dt −→ 0. 0
n ≥ 1.
(5.194)
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5 Boundary Value Problems–Hamiltonian Systems
1,p So if in (5.194) we act with the test function yn − y ∈ Wper (0, b) and then pass to the limit as n → ∞, we obtain lim A(yn ), yn − y = 0, 1,p
(0, b) . ⇒ yn −→ y in Wper Therefore from (5.194) in the limit as n → ∞, we obtain A(y) = (β − 1)εK(y),
(5.195)
with y = 1, (y = 0). Because β ∈ [0, 1], ε > 0, from (5.195) it follows that β = 1 and so A(y) = 0, hence y = ξ ∈ R, ξ = 0 (recall that y = 0). First suppose that ξ > 0. If in (5.194) we act with the test function y = ξ, we obtain
b
b
(1 − βn )εxn p−1 ξ |yn |p−2 yn dt = βn ξ (5.196) Nf (xn ) + h dt. 0
0
Because βn ∈ [0, 1], βn −→ β, yn −→ξ > 0 in C(T ), from (5.196) it follows that we can find n0 ≥ 1 such that
b
for all n ≥ n0 . (5.197) Nf (xn ) + h dt > 0 0
Note that because we have assumed that ξ > 0, for all t ∈ T we have that xn (t) −→ +∞ as n → ∞. We claim that this convergence is uniform in t ∈ T . To this end let 0 < δ < ξ. Because yn −→ ξ in C(T ), we can find n1 = n1 (δ) ≥ 1 such that for all t ∈ T and all n ≥ n1 we have |yn (t) − ξ| < δ, hence 0 < ξ − δ < yn (t). Also because 9 > 0 we can find n2 = n2 (M 9) ≥ 1 such that xn ≥ M 9 for all xn −→ ∞, given M n ≥ n2 . So for all n ≥ n2 and all t ∈ T , we have xn (t) xn (t) ≥ = yn (t) > ξ − δ = γ > 0, 9 xn M 9 ⇒ xn (t) ≥ γ M for all n ≥ n2 and all t ∈ T. 9 > 0 was arbitrary, we conclude that xn (t) −→ +∞ uniformly in t ∈ T as Because M n → ∞. This then by virtue of hypothesis H(f )3 (iv) implies that
lim sup f t, xn (t) = η+ (t) uniformly for a.a. t ∈ T. n→∞
Then by Fatou’s lemma and using (5.197), we obtain
b
b
b η+ (t)dt, − h(t)dt ≤ lim sup Nf (xn )dt ≤ n→∞
0
0
0
a contradiction to the hypotheses of the theorem. If we assume ξ < 0, then arguing in a similar fashion we reach the contradiction
b
b η− (t)dt ≤ − h(t)dt. 0
0
Therefore we can find R > 0 large enough such that
5.4 Nonlinear Eigenvalue Problems h(β, x) = x
421
for all β ∈ [0, 1] and all x = R.
Invoking the homotopy invariance property of the Leray–Schauder degree, we have
d I − h(0, ·), BR , 0 = d I − h(1, ·), BR , 0 .
Note that h(0, ·) = 0 and so d I − h(0, ·), BR , 0 = d(I, BR , 0) = 1, hence
d I − h(1, ·), BR , 0 = 1. From the solution
property of the Leray–Schauder degree, it follows that thee 1,p exists x ∈ Wper (0, b) with x ≤ R such that x = h(1, x) = Sε ◦ (εK + Nf + h)(x), ⇒
A(x) = Nf (x) + h,
⇒ − |x (t)|p−2 x (t) = f t, x(t) + h(t)
a.e. on T,
x(0) = x(b), x (0) = x (b), ⇒
1 (T ) is a solution of problem (5.174). x ∈ Cper
REMARK 5.3.11 Theorems 5.3.9 and 5.3.10 are still valid if we consider the more general situation in which instead of the ordinary p-Laplacian differential operator, we have an operator of the form x −→ ϕ(x ) , with ϕ : R −→ R a suitable homeomorphism.
5.4 Nonlinear Eigenvalue Problems In this section we study two perturbed eigenvalue problems, in which the perturbation is nonlinear. The first problem is driven by the p-Laplacian and the second by the Laplacian differential operator. For both problems we establish the existence of nontrivial smooth solutions, as the parameter λ ∈ R moves in a certain interval. In fact for the second problem (the semilinear one), we also have a negative result, showing that no positive solution exists, when λ is out of the aforementioned interval. This is done using the so-called Pohozaev’s identity, which we also prove here. In this section Z ⊆ RN is a bounded domain with C 2 -boundary ∂Z. We start with the following perturbed weighted eigenvalue problem.
⎧ ⎫ p−2 Dx(z) − λm(z)|x(z)|p−2 x(z) = f z, x(z) a.e. on Z, ⎬ ⎨ −div Dx(z) . ⎩ ⎭ x ∂Z = 0, 1 < p < ∞ (5.198) The hypotheses on the weight function m(z) and the perturbation term f (z, x) are the following. H(m): m ∈ L∞ (Z)+ ={m∈L∞ (Z)+ : m(z) ≥ 0
a.e. on Z}, m = 0.
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5 Boundary Value Problems–Hamiltonian Systems
H(f )1 : f : Z × R −→ R is a function such that f (z, 0) = 0 a.e. on Z and (i) For all x ∈ R, z −→ f (z, x) is measurable. (ii) For almost all z ∈ Z, x −→ f (z, x) is continuous. (iii) For almost all z ∈ Z and all x ∈ R, we have
and
|f (z, x)| ≤ α(z) + c|x|r−1 with α ∈ L∞ (Z)+ , c > 0 Np if p < N ∗ N −p . p 0 such that for almost all z ∈ Z and all |x| ≥ M , we have µF (z, x) ≤ xf (z, x) and (v) lim sup x→0
f (z,x) xp−2 x
ess inf F (·, M ), ess inf F (·, −M ) > 0.
≤ 0 uniformly for almost all z ∈ Z.
REMARK 5.4.1 Hypothesis H(f )1 (iv) is the so-called Ambrosetti–Rabinowitz condition (AR-condition for short). However, note that in contrast to the standard AR-condition, we do not require that F (z, x) > 0 for almost all z ∈ Z and all |x| ≥ M . Instead we only assume that ess inf F (·, M ), ess inf F (·, −M ) > 0. The following function satisfies all the hypotheses in H(f )1 . For simplicity we drop the z-dependence and so we have f (x) = |x|r−2 x − c|x|p−2 x with c ≥ 0 and p < r < p∗ . In this example the AR-condition (see hypothesis H(f )1 (iv)), is satisfied for µ = r and all M > 0. We consider the functional ϕλ : W01,p (Z) −→ R defined by
1 λ ϕλ (x)= Dxpp − m(z)|x(z)|p dz − F z, x(z) dz for all x ∈ W01,p (Z). p p Z Z We know that this functional is C 1 on W01,p (R). Next we show that it satisfies the Cerami condition (C-condition). PROPOSITION 5.4.2 If hypotheses H(m) and H(f )1 hold, then ϕλ satisfies the C-condition. PROOF: Let {xn }n≥1 ⊆ W01,p (Z) be a sequence such that |ϕλ (xn )| ≤ M and
(1 +
for some M > 0, all n ≥ 1
xn )ϕλ (xn )
−→ 0 as n → ∞.
(5.199)
5.4 Nonlinear Eigenvalue Problems
423
In the sequel by ·, · we denote the duality brackets for the pair W01,p (Z), W −1,p (Z) , (1/p) + (1/p ) = 1. Let A : W01,p (Z) −→ W −1,p (Z) be the nonlinear operator defined by
A(x), y = Dx(z)p−2 Dx(z), Dy(z) RN dz for all x, y ∈ W01,p (Z).
Z
We know that A is of type (S)+ (see Proposition 4.3.41). Moreover, we have ϕλ (xn ) = A(xn ) − λm|xn |p−2 xn − Nf (xn ),
where Nf (xn )(·) = f ·, xn (·) (the Nemitsky operator corresponding to the nonlin earity f ). Note that Nf :W01,p (Z)−→Lr (Z)⊆W −1,p (Z), where r1 + r1 = 1 (recall ∗ p < r < p ). From the choice of the sequence {xn }n≥1 ⊆ W01,p (Z)(see (5.199)), we have | ϕλ (xn ), xn | ≤ εn with εn ↓ 0,
(5.200) ⇒ −Dxn pp + λ m|xn |p dz + xn Nf (xn )dz ≤ εn . Z
Z
Also p ϕλ (xn ) ≤ pM for all n ≥ 1,
⇒ Dxn pp −λ m|xn |p dz − pF z, xn (z) dz ≤ pM . Z
(5.201)
Z
Adding (5.200) and (5.201), we obtain
− pF z, xn (z) −xn (z)f z, xn (z) dz ≤ εn + pM
Z
⇒ − µF z, xn (z) −xn (z)f z, xn (z) dz Z
+ (µ − p) F z, xn (z) dz ≤ εn + pM .
for all n ≥ 1,
(5.202)
Z
Note that
µF z, xn (z) −xn (z)f z, xn (z) dz
−
Z =−
−
µF z, xn (z) −xn (z)f z, xn (z) dz
{|xn |≥M }
µF z, xn (z) −xn (z)f z, xn (z) dz.
{|xn |<M }
(5.203) From hypothesis H(f )1 (iv), we have
− µF z, xn (z) −xn (z)f z, xn (z) dz ≥ 0. {|xn |≥M }
Also from hypothesis H(f )1 (iii) and the mean value theorem, we have
(5.204)
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5 Boundary Value Problems–Hamiltonian Systems
−
µF z, xn (z) −xn (z)f z, xn (z) dz ≥ −c1 for some c1 > 0, for all n ≥ 1.
{|xn |<M }
(5.205) Using (5.204) and (5.205) in (5.203) we obtain
− µF z, xn (z) −xn (z)f z, xn (z) dz ≥ −c1
for all n ≥ 1.
(5.206)
Z
For almost all z ∈ Z and all |x| ≥ M , the function s −→ (1/sµ )F (z, sx) is C 1 on R+ \ {0}. So we have µ 1 d d1 F (z, sx) = − F (z, sx) + F (z, sx) x. ds sµ sµ+1 sµ dx By virtue of the mean value theorem, for s > 1 we can find η ∈ (1, s) such that µ 1 d 1 F (z, sx) − F (z, x) = − µ+1 F (z, ηx) + µ F (z, ηx) x (s − 1) µ s η η dx d s − 1 = µ+1 − µF (z, ηx) + F (z, ηx) ηx η dx ≥ 0 for a.a. z ∈ Z and all |x| ≥ M, (see hypothesis H(f )1 (iv)), ⇒ sµ F (z, x) ≤ F (z, sx) for a.a. z ∈ Z, all |x| ≥ M and all s ≥ 1. (5.207) Using (5.207), we see that for almost all z ∈ Z, we have
x xµ F (z, M ) if x ≥ M M ≥ F (z, x) = F z, M Mµ
|x| |x|µ and F (z, x) = F z, F (z, −M ) if x ≤ M. (−M ) ≥ M Mµ We set ξ = (1/M µ ) min ess inf F (·, M ), ess inf F (·, −M ) > 0 (see hypothesis H(f )1 (iv)). So we have F (z, x) ≥ ξ|x|µ
for a.a. z ∈ Z and all |x| ≥ M.
(5.208)
From (5.208), hypothesis H(f )1 (iii), and the mean value theorem, we conclude that F (z, x) ≥ ξ|x|µ − c2
for some c2>0, almost all z ∈ Z, and all x ∈ R.
(5.209)
Using (5.209) and because µ > p, we obtain
for some c3 , c4 > 0, all n ≥ 1. (µ − p) F z, xn (z) dz ≥ c3 xn µ µ − c4
(5.210)
Z
Returning to (5.202) and using (5.206) and (5.210), we obtain c3 xn µ µ ≤ c5
for some c5 > 0, all n ≥ 1,
⇒ {xn }n≥1 ⊆ L (Z) is bounded, µ
⇒ {xn }n≥1 ⊆ Lp (Z) is bounded From (5.199) we have
(because p < µ).
(5.211)
5.4 Nonlinear Eigenvalue Problems µ ϕλ (xn ) − ϕλ (xn ), xn ≤ µM + εn for all n ≥ 1, µ µ p p m|xn | dz − 1 Dxn p − λ −1 ⇒ p p Z
− µF z, xn (z) −xn (z)f z, xn (z) dz ≤ µM + εn , µ Z ⇒ − 1 Dxn pp ≤ c6 for some c5 > 0, all n ≥ 1, p
425
(5.212)
(see (5.206) and (5.211)). Recall that µ > p. So from (5.212) and Poincar´e’s inequality, it follows that {xn }n≥1 ⊆ W01,p (Z) is bounded. Hence we may assume that w
and
xn −→ x
in W01,p (Z), xn −→ x in Lp (Z)
xn −→ x
in Lr (Z)
(because r < p∗ ).
From (5.199), we have
εn A(xn ), xn − x−λ m|xn |p−2 xn (xn − x)dz− f (z, xn )(xn − x)dz ≤(5.213) Z
Z
for all n ≥ 1. Note that
m|xn |p−2 xn (xn − x)dz −→ 0 λ
(because xn −→ x in Lp (Z) and
Z
because of (5.211)), f (z, xn )(xn − x)dz −→ 0
and Z
Nf (xn )
n≥1
(because xn −→ x in Lr (Z) and
⊆Lr (Z) is bounded; see hypothesis H(f )1 (iii)).
So from (5.213) it follows that lim sup A(xn ), xn − x ≤ 0.
(5.214)
Because A is of type (S)+ (see Proposition 4.3.41), from (5.214) it follows that xn −→ x in W01,p (Z).
Thus we conclude that ϕλ satisfies the C-condition.
Let u2 ∈ C01 (Z) be an eigenfunction corresponding to the second eigenvalue λ2 > 0 of − p , W01,p (Z) (see Section 4.3). Let
and
Y = span{u1 , u2 } HR = y ∈ Y : y = t1 u1 + t2 u2 , t1 ∈ R, t2 ≥ 0, y ≤ R for R > 0. (5.215)
Evidently HR is a hemisphere in Y . We consider the boundary of this hemisphere in Y . So we have 0 = y∈HR :y = t1 u1 , t1 ∈ R, y ≤ R ∪ y∈HR : y=R . (5.216) HR
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5 Boundary Value Problems–Hamiltonian Systems
0 The boundary HR is the union of two sets. The first set in this union is a circle and its interior and the second set is the dome of the hemisphere. Also we set
K = x ∈ W01,p (Z) : Dxpp = λ2 m|x|p dz . (5.217) Z
This is a closed, pointed, symmetric cone in
W01,p (Z).
For > 0, we set
D = K ∩ ∂B
where ∂B = {x ∈ W01,p (Z) : x = }. PROPOSITION 5.4.3 If hypotheses H(m), H(f )1 hold and λ ∈ [λ1 , λ2 ), then we can find R > 0 large enough such that ϕλ H 0 ≤ 0. R
PROOF: Let y ∈ Y . Then we have
1 λ m|y|p dz − F z, y(z) dz ϕλ (y) = Dypp − p p Z Z 1 p µ ≤ Dyp − ξyµ + c7 for some c7>0 independent of y ∈ Y (5.218) p (see (5.209)). Because Y ⊆ C01 (Z) is finite-dimensional, all the Lp and Sobolev norms on it are equivalent. So we can find c8 > 0 such that µ c8 Dyµ p ≤ yµ
for all y ∈ Y.
Using this in (5.218), we obtain ϕλ (y) ≤
1 Dypp − c9 Dyµ p + c7 p
for some c9 > 0 and all y ∈ Y.
(5.219)
Because µ > p, from (5.219), using Poincar´e’s inequality, we deduce that ϕλ (y) −→ −∞ as y −→ ∞, (5.220) that is, ϕλ Y is weakly anticoercive. Also for t > 0 and because u1 ∈ int C01 (Z)+ , we have
tp tp ϕλ (tu1 ) = Du1 pp − λ m|u1 |p dz − F (z, tu1 )dz p p Z Z λ tp for some c10 > 0 Du1 pp − tµ u1 µ 1− ≤ µ + c10 p λ1 (see (5.209)) tp λ ≤ for some c11 > 0. Du1 pp − tµ c11 Du1 µ 1− p + c10 p λ1 (5.221) From (5.220) and (5.221) and because λ1 ≤ λ < λ2 and p < µ, we deduce that there exists R > 0 large such that ϕλ H 0 ≤ 0 (see (5.216)). R
5.4 Nonlinear Eigenvalue Problems
427
PROPOSITION 5.4.4 If H(m), H(f )1 hold and 0 < λ < λ2 , then we can find > 0 small such that ϕλ D ≥ ξ0 > 0. PROOF: By virtue of hypothesis H(f )1 (v), given ε > 0, we can find δ = δ(ε) > 0 such that f (z, x) ≤ εxp−1
for a.a. z ∈ Z and all x ∈ [0, δ]
f (z, x) ≥ ε|x|p−2 x
and
for a.a. z ∈ Z and all x ∈ [−δ, 0].
From these inequalities, after integration we obtain F (z, x) ≤
ε p |x| p
for a.a. z ∈ Z and all x ∈ [−δ, δ].
(5.222)
On the other hand from hypothesis H(f )1 (iii) and the mean value theorem, we have F (z, x) ≤ cε |x|r
for some cε > 0, a.a. z ∈ Z and all |x| > δ.
(5.223)
Combining (5.222) and (5.223), we obtain F (z, x) ≤
ε p |x| + cε |x|r p
for a.a. z ∈ Z and all x ∈ R.
(5.224)
Let x ∈ K. Then
1 λ m|x|p dz− F z, x(z) dz Dxpp − p p Z Z
1 λ ε p p ≥ Dxp − m|x| dz − xpp − cε xrr p p Z p 1 ε λ ≥ 1− Dxpp − c14 Dxpp − cε Dxrp p λ2 p
ϕλ (x) =
(see (5.224)) (5.225)
for some c14 , cε > 0 (see (5.217)). Because λ < λ2 , choosing ε > 0 small from (5.225) we have ϕλ (x) ≥ c15 Dxpp − cε Dxrp
for some c14 > 0 and all x ∈ K.
(5.226)
Recall that r > p. So from (5.226) and Poincar´e’s inequality, it follows that we can find > 0 small such that ϕλ D ≥ ξ0 > 0, where D =K ∩ ∂B (0).
Now we can establish the existence of a nontrivial solution for problem (5.198). THEOREM 5.4.5 If H(m), H(f )1 hold and 0 < λ < λ2 , then problem (5.198) has a nontrivial solution x ∈ C01 (Z). PROOF: First assume that λ1 ≤ λ < λ2 . We choose > 0 small and R > 0 big so thatPropositions5.4.3 and 5.4.4 hold. 0 From Proposition 4.3.63, we know that the sets HR , HR , D link in W01,p (Z) via the identity map. Then Proposition 5.4.2 allows the use of Theorem 4.1.22, which gives x∈W01,p (Z) such that
428
5 Boundary Value Problems–Hamiltonian Systems ϕλ (x) ≥ ξ0 > 0 = ϕλ (0) (i.e., x = 0) and
ϕ (x) = 0.
From the last equation, we have A(x) − λm|x|p−2 x = Nf (x),
⎫
⎧ p−2 Dx(z) − λm(z)|x(z)|p−2 x(z) = f z, x(z) ⎬ ⎨ −div Dx(z) a.e. on Z, . ⇒ ⎭ ⎩ x ∂Z = 0, 1 < p < ∞ (5.227) From nonlinear regularity theory (see Theorem 4.3.35), we have that x ∈ C01 (Z) and of course it solves problem (5.198) (see (5.227)). Now assume that 0 < λ < λ1 . From the proof of Proposition 5.4.3, we know that ϕλ (tu1 ) −→ −∞
as t −→ ∞
(see (5.221)).
(5.228)
Moreover, as in the proof of Proposition 5.4.4, using (5.224), we obtain ϕλ (x) ≥
1 ε λ Dxpp − c14 Dxpp − cε Dxrp . 1− p λ1 p
Because λ < λ1 , choosing ε > 0 small, we have ϕλ (x) ≥ c16 Dxpp − cε Dxrp
for some c16 > 0, all x ∈ W01,p (Z).
Because p < r, we can find > 0 small, we have ϕλ ∂B (0) ≥ ξ0 > 0
(5.229)
(5.230)
(see (5.229) and use Poincar´e’s inequality). From (5.228), (5.230), and Proposition 5.4.2 and because ϕλ (0) = 0, we see that we can apply the mountain pass theorem (see Theorem 4.1.24) and obtain x ∈ W01,p (Z) such that ϕλ (x) ≥ ξ0 > 0 = ϕλ (0) (i.e., x = 0) and
ϕ (x) = 0.
As above it follows that x ∈ C01 (Z) and it is a nontrivial solution of problem (5.198). Next we consider the following perturbed eigenvalue problem * ) ∗ − x(z) = λx(z) + |x(z)|2 −2 x(z) a.e. on Z, . x∂Z = 0
Here ∗
2 =
2N N −2
+∞
(5.231)
if 2 < N if N = 1, 2
(the critical Sobolev exponent for 2). So we see that in this case the nonlinear perturbation in problem (5.231) has a critical growth and this causes serious difficulties
5.4 Nonlinear Eigenvalue Problems
429
dealing with the problem, because the embedding of the Sobolev space H01 (Z) into ∗ L2 (Z) is no longer compact. We are looking for positive solutions of problem (5.231). First let us consider the special case λ = 0. One way to approach problem (5.231) variationally is to try to obtain positive solutions of (5.231), as relative minima of the functional. ϕλ (x) =
∗ 1 λ 1 Dx22 − x22 − ∗ x22∗ 2 2 2
on the C 1 -manifold M = {x ∈ H01 (Z) : x2∗ = 1}. So we want to find a solution of the following minimization problem Dx2 − λx2 2 2 : x ∈ H01 (Z), x = 0 = Sλ (Z). inf 2 x2∗ If λ = 0, then we have Dx2 1 2 : x ∈ H (Z), x = 0 = S(Z), S0 (Z) = inf 0 x22∗ ∗
where S(Z) is the best Lipschitz constant for the embedding of H01 (Z) into L2 (Z). We know that S(Z) = S is independent of the domain Z and is never attained on a domain Z ⊆ RN , Z = RN (see, for example, Willem [606, Section 1.9]). So when λ = 0, problem (5.231) has no positive solution. More generally we show that this true for all λ ≤ 0 and all λ ≥ λ1 , provided that the domain Z has certain geometric structure. So we start with a definition. DEFINITION 5.4.6 A domain Z ⊆ RN containing the origin is said to be star shaped with respect to the origin, if z, n(z) RN > 0 for all z ∈ ∂Z (by n(z) we denote the unit outward normal at z ∈ ∂Z). EXAMPLE 5.4.7 Any open ball in RN centered at 0, is star-shaped with respect to the origin. To see this let z ∈ ∂Z. Then n(z) = z/R where R > 0 is the radius of the ball. So z, n(z) RN = z2 /R = R > 0. The negative result on the existence of positive solutions for problem (5.231) when λ ∈ / (0, λ1 ), is based on the following Pohozaev’s identity. PROPOSITION 5.4.8 If g : R −→ R is continuous, G(x) = C01 (Z) is a solution of
* ) − x(z) = g x(z) a.e. on Z, , x ∂Z = 0
x 0
g(r)dr, and x ∈
(5.232)
then
N
∂x 1
g x(z) x(z)dz+N G x(z) x(z)dz = 1− z, n(z) RN (z)dσ. (5.233) 2 2 ∂n Z Z ∂G
430
5 Boundary Value Problems–Hamiltonian Systems
PROOF: We multiply the equation in (5.232) with zi (∂x/∂zi ) and then integrate over Z. So we have
∂x ∂G x(z) ∂x − x(z)zi (z)dz = g x(z) zi (z)dz = zi dz ∂zi ∂zi ∂zi Z Z
Z
∂x ⇒ − x(z)zi (z)dz = − G x(z) dz (by integration by parts). ∂z i Z Z So we have
−
G x(z) dz = −
Z
N ∂ 2 x ∂x zi dz ∂zk2 ∂zi Z k=1
N N ∂x ∂ 2 x ∂x ∂x zi dz + δik dz = ∂zk ∂zk ∂zi ∂zk ∂zi Z Z k=1
k=1
N ∂x ∂x zi nk dσ − ∂zk ∂zi ∂Z k=1
N
∂x 2 1 ∂ ∂x 2 dz = zi dz + 2 Z ∂zi ∂zk Z ∂zi k=1
∂x ∂x − zi dσ ∂n ∂Z ∂zi
N
∂x 2 ∂x 2 1 dz + dz =− 2 Z ∂zk Z ∂zi +
1 2
k=1
N
∂x 2 1 ∂x ∂x zi ni dσ − zi dσ. ∂zk 2 ∂Z ∂zi ∂n ∂Z k=1
Summing over i, we obtain
N 1 Dx2 (z, n)RN dσ Dx22 + −N G x(z) dz = 1 − 2 2 ∂Z Z
∂x − (z, Dx)RN dσ. ∂Z ∂n
(5.234)
From (5.232) we have
g x(z) x(z)dz.
Dx22 =
(5.235)
Z
Moreover, because x∂Z = 0, we have Dx(z)2RN = and
z, Dx(z)
RN
∂x
2
for all z ∈ ∂Z ∂n
∂x = z, n(z) RN (z) for all z ∈ ∂Z. ∂n (z)
(5.236) (5.237)
Using (5.235) through (5.237) in (5.234), we obtain (5.233) (Pohozaev’s identity).
5.4 Nonlinear Eigenvalue Problems
431
Using Proposition 5.4.8, we can now prove a negative result concerning problem (5.231). Let Z ⊆ RN be a bounded domain with a C 2 -boundary ∂Z, which is starshaped with respect to the origin. If x ∈ H01 (Z) is a solution of (5.231), then by the strong maximum principle we have that either x = 0 or x(z) > 0 for all z ∈ Z. We show that when λ ∈ / (0, λ1 ), the second possibility can not occur. THEOREM 5.4.9 If Z ⊆ RN , N ≥ 3 is a bounded domain with a C 2 -boundary, which is star-shaped with respect to the origin, and λ ∈ / (0, λ1 ), then problem (5.231) has no nontrivial solution x ∈ H01 (Z)+ . PROOF: First suppose that λ ≥ λ1 . If x ∈ H01 (Z) \ {0} is a solution of (5.231), then using the eigenfunction u1 > 0 corresponding to λ1 > 0 as a test function, we obtain
∗ λ1 xu1 dz = (Dx, Du1 )RN dz = λ xu1 dz + |x|2 −2 xu1 dz Z Z Z
Z >λ xu1 dz
Z ≥ λ1 xu1 dz, Z
a contradiction. Next suppose that λ ≤ 0. ∗ ∗ We set g(r) = λr + |r|2 −2 r. Then G(r) = (λ/2)r2 + (1/2∗ )|r|2 . Using these 1 functions in Pohozaev’s identity (5.233) and because x ∈ C0 (Z) (regularity theory), we have
∗ ∗ Nλ N N (λx2 + |x|2 )dz + x2 dz + ∗ |x|2 dz = 1− 2 2 2 Z Z Z
∂x 2 1 (z, n)RN dσ 2 ∂Z ∂n
N ∂x 2 N 1 2∗ 2 ⇒ + 1 − |x| dz + λ λx dz = (z, n) dσ. N R 2∗ 2 2 ∂Z ∂n Z Z But
N N N N +1− = − = 0. 2∗ 2 2 2
∂x 2 1 x2 dz = (z, n)RN dσ λ 2 ∂Z ∂n Z
So we obtain
which is a contradiction if λ < 0. If λ = 0 then because Z is star-shaped with respect to the origin, we must have (∂x/∂n)(z) = 0 for z ∈ ∂Z. Hence
∗ |x|2 dz = − x(z) = 0 (by Green’s formula) Z
and so x = 0.
Z
REMARK 5.4.10 If Z ⊆ RN is not star-shaped with respect to the origin, then we can have solutions even when λ ≤ 0. In fact if Z ⊆ RN is an annular domain,
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5 Boundary Value Problems–Hamiltonian Systems
then we know that (5.231) admits a radial solution for all λ ∈ (−∞, λ1 ). Recall that x ∈ H01 (Z) is a radial solution, if x(z) = x(zRN ) for all z ∈ Z. When 0 < λ < λ1 , problem (5.231) may admit nontrivial solutions x ∈ H01 (Z)+ . In this case we have an interesting dependence on the dimension N . More precisely, we have the following theorem due to Brezis–Nirenberg [104]. As always by λ1 > 0 we denote the principal eigenvalue of − , H01 (Z) . THEOREM 5.4.11 If Z ⊆ RN , (N ≥ 3) is a bounded domain with a C 2 -boundary ∂Z, then (a) If N ≥ 4, then for any λ ∈ (0, λ1 ), there exists a positive solution for problem (5.231). (b) If N = 3, then there exists λ∗ ∈ [0, λ1 ) such that for any λ ∈ (λ∗ , λ1 ) problem (5.231) has a positive solution. (c) If N = 3 and Z = B1 (0), then λ∗ = λ1 /4 and for λ ≤ λ1 /4 problem (5.231) has no nontrivial solution in H01 (Z)+ .
5.5 Maximum and Comparison Principles The main goal of this section is to present some maximum and comparison principles for certain nonlinear differential operators involving the p-Laplacian. So let Z ⊆ RN be a bounded domain with a C 2 -boundary ∂Z. We start by considering the nonlinear differential operator Vp (x) = −div(Dxp−2 Dx) + mϑp (x),
x ∈ W01,p (Z),
(5.238)
∞
where 1 < p < ∞, m∈L (Z) (the weight function), ϑp :R−→R is the homeomorphism defined by ) p−2 |x| x if x = 0 ϑp (x) = 0 if x = 0
and ϑp :Lp (Z)−→Lp (Z), (1/p)+(1/p ) = 1 is the Nemitsky operator corresponding
to ϑp ; that is, ϑp (x)(·) = ϑp x(·) . Evidently ϑp is continuous and bounded (i.e., maps bounded sets to bounded sets). We start by recalling the following basic notions. 1,p DEFINITION 5.5.1 (a) A function x∈Wloc (Z) is said to be a weak solution of
−div(Dxp−2 Dx) + m|x|p−2 x = g,
if and only if
Dxp−2 (Dx, Dv)RN dz + m|x|p−2 xvdz = g, v Z
g ∈ W −1,p (Z)
(5.239)
for all v ∈ Cc∞ (Z)
Z
(here by·, ·we denote the duality brackets for the pair W01,p (Z), W −1,p (Z) ). 1,p (b) A function x∈Wloc (Z) is said to be an upper solution of (5.239), if
Dxp−2 (Dx, Dv)RN dz + m|x|p−2 xvdz ≥ g, v for all v ∈ Cc∞ (Z)+ . Z
Z
(5.240)
5.5 Maximum and Comparison Principles
433
1,p (c) A function x∈Wloc (Z) is said to be a lower solution of (5.239), if
Dxp−2 (Dx, Dv)RN dz + m|x|p−2 xvdz ≤ g, v for all v ∈ Cc∞ (Z)+ . Z
Z
(5.241)
REMARK 5.5.2 Recall that Cc∞ (Z) is dense in W01,p (Z) and W −1,p (Z) = W01,p (Z)∗ . So relations (5.239) through (5.241) are in fact valid for all v ∈ W01,p (Z). DEFINITION 5.5.3 (a) We say that the operator Vp (see (5.238)) satisfies the maximum principle (MP, for short), if every weak solution x ∈ W 1,p (Z) of
* ) −div Dxp−2 Dx + m|x|p−2 x = g (5.242) x∂Z ≥ 0, g ∈ W −1,p (Z) satisfies x(z) ≥ 0 a.e. on Z when g ≥ 0. (b) We say that the operator Vp (see (5.238)) satisfies the strong maximum principle (SMP, for short), if every weak solution x ∈ W 1,p (Z) of (5.242) satisfies x(z) > 0 a.e. on Z when g ≥ 0, g = 0. (c) We say that the operator Vp (see (5.238)) satisfies the weak comparison principle (WCP, for short), if Vp x1 ≤ Vp x2 in Z and x1 ∂Z ≤ x2 ∂Z with x1 , x2 ∈ W 1,p (Z), imply that x1 (z) ≤ x2 (z) a.e. on Z. To establish the maximum and comparison principles for the operator Vp , we need to know its spectrum. For this purpose the relevant nonlinear eigenvalue problem is the following one.
* ) p−2 Dx(z) + m|x(z)|p−2 x(z) = λ|x(z)|p−2 x(z) a.e. on Z, −div Dx(z) . x∂Z = 0, λ ∈ R (5.243) Following step by step the arguments used to establish the existence and proper ties of the first eigenvalue of − p , W01,p (Z) with a weight m ∈ L∞ (Z) (see Section 4.3), we obtain the following result. PROPOSITION 5.5.4 The nonlinear eigenvalue problem (5.243) has a unique eigenvalue λ = λ1 (m) defined by
λ1 (m) = inf Dxpp + m|x|p dz : x ∈ W01,p (Z), xp = 1 (5.244) Z
with the property that it has a positive eigenfunction u1 ∈ W01,p (Z). Moreover, the eigenvalue λ1 (m) is simple and isolated and the eigenfunction u1 satisfies u1 ∈ intC01 (Z)+ . The next theorem is in the direction of the nonlinear regularity results proved in Section 4.3) and is due to Di Benedetto [198] and Tolksdorf [583]. So suppose f : Z × R −→ R is a Carath´eodory function (i.e., for all x ∈ R, z −→ f (z, x) is measurable and for almost all z ∈ Z, x −→ f (z, x) is continuous). We consider the following nonlinear elliptic equation,
−div Dx(z)p−2 Dx(z) = f z, x(z) in Z, 1 < p < ∞. (5.245) In analogy to Definition 5.5.1(a), we make the following definition.
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5 Boundary Value Problems–Hamiltonian Systems
1,p DEFINITION 5.5.5 A function x ∈ Wloc (Z) is said to be a weak solution of problem (5.245), if
p−2 Dx (Dx, Dv)RN dz = f (z, x)vdz for all v ∈ Cc∞ (Z). Z
Z
The regularity result of Di Benedetto–Tolksdorf, reads as follows. x ∈ W 1,p (Z) is a weak solution THEOREM 5.5.6 If Z ⊆RN is a bounded domain, r of (5.245), and z −→ f z, x(z) belongs in L (Z) with p/(p − 1)N < r ≤ ∞, then x ∈ C 1,α (Z) for some 0 < α < 1. Another such regularity result that we need is the next theorem, which is essentially Theorem 4.3.35 but with nonhomogeneous Dirichlet boundary conditions. So we consider the following problem:
) * p−2 −div Dx + m|x|p−2 x = g in Z Dx . (5.246) ∞ x∂Z = h, 1 < p < ∞, g ∈ L (Z) The next theorem is a particular case of a more general result due to Lieberman [382]. THEOREM 5.5.7 If x∈W 1,p (Z) ∩ L∞ (Z) is a weak solution of (5.246) and h ∈ C 2 (∂Z), then x ∈ C 1 (Z). Now we can pass to the analysis of operator Vp . THEOREM 5.5.8 The following statements are equivalent. The operator Vp satisfies the MP. The operator Vp satisfies the SMP. λ1 (m) > 0 (see (5.244)). There exists a positive strict upper solution x ∈ W01,p (Z) of the equation Vp (x) = 0 and Vp (x) ∈ L∞ (Z) (i.e., Vp (x) = g in Z, g ∈ L∞ (Z)+ , g = 0). (e) For every g ∈ L∞ (Z)+ , the problem Vp (x) = g, g ∂Z = 0, has a unique weak solution x ∈ W01,p (Z)+ .
(a) (b) (c) (d)
PROOF: (a)⇒(b). Let x ∈ W01,p (Z) be a weak solution of − p x + m|x|p−2 x = g in Z, x∂Z ≥ 0,
(5.247)
where g ∈ L∞ (Z)+ , g = 0. Because by hypothesis Vp satisfies the MP, we have x(z) ≥ 0 a.e. on Z. Because g = 0, we have that x = 0. Moreover, taking into account the fact that x ∈ L∞ (Z) (see Theorem 4.3.34), from Theorem 5.5.6 we infer that x ∈ C 1 (Z). We have − p x + m∞ |x|p−2 x ≥ 0
(5.248)
(see (5.247) and recall that x ≥ 0, g ≥ 0). Then from (5.248) and the nonlinear strong maximum principle (see Theorem 4.3.37), we deduce that x(z) > 0 for all z ∈ Z. So Vp satisfies the SMP.
5.5 Maximum and Comparison Principles
435
1 (b)⇒(c). Suppose that
λ1 (m) ≤ 0 and let u1 ∈ int C0 (Z)+ be a positive eigenfunction for the operator Vp , W01,p (Z) (see Theorem 5.5.6). Then we have
Vp (−u1 ) = − p (−u1 ) + m| − u1 |p−2 (−u1 ) ≥ 0. Because Vp (u1 ) ∈ L∞ (Z) and by hypothesis Vp satisfies the SMP, we must have u1 ≤ 0, a contradiction.
(c)⇒(d). Just take x = u1 (= a positive eigenfunction of Vp , W01,p (Z) ). (d)⇒(c). As in Proposition 4.3.44 we introduce the set x y D(J) = (x, y) ∈ W01,p (Z) × W01,p (Z) : x ≥ 0, y ≥ 0, , ∈ L∞ (Z) . y x
Let u1 ∈ int C01 (Z)+ be a positive eigenfunction of Vp , W01,p (Z) . Note that because x ∈ W01,p (Z) is by hypothesis a strict upper solution of Vp (x) = 0 and Vp (x) ∈ L∞ (Z), we have x ∈ int C01 (Z)+ . So it follows that (u1 , x) ∈ D(J). In particular we have x u1 ∈ L∞ (Z). Set u1 = cu1 with c > 0 such that c ≥ x u1 ∞ . Suppose that λ1 (m) ≤ 0. Then if by ·, · we denote the duality brackets for the
pair W01,p (Z), W −1,p (Z) , (1/p) + (1/p ) = 1, we have up − xp − p x, 1 p−1 −m(up1 −xp )dz, ≥ x Z
(see (4.124) in Section 4.3) ⇒ J(u1 , x) ≤ λ1 (m)(up1 −xp )dz ≤ 0, Z
⇒ J(u1 , x) = 0 and so u1 = θx
for some θ > 0
(see (4.125) in Section 4.3).
So it follows that − p u + m|u|p−2 u = θp−1 g,
g ∈ L∞ (Z)+ , g = 0.
This contradicts the hypothesis that λ1 (m) ≤ 0. Hence we must have λ1 (m) > 0. (c)⇒(a). Suppose that x ∈ W 1,p (Z) satisfies − p x + m|x|p−2 x = g
in Z, x∂Z ≥ 0,
with g ∈ W −1,p (Z), g ≥ 0 (i.e., g, v ≥ 0 for all v ∈ W01,p (Z)+ ). Using as a test function −x− ∈ W01,p (Z) and recalling that ) −Dx(z) if x(z) < 0 , Dx− (z) = 0 if x(z) ≥ 0 we obtain Dx− pp +
m|x− |p dz = g, −x− ≤ 0
Z
⇒ λ1 (m) ≤ 0, a contradiction, unless x− = 0 in which case x ≥ 0. Thus far we have proved the equivalence of statements (a), (b), (c), (d).
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5 Boundary Value Problems–Hamiltonian Systems
(e)⇒(d). Immediate. So we are done if we show that one of the statements (a), (b), (c), or (d) implies (e). More precisely we show the following. (c)⇒(e). We consider the Euler functional ϕ : W01,p (Z) −→ R defined by
1 1 ϕ(x) = Dxpp + m|x|p dz − gxdz. p p Z Z
(5.249)
We claim that ϕ is weakly coercive. Suppose that this is not the case. Then we can find {xn }n≥1 ⊆ W01,p (Z) such that xn −→ ∞ as n → ∞ We have
and
ϕ(xn ) ≤ M
for some M > 0, all n ≥ 1.
λ1 (m) gxn dz ≤ ϕ(xn ) ≤ M for all n ≥ 1, xn pp − p Z λ1 (m) ⇒ xn pp ≤ M + cxn p for some c > 0, all n ≥ 1, p ⇒ {xn }n≥1 ⊆ Lp (Z) is bounded.
Then from the definition of ϕ (see (5.249)), we infer that {Dxn }n≥1 ⊆ Lp (Z, RN ) is bounded, ⇒ {xn }n≥1 ⊆ W01,p (Z) is bounded (by Poincar´e’s inequality), a contradiction to the fact that xn −→ ∞. Therefore ϕ is weakly coercive. Moreover, exploiting the compact embedding of W01,p (Z) into Lp (Z), we can see that ϕ is weakly lower semicontinuous on W01,p (Z). By the Weierstrass theorem, we can find x ∈ W01,p (Z) such that ϕ(x) =
inf
1,p
W0
ϕ,
(Z)
⇒ ϕ (x) = 0, ⇒ A(x) + m|x|p−2 x = g,
(5.250)
where A : W01,p (Z) −→ W −1,p (Z) as before is defined by
A(x), y = Dxp−2 (Dx, Dy)RN dz for all x, y ∈ W01,p (Z). Z
From (5.250) it follows that
* ) p−2 Dx(z) + m(z)|x(z)|p−2 x(z) = g(z) a.e. on Z, −div Dx(z) x∂Z = 0 ⇒ x ∈ C01 (Z)+ is a strong solution of the problem Vp (x) = g, x∂Z = 0. Moreover, acting on (5.250) with the test function −x− ∈W01,p (Z) and since by hypothesis λ1 (m) > 0, it follows that x ≥ 0. In fact if g = 0, by the nonlinear strong maximum principle we have x ∈ int C01 (Z)+ .
5.5 Maximum and Comparison Principles
437
Finally if x, y ∈ int C01 (Z)+ are two such solutions, when g = 0, then xp − y p xp − y p − − p y, p−1 0 ≤ J(x, y) = − p x, xp−1 y
p−1 p−1 y −x = (xp − y p )dz ≤ 0, xp−1 y p−1 Z ⇒ J(x, y) = 0 (i.e., x = cy for some c > 0), ⇒ cp−1 g = g (i.e., c = 1). Therefore x = y and we have proved the uniqueness of the solution. If g = 0, then we have A(x) + m|x|p−2 x = 0
(see (5.250)).
Acting with the test function x ∈ W01,p (Z), we obtain
m|x|p dz = 0, Dxpp + Z
⇒ λ1 (m)xpp ≤ 0 ⇒ x=0
(see (5.244)),
(because by hypothesis λ1 (m) > 0).
So when g = 0, the only possible solution is the trivial one.
THEOREM 5.5.9 If λ1 (m) > 0, xk ∈ W 1,p (Z) ∩ L∞ (Z) satisfy Vp (xk ) ∈ L∞ (Z), xk ∂Z ∈ C 2 (∂Z) k = 1, 2, * Vp (x 1 ) ≤ Vp (x 2 ) in Z (5.251) x1 ∂Z ≤ x2 ∂Z and finally Vp (x2 ) ≥ 0 in Z with x2 ∂Z ≥ 0, then x1 (z) ≤ x2 (z) for every z ∈ Z. Moreover, if in (5.251) x1 ∂Z = x2 ∂Z = 0, then the same conclusion holds under the following less restrictive assumptions, also we have
)
xk ∈ W01,p (Z), Vp (xk ) ∈ L∞ (Z)
for k = 1, 2 and Vp (x2 ) ≥ 0 in Z.
PROOF: If x2 = 0, then clearly then result is true, with x1 = 0. So suppose that x2 = 0. From Theorem 5.5.7 we have that xk ∈ C 1 (Z), k = 1, 2. Then since x2 ≥ 0 (recall λ1 (m) > 0), from the strong maximum principle, we 1 have that x2 ∈ int C0 (Z)+ . Therefore we can find c > 1 such that x1 < cx2 . Let g = Vp (x2 ), h =x2 ∂Z and consider the following problem * ) p−2 y = g in Z, − p y + m|y| . (5.252) y ∂Z = h We can easily check that y = x1
and
y = cx2
are lower and upper solutions, respectively, of (5.252). Thus employing the truncation and penalization techniques developed in Section 5.2, we can have a solution y ∈ C 1 (Z) (see Theorem 5.5.7) such that
438
5 Boundary Value Problems–Hamiltonian Systems x1 (z) ≤ y(z) ≤ cx2 (z)
for all z ∈ Z.
Because λ1 (m) > 0, y ≥ 0. But from the proof of Theorem 5.5.6 we know that (5.252) has a unique solution. So y = x2 and we have x1 ≤ x2 . Finally if we have homogeneous Dirichlet boundary conditions, then by Theorem 4.3.35, it suffices to assume that Vp (xk ) ∈ L∞ (Z), xk ∈ W01,p (Z), k = 1, 2, and Vp (x2 ) ≥ 0. The classical maximum principle (see Theorem 4.3.19), asserts that any superharmonic C 2 -function x on a smooth domain Z ⊆ RN can not achieve its minimum in the interior of Z, unless x is a constant. We saw that this result is still true if 1,p x ∈ Wloc (Z) is a p-superharmonic function such that div(Dxp−2 Dx) ∈ L2loc (Z) (see Theorem 4.3.37). From the classical maximum principle one immediately derives the strong comparison principle between two C 2 -functions xk (k = 1, 2) satisfying − x1 ≤ − x2 . Next we prove corresponding results for the p-Laplacian differential operator. Recall that Z ⊆ RN is a bounded domain with a C 2 -boundary ∂Z. 1,p PROPOSITION 5.5.10 If x, y ∈ Wloc (Z) (1 < p < ∞), f, g ∈ L∞ (Z),
−div Dx(z)p−2 Dx(z) = f (z)
and −div Dy(z)p−2 Dy(z) = g(z) a.e. on Z
x(z) ≥ y(z) and
for all z ∈ Z, f (z) ≥ g(z)
a.e. on Z
K = {x ∈ Z : x(z) = y(z)}
is compact, then K = ∅. PROOF: From nonlinear regularity theory (see Theorems 4.3.34 and 4.3.35), we know that x, y ∈ C 1,α (Z). Assume that K is nonempty compact. We can find a relatively compact open set Z1 such that K ⊆ Z1 ⊆ Z 1 ⊆ Z and y(z) < x(z)
for all z ∈ Z \ K.
Given ε > 0, let xε , yε ∈ W 1,p (Z1 ) be solutions of the equations ;
2 (p−2)/2 Dxε (z) = f (z) a.e. on Z1 , −div (ε + Dxε (z) ) (5.253) xε ∂Z = x and
1
; 2 (p−2/2 (z) ) Dy (z) = g(z) a.e. on Z , −div (ε + Dy ε ε 1 . (5.254) yε ∂Z = y
1
Such solutions can be easily obtained by direct minimization of the Euler functionals corresponding to problems (5.253) and (5.254). Moreover, Dxε ε∈(0,1] and Dyε ε∈(0,1] ⊆ Lp (Z1 ) are both bounded. Again we have xε , yε ∈ C 1,α (Z 1 ) (we can always assume that ∂Z 1 is C 2 ) and lim xε = x ε↓0
and
lim yε = y ε↓0
1,β weakly in W 1,p (Z1 ) and strongly in Cloc (Z1 ) for any β ∈ (0, α). We choose Z2 an open subset of Z such that K ⊆ Z2 ⊆ Z 2 ⊆ Z1 and set
5.5 Maximum and Comparison Principles ξ = min(x − y) > 0 ∂Z2
and
439
uε = xε − yε .
We choose ε > 0 small so that ξ 4
and
ξ y − yε L∞ (Z2 ) < . 4
for all z ∈ ∂Z2
and
xε (z) − yε (z)
ξ 2
ξ 2
for all z ∈ K.
Invoking the mean value theorem, we have −
N ∂ ε ∂uε αij =f −g ≥0 ∂zi ∂zj i,j=1
in Z2 ,
where
(p−4)/2
ε αij = ε + ti Dxε + (1 − ti )Dyε 2 δij ε + ti Dxε + (1 − ti )Dyε 2
∂xε ∂yε ∂xε ∂yε + (p − 2) ti + (1 − ti ) + (1 − ti ) ti ∂zi ∂zi ∂zj ∂zj with ti ∈ (0, 1). Set ηε = min(xε − yε ) and Kηε = z ∈ Z2 : (xε − yε )(z) = ηε . Z2
Evidently Kηε is a nonempty compact subset of Z2 and Dxε (z) = Dyε (z) for all z ∈ Kηε . So we have
(p−4)/2
∂xε ∂xε ε αij = ε + Dxε 2 δij ε + Dxε 2 + (p − 2) on Kηε . (5.255) ∂zi ∂zj ε Moreover, because αij are the entries of the Hessian matrix of the strongly
p/2 N , we have convex function v = (vk )k=1 −→ (1/p) ε + v2 N
ε αij ϑi ϑj ≥ γϑ2
(5.256)
i,j=1 N for some γ > 0 and all ϑ = (ϑk )N k=1 ∈ R . We choose an open neighborhood Zηε of Kηε such that Z ηε ⊆ Z2 , xε − yε > ηε and (5.256) still holds with γ replaced by 1 γ. Note that such a choice is possible because Dxε , Dyε ∈ C(Z 2 , RN ). Then by 2 the strong maximum principle (see Theorem 4.3.19), it follows that uε = xε − yε is constant on Z ηε , a contradiction. This proves that K = ∅.
This proposition leads to the following strong comparison principle. THEOREM 5.5.11 If x, y ∈ C01 (Z), x = 0, f, g ∈ L∞ (Z), f ≥ 0,
−div Dx(z)p−2 Dx(z) = f (z) a.e. on Z
p−2 and −div Dy(z) Dy(z) = g(z) a.e. on Z f (z) ≥ g(z) a.e. on Z and the set C = z ∈ Z : f (z) = g(z) has an empty interior, then x(z) > y(z) for all z ∈ Z and (∂x/∂n)(z) < (∂y/∂n)(z) for all z ∈ ∂Z; that is, we have that x − y ∈ int C01 (Z)+ .
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5 Boundary Value Problems–Hamiltonian Systems
PROOF: From the nonlinear strong maximum principle (see Theorem 4.3.37), we have ∂x x(z) > 0 for all z ∈ Z and (z) < 0 for all z ∈ ∂Z. ∂n − Acting with the test function −(x − y) , we can see that x(z) ≥ y(z) for all z ∈ Z. We set K = z ∈ Z : x(z) = y(z) (the coincidence set in Z). From Proposition 5.5.10 we know that K cannot be compact unless K is empty. Suppose K is nonempty. Then we can find {zn }n≥1 ⊆ K and z ∈ ∂Z such that zn −→ z. Since Dx, Dy ∈ C(Z, RN ) (see Theorems 4.3.34 and 4.3.35) and x, y ∈ C01 (Z), we have
∂y ∂x (z) = (z) = −β 2 < 0. ∂n ∂n
(5.257)
Set u = x − y. Then u ∈ C01 (Z) and it satisfies −
N ∂
∂u αij =f −g ≥0 ∂z ∂x j i i,j=1
0 (see (5.255) with ε = 0). In particular, we have with αij = αij
∂x 2 ∂x ∂x ∂x p−4 (z) (z) . δij (z) + (p − 2) αij (z) = (z) ∂n ∂n ∂zi ∂zj Exploiting the continuity of the gradient vectors Dx, Dy, we can find a ball B ⊆ Z such that z ∈ ∂B and the elliptic operator defined by the αij is strictly elliptic on B. It follows that either u = 0 in B, which is impossible because by hypothesis int C = ∅ for u > 0 on B and (∂u/∂n)(z) < 0, which contradicts (5.257). Therefore we infer that x(z) > y(z)
for all z ∈ Z.
(5.258)
for all z ∈ ∂Z.
(5.259)
Moreover, from Lemma 4.3.18 we have ∂x ∂y (z) < ∂n ∂n
Hence from (5.258) and (5.259), we conclude that x − y ∈ int C01 (Z)+ .
5.6 Periodic Solutions for Hamiltonian Systems The Hamiltonian equations in mechanics (classical and celestial), can be written in the compact form
−Jx (t) = ∇H t, x(t) a.e. on T = [0, b], (5.260)
5.6 Periodic Solutions for Hamiltonian Systems
where J=
0N −IN IN 0N
441
is the standard 2N × 2N simplectic matrix and ∇H(t, x) is the gradient of the Hamiltonian function x −→ H(t, x) defined on R2N which is assumed to be C 1 . Recall that the simplectic matrix has the following properties, J 2 = −I2N
and
(Jx, y)R2N = −(x, Jy)R2N
for all x, y ∈ R2N .
(5.261)
A basic problem in mechanics is to prove the existence of periodic trajectories for system (5.260). So first we establish the existence of periodic solutions for problem (5.260), when the Hamiltonian function is superquadratic, but does not necessarily satisfy the Ambrosetti–Rabinowitz condition (AR-condition, for short). Recall that the AR-condition on H(t, ·), says the following. (AR): there exist µ > 0 and R > 0 such for almost all t ∈ T and all x ≥ R
0 < µH(t, x) ≤ ∇H(t, x), x R2N . (5.262) Condition (AR) implies that the growth of x−→H(t, x) is superquadratic. Here instead, we make the following hypotheses concerning the Hamiltonian function H(t, x). H1 : H : T × R2N −→ R is a function such that (i) For all x ∈ R2N , t −→ H(t, x) is measurable. (ii) For almost all t ∈ T, x −→ H(t, x) is a C 1 -function. (iii) H(t, x) ≥ 0 for a.a. t ∈ T , all x ∈ R2N and lim
x→∞ H(t,x) 2 x→0 x
(iv) lim
H(t, x) = +∞ x2
uniformly for a.a. t ∈ T.
= 0 uniformly for a.a. t ∈ T .
, c1 , c2 > 0, and M > 0 such (v) There exist constants 1 < r and 1 < ϑ < 1 + r−1 r that
c1 xr ≤ ∇H(t, x), x R2N −2H(t, x) and
∇H(t, x) ≤ c2 xϑ
for a.a. t ∈ T and all x ≥ M.
q
EXAMPLE 5.6.1 The function H(x) = x2 ln(1 + xp ) with p, q > 1 satisfies hypotheses H1 but not the AR-condition (see (5.262)). < Let S 1 =R (2π/b)Z. We consider the Hilbert space V =W 1/2,2 (S 1 , R2N ) of all b-periodic functions x(t) =
k∈Z
exp
2πk
tJ xk ∈ L2 [0, 1], R2N b
with Fourier coefficients xk ∈ R2N that satisfy
442
5 Boundary Value Problems–Hamiltonian Systems x2V = x0 2R2N +
2π |k|xk 2R2N . b k∈Z
The inner product on the Hilbert space V is given by (x, y)V = (x0 , y0 )R2N +
2π |k|(xk , yk )R2N . b k∈Z
Evidently the inner product (·, ·)V generates the norm · V . We define the operator A∈L(V ) by
b
(Ax, y)V =
− Jx (t), y(t)
0
RN
dt
for all x, y ∈ V.
Using the properties of the simplectic matrix (see (5.261)), we see that A is a bounded self-adjoint √ operator and ker A = R2N . Note that W 1/2,2 (S 1 , R2N ) = 1/2 D(|A| ), where |A| = A2 (since A is self-adjoint). Also we consider the energy functional ϕ : W 1/2,2 (S 1 , R2N )−→R defined by ϕ(x) =
1 (Ax, x)V − 2
b
H t, x(t) dt
for all x ∈ W 1/2,2 (S 1 , R2N ).
0
Clearly ϕ ∈ C 1 (V ) and the critical points of ϕ are the solutions of )
* −Jx (t) = ∇H t, x(t) a.e. on [0, b], . x(0) = x(b), x (0) = x (b)
(5.263)
We consider the orthogonal direct sum decomposition V = V − ⊕V0 ⊕V + , where
and
V −= {x ∈ V : xk = 0
for k ≥ 0}
V0 = {x ∈ V : xk = 0
for k = 0}
V = {x ∈ V : xk = 0
for k ≤ 0}.
+
So every x ∈ V may be uniquely written as x = x + x0 + x with x ∈ V − , x0 ∈ V0 and x ∈ V + . Note that the self-adjoint operator A ∈ L(V ) has eigenspaces 2πk Ek = exp tJ xk : xk ∈ R2N b with corresponding eigenvalues 2πk, k∈Z. Moreover, dim Ek = 2N . Hence V − = the negative definite subspace of A V0 = ker A = R2N and
V + = the positive definite subspace of A.
PROPOSITION 5.6.2 If hypotheses H1 hold, then ϕ satisfies the P S-condition.
5.6 Periodic Solutions for Hamiltonian Systems
443
PROOF: Let {xn }n≥1 ⊆ V be a sequence such that |ϕ(xn )| ≤ M
for some M > 0, all n ≥ 1 and ϕ (xn ) −→ 0 as n → ∞.
(5.264)
Because of hypotheses H1 , we have that |H(t, x)| ≤ c3 + c4 xϑ+1
for a.a. t ∈ T, all x ∈ R2N with c3 , c4 > 0.
This growth condition combined with hypothesis H1 (v), imply that
c1 xr − c5 ≤ ∇H(t, x), x R2N −2H(t, x)
(5.265)
for a.a. t ∈ T , all x ∈ R2N , with c5 > 0. Then we have
2ϕ(xn ) − ϕ (xn ), xn V =
b
∇H(t, xn ), xn
0
R2N
− 2H(t, xn ) dt
b
≥ c1
xn r dt − c5 b
(see (5.265)).
0
(5.266) We claim that {xn }n≥1 ⊆ V is bounded. Suppose that this is not the case. We may assume that xn V −→ ∞. Then from (5.264) and (5.266), it follows that 1 xn V
b
xn r dt −→ 0
as n → ∞.
(5.267)
0
By hypothesis H1 (v), we have 1 < ϑ < 1 + (r − 1)/r. Let ξ = (r − 1) r(ϑ − 1) . We have 1 ξ > 1 and ξϑ − 1 = ξ − . (5.268) r Hypothesis H1 (v) implies that ∇H(t, x)ξ ≤ c6 xϑξ + c7
for a.a. t ∈ T, all x ∈ R2N , with c6 , c7 > 0. (5.269)
Recall that xn = xn + x0n + xn with xn ∈ V , x0n ∈ V0 ad xn ∈ V . Exploiting the orthogonality of the component spaces, we have
ϕ (xn ), xn V = (Axn , xn )V −
b
∇H(t, xn ), xn
0
R2N
b
≥ (Axn , xn )V −
∇H(t, xn )xn dt 0
≥ (Axn , xn )V − c8
0
for some c8 > 0, all n ≥ 1. Using (5.269), we have
dt
1/ξ
b
∇H(t, xn )ξ dt
xn dt (5.270)
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5 Boundary Value Problems–Hamiltonian Systems
1/ξ
b
∇H(t, xn )ξ dt
b
≤
(c6 xn ϑξ + c7 )dt
0
0
≤ c9
1/r
b
b
xn r dt 0
r
1−1/r
xn (ξϑ−1) r−1 dt 0
+ c10 , c9 , c10 > 0 1/r b (ξϑ−1) ≤ c11 xn r dt xn V + c10 , c11 > 0. 0
(5.271) If we combine (5.267), (5.268), and (5.271), we see that
1/ξ 1 b ∇H(t, xn )ξ dt −→ 0 as n → ∞. xn V 0 Then from (5.264) and (5.270), we infer that
1/ξ ϕ (xn )V xn V (Axn , xn )V 1 b ≤ + c8 ∇H(t, xn )ξ dt xn V xn V xn V xn V xn V 0 −→ 0
as n → ∞,
xn V −→ 0 as n → ∞. ⇒ xn V
(5.272)
In a similar fashion, we show that xn −→ 0 xn
as n → ∞.
From hypothesis H1 (v), we have
c12 x − c13 ≤ ∇H(t, x), x R2N −2H(t, x)
(5.273)
for a.a. t ∈ T, all x ∈ R2N ,
c12 , c13 > 0. So it follows that
2ϕ(xn ) − ϕ (xn ), xn V =
b
∇H(t, xn ), xn
0
0
R2N
− 2H(t, xn ) dt
b
≥
(c12 xn − c13 )dt b
≥
(c12 x0n − c12 xn − c12 xn − c13 )dt 0
≥ c13 x0n V − c14 (xn V + xn V ) − c15 , c13 , c14 , c15 > 0, because V0 =R2N . Therefore from (5.272) through (5.274), we deduce that x0n V −→ 0 xn V So finally
as n → ∞.
(5.274)
5.6 Periodic Solutions for Hamiltonian Systems 1=
xn V + x0n V + xn V xn V ≤ −→ 0 xn V xn V
445
as n → ∞,
a contradiction. This proves that {xn }n≥1 ⊆ V is bounded. So we may assume that w
xn −→ x
in X
and
xn −→ x
in L2 (T, R2N )
(recall that W (1/2),2 (S 1 , R2N ) is embedded compactly in L2 (T, R2N )). From (5.264), we have
b
1 as n → ∞. ∇H(t, xn ), xn − x R2N dt −→ 0 (Axn , xn − x)V − 2 0 Evidently
b
∇H(t, xn ), xn − x
0
R2N
dt −→ 0.
So we have (Axn , xn − x)V −→ 0, ⇒ xn V −→ xV
as n → ∞.
w
Because xn −→ x in H and H is a Hilbert space, we conclude that xn −→ x in V , therefore ϕ satisfies the P S-condition. THEOREM 5.6.3 If hypotheses H1 hold, then problem (5.263) admits a nonconstant solution x ∈ C 1 (T, R2N ). PROOF: Hypotheses H1 imply that for almost all t ∈ T and all x ∈ RN , we have |H(t, x)| ≤ c1 + c2 xϑ+1
with c1 , c2 > 0.
(5.275)
Also because of hypotheses H1 (iv), given ε > 0, we can find δ = δ(ε) > 0 such that H(t, x) ≤ εx2 for a.a. t ∈ T and all x ≤ δ. (5.276) From (5.275) and (5.276) it follows that we can find cε such that H(t, x) ≤ εx2 + cε xϑ+1
for a.a. t ∈ T and all x ∈ R2N .
(5.277)
Therefore by choosing ε > 0 small, for every x ∈ V + we have
b
1 b (−Jx , x)R2N dt − H t, x(t) dt ϕ(x) = 2 0 0 ≥ c3 x2V − c4 xϑ+1 V
(5.278)
(see (5.277) and recall that W (1/2),2 (S 1 , R2N ) is embedded compactly in Lϑ+1 (S 1 , R2N )). Because of (5.278), we can find > 0 small such that ϕ(x) ≥ β > 0
for all x ∈ V + with xV = .
(5.279)
Let W = V − ⊕ V 0 and consider e ∈ V + with eV = 1 and E = W ⊕ Re. We introduce
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5 Boundary Value Problems–Hamiltonian Systems
and
µ = inf (Ax, x)V : x ∈ V − , xV = 1
BE = {x ∈ E : xV = 1},
1 1/2 γ= . A|L µ
(5.280)
For x ∈ BE we write x = x + x0 + x ∈ E. (a) If xV > γx0 + xV , then for any η > 0 we have
b 1 1 ϕ(ηx) = (Aηx, x)V + (Aη x, x)V − H(t, ηx)dt 2 2 0 1 µ ≤ − η 2 xV + AL η 2 x2V (see (5.280) and recall that H ≥ 0) 2 2 ≤0 (because xV > γx0 + xV ). (b) If xV ≤ γx0 + xV , then we have 1 = x2V = x2V + x0 + x2V ≤ (γ 2 + 1)x0 + x2V 1 ⇒ 0< ≤ x0 + x2V . 1 + γ2 $E = x ∈ BE : xV ≤ γx0 + x2V . Set B $E we have Claim: We can find ε1 > 0 such that for all x ∈ B t ∈ T : x(t) ≥ ε1 ≥ ε1 . 1
(5.281)
(5.282)
Here by | · |1 we denote the Lebesgue measure on T . $E Suppose that the claim is not true. Then for any n ≥ 1, we can find xn ∈ B such that t ∈ T : xn (t) ≥ 1 < 1 . n 1 n We set xn = xn + x0n + xn ∈ E. Because dim(V0 ⊕ Re) < ∞ and x0n + xn V ≤ 1, by passing to a subsequence if necessary, we may assume that x0n + xn −→ x0 + x ∈ V0 ⊕ Re as n → ∞, 1 ⇒ 0< ≤ x0 + x2V (see (5.281)). 1 + γ2
(5.283)
Also xn V ≤ 1 for all n ≥ 1 and so because V is a Hilbert space, by the Eberlein– w Smulian theorem, we may assume that xn −→ x in V − as n → ∞. So finally w
xn −→ x = x + x0 + x ⇒ xn −→ x
in L (S , R 2
1
2N
in V as n → ∞, ),
(because V is embedded compactly in L2 (S 1 , R2N )), ⇒ xV > 0
(see (5.283)),
⇒ x2 > 0. Therefore we can find δ2 ≥ δ1 > 0 such that t ∈ T : x(t) ≥ δ1 ≥ δ2 . 1
(5.284)
5.6 Periodic Solutions for Hamiltonian Systems
447
Indeed, if (5.284) is not true, then for all n ≥ 1, we must have t ∈ T : x(t) ≥ 1 = 0, n 1 1 = 1, ⇒ t ∈ T : x(t) < n 1
b 1 ⇒ 0< x(t)2 dt < 2 −→ 0 n 0
as n → ∞,
a contradiction. So we see that (5.284) is true. We set T = t ∈ T : x(t) ≥ δ1 , Tn = t ∈ T : xn (t) < 1/n for all n ≥ 1 and c Tn = T \ Tn . Because of (5.282) and (5.284), we have T ∩ Tn = T \ (T ∩ Tnc ) ≥ |T |1 − |T ∩ Tnc |1 ≥ δ2 − 1 . 1 1 n Let n ≥ 1 be large enough so that δ2 − 1/n ≥ δ2 /2 and δ1 − 1/n ≥ δ1 /2. We have
1 2 δ1 2 ≥ xn (t) − x(t)2 ≥ δ1 − n 2 So it follows that for n ≥ 1 large
b
xn (t) − x(t)2 dt ≥
δ1 2 |T ∩ Tn |1 2
δ1 2
1 ≥ δ2 − 2 n
δ1 3 > 0. ≥ 2
xn (t) − x(t)2 dt ≥
∩Tn T
0
for all t ∈ T ∩ Tn .
(5.285)
But recall that xn −→ x in L2 (S 1 , R2N ). Comparing this with (5.285), we reach a contradiction. This proves the claim (see (5.282)). $E , let T (x) = {t ∈ T : x(t) ≥ ε1 }. Hypothesis H1 (iii) For x = x + x0 + x ∈ B implies that for β0 = AL ε31 > 0, we can find M0 > 0 such that H(t, x) ≥ β0 x2
for a.a. t ∈ T and all x ≥ M0 .
Choose η0 ≥ M0 /ε1 . Then for η ≥ η0 , we have
H t, ηx(t) ≥ β0 ηx(t)2 ≥ β0 r2 ε21 for a.a. t ∈ T (x). Hence ϕ(ηx) = ≤ ≤ ≤ ⇒ ϕ(ηx) ≤
b
1 2 1 H t, ηx(t) dt η (Ax, x)V + η 2 (Ax, x)V − 2 2 0
1 2 H t, ηx(t) dt (because H ≥ 0) η AL − 2 T (x) 1 2 η AL − β0 η 2 ε21 |T (x)|1 2 1 2 1 η AL − β0 η 2 ε31 = − η 2 AL < 0, 2 2 0 for all x ∈ BE and all η ≥ η0 .
(5.286)
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5 Boundary Value Problems–Hamiltonian Systems
Set C = x ∈ W : xV ≤ 2η0 ⊕ ηe : 0 ≤ η ≤ 2η0 and C0 = ∂C. Then from (5.286) we have ϕC ≤ 0. (5.287) 0
This together with (5.279) and Proposition 5.6.2, permit the use of the generalized mountain pass theorem (see Theorem 4.1.26), which gives x ∈ V such that ϕ(x) ≥ β > 0 = ϕ(0)
and
ϕ (x) = 0.
It follows that x = 0 and it is a solution of (5.263). Moreover, because H ≥ 0 we see that x is nonconstant. Finally, it is clear from (5.263) that x ∈ C(T, R2N ). Using the same technique we can also have an existence theorem for Hamiltonian systems for which the function H(t, x) satisfies the (AR)-condition. In fact in this case the proofs are simplified. So our hypotheses on the Hamiltonian H(t, x) are the following. H2 : H : T × R2N −→ R is a function such that (i) For all x ∈ R2N , t −→ H(t, x) is measurable. (ii) For almost all t ∈ T, x −→ H(t, x) is a C 1 -function. (iii) H(t, x) ≥ 0 for a.a. t ∈ T , all x ∈ R2N , and lim a.a. t ∈ T .
H(t,x) 2 x→0 x
= 0 uniformly for
(iv) Condition (AR) is satisfied (see (5.262)). REMARK 5.6.4 Integrating the (AR)-condition (hypothesis H2 (iv)), we see that for a.a. t ∈ T and all x ∈ R2N , we have c1 xµ − c2 ≤ H(t, x) with c1 , c1 > 0. Hence H(t, )˙ is superquadratic. Following the reasoning of the proof of Theorem 5.6.3, via the generalized mountain pass theorem (see Theorem 4.1.26), we can have the following theorem. The reader can fill the details. THEOREM 5.6.5 If hypotheses H2 hold, then problem (5.263) admits a nonconstant solution x ∈ C 1 (T, R2N ). Next we consider a problem of a seemingly different nature. So we consider a Hamiltonian system but instead of fixing the period, the energy level is prescribed. So we consider the following autonomous Hamiltonian system:
−Jx (t) = ∇H x(t) for all t ∈ T. (5.288) Let x ∈ C 1 (T, R2N ) be a solution of (5.288). Taking inner product of (5.288) with x (t), we obtain
∇H x(t) , x (t) R2N d
⇒ H x(t) = 0 for all t ∈ T, dt
⇒ H x(t) = constant for all t ∈ T.
5.6 Periodic Solutions for Hamiltonian Systems
449
This means that the energy is conserved. It is therefore natural to seek solutions of (5.288) and in particular periodic solutions with a prescribed energy level. The difficulty in dealing with this problem is that the period, and consequently the underlying solution space, is not a priori known. Nevertheless, with some assumptions on the Hamiltonian H(x), we are able to reduce the fixed energy case to the fixed period case. We start with a result, which says that under some conditions on the gradient map ∇H, the trajectories of (5.288) on the energy manifold S, are actually independent of the Hamiltonian H and depend only on the manifold S. PROPOSITION 5.6.6 If Hk ∈ C 1 (R2N ) and ck ∈ R, k = 1, 2, are such that
and
S = Hk−1 (ck ),
k = 1, 2
∇Hk (x) = 0
for all x ∈ S, k = 1, 2,
then the trajectories of the Hamiltonian systems
for all t ∈ T, −Jx (t) = ∇Hk x(t)
k = 1, 2,
are the same. of system correspondPROOF: Let x1 ∈C 1 (T, R2N ) be a solution the Hamiltonian
ing to H1 . Note that the vectors ∇H1 x1 (t) , ∇H2 x1 (t) , t ∈ T , are normal to S. So we can find µ : T −→ R such that
∇H2 x1 (t) = µ(t)∇H1 x1 (t) for all t ∈ T. (5.289)
By hypothesis ∇H1 x1 (t) , ∇H2 x1 (t) = 0 for all t ∈ T and so µ(t) = 0 for all t ∈ T . Moreover, because t −→ ∇Hk x1 (t) is continuous for k = 1, 2, it follows that t −→ µ(t) is continuous. Therefore t −→ µ(t) has constant sign on T . Let
t
ϑ(t) = 0
1 ds. µ(s)
Then ϑ is continuous and strictly monotone. We set x2 = x1 ◦ ϑ−1 and we have Jx2 (t) = J(x1 ◦ ϑ−1 ) (t) 1 = J(x1 ◦ ϑ−1 )(t) ◦ ϑ−1 (t) (by the chain rule) ϑ
= −∇H1 x1 ϑ−1 (t) µ ϑ−1 (t)
= −∇H2 x1 ϑ−1 (t) for all t ∈ ϑ(T ) (see 5.289))
for all t ∈ ϑ(T ), = −∇H2 x2 (t)
⇒ x2 ∈ C 1 ϑ(T ), R2N is a solution of − Jx (t) = −∇H2 x(t) , for all t ∈ ϑ(T ).
REMARK 5.6.7 Evidently if x1 ∈ C 1 (T, R2N ) is periodic, then so is x2 ∈ C 1 (T, R2N ).
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5 Boundary Value Problems–Hamiltonian Systems
Now we use Proposition 5.6.6 and some convexity properties of H, to replace H by another Hamiltonian H that satisfies hypotheses H1 and so Theorem 5.6.3 can be used to produce a periodic trajectory. The hypotheses on the time-invariant (autonomous) Hamiltonian function H(x), are the following. H3 : H ∈ C 1 (R2N ), the energy surface S = H −1 is the boundary of a relatively compact, star-shaped neighborhood of the origin and ∇H(x) = 0 for all x ∈ S. THEOREM 5.6.8 If hypotheses H3 hold, then system (5.288) admits a periodic trajectory. PROOF: Because S = ∂U with U a relatively compact, star-shaped neighborhood of the origin in R2N , for every x ∈ R2N , we can find a unique ξ(x) ∈ S and µ(x)>0 such that x = µ(x)ξ(x) (in fact µ(x) = xn /ξ(x)). Evidently µ is 1-homogeneous and C 1 on R2N \ {0}. We define ) 0 if x = 0 . H(x)= µ(x)4 if x = 0
−1 Then H ∈ C 1 (R2N ), H (1) = S, H ≥ 0, lim H(x) x2 = 0 and since H is 4x→0
homogeneous we have that ∇H(x), x R2N = 4H(x) for all x ∈ R2N . Therefore in this case H satisfies the global (AR)-condition and so hypotheses H3 are satisfied and we can apply Theorem 5.6.5. So for b = 2π (for example), we can find a nonconstant
2π-periodic solution u ∈ C 1 (T, R2N ), T = [0, 2π]. We need not have H u(t) = 1
for all t ∈ T . Nevertheless, we have that H u(t) = λ = 0 (because u = 0). Set
1 . Then H νu(t) = 1 for all t ∈ T . Moreover, ν = λ1/4
1 for all t ∈ T. (5.290) −J νu (t) = ν∇H u(t) = 2 ∇H νu(t) ν
Set τ = ν −2 t and x(τ ) = νu(t). Then x ∈ C 1 ν12 T, R2N is a 2π -periodic map ν2 which by virtue of (5.290) satisfies −Jx (τ ) =
dt
1 ∇H νu(ν 2 τ ) (τ ) = ∇H x(τ ) . ν2 dτ
5.7 Remarks 5.1: Hypotheses H(f )1 are such that when p = 2 (semilinear problem), they incorporate in our framework the so-called asymptotically linear problems at 0+ and at +∞. Since the appearance of the pioneering work of Amann–Zehnder [17], these problems have attracted a lot of interest and several papers dealing with them have appeared. Indicatively we mention the works of Stuart–Zhou [565], Tehrani [579], and Zhou [627] (for p = 2), and Costa–Magalhaes [159], Li–Zhou [379], Fan–Zhao–Huang [237], Huang–Zhou [318], and Hu–Papageorgiou [317] (for p > 1). In the last work
5.7 Remarks
451
the potential function F (z, x) is nonsmooth, locally Lipschitz in x ∈ R, and so the right-hand side nonlinearity in problem (5.1) is multivalued, namely the generalized subdifferential of the function x −→ F (z, x). From the aforementioned nonlinear works with the exception of Huang–Zhou [318] and Hu–Papageorgiou [317], the rest deal with the situation
considered here in hypotheses H(f )1 , namely they assume that the slope f (z, x) (xp−1 ) stays below λ1 > 0 asymptotically as x −→ 0+ and stays above λ1 > 0 asymptotically as λ → +∞. However, their hypotheses are more restrictive than the ones used here. Huang–Zhou [318] and Hu–Papageorgiou [317] deal with the opposite situation. Compared with the Dirichlet problem, the study of nonlinear Neumann problems involving the p-Laplacian differential operator is lagging behind. We mention the works of Binding–Drabek–Huang [76], Faraci [238], Filippakis–Gasi´ nski– Papageorgiou [246], Motreanu–Papageorgiou [440], and Papalini [474]. When A = 0, problem (5.30) has been studied extensively and many solvability conditions are given, such as the coercivity condition (see Berger–Schechter [69]), the convexity condition (see Mawhin [414]), the subadditivity condition (see Tang [576]), and the sublinear condition (see Tang [577]). The case where A(t) = k2 ω 2 IN , with k ∈ N , ω = 2π/b, and IN the N×N identity matrix, was considered by Mawhin– Willem [415], under the assumption that the potential function x −→ F (t, x) is a C 1 convex function (hence x −→ ∇F (t, x) is a continuous, monotone, hence maximal monotone operator from RN into itself). The approach of Mawhin–Willem [415, p. 61], is based on a variant of the dual action principle. Also in the book of Mawhin–Willem [415, p. 88], we find the general problem, with the right-hand side nonlinearity F (t, x) satisfying |F (t, x)| ≤ h(t)
and
∇F (t, x) ≤ h(t)
for almost all t ∈ T , all x ∈ R , and with h ∈ L1 (T )+ . The potential function x −→ F (t, x) is not assumed to be convex. The work of Mawhin–Willem [415], was extended recently by Tang–Wu [578], who considered systems with a general Carath´eodory, strictly sublinear nonlinearity. The opposite situation, of a strictly superlinear nonlinearity, was studied by Motreanu–Motreanu–Papageorgiou [438]. We should mention that by virtue of hypothesis H(F )(iv), problem (5.31) is classified as a strongly resonant (at the zero eigenvalue) problem. Such problems exhibit a certain partial lack of compactness, which is evident in Proposition 5.1.16. Finally we should mention that uses of the second deformation theorem (see Theorem 4.6.1) to obtain multiplicity results for boundary value problems, can be found in Papageorgiou–Papageorgiou [484] and Kyritsi–Motreanu–Papageorgiou [369]. Multiplicity results for resonant nonlinear elliptic problems with the p-Laplacian can be found in Costa–Magalhaes [159], Li–Zhou [379], and Liu–Su [390]. Finally Theorem 5.1.27, was first proved for p = 2 by Brezis–Nirenberg [106]. N
5.2: The method of upper–lower solutions, provides an effective tool to produce existence theorems for first- and second-order initial and boundary value problems and to generate monotone iterative techniques which provide constructive methods (amenable to numerical treatment) to obtain these solutions. The question of existence of multiple solutions for problem (5.1) under condition of superlinear behavior of the nonlinearity f (z, x) both at zero and ±∞, has been studied in the past only in the context of semilinear problems (i.e., p = 2). First Ambrosetti–Mancini [22] proved that if λ > λ1 , then the problem has two nontrivial
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5 Boundary Value Problems–Hamiltonian Systems
solutions of constant sign (one positive and one negative). Soon thereafter Struwe [563] improved their result by showing that if λ > λ2 , then problem (5.1) (with p = 2) has at least three nontrivial solutions. Ambrosetti–Lupo [23] slightly improved the result of Struwe and they also presented an approach based on Morse theory. This required that f (z, ·) ∈ C 1 (R). The most general result for the semilinear problem (p = 2), can be found in Struwe [564, p. 132], who succeeded in eliminating the differentiability condition on the nonlinearity f and simplified the argument of Ambrosetti–Lupo. Still though Struwe [564] requires that f is Lipschitz continuous in x ∈ R. So even when p = 2, Theorem 5.2.10 is more general than the result of Struwe [564]. To our knowledge no other such multiplicity result for problem (5.1) with p > 1, can be found in the literature. Problem (5.33) has the interesting feature that it presents a compact formulation for the Dirichlet, Neumann, and Sturm–Liouville problems. Moreover, the method of proof for problem (5.33), can be repeated with only very minor changes in the case of the periodic problem (see problem (5.145)). Even more general nonlinear, multivalued boundary conditions can be found in the works of Halidias–Papageorgiou [282] and Gasi´ nski–Papageorgiou [258]. The method of upper and lower solutions was used to study such problems by Bader–Papageorgiou [50] and Douka–Papageorgiou [204]. Further uses of the method to initial and boundary value problems can be found in the book of Heikkila–Lakshmikantham [287]. 5.3: Degree theory has always been a powerful tool for the study of boundary value problems, especially for ordinary differential equations. The monograph of Mawhin [412] contains many such examples. The relatively recent generalizations of degree theory to nonlinear operators of monotone type (see Section 3.3) paved the way for the use of degree-theoretic methods to boundary value problems of nonlinear partial differential equations and to problems with unilateral constraints (such as variational and hemivariational inequalities). Our approach here in dealing with problem (5.146) is close to Amann [16] (where p = 2) and Ambrosetti–Garcia Azorero–Peral Alonso [24]. However, instead of the Leray–Schauder degree, here we employ the degree for operators of type (S)+ (see Section 5.3). This permits us to assume a nonsymmetric structure for our problem, in contrast to Ambrosetti–Garcia Azorero–Peral Alonso [24] (see hypothesis H(f )1 ). Other uses of degree-theoretic methods in the study of nonlinear elliptic problems with unilateral constraints can be found in Fillipakis–Papageorgiou [245] and Aizicovici–Papageorgiou–Staicu [6]. For the periodic problem (5.180), hypotheses H(f )2 assume a nonuniform nonresonance condition between two successive eigenvalues of the negative scalar pLaplacian with periodic boundary conditions. Analogous situations for p = 2 (semilinear problems) were considered by Fonda–Mawhin [251], and Habets–Metzen [280], and for p > 1 (nonlinear problems) and Dirichlet boundary value conditions by Zhang [623]. In hypotheses H(f )3 the asymptotic at 0 and ±∞ nonuniform nonresonance conditions (see hypotheses H(f )3 (iv) and (v)), are replaced by Landesman–Lazer type conditions (see hypothesis H(f )2 (iv)). In this direction results for p = 2 were obtained by Cesari–Kannan [141], Iannacci–Nkashama [322]; also see the references therein. The results on problem (5.180) here are a particular case of those by Aizicovici–Papageorgiou–Staicu [5]. 5.4: Problem (5.198) was studied for p = 2 (semilinear case) by Rabinowitz [508]. Problem (5.198) with p > 1 and m = 1, was examined by Fan–Li [236]. They also used
5.7 Remarks
453
the Ambrosetti–Rabinowitz condition (see hypothesis H(f )1 (iv)) and in addition they assumed that near the origin the right-hand side nonlinearity of the problem grows like |x|r−1 with r > p. We should also mention the work of Garcia Melian– Sabina de Lis [257], who obtained bifurcation results but under radial symmetry and with a C 2 -right-hand side nonlinearity. Proposition 5.4.8 is due to Pohozaev [497]. In problem (5.231) the term causing ∗ problems is the last term |x|2 −2 x. Recall that in the Sobolev embedding theorem ∗ the case r = 2 is the limiting case for the inclusion H01 (Z) ⊆ Lr (Z) and it is not compact. Thus for s = (N + 2) (N − 2) − 2∗ − 1 the map x −→ xs is not compact from H01 (Z) into H −1 (Z). For this reason compactness arguments do not work for problem (5.231), the linear term λu may be replaced by other compact perturbations (see Brezis–Nirenberg [104]). Additional results for problems with critical nonlinearities can be found in the works of Ambrosetti–Struwe [21], Cerami– Fortunato–Struwe [137], Guedda–Veron [275], Miyagaki [437], and Willem [606]. 5.5: Maximum principle–type results for the p-Laplacian can be found in Alegretto– Huang [12] and Garcia Melian–Sabina de Lis [257] who proved Theorem 5.1.8. Analogous results for some general nonlinear differential operators, can be found in Damascelli [172]. Also, we should mention the so-called antimaximum principle. So consider the following problem.
) * −div Dx(z)p−2 Dx(z) = λ|x(z)|p−2 x(z) + h(z) a.e. on Z, . (5.291) Bx = 0 Here B stands for the Dirichlet or Neumann boundary conditions, 1 < p < ∞ and h ∈ L∞ (Z). THEOREM 5.7.1 If h ∈ L∞ (Z), h(z) ≥ 0 for almost all z ∈ Z, h = 0, then (5.291) there exists δ = δ(h) such that for all λ ∈ (λ1 , λ1 + δ) any solution x ∈ C 1 (Z) of (5.291) satisfies x(z) < 0 for all z ∈ Z and (∂x/∂n)(z) > 0 for all z ∈ ∂Z. REMARK 5.7.2 It can be shown that δ > 0 can be chosen independently of h for the scalar (i.e., N = 1) Neumann problem (uniform antimaximum principle). For p = 2, the antimaximum principle was investigated by Clement–Peletier [155] and Godoy–Gossez–Paczka [267]. For p = 2 we refer to Godoy–Gossez–Paczka [268]. For comparison results involving the p-Laplacian, we refer to Garcia Melian– Sabina de Lis [257] (they obtained Theorem 5.5.9) and Guedda–Veron [275] (who proved Theorem 5.5.11). Further results in this direction can be found in Alegretto– Huang [12] and Damascelli [172]. 5.6: Periodic solutions with prescribed minimal period were obtained by Clarke [150] (who introduced the dual action technique), Clarke–Ekeland [151], Girardi– Matzeu [265], Mawhin–Willem [413] and Rabinowitz [506] (who employed the (AR)condition, see (5.262)). Periodic solutions on a given energy surface were first proved by Rabinowitz [506] and Weinstein [603]. Additional existence and multiplicity results in this direction can be found in Berestycki–Lasry–Mancini–Ruf [66], Ekeland– Hofer [225], and Rabinowitz [507]. Detailed studies of Hamiltonian systems can be
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5 Boundary Value Problems–Hamiltonian Systems
found in the books of Chang [143], Ekeland [227], Mawhin–Willem [415], and Struwe [564].
6 Multivalued Analysis
Summary. *Multivalued analysis deals with the study of maps whose values are sets. Multivalued analysis is closely related to nonsmooth analysis and a symbiotic relationship exists between them which feeds both with new ideas and directions to grow. This chapter presents in detail the main aspects of nonsmooth analysis. We study the continuity and measurability properties of multifunctions (set-valued functions) and we present the main selection theorems, for both continuous and measurable selectors (Michael’s theorem and the Kuratowski–Ryll Nardzewski and Yanlov–von Neumann–Aumann theorems). Then we deal with the set of integrable selectors of a multifunction and develop the main properties of the set-valued integral. We also prove fixed point theorems and study Carath´eodory multifunctions. Finally we examine the different modes of convergence of sets and of multifunctions.
Introduction Multivalued analysis deals with the study of the properties of the maps whose values are sets (elements in a hyperspace). The need for set-valued maps was recognized early in the twentieth century, but a systematic study started only in the mid-1960s. It is closely related to nonsmooth analysis (see Chapter 1). In fact the real explosion on multivalued analysis occurred at the exact same time that nonsmooth analysis made its appearance. Since then the two fields have moved in sychronization and provided each other with new tools, ideas, concepts, and results. This symbiotic relationship sustains their remarkable growth. In this chapter we attempt a survey of some of the basic aspects of the theory of multivalued analysis. Needless to say our presentation omits certain parts of the theory. A more comprehensive treatment can be found in the two volumes of Hu–Papageorgiou [313, 316]. In Section 6.1, which is topological in nature, we introduce various continuity notions. We investigate their properties and also examine how they are related among them. Section 6.2 is measure-theoretic in nature and deals with measurability properties of multifunctions. N.S. Papageorgiou, S.Th. Kyritsi-Yiallourou, Handbook of Applied Analysis, Advances in Mechanics and Mathematics 19, DOI 10.1007/b120946_6, © Springer Science+Business Media, LLC 2009
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In Section 6.3 we address the fundamental problem of existence of measurable selectors for multifunctions. So we prove the Michael selection theorem (for continuous selectors) and the Kuratowski–Ryll Nardzewski and the Yankov–von Neumann– Aumann selection theorems. In Section 6.4 we examine decomposable sets and the resulting set-valued integration theory. It turns out that in many situations decomposability is an effective substitute of convexity. Section 6.5 contains some major fixed point theorems. Also we conduct a brief investigation of multifunctions of two variables (Carath´eodory multifunctions). Finally in Section 6.6 we introduce and study various modes of convergence of sets. We also obtain Fatou-type results for set-valued integrals and for sets of integrable selectors.
6.1 Continuity of Multifunctions We start by fixing the notation. So let X be a Hausdorff topological space. We introduce the following hyperspaces, Pf (X) = {A ⊆ X : A is nonempty and closed} and
Pk (X) = {A ⊆ X : A is nonempty and compact}.
If X is a normed space, we also consider the following hyperspaces Pf c (X) = {A ∈ Pf (X) : A is convex} P(w)kc (X) = {A ⊆ X : A is nonempty, (weakly-) compact and convex} Pbf (c) (X) = {A ∈ Pf (X) : A is bounded (and convex)}. In what follows for X a Hausdorff topological space and x ∈ X, by N (x) we denote the filter of all neighborhoods of x. If (X, d) is a metric space, x ∈ X, and r > 0, then Br (x) = {y ∈ X : d(x, y) < r} (= the open r-ball with center at x ∈ X) and B r (x) = {y ∈ X : d(x, y) ≤ r} (= the closed r-ball with center at x ∈ X). When X is a normed space and x = 0, then as before we write Br = Br (0) and B r = B r (0). For sets X, Y and a multifunction (set-valued function) F : X −→ 2Y , given a set C ⊆ Y , we have two types of inverse images of C under the action of F , namely F + (C) = {x ∈ X : F (x) ⊆ C} and
−
(the strong inverse image of C)
F (C) = {x ∈ X : F (x) ∩ C = ∅}
(the weak inverse image of C).
−
Evidently we have F (C) ⊆ F (C) ⊆ X. It is straightforward to check that these inverse images obey the following calculus rules. +
PROPOSITION 6.1.1 Suppose that X, Y, Z are nonempty sets. (a) If F, G :X −→2Y are two multifunctions and (F ∪ G)(x) = F (x) ∪ G(x), (F ∩ G)(x) = F (x) ∩ G(x) for all x ∈ X, then (F ∪ G)+ (C) = F + (C) ∪ G+ (C),
(F ∩ G)+ (C) ⊇ F + (C) ∩ G+ (C)
(F ∪ G)− (C) = F − (C) ∪ G− (C),
(F ∩ G)− (C) ⊆ F − (C) ∩ G− (C)
for all C ⊆ Y.
6.1 Continuity of Multifunctions
457
(b) If F : X −→ 2Y , G : X −→ 2Z and x −→ (G ◦ F )(x) = G F (x) = G(y) for y∈F (x)
all x ∈ X, then (G ◦ F )+ (C) = F + G+ (C) , (G ◦ F )− (C) = F − G− (C) for all C ⊆ Y . (c) If {Ci , C}i∈I are subsets of Y (I being an arbitrary index set), then ! + # +
!
# F (Ci ) ⊆ F + Ci , F (Ci ) ⊆ F + Ci i∈I
!
i∈I
−
F (Ci ) =
! i∈I
i∈I −
F (Ci ),
i∈I
F
−
# i∈I
Ci ⊆
#
i∈I
F − (Ci ).
i∈I
(d) If F : X −→ 2Y , G : X −→ 2Z are two multifunctions and F ×G : X −→ 2Y ×Z is defined by (F × G)(x) = F (x) × G(x) for all x ∈ X, then (F × G)+ (C × D) = F + (C) ∩ G+ (D), (F × G)− (C × D) = F − (C) ∩ G− (D) for all C ⊆ Y , D ⊆ Z; these equalities are still true for arbitrary products of multifunctions. Now we introduce the first continuity notions for multifunctions. In what follows X, Y are Hausdorff topological spaces. Additional hypotheses are introduced as needed. DEFINITION 6.1.2 Let F : X −→ 2Y be a multifunction. (a) We say that F is upper semicontinuous at x0 ∈ X (usc at x0 for short), if for all V ⊆ Y open such that F (x0 ) ⊆ V , we can find U ∈ N (x0 ) such that F (U ) ⊆ V . If this true at every x0 ∈ X, we say that F is upper semicontinuous (usc for short). (b) We say that F is lower semicontinuous at x0 ∈ X (lsc at x0 for short), if for all V ⊆ Y open such that F (x0 ) ∩ V = ∅, we can find U ∈ N (x0 ) such that F (x) ∩ V = ∅ for all x ∈ U . If this is true at every x0 ∈ X, we say that F is lower semicontinuous (lsc for short). (c) We say that F is continuous or (Vietoris continuous) at x0 ∈ X, if it is both usc and lsc at x0 ∈ X. If this is true at every x0 ∈ X, then we say that F is continuous or Vietoris continuous. The following propositions are immediate consequences of the above definitions. PROPOSITION 6.1.3 Given a multifunction F : X −→ 2Y , the following statements are equivalent. (a) F is usc. (b) For every C ⊆ Y closed, F − (C) is closed in X. (c) If x ∈ X, {xα }α∈J ⊆ X is a net in X, xα −→ x, V ⊆ Y is an open set with F (x) ⊆ V , then we can find α0 ∈ J such that F (xα ) ⊆ V for all α ∈ J with α ≥ α0 . PROPOSITION 6.1.4 Given a multifunction F : X −→ 2Y , the following statements are equivalent. (a) F is lsc. (b) For every C ⊆ Y closed, F + (C) is closed in X.
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6 Multivalued Analysis
(c) If x ∈ X, {xα }α∈J ⊆ X is a net in X, xα −→ x, and V ⊆ Y is an open set with F (x) ∩ V = ∅, then we can find α0 ∈ J such that F (xα ) ∩ V = ∅ for all α ∈ J with α ≥ α0 . (d) If x ∈ X, {xα }α∈J ⊆ X is a net in X, xα −→ x, and y ∈ F (x), then for every α ∈ J we can find yα ∈ F (xα ) such that yα −→ y in X. REMARK 6.1.5 It is clear from Definition 6.1.2, that when F is single-valued, the notions of upper and lower semicontinuity coincide with the usual notion of continuity of a map between two Hausdorff topological spaces. Also because of Proposition 6.1.1(c), we see that in the definition of lower semicontinuity (see Definition 6.1.2(b)), we can take V to be a basic open set in Y . In general the notions of upper and lower semicontinuity are distinct. Upper semicontinuity allows upward jumps (in the sense of inclusion), and lower semicontinuity allows downward jumps (in the sense of inclusion); see Propositions 6.1.3(c) and 6.1.4(c), respectively. For example [0, 1] if x = 0 F1 (x)= 1 if x = 0 is usc but not lsc, whereas F2 (x)=
{0} [0, 1]
if x = 0 if x = 0
is lsc but not usc. Note that if F : R −→ 2R is defined by F (x) = [ψ(x), ϕ(x)] = {y ∈ R : ψ(x) ≤ y ≤ ϕ(x)} and ψ is lower semicontinuous, ϕ is upper semicontinuous, then F is usc, whereas if ψ is upper semicontinuous, and ϕ is lower semicontinuous, then F is lsc. Combining Propositions 6.1.3 and 6.1.4, we have the following. PROPOSITION 6.1.6 Given a multifunction F : X −→ 2Y , the following statements are equivalent. (a) F is continuous. (b) For every C ⊆Y closed, F + (C) and F − (C) are both closed in X. (c) If x ∈ X, {xα }α∈J ⊆ X is a net in X, xα −→ x, and V ⊆ Y is an open set with F (x) ⊆ V or F (x) ∩ V = ∅, then we can find α0 ∈ J such that for all α ∈ J, α ≥ α0 we have F (xα ) ⊆ V or F (xα ) ∩ V = ∅. DEFINITION 6.1.7 Given a multifunction F : X −→ 2Y , its graph is the set GrF = {(x, y) ∈ X × Y : y ∈ F (x)}. It is well known that a continuous map between two Hausdorff topological spaces, has a closed graph. The same is true for usc multifunctions. PROPOSITION 6.1.8 If Y is a regular topological space and F :X −→Pf (Y ) is usc, then GrF is closed in X × Y .
6.1 Continuity of Multifunctions
459
PROOF: Let {(xα , yα )}α∈J ⊆ GrF ⊆ X ×Y be a net and suppose that (xα , yα ) −→ (x, y) in X × Y . Arguing by contradiction, suppose that y ∈ / F (x). Then exploiting the regularity of the space Y , we can find V1 ∈ N (y) and V2 an open neighborhood of F (x) such that V2 ⊇ F (x) and V1 ∩ V2 = ∅. But because yα −→ y and using Proposition 6.1.3, we can find α0 ∈ J such that for all α ∈ J, α ≥ α0 we have yα ∈ V1 and F (xα ) ⊆ V2 , hence yα ∈ / F (xα ), a contradiction. REMARK 6.1.9 It is clear from the above proof that if F has values in Pk (Y ), then in the above proposition we can drop the requirement that Y is regular. This is consistent with the fact that for single-valued maps continuity implies closedness of the graph without any additional conditions on the space Y . Even for single-valued maps, the converse of Proposition 6.1.8 is not in general true. PROPOSITION 6.1.10 If F : X −→ Pk (Y ) has a closed graph and it is locally compact (i.e., for every x ∈ X we can find U ∈ N (x) such that F (U ) is compact in Y ), then F is usc. PROOF: Given C ⊆ Y we show that F − (C) is closed and this by virtue of Proposition 6.1.3 implies that F is usc. So let {xα }α∈J ⊆ F − (C) be a net and assume that xα −→ x. By hypothesis we can find U ∈ N (x) such that F (U ) is compact in Y . We can find α0 ∈ J such that if α ∈ J, α ≥ α0 , then xα ∈ U and so if yα ∈ F (xα ), α ≥ α0 , we can find a subnet {yβ }β∈I of {yα }α∈J and y ∈ F (U ) such that yβ −→ y. Evidently (x, y) ∈ GrF ∩ (X × C); that is, y ∈ F (x) and y ∈ C. Therefore x ∈ F − (C) and so F − (C) is closed. DEFINITION 6.1.11 We say that a multifunction F : X −→ 2Y is closed (resp., sequentially closed ), if GrF ⊆X×Y is closed (resp., sequentially closed). REMARK 6.1.12 Evidently a closed multifunction F : X −→ 2Y \{∅} has values in Pf (Y ). PROPOSITION 6.1.13 If F : X −→ Pk (Y ) is usc and K ⊆ X is compact, then F (K) ⊆ Y is compact. PROOF: Let {yα }α∈J ⊆ F (K) be a net. Then yα ∈ F (xα ), xα ∈ K for all α ∈ J. By virtue of the compactness of K we can find {xβ }β∈I a subnet of {xα }α∈J such that xβ −→ x. We claim that {yβ }β∈I has a cluster point in F (x). We argue indirectly. So suppose that for all y ∈ F (x), we can find β0 (y) ∈ J and V (y) ⊆ N (y) such that yβ ∈ / V (y) for all β ∈ I, β ≥ β0 (y). Note that {V (y)}y∈F (x) is an open cover of the compact set F (x). So we can find {V (y)}N k=1 an open subcover. Set N
V = ∪ V (yk ) ∈ N (y). Then we can find β ∈ I such that for all β ∈ I, β ≥ β1 , we k=1
N
have yβ ∈ / V = ∪ V (yk ) ⊇ F (x), which contradicts Proposition 6.1.3(c). Therefore k=1
we can find a subnet {yr }r∈S of {yβ }β∈I such that yr −→ y ∈ F (x) ⊆ F (K). This proves the compactness of F (K). The following two functions associated with a set A ⊆ X play a central role in what follows.
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6 Multivalued Analysis
DEFINITION 6.1.14 (a) Let (X, d) be a metric space and A ⊆ X. For every x ∈ X, we define d(x, A) = inf[d(x, a) : a ∈ A]. As usual, we adopt the convention that inf = +∞. The distance function ∅
x −→ d(x, A) is a contraction. (b) Let X be a normed space. By X ∗ we denote the topological dual of X and by ·, · the duality brackets for the pair (X, X ∗ ). Given A ⊆ X, for every x∗ ∈ X ∗ we define σ(x∗ , A) = sup[x∗ , a : a ∈ A]. As usual, we adopt the convention that sup = −∞.The function x∗ −→ σ(x∗ , A) ∅
from X ∗ into R ∪ {±∞} is called the support function of the set A. PROPOSITION 6.1.15 Let F : X −→ 2Y \{∅} be a multifunction. (a) If (Y, d) is a metric space, then F (·) is lsc if and only if for every y ∈ Y , x −→ d y, F (x) is upper semicontinuous.
(b) If (Y, d) is a metric space and F is usc, then for every y ∈ Y x −→ d y, F (x) is lower semicontinuous; the converse is true if F is locally compact (see Proposition 6.1.10). (c) If Y is a normed space with the weak topology and F is usc, then for
furnished all y ∗ ∈ Y ∗ x −→ σ y ∗ , F (x) is an upper semicontinuous function from X into R = R ∪ {+∞}. PROOF: (a) ⇒ : Suppose F is lsc. We
need to show that for every λ ∈ R the upper level set Uλ = x ∈ X : ϑy (x) = d y, F (x) ≥ λ is closed. So suppose that {xα }α∈J ⊆ Uλ is a net and assume that xα −→ x. Given ε > 0 we can find v ∈ F (x) such that d(y, v) < ϑy (x) + ε. Because F is lsc, we can find α0 ∈ J such that if α ∈ J, α ≥ α0 we have F (xα ) ∩ Bε (v) = ∅. So we can find yα ∈ F (xα ), α ≥ α0 , such that d(y, yα ) ≤ ϑy (x) + 2ε and so ϑy (xα ) ≤ ϑy (x) + 2ε, hence λ ≤ ϑy (x) + 2ε. Because ε > 0 was we let ε ↓ 0 to obtain λ ≤ ϑy (x). Therefore Uλ is closed
arbitrary, and so x −→ d y, F (x) is upper semicontinuous. ⇐: Let V ⊆ Y be open and let x ∈ F − (V ). We choose y ∈ F (x) ∩ V . Let ε > 0 such that Bε (y) ⊆ V . Because ϑy (·) is upper semicontinuous, we can find U ∈ N (x) such that ϑy (x ) < ϑy (x) + ε = ε ⇒ F (x ) ∩ Bε (y) = ∅
⇒ F (x ) ∩ V = ∅
for all x ∈ U, for all x ∈ U,
for all x ∈ U.
This means that F is lsc (see Definition 6.1.2(b)). (b) We need that for every λ ∈ R, the lower level set Lλ = {x ∈ X :
to show ϑy (x) = d y, F (x) ≤ λ} is closed. To this end let {xα }α∈J ⊆ Lλ be a net such that xα −→ x. Because F is usc, given ε > 0, we can find α0 ∈ J such that if α ∈ J, α ≥ α0 , then
F (xα ) ⊆ F (x)ε = {v ∈ Y : d v, F (x) < ε}, ⇒ ϑy (x) < ϑy (xα ) + ε, ⇒ ϑy (x) < λ + ε.
6.1 Continuity of Multifunctions
461
Because ε > 0 was arbitrary, we let ε ↓ 0 to conclude that ϑy (x) ≤ λ and so x ∈ Lλ . This proves the lower semicontinuity of the distance function x −→ ϑy (x) = d y, F (x) . Now suppose that F : X −→ Pk (Y ) is locally compact and the distance function x −→ ϑy (x) is lower semicontinuous for all y ∈ Y . We show that F is usc. By virtue of Proposition 6.1.10, it suffices to show that Gr F is closed in X × Y . To this end let {(xα , yα
)}α∈J ⊆ Gr F be a net and assume that (xα , yα ) −→ (x, y) in X × Y . We have d y, F (x) ≤ d(y, yα ) −→ 0. Because ϑy (·) is lower semicontinuous
we also have d y, F (x) ≤ lim inf d y, F (xα ) . Therefore it follows that d y, F (x) = 0 (i.e., (x, y) ∈ Gr F ).
α∈J
(c) Fix y ∗ ∈ Y ∗ and ε > 0 and define W (y ∗ , ε) = {y ∈ Y : y ∗ , y < ε}. This is a weak neighborhood of the origin. By hypothesis F is usc from X into Yw (= space Y endowed with the weak topology). So we can find U ∈ N (x) such that for all u ∈ U, F (u) ⊆ F (x) + W (y ∗ , ε)
for all y ∈ U, ⇒ σ y ∗ , F (u) ≤ σ y ∗ , F (x) + ε
∗ ⇒ x −→ σ y , F (x) is upper semicontinuous. EXAMPLE 6.1.16 (a) In general lower semicontinuity of the distance function does not imply upper semicontinuity of the multifunction. To see this let X = R, Y = R 2 , and F : X −→ Pf (Y ) is defined by F (x) = (t, xt) : t ∈ R . Clearly ϑy (x)=d y, F (x) is continuous in x ∈ R, but F is not usc. (b) The converse of Proposition 6.1.15(c) is not in general true. To see this let X =R+ , Y =R and consider F :X −→Pkc (Y ) defined by F (x)=
{−1, 1} [0, x]
if x = 0 . if x = 0
For every y ∗ ∈Y ∗ =R we have that x −→ σ y ∗ , F (x) is upper semicontinuous, but x −→ F (x) is not usc at x = 0. However, if we strengthen the hypotheses on the multifunction F , we can have a converse to Proposition 6.1.15(c). For a proof of this result we refer to Hu– Papageorgiou [313, pp. 47–48]. PROPOSITION 6.1.17 If Y is a normed space and F : X −→ Pwkc (Y ) is a multifunction such that for all y ∗ ∈ Y ∗ the function x −→ σ y ∗ , F (x) from X into R = R ∪ {+∞} is upper semicontinuous, then the multifunction F is usc from X into Yw .
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The next result is useful in many applications. Consider a function u : X ×Y −→ R ∪ {±∞} and a multifunction F : Y −→ 2X \ {∅}. We consider the following parametric maximization problem, v(y) = sup[u(x, y) : x ∈ F (y)]. We call v(·) the value function depending on the parameter y ∈ Y . In addition to the value function, we also consider the solution multifunction y −→ S(y) defined by S(y) = {x ∈ F (y) : u(x, y) = v(y)}. The next result, useful in a variety of applications, is often called the Berge maximum theorem. THEOREM 6.1.18 Let X, Y, F, v(·), and S(·) be as above and assume that for every y ∈ Y we can find x ∈ F (y) such that u(x, y) ∈ R. (a) If u is lower semicontinuous and F is lower semicontinuous, then the value function v : Y −→ R = R ∪ {+∞} is lower semicontinuous. (b) If u is upper semicontinuous and F is upper semicontinuous with values in Pk (X), then the value function v : Y −→ R = R∪{+∞} is upper semicontinuous. (c) If u : X × Y −→ R is continuous and F is continuous with values in Pk (X), then the value function v : Y −→ R is continuous and the solution multifunction S : Y −→ Pk (X) is upper semicontinuous. PROOF: (a) For every λ ∈ R, we need to show the level set Lλ = {y ∈ Y : v(y) ≤ λ} is closed. So let {yα }α∈J ⊆ Lλ be a net such that yα −→ y. By virtue of Proposition 6.1.4(d), for every α ∈ J we can find xα ∈ F (yα ) such that xα −→ x in X. We have u(xα , yα ) ≤ v(yα ) ≤ λ for all α ∈ J and because u is lower semicontinuous, we obtain u(x, y) ≤ λ. Because x ∈ F (y) was arbitrary, it follows that v(y) ≤ λ, hence y ∈ Lλ . This proves the lower semicontinuity of v(·). (b) For every λ ∈ R, we need to show that the upper level set Uλ = {y ∈ Y : v(y) ≥ λ} is closed. So let {yα }α∈J be a net such that yα −→ y. Since F has values in Pk (X), for every α ∈ J, we can find xα ∈ F (yα ) such that v(yα ) = u(xα , yα ) (Weierstrass theorem). As in the proof of Proposition 6.1.13, we can show that {xα }α∈J has a cluster point in F (y). Therefore we can find x ∈ F (y) and a subnet {xβ }β∈I of {xα }α∈J such that xβ −→ x. We have λ ≤ v(yβ ) = u(xβ , yβ )
for all β ∈ I and lim sup u(xβ , yβ ) ≤ u(x, y), β∈I
⇒ λ ≤ u(x, y) ≤ v(y)
(see x ∈ F (y)).
This proves that Uλ is closed and so v is upper semicontinuous. (c) The continuity of v follows from parts (a) and (b). So it remains to show that S : Y −→ Pk (X) is usc. Because of Proposition 6.1.3, given A ⊆ X closed, we need to show that S − (A)={y ∈ Y :S(y) ∩ A = ∅} is closed. So let {yα }α∈J ⊆ S − (A) be a net such that yα −→ y in Y . For each α ∈ J, we can find xα ∈ F (yα ) ∩ A such that u(xα , yα ) = v(yα ). As before {xα }α∈J has a cluster point x ∈ F (y). Thus we can find a subnet {xβ }β∈I of {xα }α∈J such that xβ −→ x. Evidently x ∈ F (y) ∩ A
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and u(x, y) = v(y) (because u and v are continuous). Therefore y ∈ S − (A) and we have that S − (A) is closed in Y , hence S is usc. Given a multifunction F : X −→ 2Y , note that for V ⊆ Y open, F (x) ∩ V = ∅ if and only if F (x) ∩ V = ∅. So we can state the following result. PROPOSITION 6.1.19 A multifunction F : X −→ 2Y \ {∅} is lsc if and only if x −→ F (x) = F (x) is lsc. The above result is not true for usc multifunctions. EXAMPLE 6.1.20 Proposition 6.1.19 fails for usc multifunctions. To see this let X = Y = R and let F : R −→ 2R \ {∅} be defined by F (x)=(x − 1, x + 1). Note that F + (−1, 1) ={0} and so F is not usc. On the other hand F (x)=[x − 1, x + 1] and by virtue of Remark 6.1.5 F is continuous. For usc multifunctions we have the following result. PROPOSITION 6.1.21 If Y is normal and F : X −→ 2Y \ {∅} is usc, then so is F (·). PROOF: Let {xα }α∈J be a net such that xα −→ x and let V ⊆ Y be an open set such that F (x) ⊆ V . Because of the normality of Y , we can find V0 ⊆ Y another open set such that F (x) ⊆ F (x) ⊆ V0 ⊆ V 0 ⊆ V. (6.1) Because F is usc, we can find α0 ∈ J such that for α ∈ J, α ≥ α0 we have F (xα ) ⊆ V0 ⇒ F (xα ) ⊆ V ⇒ F
(see Proposition 6.1.3), for all α ≥ α0
(see (6.1)),
is usc (see Proposition 6.1.3).
PROPOSITION 6.1.22 If Y is normal and F1 , F2 : X −→ Pf (Y ) are both usc multifunctions such that for every x ∈ X, (F1 ∩ F2 )(x) = F1 (x) ∩ F2 (x) = ∅, then x −→ (F1 ∩ F2 )(x) is usc too. PROOF: We need to show that if V ⊆ Y is open, then (F1 ∩ F2 )+ (V ) is open. By definition (F1 ∩ F2 )+ (V ) = {x ∈ X : F1 (x) ∩ F2 (x) ∩ V c = ∅}. (6.2) Let x ∈ (F1 ∩ F2 )+ (V ). Then because of (6.2) the sets F1 (x) and F2 (x) ∩ V c are disjoint closed sets. Due to the normality of Y we can find V1 , V2 ⊆ Y disjoint open sets such that F1 (x) ⊆ V1 and F2 (x) ∩ V c ⊆ V2 . Let V3 = V2 ∪V . Then F2 (x) ⊆ V3 . Because F1 , F2 are usc, we can find U ∈ N (x) such that for all x ∈ U , we have F1 (x ) ⊆ V1 and F2 (x ) ⊆ V3 . Therefore F1 (x ) ∩ F2 (x ) ⊆ V1 ∩ V3 = V1 ∩ (V2 ∪ V ) ⊆ V ⇒ (F1 ∩ F2 )+ (V ) is open (i.e., F1 ∩ F2 is usc).
for all x ∈ U,
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Another result in this direction is given below. PROPOSITION 6.1.23 If F1 :X −→Pf (Y ) is closed (see Definition 6.1.11), F2 : X −→ Pk (Y ) is usc and for all x ∈ X, (F1 ∩ F2 )(x) = F1 (x) ∩ F2 (x) = ∅, then x −→ (F1 ∩ F2 )(x) is usc. PROOF: We need to show that if V ⊆ Y is open, then (F1 ∩ F2 )+ (V ) is open. Let x ∈ (F1 ∩ F2 )+ (V ). From (6.2) we have that the sets F1 (x) and F2 (x) ∩ V c are disjoint. Note that F2 (x) ∩ V c is compact in Y . Let y ∈ F2 (x) ∩ V c . Then (x, y) ∈ / Gr F1 and because Gr F1 is closed, we can find Uy ∈ N (x) and Wy ∈ N (y) such that (Uy × Wy ) ∩ Gr F1 = ∅. Hence for all x ∈ Uy we have F1 (x ) ∩ Wy = ∅. As we already observed, the set F2 (x) ∩ V c is compact. So we N N c Wyk =V1 ∈N (y). Set U1 = Uyk ∈ N (x). If can find {yk }N k=1 ⊆ F2 (x) ∩ V ⊆ k=1
k=1
x ∈ U1 , then F1 (x ) ⊆ V1c . Let V2 = V ∪ V1 and U2 ∈ N (x) be such that if x ∈ U2 , then F2 (x ) ⊆ V2 (recall that F2 is usc). Then if x ∈ U1 ∩ U2 ∈ N (x), we have (F1 ∩ F2 )(x ) = F1 (x ) ∩ F2 (x ) ⊆ V1c ∩ V2 ⊆ V , which proves that (F1 ∩ F2 )+ (V ) is open, hence x −→ (F1 ∩ F2 ) is usc. What about the intersection of two lsc multifunctions? In this case the situation is more delicate. First note that, if F : X −→ 2Y \{∅} is lsc and V ⊆Y a nonempty open set such that F (x) ∩ V = ∅ for all x ∈ X, then x −→ F (x) ∩ V is lsc. Moreover, if C ⊆ X is closed and F (x) ∩ V for x ∈ C , F (x) = F (x) for x ∈ C c then it is straightforward to check that F (·) is lsc. More generally, we have the following result. PROPOSITION 6.1.24 If F1 :X −→2Y \{∅} is lsc, F2 :X −→2Y \{∅} has an open graph, and for every x ∈ X, (F1 ∩F2 )(x) = F1 (x)∩F2 (x) = ∅, then x −→ (F1 ∩F2 )(x) is lsc. PROOF: Let V ⊆Y be nonempty open, x∈(F1 ∩F2 )− (V ), and y∈F1 (x)∩F2 (x)∩V . Then (x, y) ∈ Gr F2 ∩ (X × V ). Since Gr F2 is open we can find U1 (x) ∈ N (x) and V1 (y) ∈ N (y) such that U1 (x)×V1 (y) ⊆ Gr F2 ∩(X ×V ). Note that F1 (x)∩V1 (y) = ∅ and because F1 is lsc, we can find U2 (x) ∈ N (x) such that F1 (x ) ∩ V1 (y) = ∅ for all x ∈ U2 (x). Set U (x) = U1 (x) ∩ U2 (x) ∈ N (x). Then for all x ∈ U (x) we have F1 (x ) ∩ V1 (y) = ∅ and U (x) × V1 (y) ⊆ Gr F2 ∩ (X × V ). Therefore for all x ∈ U (x), we have F1 (x ) ∩ F2 (x ) = ∅ which means that (F1 ∩ F2 )− (V ) is open and so x −→ (F1 ∩ F2 )(x) is lsc. REMARK 6.1.25 Note that if Gr F is open, then F has open values. However, Proposition 6.1.24 fails if F2 instead of an open graph has only open values. If Y =RN , then the situation improves and we have the following two propositions whose proof can be found in Hu–Papageorgiou [313, pp. 55–56].
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PROPOSITION 6.1.26 If F1 , F2 : X −→ RN \{∅} are lsc multifunctions, F2 has open convex values, and for all x ∈ X, (F1 ∩ F2 )(x) = F1 (x) ∩ F2 (x) = ∅, then x −→ (F1 ∩ F2 )(x) is lsc. PROPOSITION 6.1.27 If F1 , F2 :X −→RN\{∅} are lsc with convex values and for every x ∈ X, (F1 ∩ int F2 )(x) = ∅, or (int F1 ∩ F2 )(x) = ∅, then x −→ (F1 ∩ F2 )(x) is lsc. REMARK 6.1.28 It is easy to check that if Y is a Banach space and F : X −→ 2Y \ {∅} is lsc, then so are the multifunctions x −→ conv F (x), x −→ conv F (x). Similarly if F : X −→ Pk (Y ) is usc, then so is the multifunction x −→ conv F (x). Finally for X, Y Hausdorff topological spaces, then both upper and lower semicontinuity are preserved by the set-theoretic operation of union. When X is a metric space, then we can exploit its metric structure, to introduce some additional continuity notions for multifunctions. For this purpose, we introduce the following notions. DEFINITION 6.1.29 Let (X, d) be a metric space and A, C ⊆ X. We set h∗ (A, C) = sup d(a, C) : a ∈ A = inf ε > 0 : A ⊆ Cε , where Cε = {x ∈ X : d(x, C) < ε} (the open ε-enlargement of C). We call h∗ (A, C) the excess of A over C. Then the Hausdorff distance between A and C is defined by h(A, C) = max h∗ (A, C), h∗ (C, A) = inf ε > 0 : A ⊆ Cε and C ⊆ Aε . It is easy to see that h(A, C) = 0 if and only if A = C and so
REMARK 6.1.30 Pf (X) ∪ {∅}, h is a (generalized) metric space. The empty set is an isolated point in this metric space and the metric h(·, ·) is called the Hausdorff metric. The next proposition is a straightforward consequence of Definition 6.1.29. PROPOSITION 6.1.31 Let (X, d) be a metric space and A, C ⊆ X nonempty. (a) h∗ (A, C) = sup d(x, C) − d(x, A) : x ∈ X (b) h(A, C) = sup |d(x, C) − d(x, A)| : x ∈ X ∗ (c) If X is a normed space and A, C ∈ Pbf c (X), then h(A, C) = sup |σ(x ; A) − ∗ ∗ ∗ ∗ σ(x ; C)| : x ∈ X , x ≤ 1 (H¨ ormander’s formula) The next proposition summarizes the basic facts about this metric. Detailed proofs can be found in Hu–Papageorgiou [313, p. 8]. PROPOSITION 6.1.32 (a) If (X, d) is a complete metric space, then so is (Pf (X), h).
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(b) Pbf (X) is a closed subset of (Pf (X), h) and if X is separable, then so is (Pk (X), h). (c) If X is a normed space, then
Pkc (X) ⊆ Pbf c (X) ⊆ Pf c (X) and Pk (X) ⊆ Pbf (X) are all closed subspaces of Pf (X), h . Now we are ready to introduce the new continuity concepts for multifunctions. DEFINITION 6.1.33 Let X be a Hausdorff topological space, (Y, d) a metric space, and F : X −→ 2Y \{∅} a multifunction. (a) We say that F is h-upper semicontinuous at x0 ∈ X (h-usc for short), if the function x −→ h∗ F (x), F (x0 ) is continuous at x0 ∈ X. If F (·) is h-usc at every x0 ∈ X, then we say that F is h-upper semicontinuous (h-usc for short). (b) We say that F is h-lower semicontinuous at x0 ∈ X (h-lsc for short), if the function x −→ h∗ F (x0 ), F (x) is continuous at x0 ∈ X. If F (·) is h-lsc at every x0 ∈ X, then we say that F is h-lower semicontinuous (h-lsc for short). (c) We say that F is h-continuous at x0 ∈ X, if it is both h-usc and h- lsc at x0 ∈ X. If F (·) is h-continuous at every x0 ∈ X, then we say that F is h-continuous. REMARK 6.1.34 Note that h-continuity of F , is continuity from X into the pseudometric space 2Y \{∅}, h . It is natural to ask how these new continuity notions compare to those of Definition 6.1.2. In what follows X is a Hausdorff topological space and (Y, d) a metric space. PROPOSITION 6.1.35 If F : X −→ 2Y \{∅} is usc, then F is h-usc.
PROOF: Because F is usc, for every ε > 0 and every x ∈ X we have F + F (x)ε = Ux ∈ N (x) (recall that F (x)ε ={y ∈Y
: d y, F (x) < ε}). For every x ∈Ux we have F (x ) ⊆ F (x)ε , which means that h F (x ), F (x) < ε and so we conclude that F is h-usc. The converse of the above proposition is not in general true as the following example illustrates. EXAMPLE 6.1.36 h-usc usc: Let X = [0, 1], Y = R and consider the multifunction F : X −→ 2Y \ {∅} defined by F (x) =
[0, 1] [0, 1)
if 0 ≤ x < 1 . if x = 1
Clearly F is h-usc, but it is not usc because F + (−1, 1) ={1}. PROPOSITION 6.1.37 If F : X −→ 2Y \{∅} is h-lsc, then F is lsc.
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PROOF: We need to show that for every C ⊆ Y closed, the set F + (C) is closed. To this end, let {xα }α∈J ⊆ F + (C) be a net and assume that xα −→ x. We have F (xα ) ⊆ C for all α ∈ J. Because F is h-lsc, given ε > 0, we can find α0 ∈ J such that for all α ∈ J, α ≥ α0 we have F (x) ⊆ F (xα ) ⊆ Cε . Let ε ↓ 0 to conclude that F (x) ⊆ C (i.e., x ∈ F + (C)). This proves that F + (C) is closed and so F is lsc. Again the converse of this proposition, in general fails. 2 Y EXAMPLE 6.1.38 lsc h-lsc: Let X = [0, 1], Y = R+ , and let F : X −→ 2 \{∅} be defined by F (x) = [t, xt] : t > 0 . Then F is lsc but not h-lsc.
However, for Pk (Y )-valued multifunction the situation is better. PROPOSITION 6.1.39 If F : X −→ Pk (Y ), then (a) F is usc if and only if F is h-usc. (b) F is lsc if and only if F is h-lsc. PROOF: (a)⇒ : This implication is Proposition 6.1.35. ⇐ : We need to show that for every C ⊆ Y closed, F − (C) is closed. To this end let {xα }α∈J ⊆ F − (C) be a net such that xα −→ x. Let yα ∈ F (xα ) ∩ C for all α ∈ J. We have
d yα , F (x) ≤ h∗ F (xα ), F (x) −→ 0
(because F is h-usc).
(6.3)
Because F (x) ∈ Pk (Y ), for every α ∈ J we can find uα ∈ F (x) such that d yα , F (x) = d(yα , uα ). Due to the compactness of F (x), we can find {uβ }β∈I a subnet of {uα }α∈J such that uβ −→ y ∈ F (x). Then because of (6.3), we see that yβ −→ y ∈ F (x) ∩ C (because C is closed). Therefore x ∈ F − (C) and we conclude that F − (C) is closed, hence F is usc. F (b)⇒ : Suppose {xα }α∈J ⊆X is a net such that xα −→ x. Because
(x) ∈ Pk (X), for every α ∈ J we can find uα ∈ F (x) such that d uα , F (xα ) = h∗ F (x), F (xα ) . Also we can find a subnet {uβ }β∈I of {uα }α∈J such that uβ −→ y. Since F is lsc, given ε > 0, we can find β0 ∈ I such that for all β ∈ I, β ≥ β0 , we have F (xβ ) ∩ Bε/2 (y) = ∅ and uβ ∈ Bε/2 (y). So for β ≥ β0 , we have
h∗ F (x), F (xβ ) ≤ d(uβ , y) + d y, F (xβ ) < ε,
⇒ lim h∗ F (x), F (xβ ) = 0. β∈I
(6.4)
Because every α }α∈J has a further subnet so that (6.4) holds, we
subnet of {x conclude that h∗ F (x), F (xβ ) −→ 0, hence F is h-lsc. ⇐ : This implication is Proposition 6.1.37.
COROLLARY 6.1.40 If F :X −→Pk (Y ), then F is continuous if and only if it is h-continuous.
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REMARK 6.1.41 If a multifunction F : X −→ 2Y \{∅} is h-usc (resp., h-lsc, hcontinuous), then so are the multifunctions x −→ F (x), x −→ conv F (x), x −→ conv F (x). Moreover, if G : X −→ 2Y \ {∅} is another multifunction which too is h-usc (resp., h-lsc, h-continuous), then so is x −→ (F ∪ G)(x). The situation is more complicated with the operation of intersection. To deal with the h-continuity properties of the intersection of two multifunctions, we need the following lemma, known as the cancellation law lemma. LEMMA 6.1.42 If V is a locally convex space, D ⊆ V is nonempty bounded, and C ⊆ V is convex, then A + D ⊆ C + D implies A ⊆ C. PROOF: First we assume that C ∈ Pf c (V ) and A + D ⊆ C + D. Let a ∈ A and d1 ∈ D. By induction we can choose ck ∈ C and dk+1 ∈ D such that a + dk = ck + dk+1 for all k ≥ 1. Then ck = a + dk − dk+1 . Summing up to n and then dividing by n, we obtain a+
n d1 dn+1 1 ck . − = n n n k=1
Because by hypothesis D is bounded, we have n
d1 dn+1 1 a = lim a + ck ∈ C, − = lim n→∞ n→∞ n n n k=1
because we have assumed that C ∈ Pf c (V ). Now we consider the general case. Let U ∈ N (0) be convex and choose U1 ∈ N (0) convex such that U1 + U1 ⊆ U . We have A + D ⊆ C + D ⊆ C + D + U1 ⊆ C + U1 + D. Note that C + U1 ∈ Pf c (V ). So we are in the first part of the proof, from which it follows that A ⊆ C + U1 ⊆ C + U1 + U1 ⊆ C + U . Because U ∈ N (0) was arbitrary, we conclude that A ⊆ C. LEMMA 6.1.43 If V is a normed space and A ⊆ V is convex bounded with int A = ∅, then for every ε > 0 there exists C ⊆ int A and δ > 0 such that Cδ ⊆ A ⊆ Cε . PROOF: By translating things if necessary, we may assume that 0 ∈ int A. Because A is bounded, given ε > 0 we can find λ ∈ (0, 1) such that λ int A ⊆ Bε/2 . Also because 0 ∈ int A, we can find δ > 0 such that Bδ ⊆ λ int A. Set C = (1 − λ)int A. Then C + Bδ = Cδ ⊆ A. On the other hand because A is convex, A = int A and so A ⊆ C + Bε = Cε . REMARK 6.1.44 This lemma fails if A is not bounded. PROPOSITION 6.1.45 If Y is a normed space and F1 , F2 : X −→ Pbf c (Y ) are h-lsc and for all x ∈ X we have int (F1 ∩ F2 )(x) = ∅, then x −→ F (x) = (F1 ∩ F2 )(x) = F1 (x) ∩ F2 (x) is h-lsc.
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PROOF: Fix x ∈ X and let ε > 0 be given. Because of Lemma 6.1.43, we can find C ⊆ int(F1 ∩ F2 )(x) and δ > 0 such that Cδ ⊆ F (x) ⊆ Cε . Since F1 , F2 are h-lsc, we can find U ∈ N (x) such that Fk (x) ⊆ Fk (x )δ
for all x ∈ U and for k = 1, 2, for all x ∈ U.
⇒ Cδ ⊆ Fk (x )δ
(6.5)
From (6.5) and Lemma 6.1.42, it follows that C ⊆ (F1 ∩ F2 )(x ) = F (x ) ⇒ F (x) ⊆ Cε ⊆ F (x )ε
for all x ∈ U,
for all x ∈ U,
⇒ F is h-lsc. There is an analogous result for h-usc multifunctions. PROPOSITION 6.1.46 If F1 :X−→Pf (Y ) and F1 :X−→Pk (Y ) are h-usc and for all x ∈ X, F1 (x) ∩ F2 (x) = ∅, then x −→ (F1 ∩ F2 )(x) = F1 (x) ∩ F2 (x) is h-usc. PROOF: It is easy to see that a Pf (Y )-valued, h-usc multifunction is closed. Then the proposition follows by combining Propositions 6.1.23 and 6.1.39(a). For h-continuity, if Y is an infinite-dimensional space, we cannot simply combine Propositions 6.1.45 and 6.1.46, because a normed compact set has empty interior. Nevertheless the result is still true for h-continuity (see Hu–Papageorgiou [313, p.66]). PROPOSITION 6.1.47 If Y is a normed space, F1 , F2 : X −→ Pbf c (Y ) are hcontinuous and for all x ∈ X int(F1 ∩ F2 )(x) = ∅, then x −→ F (x) = (F1 ∩ F2 )(x) is h-continuous. We conclude this section with two weak versions of lower semicontinuity and of h-lower semicontinuity. DEFINITION 6.1.48 Let X, Y be Hausdorff topological spaces and F : X −→ 2Y \{∅} a multifunction. (a) We say that F is almost lower semicontinuous (a-lscfor short), if for every x ∈ X and every ε > 0, we can find U ∈ N (x) such that F (x )ε = ∅. x ∈U
(b) We say that F is weakly h-lower semicontinuous (hw -lsc for short), if for every x ∈ X and every U ∈ N (x), we can find V ∈ N (x), V ⊆ U and a point x ∈ V such that F (x ) ⊆ F (u)ε for all u ∈ V . REMARK 6.1.49 If in Definition 6.1.48(b), x = x , then we recover the definition of h-lower semicontinuity. Clearly h-lower semicontinuity implies hw -lower semicontinuity and hw -lower semicontinuity implies almost lower semicontinuity.
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6.2 Measurability of Multifunctions In the previous section we focused our attention to the topological properties of multifunctions. In this section we shift our investigation to the measure theoretic properties. Throughout this section (Ω, Σ) is a measurable space and (X, d) a metric space. Additional hypotheses are introduced as needed. DEFINITION 6.2.1 Let F : Ω −→ 2X be a multifunction. (a) We say that F is measurable, if for all U ⊆ X open, we have F − (U ) = {ω ∈ Ω : F (ω) ∩ U = ∅} ∈ Σ. (b) We say that F is graph measurable, if Gr F = {(ω, x) ∈ Ω × X : x ∈ F (ω)} ∈ Σ × B(X) with B(X) being the Borel σ-field of X. (c) If X is a separable Banach space,
we say that F is scalarly measurable, if for all x∗ ∈ X ∗ , the function ω −→ σ x∗ , F (ω) is Σ-measurable. REMARK 6.2.2 We define the domain of a measurable multifunction F : Ω −→ 2X to be the set dom F = {ω ∈ Ω : F (ω) = ∅}. It is clear from Definition 6.2.1(a) that dom F ∈ Σ. So in the sequel without any loss of generality, for measurable multifunctions, we always assume that dom F = Ω. PROPOSITION 6.2.3 If F : Ω −→ 2X \{∅} and for every C ⊆ X closed we have F − (C) ∈ Σ, then F is measurable. PROOF: Recall that in a metric space, an open set U is Fσ . So U = Cn with n≥1 − Cn ⊆ X closed for all n ≥ 1. Hence F − (U ) = F − ( Cn ) = F (Cn ) ∈ Σ (see n≥1
Proposition 6.1.1(c)). Therefore F is measurable.
n≥1
PROPOSITION 6.2.4 A multifunction F : Ω −→ 2X \ {∅} is measurable if and only if for all x ∈ X, ω −→ d x, F (ω) is Σ-measurable.
PROOF: ⇒: For every λ > 0, we need to show that Lλ (x) = {ω ∈Ω : d x, F (ω) <
λ}∈Σ. Note that Lλ (x)=F − Bλ (x) ∈Σ.
− ⇐: For every x ∈ X and every λ > 0, we have F Bλ (x) ∈ Σ. If U ⊆ X is open, then U = Bλn (xn ) (recall that X is separable, hence second countable). Then n≥1
F − (U )=F − Bλn (xn ) = F − Bλn (xn ) ∈ Σ. n≥1
n≥1
In earlier sections we already used the notion of the Carath´eodory function. Because it is central in our subsequent considerations, for easy reference we recall here its definition (in a general setting) and some fundamental properties of such functions. DEFINITION 6.2.5 Let (Ω, Σ) be a measurable space and V, Y Hausdorff topological spaces. A function ϕ : Ω × V −→ Y is said to be a Carath´ eodory function, if
6.2 Measurability of Multifunctions
471
(a) For every v ∈ V, ω −→ ϕ(ω, v) is Σ, B(Y ) -measurable and (B(Y ) is the Borel σ-field of Y ). (b) For every ω ∈ Ω, v −→ ϕ(ω, v) is continuous. The two theorems that follow reveal two basic and useful properties of Carath´eodory functions. For their proof we refer to Denkowski–Mig´ orski– Papageorgiou [194, pp. 188–189]. THEOREM 6.2.6 If (Ω, Σ) is a measurable space, V is a separable metric space, Y a metric space, and ϕ : Ω × V −→ Y a Carath´ eodory function, then ϕ is Σ × B(V )measurable. REMARK 6.2.7 This theorem implies that ϕ is superpositionally measurable (sup-measurable for short), namely if u : Ω −→ V is Σ-measurable, then ω −→
ϕ ω, u(ω) is Σ-measurable (i.e., the Nemitsky operator corresponding to ϕ maps measurable maps to measurable ones). The second theorem is known in the literature as the Scorza–Dragoni theorem and is a parametric version of Lusin’s theorem. First a definition. DEFINITION 6.2.8 A Hausdorff topological space (V, τ ) (τ being the Hausdorff topology of V ), is said to be a Polish space, if it is separable and there exists a metric on V for which the topology τ is complete. THEOREM 6.2.9 If T, V are Polish spaces, Y is a separable metric space, µ a tight Borel measure on T , and ϕ : T × V −→ Y a Carath´ eodory function, then for every ε > 0, we can find Tε ⊆ T a compact subset with µ(T \ Tε ) < ε such that ϕT ×V is continuous. ε
Returning to the measurability of multifunctions, we have the following result. PROPOSITION 6.2.10 If F : Ω −→ Pf (X) is measurable, then F is graphmeasurable.
PROOF: Note that Gr F = (ω, x) ∈ Ω × X
: d x, F (ω) = 0 . From Proposition 6.2.4, we have that (ω, x) −→ ϕ(ω, x) = d x, F (ω) is a Carath´eodory function. Then by virtue of Theorem 6.2.9 ϕ(ω, x) is Σ × B(X)-measurable and so Gr F ∈ Σ × B(X) (i.e., F is graph-measurable). The converse of the previous proposition is not in general true. EXAMPLE 6.2.11 Graph-measurable measurable: Let Ω = [0, 1] with Σ = B([0, 1]) (the Borel σ-field of [0, 1]) and X = R\ Q = the set of irrationals. Recall that X is a Polish space. Consider C a closed subset of Ω × X such that projΩ C ∈ / Σ and projX C = X. Choose x0 ∈ X \ projX C and let F : Ω −→ Pf (X) be a multifunction defined by F (ω) = C(ω) ∪ {x0 }. Then clearly F has closed graph, hence it is graph-measurable. On the other hand, if U is a neighborhood of projX C not containing x0 , then F − (U ) = projΩ C ∈ / Σ and so F is not measurable.
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For every open set U ⊆ X, we have A ∩ U = ∅ if and only if A ∩ U = ∅. This leads at once to the following proposition. PROPOSITION 6.2.12 F : Ω −→ 2X \{∅} is measurable if and only if F : Ω −→ Pf (X) is measurable. In the next result we produce conditions under which the converse of Proposition 6.2.3 holds. PROPOSITION 6.2.13 If F : Ω −→ Pk (X), then F is measurable if and only if F − (C) ∈ Σ for all C ⊆ X closed. PROOF: ⇒: Let C ⊆ X be closed and set U =X \C. Then we know that U = Cn n≥1 with Cn = x ∈ X : d(x, C) ≥ 1/n (i.e., U is an Fσ -set). Then F − (C) = Ω\F + (U ) =
Ω\F + Cn . From Proposition 6.1.1(c) we know that n≥1
F+
!
! + F (Cn ). Cn ⊇
n≥1
(6.6)
n≥1
We show that opposite inclusion is also true. Indeed, if this is not true, we can the find ω ∈ F + Cn such that F (ω)∩Cnc = ∅ for all n ≥ 1. Let xn ∈ F (ω)∩Cnc , n ≥ 1. n≥1
Because F (ω) ∈ Pk (X), by passing to a suitable subsequence if necessary, we may assume that xn −→ x and x ∈ F (ω) ⊆ U . On the other hand xn ∈ Cnc for all n ≥ 1 and so d(xn , C) < 1/n. Therefore in the limit as n → ∞, we obtain d(x, C) = 0; that is, x ∈ C = X \ U , a contradiction. So the opposite inclusion in (6.6) holds and we have
! ! + F+ F (Cn ) Cn = n≥1
n≥1
=
!
Ω \ F − (Cnc ) ∈ Σ
n≥1 −
⇒ F (C) ∈ Σ. ⇐: This implication is Proposition 6.2.3.
PROPOSITION 6.2.14 If F : Ω −→ Pf (X) is measurable, then F − (K) ∈ Σ for all K ⊆ X compact. PROOF: Because X is a separable metric space, we can think of X as a dense subspace of a compact metric space Y (in fact it is homeomorphic to a subset of the Hilbert cube [0, 1]N which is compact by Tychonov’s theorem). Let G : Ω −→ Pk (X) be defined by G(ω) = F (ω). From Proposition 6.2.12 we have that G is measurable. Now let K ⊆ X be compact. We have F − (K) = {ω ∈ Ω : F (ω) ∩ K = ∅} = {ω ∈ Ω : G(ω) ∩ X ∩ K = ∅} = G− (K).
(6.7)
6.2 Measurability of Multifunctions
473
But G is Pk (Y )-valued. So Proposition 6.2.13 implies that G− (K) ∈ Σ, hence F − (K) ∈ Σ (see (6.7)). If we enrich the structure of the space X, then we can strengthen our conclusions. PROPOSITION 6.2.15 If X is a σ-compact metric space and F : Ω −→ Pf (X), then the following statements are equivalent. (a) F − (C) ∈ Σ for every C ⊆ X closed. (b) F is measurable. (c) F − (K) ∈ Σ for every K ⊆ X compact. PROOF: (a)⇒ (b): This implication is Proposition 6.2.3. (b)⇒ (c): This implication is Proposition 6.2.14. (c)⇒ (a): Because X is σ-compact, we have X = Kn with Kn ⊆X is compact. Let n≥1
C ⊆X be a closed set. Then we have F − (C)=F − Kn ∩ C = F − (Kn ∩ C) ∈ Σ (because Kn ∩ C is compact for every n ≥ 1).
n≥1
n≥1
Next we introduce a class of spaces, broader than the class of Polish spaces, because they are not necessarily metrizable. This makes them useful in many concrete situations, for example, when we deal with certain Banach spaces furnished with their weak topology. DEFINITION 6.2.16 A Hausdorff topological space X is said to be a Souslin space, if there exists a Polish space Y and a continuous surjection u : Y −→ X. REMARK 6.2.17 Souslin subspaces of a Polish space, are known in the literature as analytic sets. A Souslin space is always separable but need not be metrizable. For example, consider an infinite-dimensional separable Banach space endowed with the weak topology. Evidently this is a nonmetrizable Souslin space. It can be shown that closed and open subsets of a Souslin space are Souslin, countable products of Souslin spaces are Souslin, and countable intersections and unions of Souslin subspaces of a Hausdorff topological space are Souslin. From these facts, it follows ∗ ∗ that if X is a separable Banach space and Xw of X endowed ∗ denotes the dual X ∗ ∗ ∗ with the w∗ –topology, then Xw nB 1 , where ∗ is Souslin (indeed note that X = n≥1 ∗ ∗ B 1 = x∗ ∈ X ∗ : x∗ ≤ 1 and recall that (B 1 , w∗ ) is compact metrizable, hence Polish, in particular then Souslin). It is well known that a Borel set in R2 does not in general project to a Borel set on R. The next theorem determines more precisely the structure of this projection. The result is known as the Yankov–von Neumann–Aumann projection theorem and its proof can be found in Hu–Papageorgiou [313, p. 149]. Recall that if (Ω, Σ) is a measurable space, the universal σ-field corresponding to Σ is defined by Σ = Σµ , µ
where µ ranges over all finite measures on Σ and Σµ denotes the µ-completion of Σ. If (Ω, Σ, µ) is a σ-finite complete measure space, then Σ = Σ.
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THEOREM 6.2.18 If (Ω, Σ) is a measurable space, X is a Souslin space, and A ∈ Σ × B(X), then proj A ∈ Σ. Using this theorem, we can have a partial converse to Proposition 6.2.10. PROPOSITION 6.2.19 If (Ω, Σ) is a complete measurable space (i.e., Σ = Σ), X is a Souslin space, and F : Ω −→ 2X \ {∅} is a multifunction such that Gr F ∈ Σ × B(X), then for every D ∈ B(X), we have F − (D) ∈ Σ. PROOF: By definition F − (D) = ω ∈ Ω : F (ω) ∩ D = ∅ = proj GrF ∩ (Ω × D) ∈ Σ = Σ (see Theorem 6.2.18). Now we can state a theorem summarizing the situation concerning the measurability of a multifunction with values in a separable metric space. THEOREM 6.2.20 Let (Ω, Σ) be a measurable space, X a separable metric space, and F : Ω −→ Pf (X). We consider the following properties. (a) (b) (c) (d) (e)
For every D ∈ B(X), F − (D) ∈ Σ. For every C ⊆ X closed, F − (C) ∈ Σ. F is measurable.
For every x ∈ X, ω −→ d x, F (ω) is Σ-measurable. F is graph-measurable.
Then the following implications are true. (1) (a)⇒(b)⇒(c)⇔ (d)⇒(e). (2) If X is σ-compact, then (b)⇔ (c). (3) If Σ = Σ and X is complete (i.e., X is a Polish space), then statements (a) to (e) are equivalent. It is easy to see that countable unions preserve measurability and graph measurability. This is no longer true with intersections. Additional conditions are needed to guarantee measurability of countable intersections. PROPOSITION 6.2.21 If Fn : Ω −→ Pf (X), n ≥ 1, are measurable multifunctions and for some n0 ≥ 1 Fn0 has values in Pk (X), then ω −→ F (ω) = Fn (ω) is n≥1
measurable. PROOF: " First we assume that all multifunctions Fn have values in Pk (X). We set P(ω)= Fn (ω). It is easy to see that P is measurable. Because P is compactn≥1
valued, from Proposition 6.2.13 we have that P − (C) ∈ Σ for every C ⊆ X closed. N Let C ⊆ X be closed diagonal set of the product space. Then and ∆ ⊆ X be the F − (C)= ω ∈ Ω: Fn (ω) ∩ C = ∅ = ω ∈ Ω : P(ω) ∩ ∆ ∩ C N = ∅ ∈ Σ (because n≥1
∆ ∩ C N is closed in X N ). Next we assume that only one of the Fn ’s has compact values. Let Y be a metrizable compactification of X (recall that X is homeomorphic to a subset of the Hilbert cube H = [0, 1]N ). Let Fn : Ω −→ Pk (Y ) be defined by Fn (ω) = F n (ω) (the closure in Y ), for all n ≥ 1. We know that Fn is measurable (see Proposition
6.3 Continuous and Measurable Selectors 6.2.14). So from the first part of the proof, we know that if F (ω) =
475
Fn (ω); then
n≥1
F − (C) ∈ Σ for all C ⊆ Y closed. Because Fn = Fn for some n ≥ 1, we have F = F and so from Proposition 6.2.3 we conclude that F is measurable. In order to study scalarly measurable multifunctions (see Definition 6.2.1(c)), we need to produce some results on the existence of measurable selectors for multifunctions. This is done in the next section, where we prove theorems on the existence of continuous and measurable selectors.
6.3 Continuous and Measurable Selectors Given two sets X, Y and a multifunction F : X −→ 2Y \{∅}, a selector of F is a single-valued map f : X −→ Y such that f (x) ∈ F (x) for all x ∈ X. When X, Y both have topological structure, it is natural to look for continuous selectors. If X = Ω has a measure-theoretic structure, then we seek to produce measurable selectors. In this section we study both cases. We start with continuous selectors. First a negative observation directs our efforts to the appropriate class of multifunctions. EXAMPLE 6.3.1 An usc multifunction need not have a continuous selector. We consider the usc multifunction F : R −→ Pf c (R) defined by ⎧ ⎪ ⎨ −1 F (x) = [0, 1] ⎪ ⎩ 1
if x < 0 if x = 0 . if x > 0
It is clear that F cannot have a continuous selector. Note that F (x) = ∂ϕ(x) where ϕ(x) = |x| (the subdifferential is in the sense of convex analysis, see Definition 1.2.28). This example suggests that usc multifunctions are not the right class to consider. The next proposition indicates where we should look in order to produce continuous selectors. In what follows, for the study of continuous selectors of a multifunction we assume that X, Y are Hausdorff topological spaces. Additional hypotheses are introduced as needed. PROPOSITION 6.3.2 If F : X −→ 2Y \{∅} and for every (x, y) ∈ Gr F we can find U ∈ N (x) and f a continuous selector of F U such that f (x) = y, then F is lsc. PROOF: Given V ⊆ Y open set, we need to show that F − (V ) is open. Let (x, y) ∈ Gr F ∩ (X × V ). By hypothesis we know that there exist U ∈ N (x) and a continuous function f : U −→ Y such that f (u) ∈ F (u) for all u ∈ U and f (x) = y. Set U ={x ∈ U : f (x ) ∈ V } ∈ N (x). Then U ⊆ F − (V ) and so F − (V ) is open. REMARK 6.3.3 Multifunctions that satisfy the hypotheses of the above proposition are called locally selectionable.
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PROPOSITION 6.3.4 If X is paracompact, Y is a topological vector space, and F : X −→ 2Y \{∅} is a multifunction with convex values such that F − (y) = {x ∈ X : y ∈ F (x)} is open for all y ∈ Y , then F admits a continuous selector f (i.e., there exists a continuous map f : X −→ Y such that f (x) ∈ F (x) for all x ∈ X). PROOF: For every (x, y) ∈ Gr F , we can take U = F − (y) ∈ N (x) and f : U −→Y defined by f (x ) = y for all x ∈ U . So F is locally selectionable (see Remark 6.3.3). Therefore for every (x, y) ∈ Gr F we can find Ux ∈ N (x) and fx : Ux −→ Y continuous. Because of paracompactness let {Ui }i∈I be a locally finite refinement of the cover {Ux }x∈X . Let {gi }i∈I be a continuous partition of unity subordinate to the cover {Ui }i∈I . We define the map gi (x)fxi (x) for all x ∈ X. (6.8) f (x) = i∈I
Here xi ∈X is such that Ui ⊆Uxi (recall that {Ui }i∈I is a refinement of {Ux }x∈X ). Note that because {Ui }i∈I is locally finite, at every x ∈ X, the summation in (6.8) is finite. Therefore f is continuous and because F is convex-valued, for every x ∈ X we have f (x) ∈ F (x). REMARK 6.3.5 If Gr F is open in X×Y , then for every y ∈ Y F − (y) is open. Proposition 6.3.2 suggests that we should focus on lsc multifunctions. This leads to the celebrated Michael’s selection theorem. THEOREM 6.3.6 If X is paracompact, Y is a Banach space, and F : X −→ Pf c (Y ) is lsc, then F admits a continuous selector. PROOF: In what follows B1 = {y ∈ Y : y < 1}. First we produce a continuous function f : X −→ Y such that f (x) ∈ F (x) + εB1
for all x ∈ X, with ε > 0.
(6.9)
For every x ∈ X, we choose yx ∈ F (x). Then due to the lower semicontinuity of F , the set F − (yx + εB1 ) is open and so {F − (yx + εB1 )}x∈X is an open cover of X. Due to the paracompactness of X, we can find F − (yi + εB1 ) i∈I a locally finite refinement of F − (yi + εBi ) i∈I . We set f (x) =
gi (x)ui
for all x ∈ X,
i∈I
where ui ∈ (yi +εBi ) ⊆ (yxi +εB1 ). As before, at every x ∈ X the above sum is finite and so f is continuous and because of the convexity of the values of x −→ F (x)+εB1 , we have f (x) ∈ F (x) + εB1 for all x ∈ X. Next, inductively we generate a sequence {fn }n≥1 of continuous functions fn : X −→ Y such that fn (x) ∈ F (x) + and
1 B1 2n
fn+1 (x) − fn (x)
0 there exists fε : X −→ Y a continuous map such that fε (x) ∈ F (x) + εB1 for every x ∈ X) if and only if F (·) is a-lsc (see Definition 6.1.48(a)). However, there exist multifunctions with nonempty, closed, and convex values values which are a-lsc and do not admit continuous selectors. Deutsch–Kenderov [196] and De Blasi–Myjal [179], produced examples in this direction. In addition De Blasi–Myjal [179] proved the following theorem. THEOREM 6.3.14 If X is a paracompact space, Y is a Banach space, and F : X −→ Pf c (Y ) is an hw -lsc multifunction, then F admits a continuous selector. REMARK 6.3.15 Because a multifunction F : X −→ 2Y \{∅} that is lsc is not necessarily hw -lsc and vice-versa, we see that Theorem 6.3.14 is distinct from Theorem 6.3.6. Before passing to measurable selectors, let us have a look at what happens with an usc multifunction. Recall that we started this section with a simple example illustrating that in general we do not expect to have continuous selectors for an usc multifunction. However, we can have some kind of approximate continuous selector (verify this with the multifunction given in Example 6.3.1). PROPOSITION 6.3.16 If X is a metric space, Y is a Banach space, and F : X −→ 2Y \{∅} is usc with convex values, then given ε > 0, we can find a locally Lipschitz function fε : X −→ Y such that fε (X) ⊆ conv F (X)
and
h∗ (Gr fε , Gr F ) < ε.
PROOF: Fix ε > 0. Because F is usc, for every x ∈ X we can find 0 < δ < δ(ε, x) such that F (x ) ⊆ F (x)+(ε/2)B1 for all x ∈ Bδ(x) (x). The collection Bδ/4 (x) x∈X is an open cover of X. The space X being a metric space, it is paracompact and so we can find {Uα }α∈J a locally finite refinement and {gα }α∈J a corresponding locally Lipschitz partition of unity subordinate to it. For each α ∈ J we choose
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(uα , yα ) ∈ Gr F ∩ (Uα × Y ) and set fε (x) =
gα (x)yα . Clearly fε is locally
α∈J
Lipschitz and fε (X) ⊆ conv F (X). For x ∈ X we can find J(x) ⊆ J finite such that gα (x) > 0 if α ∈ J(x). For every α ∈ J(x), let xα ∈ X be such that Uα ⊆ Bδα /4 (xα ) with δα = δ(xα ). Let β ∈ J(xα ) and set δβ = max{δα : α ∈ J(x)}. We have xα ∈ Bδβ /2 (xβ ) and so Uα ⊆ Bδβ (xβ ). Hence for any α ∈ J(x), we have yα ∈ F (Uα ) ⊆ F (xβ ) + (ε/2)B1 and so fε (x) ⊆ F (xβ ) + (ε/2)B1 . Because x ∈ X was arbitrary we conclude that h∗ (Gr fε , Gr F ) < ε. Next we turn our attention to measurable selectors of measurable multifunctions. The first result in this direction is the so-called Kuratowski–Ryll Nardzewski selection theorem. The technique of its proof is similar to the proof of Michael’s selection theorem (Theorem 6.3.6). THEOREM 6.3.17 If (Ω, Σ) is a measurable space, X is a Polish space, and F : Ω −→ Pf (X) is measurable, then F admits a measurable selector. PROOF: Let d be a compatible metric on X and let {xn }n≥1 be a dense subset of X. For every ω ∈ Ω let n ≥ 1 be the smallest integer such that F (ω)∩B1 (xn ) = ∅. We set f0 (ω) = xn . Because by hypothesis F is measurable, the function f0 : Ω −→ X is Σ-measurable. Moreover, we have
d f0 (ω), F (ω) < 1 for all ω ∈ Ω. Suppose that we have constructed measurable maps fk : Ω −→ X, k = 0, 1, . . . , m such that fk (Ω) ⊆ {xn }n≥1 and 1
d fk (ω), F (ω) < k 2
d fk (ω), fk+1 (ω)
λ if and only if we can find x ∈ F (ω) such that ϕ(ω, x) > λ. Hence {ω ∈ Ω : m(ω) > λ} = projΩ {(ω, x) ∈ Ω × X : ϕ(ω, x) > λ}. Because ϕ is jointly measurable, we have {(ω, x) ∈ Ω × X : ϕ(ω, x) > λ} ∈ Σ × B(X). Invoking Theorem 6.2.18, we obtain projΩ {(ω, x) ∈ Ω × X : ϕ(ω, x) > λ} ∈ Σ, ⇒ {ω ∈ Ω : m(ω) > λ} ∈ Σ. Because λ ∈ R was arbitrary, we conclude that ω −→ m(ω) is Σ-measurable. (b) Let ψ(ω, x) = m(ω) − ϕ(ω, x). Evidently ψ is Σ × B(X)-measurable (see part (a)). Then Gr S = Gr F ∩ {(ω, x) ∈ Ω × X : ψ(ω, x) = 0} ∈ Σ × B(X). REMARK 6.3.25 Because of Theorem 6.3.24(b) and Theorem 6.3.20 we can have a Σ-measurable selector of S. If S has empty values, then we can look for εmaximizers. More precisely, if m is R-valued and ε > 0, then the multifunction ω −→ Sε (ω) = {x ∈ F (ω) : ϕ(ω, x) ≥ m(ω) − ε} has always nonempty values, its graph belongs in Σ × B(X) and so admits a Σ-measurable selector. PROPOSITION 6.3.26 If (Ω, Σ) is a measurable space, X is a separable Banach space, u : Ω −→ X is a Σ-measurable
map, and r : Ω −→ R+is a Σ-measurable function, then ω −→ B r(ω) u(ω) = x ∈ X : x − u(ω) ≤ r(ω) is measurable. PROOF: Let {xn }n≥1 be dense in the closed unit ball of X. Set vn (ω) = u(ω) + r(ω)xn ,
n ≥ 1.
Evidently for every n ≥ 1, vn is Σ-measurable and
B r(ω) u(ω) = {vn (ω)}n≥1
⇒ ω −→ B r(ω) u(ω) is measurable (see Theorem 6.3.18(2)).
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The measurable selection theorems also provide the necessary tools to deal with scalarly measurable multifunctions (see Definition 6.2.1(c)). PROPOSITION 6.3.27 If (Ω, Σ) is a complete measurable space, X is a separable Banach space, and F : Ω −→ 2X \{∅} is graph-measurable, then F is scalarly measurable. PROOF: By Theorem 6.3.20 we can find a sequence {fn }n≥1 of Σ-measurable selectors of F such that F (ω) ⊆ {fn (ω)}n≥1 for all ω ∈ Ω. Then for every x∗ ∈ X ∗ , we have
σ x∗ , F (ω) = sup x∗ , fn (ω) , n≥1
⇒ ω −→ σ x∗ , F (ω) is Σ-measurable, ⇒ F
is scalarly measurable.
If F is Pwkc -valued, then measurability and scalar measurability are equivalent notions. To prove this, we need to do some preparatory work. We start with a lemma. LEMMA 6.3.28 If V is a normed space and C ⊆ V is nonempty and convex, then for every v ∈ V , we have d(v, C) = sup v ∗ , v − σ(v ∗ , C) : v ∗ ≤ 1 . PROOF: Let ϕ(v) = d(v, C). We know that ϕ is continuous convex. From Example ∗ 1.2.27(b), we know that ϕ∗ =σ(·, C) + iB ∗1 , where B 1 = v ∗ ∈ V ∗ : v ∗ ≤ 1 and ∗
iB ∗1 (v ) =
0 +∞
∗
if v ∗ ∈ B 1 ∗ if v ∗ ∈ / B1
for all v ∗ ∈ V ∗ .
Because ϕ is continuous convex, we have that ϕ = ϕ∗∗ = (ϕ∗ )∗ (see Theorem 1.2.21). Hence ϕ(v) = sup v ∗ , v − σ(v ∗ , C) − iB ∗1 (v ∗ ) : v ∗ ∈ V ∗ = sup v ∗ , v − σ(v ∗ , C) : v ∗ ≤ 1 . Also we use a particular topology on X ∗ , which for convenience of the reader we recall in the next definition. DEFINITION 6.3.29 Let V be a locally convex space. The locally convex topology on V of uniform convergence on w∗ -compact, convex balanced sets in V ∗ is called the Mackey topology on V . Recall that C ⊆ X ∗ is balanced, if λC ⊆ C for all |λ| ≤ 1.
6.3 Continuous and Measurable Selectors
485
REMARK 6.3.30 The strongest locally convex topology τ on V for which we have (V, τ )∗ = V ∗ , is the Mackey topology. So, if V is a Banach space and we consider the strongest topology τ on V ∗ such that (V ∗ , τ )∗ = V , this is the Mackey topology, whereas the weakest topology on V ∗ for which this is true is the weak∗ -topology. Of course in this case the Mackey topology is strictly weaker than the strong (norm) topology. However, on the Banach space V the Mackey and strong topologies coin∗ cide. Finally note that in a locally convex
∗ space V ∗a set C ⊆ V is w(V, V ∗)–compact ∗ if and only if the function v −→ σ v , C is m(V , V )–continuous (m(V , V ) is the Mackey topology on V ∗ for the pair (V ∗ , V )). PROPOSITION 6.3.31 If (Ω, Σ) is a measurable space, X is a separable Banach space, and F : Ω −→ Pwkc (X), then F is measurable if and only if it is scalarly measurable. PROOF: ⇒: From Theorem 6.3.19(2), we know that we can find a sequence {fn }n≥1 of Σ-measurable selectors of F such that F (ω) = {fn (ω)}n≥1 for all ω ∈ Ω. Then
for all x∗ ∈ X ∗ we have σ x∗ , F (ω) = sup x∗ , fn (ω) and so we conclude that n≥1
ω−→σ x∗ , F (ω) is Σ-measurable. ∗ ∗ endowed with ⇐: Since X is separable, the space Xw ∗ (= the dual Banach space X ∗ the w -topology) is separable. It follows that for every other topology τ on X ∗ for which we have (Xτ∗ )∗ = X, the space Xτ∗ is separable. In particular this is true if τ = m(X ∗ , X). Because F is Pwkc (X)-valued, from Remark 6.3.30 we know that x∗ −→ σ x∗ , F (ω) is m(X ∗ , X)-continuous on X ∗ . From Lemma 6.3.28 for every x ∈ X and every ω ∈ Ω, we have
d x, F (ω) = sup x∗n , x − σ x∗n , F (ω) , (6.14) n≥1
∗
where {x∗n }n≥1 is dense in B 1 ={x∗ ∈X ∗ : x∗ ≤ 1} for the Mackey topology. From (6.14) and Theorem 6.3.19(2), we conclude that F is measurable. Now we can state a version of Theorem 6.3.19 for nonmetrizable spaces. THEOREM 6.3.32 If (Ω, Σ) is a complete measurable space, X is a regular Souslin space, F : Ω −→ Pf (X) is a multifunction, and we consider the following statements, (a) (b) (c) (d)
For every D ∈ B(X), F − (D) ∈ Σ; For every C ⊆ X closed, F − (C) ∈ Σ; For every U ⊆ X open, F − (U ) ∈ Σ; There exists a sequence {fn }n≥1 of Σ-selectors of F such that F (ω) = {fn (ω)}n≥1
for all ω ∈ Ω;
(e) Gr F ∈ Σ × B(X); (f) For every continuous function u : X −→ R, the function ω −→ m(ω) = sup[u(x) : x ∈ F (ω)] is Σ-measurable, then
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6 Multivalued Analysis
(1) (a)⇔(d)⇔(e)⇔(f)⇒(b) ⇒(c). (2) If X is second countable, then (a)−→(f) are all equivalent. REMARK 6.3.33 The proof of this theorem is based on the fact that there exists a metric d on X defining a Souslin metric topology finer that the original topology on X (see Saint-Pierre [536]). Then B(X) = B(Xd ) and we can apply Theorem 6.3.19 to the multifunction F : Ω −→ Pf (Xd ) (Xd is the space X with the new Souslin metric topology). Now, if X is a separable Banach space, then Xw∗ (= the dual Banach space X ∗ with the w∗ -topology) is a Souslin space (see Remark 6.2.17) which is regular and second countable. Then using Theorem 6.3.32 we can have the following “dual” version of Proposition 6.3.31. PROPOSITION 6.3.34 If (Ω, Σ) is a complete measurable space, X is a separable Banach space, and F : Ω −→ 2X \ {∅} is a multifunction with w∗ -compact convex values, then for every U ⊆ X ∗ open F − (U ) ∈ Σ if and only if for every x ∈ X,
ω −→ σ x, F (ω) is Σ-measurable. Let (Ω, Σ) be a measurable space and X a separable Banach space. Given a multifunction F : Ω −→ 2X \ {∅}, by ext F we denote the multifunction which to each ω ∈ Ω assigns the set ext F (ω) of extreme points of the set F (ω). In the next proposition we establish the measurability properties of the multifunction ω −→ ext F (ω). PROPOSITION 6.3.35 If (Ω, Σ) is a measurable space, X is a separable Banach space, and F : Ω −→ Pwkc (X) is a measurable multifunction, then ω −→ ext F (ω) is graph-measurable. PROOF: From the Krein–Milman theorem we know that for all ω ∈ Ω, ext F (ω) = ∅. Recall that X ∗ with the Mackey topology m(X ∗ , X) is separable. Let {x∗n }n≥1 ⊆ ∗ B 1 be m(X ∗ , X)-dense and consider the function ϕF : Ω × X −→ R+ defined by ⎧ ∗ 2 xn ,x ⎨ if x ∈ F (ω) 2n . ϕF (ω, x) = n≥1 ⎩ +∞ otherwise Evidently ϕF is Σ × B(X)-measurable and for every ω ∈ Ω, ϕF (ω, ·)F (ω) is continuous. Let α be the set of all continuous affine function α : X −→ R. We consider ϕF (ω, x) = inf α(x) : α ∈
α, α(v) > ϕF (ω, v)
for all v ∈ F (ω) .
Let un : Ω −→ X, n ≥ 1, be a sequence of Σ-measurable selectors of F such that F (ω) = {un (ω)}n≥1
(see Theorem 6.3.19).
For every (ω, x∗ ) ∈ Ω × X ∗ , let rx∗ (ω) = sup[ ϕF (ω, v) − x∗ , v : v ∈ F (ω)].
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487
We have that rx∗ (ω) < +∞, for every ω ∈ Ω x∗ −→ rx∗ (ω) is continuous on X ∗ and rx∗ (ω) = sup ϕF (ω, x) − x∗ , x : x ∈ F (ω) ,
⇒ rx∗ (ω) = sup ϕF ω, un (ω) − x∗ , un (ω) , n≥1
∗
⇒ (ω, x ) −→ rx∗ (ω) is Σ × B(X ∗ )-measurable. If we set
ϕF (ω, x) = inf x∗ , x + rx∗ (ω) : x∗ ∈ X ∗ ,
∗ }m≥1 ⊆ X ∗ m(X ∗ , X)-dense set, we have then for {vm ∗ ∗ (ω)], ϕF (ω, x) = inf vm , x + rvm m≥1
⇒ (ω, x) −→ ϕF (ω, x) is Σ × B(X)-measurable. From Choquet [144, Chapter 6], we know that ext F (ω) = {x ∈ X : ϕF (ω, x) = ϕF (ω, x)}, ⇒ Gr F ∈ Σ × B(X). REMARK 6.3.36 If we set ξF (ω, x) = ϕF (ω, x) − ϕF (ω, x), then ξF is Σ × B(X)measurable, for every ω ∈ Ω, ξF (ω, ·) is strictly concave on F (ω), concave on X, and upper semicontinuous. The function ξF (ω, ·) is known in the literature as the Choquet function of the set F (ω).
6.4 Decomposable Sets and Set-Valued Integration Throughout this section the standing hypotheses are: (Ω, Σ, µ) is a σ-finite measure space and X is a separable Banach space. By L0 (Ω, X) we denote the space of all equivalence classes in the set of all Σ-measurable maps from Ω into X, for the equivalence relation of equality almost everywhere. DEFINITION 6.4.1 A set K ⊆ L0 (Ω, X) is said to be decomposable, if for every triple (A, f1 , f2 ) ∈ Σ × K × K we have χA f1 + χAc f2 = χA f1 + (1 − χA )f2 ∈ K. EXAMPLE 6.4.2 If F : Ω−→2X \ {∅} is a multifunction and SF ={f ∈ L0 (Ω, X) : f (ω) ∈ F (ω) µ-a.e.}, then SF is decomposable (possibly empty). The same is true for the sets SFp = SF ∩Lp (Ω, X), 1 ≤ p ≤ ∞. We show in the sequel that within closure all decomposable sets are of this form. The condition of decomposability looks similar to convexity but only formally, because χA is not a constant and does not assume values between zero and one. However, as we show in this section, decomposability has some implications analogous to those of convexity.
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PROPOSITION 6.4.3 If F : Ω−→2X \{∅} is graph-measurable and SFp = ∅, 1 ≤ p ≤ ∞, then we can find a sequence {fn }n≥1 ⊆ SFp such that F (ω) ⊆ {fn (ω)}n≥1 µa.e. on Ω. PROOF: From Theorem 6.3.19 (see also Remark 6.3.21), we can find a sequence {gm }m≥1 of Σ-measurable selectors of F such that F (ω) ⊆ {gm (ω)}m≥1 µ-a.e. on Ω. Since µ is σ-finite on Ω, we can find {Ak }k≥1 ⊆ Σ a countable partition of Ω with µ(Ak ) < +∞ for all k ≥ 1. Let f ∈ SFp and define Cmki = {ω ∈ Ω : i − 1 ≤ gm (ω) < i} ∩ Ak for all m, k, i ≥ 1.
c fmki = χCmki gm + χCmki f
Evidently {fmki }m,k,i≥1 ⊆ SFp and F (ω) ⊆ {fmki (ω)}m,k,i≥1 µ-a.e. on Ω.
COROLLARY 6.4.4 If F1 , F2 : Ω−→2X \{∅} are graph measurable and for some 1 ≤ p ≤ ∞ we have SFp 1 =SFp 2 = ∅, then F1 (ω) = F2 (ω) µ-a.e. on Ω. LEMMA 6.4.5 If F : Ω −→ 2X \ {∅} is graph-measurable, 1 ≤ p ≤ ∞, {fn }n≥1 ⊆ SFp satisfies F (ω) ⊆ {fn (ω)}n≥1 µ-a.e. f ⊆ SFp , and ε > 0, then we can find a finite Σ-partition {Ck }N k=1 of Ω such that N f − χCk fk p < ε. k=1
PROOF: Without any loss of generality we assume that f (ω) ∈ F (ω) for all ω ∈ Ω. Let ϑ ∈ L1 (Ω), ϑ(ω) > 0 for all ω ∈ Ω and Ω ϑdµ < εp /3. We can find {Dn }n≥1 a Σ-partition of Ω such that f (ω) − fn (ω)p < ϑ(ω)
for all ω ∈ Dn , n ≥ 1.
We choose N ≥ 1 large enough so that εp f (ω)p dµ < and 3.2p Dn n≥N +1
f1 (ω)p dµ
0. From Lemma 6.4.5, we know that we can find a finite Σ-partition N {Ck }N k=1 of Ω and {gk }k=1 ⊆{hnm (ω)}n,m≥1 such that N f − χCk gk p < ε. k=1
Due to the decomposability of K we have N
χCk gk ∈ K,
k=1
⇒ f ∈K ⇒ Suppose that
K = SFp .
SFp
(because K is closed),
⊆ K.
(6.15)
Then we can find f ∈ K, A ∈ Σ with µ(A) > 0 and δ > 0 such
f (ω) − hnm (ω) ≥ δ
for all ω ∈ A and all n, m ≥ 1.
For what follows we fix n ≥ 1 so that δ E = A ∩ ω ∈ Ω : f (ω) − fn (ω) < 3
(6.16)
(6.17)
has a positive measure. We define gm = χE f + χE c hnm ,
m ≥ 1.
We have gm ∈ K and hnm (ω) − fn (ω) ≥ hnm (ω) − f (ω) − f (ω) − fn (ω) 2δ δ for all ω ∈ E (see (6.16) and (6.17)). ≥δ− = 3 3 (6.18) Therefore fn − hnm pp − ξnp ≥ fn − hnm pp − fn − gm pp
≥ fn − hnm p − fn − f p dµ E
2δ δp p − p µ(E) > 0, ≥ 3 3
m≥1
(see (6.18)).
(6.19)
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6 Multivalued Analysis
If in (6.19) we let m −→ ∞, we have a contradiction. Hence K ⊆ SFp . From (6.15) and (6.20), we conclude that K = SFp .
(6.20)
REMARK 6.4.7 The result is also true for p=+∞, provided that K is boundedly closed; that is, if {fn }n≥1 ⊆ K, sup fn ∞ < +∞ and fn (ω) −→ f (ω) µ-a.e. on Ω, then f ∈ K.
n≥1
Bounded decomposable sets in L1 (Ω, X) are in fact uniformly integrable. To show this we need the following lemma, whose proof can be found in Hu– Papageorgiou [313, p. 178], or Neveu [458, p. 121]. LEMMA 6.4.8 If (Ω, Σ, µ) is a finite space and R is a family of Σ-measurable, R+ = R ∪ {+∞}-valued functions on Ω, then we can find a unique (modulo µ-a.e. equality) Σ-measurable function h : Ω −→ R+ such that (a) For every f ∈ R, we have f (ω) ≤ h(ω) µ-a.e. on Ω. (b) If h : Ω −→ R+ is another Σ-measurable function such that f (ω) ≤ h(ω) µ-a.e. on Ω for all f ∈ R, then h(ω) ≤ h(ω) µ-a.e. on Ω. Moreover, we can find a sequence {fn }n≥1 ⊆ R such that h(ω) = sup fn (ω) n≥1
µ-a.e. on Ω.
Finally, if R is directed upwards (namely if for every f1 , f2 ∈ R, we can find f3 ∈ R such that f1 (ω), f2 (ω) ≤ f3 (ω) µ-a.e. on Ω), then {fn }n≥1 can be chosen to be increasing. REMARK 6.4.9 The function h is denoted by ess sup R and is the least upper bound of R in the sense of inequality µ-a.e. The essential supremum coincides with the supremum modulo µ-null sets for countable families, but it is not the same for uncountable families. To see this let A ⊆ [0, 1] and R = {χ{α} : α ∈ A}. Then ess sup R = 0, but sup χ{α} = χA which need not be measurable if A is not or even if A is measurable but of positive Lebesgue measure, then χA = 0. In a similar way we can define the essential infimum of R, denoted by ess inf R. PROPOSITION 6.4.10 If (Ω, Σ, µ) is a finite measure space, Y is a Banach space, and K ⊆ L1 (Ω, Y ) is bounded and decomposable, then K is uniformly integrable. PROOF: We introduce the set |K| = {f (·) : f ∈ K} ⊆ L1 (Ω) and let h = ess sup |K| (see Remark 6.4.9). From Lemma 6.4.8, we know that we can find {fn }n≥1 ⊆ K such that h(ω) = sup fn (ω) µ-a.e. on Ω. Moreover, the decomn≥1
posability of K implies that |K| is directed upwards and so {fn }n≥1 can be chosen such that fn (ω) ↑ h(ω) µ-a.e. on Ω as n → ∞. From the monotone convergence theorem and because by hypothesis K is bounded, it follows that h ∈ L1 (Ω). Because f (ω) ≤ h(ω) µ-a.e. on Ω for all f ∈ K, we conclude that K is uniformly integrable.
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491
An immediate consequence of this proposition and of the Dunford–Pettis theorem, is the following result. COROLLARY 6.4.11 If (Ω, Σ, µ) is a finite-measure space, Y is a reflexive Banach space, and K ⊆ L1 (Ω, Y ) is a bounded decomposable set, then K is relatively weakly compact in L1 (Ω, Y ). To prove the next proposition, we need to recall the following important result of measure theory, which has remarkable applications in control theory. The result is known as Lyapunov’s convexity theorem and for a proof of it we refer to Diestel–Uhl [199, pp. 264 and 266]. THEOREM 6.4.12 Let (Ω, Σ) be a measure space. (a) If m : Σ −→ RN is a nonatomic vector measure, then m(Σ) is compact and convex in X. (b) If X is a Banach space with the RNP (Radon–Nikodym property) and m : Σ −→ X is a nonatomic measure of bounded variation, then m(Σ) is compact and convex. Using this theorem, we can prove the next proposition, which is another indication that decomposability and convexity are closely related notions. PROPOSITION 6.4.13 If (Ω, Σ, µ) is a finite nonatomic measure space, Y is a Banach space, K ⊆ Lp (Ω, Y ) 1 ≤ p < ∞ is nonempty, and decomposable, V is a finite-dimensional Banach space, and L ∈ L Lp (Ω, Y ), V , then L(K) is convex. PROOF: Let v1 , v2 ∈ L(K). Then v1 = L(f1 ) and v2 = L(f2 ) with f1 , f2 ∈ K. Consider the vector-valued set function m : Σ−→V defined by m(A) = L χA (f1 − f2 ) for all A ∈ Σ. We claim that m is a vector measure. To this end, let {An }n≥1 ⊆ Σ be mutually disjoint sets and let A= An , Cn = Ak ∈ Σ. We have n≥1
k≥n+1
m(A) = L χA (f1 − f2 ) n
= L χAk (f1 − f2 ) + L χCn (f1 − f2 ) , k=1 n n
⇒ m(A) − L χAk (f1 − f2 ) V = m(A) − m(Ak ) V k=1
k=1
= L χCn (f1 − f2 ) V .
p
Note that C n ↓ ∅ as n → ∞ and so χCn (f1 − f2 ) −→ 0 in L (Ω, Y ) and so L χCn (f1 − f2 ) −→ 0 in V . This proves that m is a vector measure. Because µ is nonatomic and m ! µ, m is nonatomic too. Therefore we can apply Theorem 6.4.12(a) and deduce that m(Σ) is convex. Then D = m(Σ) + L(f2 ) is convex in V . Note that v1 , v2 ∈ D and so for every λ ∈ [0, 1] we have λv1 +(1−λ)v2 ∈ D. Because χA (f1 − f2 ) + f2 = χA f1 + χAc f2 ∈ K (due to the decomposability of K), we infer that D ⊆ L(K). Therefore λv1 + (1 − λ)v2 ∈ L(K) for all λ ∈ [0, 1]. Since v1 , v2 ∈ L(K) were arbitrary, we conclude that L(K) is convex.
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6 Multivalued Analysis
PROPOSITION 6.4.14 If (Ω, Σ, µ) is a finite measure space, Y is a Banach space, and K ⊆ Lp (Ω, Y ) 1 ≤ p < ∞ is nonempty, decomposable and w-closed, then L(K) is convex. PROOF: Let g = conv K. Then g =
n
λk fk with λk ∈ [0, 1],
k=1
n
λk = 1 and
k=1
fk ∈ K for all k ∈ {1, . . . , n}. Let (·, ·)p denote the duality brackets for the pair
p L (Ω, Y ), Lp (Ω, Yw∗∗ ) , p1 + p1 = 1 (see Ionescu–Tulcea [328, p. 99]). Consider the basic weak neighborhood of g defined by V (g) = h ∈ Lp (Ω, Y ) : |(uk , h − g)p | < ε, k ∈ {1, . . . , N } ,
where N ≥ 1, uk ∈ Lp (Ω, Yw∗∗ ), k ∈ {1, . . . , N } and ε > 0. Let L : Lp (Ω, Y ) −→ RN be defined by N L(h) = (uk , h) k=1 . From Proposition 6.4.13 we know that L(K) is convex. Because L(conv K) = conv L(K) = L(K), we can find f ∈ K such that L(g) = L(f ), hence V (g) ∩ K = ∅ w and so g ∈ K = K. Therefore K = conv K. REMARK 6.4.15 Both Propositions 6.4.13 and 6.4.14 are also valid for σ-finite measure spaces (Ω, Σ, µ). In this case we work on the finite measure components of Ω. The next theorem is a basic tool in what follows. It is also very useful in various optimization problems. THEOREM 6.4.16 If (Ω, Σ, µ) is a σ-finite measure space, X is a separable Banach space, ϕ : Ω × X −→ R∗ = R ∪ {±∞} is a Σ × B(X)-measurable function, F : Ω −→ 2X \ {∅} is graph-measurable, and the integral functional
Iϕ (u) = ϕ ω, u(ω) dµ(ω), u ∈ Lp (Ω, X), 1 ≤ p ≤ ∞, Ω
is defined for all u ∈ SFp = {u∈Lp (Ω, X) : u(ω) ∈ F (ω) µ-a.e. on Ω}, exists and there p u0 ∈ Lp (Ω, X) such that I (u ) > −∞, then sup I (u) : u ∈ S sup ϕ(ω, x) : = ϕ 0 ϕ F Ω x ∈ F (ω) dµ. PROOF: Let m(ω) = sup ϕ(ω, x) : x ∈ F (ω) . From Theorem 6.3.24, we know
p for every u ∈ S we have ϕ ω, u(ω) ≤ m(ω) µ-a.e. that m is Σµ -measurable. Also F
on Ω. In particular then ϕ ω, u0 (ω) ≤ m(ω) and so we see that Ω mdµ exists (possibly equals +∞). We have
sup Iϕ (u) : u ∈ SFp ≤ mdµ. Ω
If Iϕ (u0 ) = +∞, then we are done.
So assume that in Iϕ (u0 ) ∈ R. Then the function ω −→ ϕ ω, uo (ω) belongs 1 L (Ω). Let ξ < Ω mdµ be given. We show that ξ < Iϕ (u) for some u ∈ SFp . Let {Cn }n≥1 ⊆ Σ be such that Cn ↑ Ω and µ(Cn ) < ∞ and consider a strictly positive function ϑ ∈ L1 (Ω). We set
6.4 Decomposable Sets and Set-Valued Integration
493
Dn = Cn ∩ {ω ∈ Ω : ϕ ω, uo (ω) ≤ n} ⎧ ϑ(ω) ⎪ if ω ∈ Dn and ϑ(ω) ≤ n ⎨ m(ω) − n ϑ(ω) mn (ω) = n − n if ω ∈ Dn and ϑ(ω) > n . ⎪ ⎩
if ω ∈ Dnc ϕ ω, uo (ω) − ϑ(ω) n
and
Clearly {mn }n≥1 ⊆ L1 (Ω) and mn ↑ m as n → ∞. Therefore by the monotone convergence theorem, we can find n0 ≥ 1 such that ξ < Ω mn0 dµ. Set η = mn0 . Then ξ < Ω ηdµ and η(ω) < m(ω) µ-a.e. on Ω. Define G(ω) = F (ω) ∩ {x ∈ X : η(ω) ≤ ϕ(ω, x)} = ∅
for all ω ∈ Ω.
(6.21)
Evidently Gr G ∈ Σ × B(X) and so by Theorem 6.3.20, we can find u : Ω −→ X a Σ–measurable function such that u(ω) ∈ G(ω) for all ω ∈ Ω. We define En = Cn ∩ {ω ∈ Ω : u(ω) ≤ n} vn = χEn u + χEnc u0 , n ≥ 1.
and
Evidently {vn }n≥1 ⊆ SFp and we have
ϕ ω, u(ω) dµ + Iϕ (vn ) = En
≥
η(ω)dµ + En
η(ω)dµ +
= Ω
Because have
Ω
c En
c En
c En
ϕ ω, u0 (ω) dµ
ϕ ω, u0 (ω) dµ
(see (6.21))
ϕ ω, u0 (ω) − η(ω) dµ.
(6.22)
η(ω)dµ > ξ and En ↑ Ω, from (6.22) it follows that for n ≥ 1 large we Iϕ (vn ) > ξ.
In what follows we study the set SFp 1 ≤ p ≤ ∞ for a multifunction F . Throughout this study (Ω, Σ, µ) is a σ-finite measure space and X a separable Banach space. As always additional hypotheses are introduced as needed. PROPOSITION 6.4.17 If F : Ω −→ 2X \ {∅} is graph-measurable and SFp = ∅, p 1 ≤ p < ∞, then conv SFp = SconvF .
p PROOF: It is easy to see that SconvF ∈ Pf c Lp (Ω, X) . So we have p conv SFp ⊆ SconvF .
(6.23)
p Suppose that the conclusion in (6.23) is strict. So we can find f ∈ SconvF p such that f ∈ / convSF . Then from the strong separation theorem we can find ∗ p ∗ u∗ ∈ Lp (Ω, Xw ∗ )=L (Ω, X) , (1/p) + (1/p ) = 1 such that
(6.24) σ u∗ , conv SFp < u∗ , f p .
From the definition of the support function (see Definition 6.1.14(b)), we have
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6 Multivalued Analysis
σ u∗ , conv SFp = σ u∗ , SFp ∗ = sup u , hp : h ∈ SFp
u∗ (ω), h(ω) dµ : h ∈ SFp = sup Ω
= sup u∗ (ω), x : x ∈ F (ω) dµ Ω
= σ u∗ (ω), F (ω) dµ Ω
= σ u∗ (ω), conv F (ω) dµ.
(see Theorem 6.4.16)
(6.25)
Ω p , we have From (6.24) and (6.25) and because f ∈ SconvF
σ u∗ (ω), conv F (ω) dµ < u∗ (ω), f (ω) dµ Ω Ω
≤ σ u∗ (ω), conv F (ω) dµ, Ω
a contradiction. This proves that in (6.23) we have equality.
A useful byproduct of the previous proof is the following result. COROLLARY 6.4.18 If F : Ω −→ 2X \ {∅} is graph-measurable and SFp = ∅, ∗ p ∗ 1 1 ≤ p < ∞, then for every u∗ ∈ Lp (Ω, Xw + p1 = 1, we have ∗ ) = L (Ω, X) , p σ(u∗ , SFp )=
σ u∗ (ω), F (ω) dµ.
Ω
PROPOSITION 6.4.19 If (Ω, Σ, µ) is a nonatomic, σ-finite measure space, F : w p Ω −→ 2X \{∅} is graph-measurable, and SFp = ∅, 1 ≤ p < ∞, then SFp = SconvF (here w we denote the closure in the weak topology). by w
PROOF: Evidently SFp is nonempty, decomposable, and weakly closed. Therefore w w by Proposition 6.4.14 SFp is convex. It follows that conv SFp = SFp . Invoking Proposition 6.4.17, we conclude that w
p SconvF = SFp .
PROPOSITION 6.4.20 If F : Ω −→ 2X \ {∅} is graph-measurable, and SFp = ∅, 1 ≤ p < ∞, then SFp =SFp .
PROOF: Because SFp ∈ Pf Lp (Ω, X) , we have SFp ⊆ SFp . Because µ is σ-finite, we can find {An }n≥1 ⊆ Σ of finite µ-measure suchp that An ⊆ An+1 for all n ≥ 1 and An =Ω. Then we have µ(An ) ↑ +∞. Let f ∈ SF and for every n ≥ 1, we consider n≥1
the multifunction
6.4 Decomposable Sets and Set-Valued Integration Hn (ω) = x ∈ F (ω) : f (ω) − x
0, we can find Kε ∈ Pk (X) such that for every f ∈ W there exists Sε,f ⊆ T -measurable subset with λ(T \ Sε,f ) < ε such that f (t) ∈ Kε for all t ∈ Sε,f . REMARK 6.4.32 If W ⊆ L1 (T, X) has property U , then W is relatively weakly compact in L1 (T, X). Using property U , we can compare the · w -norm and weak topologies on L1 (T, X). PROPOSITION 6.4.33 If W ⊆ L1 (T, X) has property U , then the · w -norm and weak topologies coincide on the set W . The second continuous selection theorem uses the · w -norm topology on L1 (T, X). This theorem concerns multifunctions F (t, x) of two variables (t, x) ∈ T × X. The hypotheses on F are the following. H(F): F : T ×Y −→ Pwkc (X) is a multifunction such that (i) For every y ∈ Y, t −→ F (t, y) is measurable. (ii) For almost all t ∈ T, y −→ F (t, y) is h-continuous. (iii) For every C ∈ Pk (Y ), we can find αC ∈ L1 (T )+ such that for a.a. t ∈ T , all y ∈ C, and all u ∈ F (t, y), we have u ≤ αC (t).
6.4 Decomposable Sets and Set-Valued Integration
499
REMARK 6.4.34 In the next section we discuss in some detail multifunctions F (t, x) of two variables. For the moment it suffices to mention that if F satisfies hypotheses H(F )(i), (ii), then F is jointly measurable, in particular then supmeasurable (compare with Theorem 6.2.6). This fact combined with hypothesis = ∅. H(F )(iii), implies that if y ∈ C(T, Y ), then S 1
F ·,y(·)
Now let K ⊆ C(T, Y ) be nonempty, and compact and consider the multifunc . Let CSΓw (resp., tion Γ : K −→ Pwkc L1 (T, X) defined by Γ(y) = S 1
F ·,y(·)
w CSextΓ ) be the set of selectors of Γ (resp., of ext Γ) that are continuous from K into L1 (T, X) endowed with the weak norm (denoted by L1w (T, X), not to be confused with L1 (T, X)w which is the Lebesgue–Bochner space L1 (T, X) with the weak topology).
THEOREM 6.4.35 If F (t, x) satisfies hypotheses H(F ), K⊆C(T, Y ) is compact, ·w w for all y ∈ K, then CSΓw = CSextΓ . and Γ(y) = S 1
F ·,y(·)
REMARK 6.4.36 Theorems 6.4.28 and 6.4.35 are indispensable tools in the study of nonconvex differential inclusions (see Hu–Papageorgiou [316]). Now we pass to set-valued integration. First of all the definition of the integral of a multifunction that we use is a straightforward continuous extension of the Minkowski sum of sets. Throughout this last part of this section (Ω, Σ, µ) is a σfinite measure space, X is a separable Banach space, and F : Ω −→ 2X \ {∅} is a multifunction with SF1 = ∅. REMARK 6.4.37 For a graph-measurable multifunction F : Ω −→ 2X \ {∅}, a straightforward measurable selection argument involving Theorem 6.3.20, reveals that SF1 = ∅ if and only if there exists h ∈ L1 (Ω)+ such that inf{u : u ∈ F (ω)} ≤ h(ω). DEFINITION 6.4.38 The set-valued integral of a multifunction F : Ω −→ 2X \{∅} with SF1 = ∅, is defined by
F dµ = f dµ : f ∈ SF1 . Ω
Ω
The following is an immediate consequence of Theorem 6.4.16. X measurable and SF1 = ∅, then PROPOSITION 6.4.39 If F : Ω −→ 2 \{∅} is graph ∗ ∗ ∗ ∗ for every x ∈ X , we have σ(x , Ω F dµ) = Ω σ(x , F )dµ.
Also from Proposition 6.4.17, we obtain the following. X 1 PROPOSITION 6.4.40 If F : Ω −→ 2 \{∅} is graph-measurable and SF = ∅, then cl Ω conv F dµ = conv Ω F dµ = cl Ω conv F dµ.
The set-valued integral has some remarkable intrinsic convexity properties.
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6 Multivalued Analysis
X THEOREM 6.4.41 If µ is nonatomic and F : Ω −→ 2 \{∅} is graph-measurable with SF1 = ∅, then cl Ω F dµ is convex. PROOF: Let v1 , v2 ∈ Ω F dµ. Then by Definition 6.4.38, we have
v1 = f1 dµ and v2 = f2 dµ, with f1 , f2 ∈ SF1 . Ω
Ω
Consider the vector measure m : Σ −→ X × X defined by
m(A) = f1 dµ, f2 dµ , A ∈ Σ. A
A
Invoking Theorem 6.4.12(b), we see that m(Σ) is convex. We have m(∅) = 0
and
m(Ω) = (v1 , v2 ).
So given ε > 0 and t ∈ [0, 1], we can find A ∈ Σ such that
ε t fk dµ − fk dµ < , k = 1, 2. 2 Ω A Set f = χA f1 + χAc f2 . Then f ∈ SF1 and we have
tv1 + (1 − t)v2 − f dµ < ε, Ω
⇒ cl F dµ is convex. Ω
COROLLARY 6.4.42 If µ is nonatomic and F : Ω −→ 2X\{∅} is graph-measurable 1 with SF = ∅, then cl Ω conv F dµ = cl Ω F dµ. REMARK 6.4.43 So when µ is nonatomic, convexification of the multifunction F essentially does not add anything new to the set-valued integral. If in Theorem 6.4.41, dim X 0 such that Bε ⊆ Ω F dµ. For
this purpose let ξF (ω) = d 0, F (ω)c . Let λ > 0 and consider the set Lλ = {ω ∈ Ω : ξF (ω) < λ}. If G = {(ω, x) ∈ Ω × X : x ∈ Bλ ∩ F (ω)c } = (Ω × Bλ ) ∩ GrF c ∈ Σ × B(X), then Lλ = projΩ G ∈ Σµ (see Theorem 6.2.18). This proves that the function ξF is Σµ -measurable. Because F has open values, we have ξF (ω) > 0 for all ω ∈ Ω and so we can find ε > 0 and A ∈ Σ, with µ(A) > 0 such that ξF (ω) ≥ ε for all ω ∈ A. This means that Bε ⊆ F (ω) for all ω ∈ A and so µ(A)Bε ⊆ A F dµ ⊆ Ω F dµ (because 0 ∈ F (ω) for all ω ∈ Ω). In fact under some reasonable conditions, we can establish that the interior and integral operators commute. To show this we need two auxiliary results. LEMMA 6.4.50 If V is a Banach space and U1 ⊆ U2 ⊆ X are nonempty open sets with U1 convex and dense in U2 , then U1 = U2 . PROOF: We have U2 ⊆ int U 1 and because U1 is convex U1 = int U 1 . Therefore U1 = U2 . LEMMA 6.4.51 If F : Ω −→ 2X \{∅} is graph-measurable and intF (ω) = ∅ for all ω ∈ Ω, then Gr int F ∈ Σµ × B(X). PROOF: For every ω ∈ Ω we have int F (ω) = F (ω) \ ∂F (ω) with ∂F (ω) being the c boundary of the set F (ω). Since ∂F (ω) = F (ω) ∩ F (ω) , we infer that Gr ∂F ∈ Σµ ×B(X) (see the proof of Proposition 6.4.49). Hence Gr int F = Gr F ∩(Gr F )c ∈ Σµ × B(X). Using these auxiliary results, we can prove a theorem on the commutation of the interior and integral operators.
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6 Multivalued Analysis
THEOREM 6.4.52 If µ is finite, F : Ω −→ 2X \{∅} is graph-measurable, SF1 = ∅, F (ω) is convex for all ω ∈ Ω, and int F (ω) = ∅ µ-a.e., then for all A ∈ Σ we have int A F dµ = A int F dµ. PROOF: Clearly without any loss of generality, we may assume that int F (ω) = ∅ for all ω ∈ Ω. We fix A ∈ Σ with µ(A) > 0. We show that A int F dµ. To this end let f ∈ SF1 and ε > 0. We introduce the multifunction Gε : Ω −→ 2X defined by Gε (ω) = x ∈ intF (ω) : x − f (ω)
0. If no such x1 ∈ F (x0 ) can be found, then x0 ∈ F (x0 ) and we are done. Then
d x1 , F (x1 ) ≤ h F (x0 ), F (x1 ) ≤ kd(x0 , x1 ) < k1 d(x0 , x1 ). Hence we can find x2 ∈ F (x1 ) such that d(x1 , x2 ) < k1 d(x0 , x1 ). By induction we can generate a sequence {xn }n≥1 ⊆ X such that
6.5 Fixed Points and Carath´eodory Multifunctions xn+1 ∈ F (xn ) and d(xn , xn+1 ) < k1n d(x0 , x1 )
for all n ≥ 1.
505 (6.28)
From the inequality in (6.28) and since k1 < 1, we deduce that {xn }n≥1 ⊆ X is Cauchy. Hence xn −→ x ∈ X for some x ∈ X. From the inclusion in (6.28) and if we pass to the limit as n → ∞, we obtain x ∈ F (x). REMARK 6.5.3 In contrast to Theorem 3.4.3, in this case the fixed point is not unique. Indeed, if F is the trivial constant multifunction F (x) = X for all x ∈ X, then every point in X is a fixed point of F . If Fix(F ) = {x ∈ X : x ∈ F (x)} (the set of fixed points of F ), then it is easily seen that Fix(F ) is closed in X. The next proposition proves a remarkable stability property of the set Fix(F ) with respect to the multifunction F . PROPOSITION 6.5.4 If (X, d) is a complete metric space, F1 , F2 : X −→ Pbf (X) are h-contractions
with the same constant 0 < k < 1, then h Fix(F1 ), Fix(F2 ) ≤ 1/(1 − k) sup h F1 (x), F2 (x) . x∈X
PROOF: Let ε > 0 and also choose ξ > 0 such that ξ nkn < 1. We set ε1 = n≥1
ξε 1/(1 − k) , pick x0 ∈ Fix(F1 ), and then we choose x1 ∈ F2 (x0 ) such that
(6.29) d(x0 , x1 ) ≤ h F1 (x0 ), F2 (x0 ) + ε. We can find x2 ∈ F2 (x1 ) such that d(x2 , x1 ) ≤ k d(x0 , x1 ) + kε1 and then inductively we obtain {xn }n≥1 ⊆ X such that xn+1 ∈ F2 (xn ) ⇒
n≥m
and
d(xn+1 , xn ) ≤ kn d(x0 , x1 ) + nkn ε1
for all n ≥ 1, (6.30)
km d(xn+1 , xn ) ≤ nkn , d(x0 , x1 ) + ε1 1−k n≥m
⇒ {xn }n≥1 ⊆ X is a Cauchy sequence and so xn −→ x ∈ X. Then from (6.30) in the limit as n → ∞ we obtain x ∈ Fix(F2 ). Moreover, d(x0 , x) ≤
1 nkn d(x0 , x1 ) + ε1 1−k n≥m 1
≤ h F1 (x0 ), F2 (x0 ) + 2ε 1−k
d(xn , xn+1 ) ≤
n≥0
(see (6.29)).
Reversing the roles of F1 and F2 in the above argument, we obtain
h Fix(F1 ), Fix(F2 ) ≤
1 sup h F1 (x), F2 (x) . 1 − k x∈X
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6 Multivalued Analysis
COROLLARY 6.5.5 If (X, d) is a complete metric space, Fn , F : X −→ Pbf (X) for n ≥ 1 are h-contractions all with the
same constant 0 < k < 1, and sup h Fn (x), F (x) −→ 0 as n → ∞, then h Fix(Fn ), Fix(F ) −→ 0 as n → ∞. x∈X
There is also a multivalued analogue of Theorem 3.4.23. DEFINITION 6.5.6 Let (X, d) be a metric space and h the corresponding Hausdorff
metric on Pf (X). A multifunction F : X −→ Pf (X) is said to be nonexpansive, if h F (x), F (y) ≤ d(x, y) for all x, y ∈ X. The multivalued generalization of Theorem 3.4.23 is due to Lim [383]. THEOREM 6.5.7 If X is a uniformly convex Banach space, C ∈ Pbf c (X), and F : C −→ Pf (C) is nonexpansive, then F has a fixed point. Now we turn our attention to topological fixed point theorems for multifunctions. DEFINITION 6.5.8 Let X be a vector space. (a) A subset C ⊆ X is said to be finitely closed, if C ∩ Y is closed for every finitedimensional affine subspace Y in X, when on Y we consider its Euclidean topology (recall that Y is an affine subspace of X, if Y = y0 + Y0 for some y0 ∈ X and some finite-dimensional subspace of X). (b) A family {Ci }i∈I of nonempty sets in X is said to have the finite intersection property, if the intersection of every finite subfamily is nonempty. (c) Let C ⊆ X be nonempty. A multifunction F : C −→ 2X is a Knaster– Kuratowski–Mazurkiewicz multifunction (a KKM-multifunction for short), if for every finite set {xk }m k=1 ⊆ C, we have conv{xk }m k=1 ⊆
m !
F (xk ).
k=1
(see also Definition 2.3.6). The next theorem gives the basic property of KKM-multifunctions. It extends Theorem 2.3.7. THEOREM 6.5.9 If X is a vector space, C ⊆ X is nonempty, and F : C −→ 2X is a KKM-multifunction with values that are finitely closed, then the family {F (x)}x∈C has the finite intersection property (see Definition 6.5.8(b)). PROOF: We argue by contradiction. So suppose that
n
F (xk ) = ∅. Let Y =
k=1
span{xk }n k=1 ,
let d be the Euclidean metric on Y and D = conv{xk }n k=1 ⊆ Y . We n
know that Y ∩ F (xk ) is closed for all k ∈ {1, . . . , n}. We have Y ∩ F (xk ) = ∅ k=1
n
and so the function ξ(x) = d x, Y ∩ F (xk ) satisfies ξ(x) > 0 for all x ∈ C. Then
ϑ : D −→ D defined by
k=1
6.5 Fixed Points and Carath´eodory Multifunctions ϑ(x) =
507
n 1
d x, Y ∩ F (xk ) xk ξ(x) k=1
is continuous and so by Theorem 3.5.3 (Brouwer’s fixed point theorem), we can find x0 ∈ D such that ϑ(x0 ) = x0 . Let J = k ∈ {1, . . . , n} : d x0 , Y ∩ F (xk ) = 0 . Then x0 ∈ / F (xk ). But k∈J
ϑ(x0 ) = x0 ∈ conv{xk }k∈J ⊆
!
F (x)
k∈J
(because F is a KKM-multifunction), a contradiction.
An immediate consequence of the previous theorem is the following result (see also Corollary 2.3.9). THEOREM 6.5.10 If X is a Hausdorff topological vector space, C ⊆ X is nonempty, F : C −→ 2X is a KKM-multifunction with closed values, and for at least one x0 ∈ C, F (x0 ) is compact, then F (x) = ∅. x∈C
The same conclusion can be reached in another way, which avoids any topological structure and any topological hypotheses and instead uses an auxiliary multifunction. THEOREM 6.5.11 If X is a vector space, C ⊆ X is nonempty, F : C −→ 2X is a KKM-multifunction, G : C −→ 2X is another multifunction such that # # G(x) ⊆ F (x) for all x ∈ C and G(x)= F (x), x∈C
x∈C
and for some topology on X, G has compact values, then
F (x) = ∅.
x∈C
PROPOSITION 6.5.12 If X is a vector space, C ⊆ X is nonempty, and F : C −→ 2X is a multifunction such that x∈ / conv F (x)
for all x ∈ C,
then x −→ G(x) = X \F −1 (x) is a KKM-multifunction. PROOF: Let {xk }n k=1 ⊆ X. We have X\
n ! k=1
G(xk ) =
n #
F −1 (xk ).
k=1
Therefore we have y∈ /
n ! k=1
and this is equivalent to
G(xk )
if and only if y ∈
n # k=1
F −1 (xk )
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6 Multivalued Analysis y∈C
xk ∈ F (y)
and
Next let y ∈ then from (6.31) we have
conv{xk }n k=1 .
for all k ∈ {1, . . . , n}. (6.31) n We claim that y ∈ k=1 G(xk ). If this is not true,
conv {xk }n k=1 ⊆ conv F (y) ⇒ y ∈ conv F (y), a contradiction. Therefore, we have conv{xk }n k=1 ⊆
n !
G(xk ),
k=1
⇒ G is a KKM-multifunction. This proposition can be used to produce maximal elements with respect to certain irreflexive binary relations. The result is of importance in mathematical economics where the binary relation represents preference among pairs of goods. THEOREM 6.5.13 If X is a Hausdorff topological vector space, C ⊆ X is a nonempty, compact, convex set, ≺ is an irreflexive binary relation on C, and (i) For every x ∈ C, x ∈ / conv{y ∈ C : x ≺ y}. (ii) For every x ∈ C, the lower section {y ∈ C : y ≺ x} is open in C, then the set of ≺-maximal elements in C, namely the set {x ∈ C : x ⊀ y for all y ∈ C} is nonempty and compact. PROOF: We consider the multifunction x −→ F (x) = {y ∈ C : x ≺ y}
for all x ∈ C.
By virtue of hypothesis (i) and Proposition 6.5.12, we have that x −→ G(x) = X \F −1 (x) is a KKM-multifunction. Hence H : C −→ 2C defined by x −→ H(x) = C ∩ G(x) = C \F −1 (x)
is a KKM-multifunction too.
Moreover, because of hypothesis (ii) the set F −1 (x) is open and so H is compactvalued. So Theorem 6.5.10 implies that # #
H(x) = C \ F −1 (x) = ∅ x∈C
x∈C
and of course it is compact. But this is the set of ≺-maximal elements.
This theorem leads to a fundamental existence theorem for variational inequalities.
6.5 Fixed Points and Carath´eodory Multifunctions
509
THEOREM 6.5.14 If X is a locally convex space, C ⊆ X is nonempty, compact, and convex, and u : C −→ X ∗ is a map such that (x, y) −→ u(x), y is jointly continuous on C×C (by ·, · we denote the duality brackets for the dual pair (X, X ∗ )), then we can find x ∈ C such that u(x), y − x ≥ 0 for all y ∈ C. PROOF: On C we define the irreflexive binary relation ≺ by x≺y
if and only if u(x), x − y > 0.
Note that for every x ∈ C, we have x∈ / conv{y ∈ C : x ≺ y} = {y ∈ C : x ≺ y} = y ∈ C : u(x), x − y > 0 . Also note that because of the hypothesis on u, the set {y ∈ C : y ≺ x} = y ∈ C : u(x), x − y < 0 is open. So we can apply Theorem 6.5.13 and obtain a ≺-maximal element x ∈ C. Hence u(x), x − y ≥ 0 for all y ∈ C. Next we use the results on KKM-multifunctions to deduce some basic topological fixed point theorems for set-valued maps. DEFINITION 6.5.15 Let X be a vector space and C ⊆ X a nonempty set. (a) The inward set of x ∈ C with respect to C is defined by IC (x) = {x + λ(y − x) : λ ≥ 0 and y ∈ C}. (b) A multifunction F : C −→ 2X \ {∅} is said to be weakly inward if F (x) ∩ IC (x) = ∅
for all x ∈ C.
REMARK 6.5.16 If F (C) ⊆ C, then F (x) ⊆ IC (x) for all x ∈ C. Indeed, if y ∈ F (x), then for λ = 1, y = x + λ(y − x) = x + (y − x) = y. Multifunctions satisfying F (x) ⊆ IC (x) are often called strongly inward . For single-valued maps the geometric notion of weak inwardness hasa metric equivalent,namely f : C −→ X is
weakly inward if and only if lim (1/λ)d x + λ f (x) − x , C = 0 for all x ∈ C. λ→0+
THEOREM 6.5.17 If X is a locally convex space, C ⊆ X is nonempty, compact, and convex, and F : C −→ Pf c (X) is usc and weakly inward, then F admits a fixed point; that is, there exists x ∈ C such that x ∈ F (x). PROOF: We argue indirectly. So suppose that F has no fixed point. Then for any given x ∈ C, we have 0 ∈ / x − F (x). Invoking the strong separation theorem for convex sets, we can find x∗ ∈ X ∗ \{0} such that x∗ , x − y < 0 for all y ∈ F (x),
∗ ⇒ σ x , x − F (x) < 0.
510
6 Multivalued Analysis We set
U (x∗ ) = x ∈ C : σ x∗ , −F (x) < 0
for all x∗ ∈ X ∗ \ {0}.
Because F is usc, we have that U (x∗ ) is open (see Proposition 6.1.15(c)) and {U (x∗ )}x∗ ∈X ∗ \{0} is an open cover of C. Inasmuch as C is compact, we can find {U (x∗k )}m k=1 a finite subcover and a corresponding continuous partition of unity {pk }m . k=1 We define m u(x) = pk (x)x∗k . k=1
Clearly the function (x, y) −→ u(x), y is continuous on C × C (as before by ·, · we denote the duality brackets for the dual pair (X, X ∗ )). So we can apply Theorem 6.5.17 and obtain x0 ∈ C such that u(x0 ), y − x0 ≥ 0
for all y ∈ C.
By hypothesis we can find y ∈ F (x0 ) ∩ IC (x0 ). Then y = x0 + lim λn (yn − x0 ), n→∞
with λn ≥ 0 and yn ∈ C,
⇒ y − x0 = lim λn (yn − x0 ), n→∞
⇒ u(x0 ), y − x0 = lim λn u(x0 ), y − x0 ≥ 0. n→∞
But this contradicts the fact that
σ u(x), x − F (x) < 0
for all x ∈ C.
REMARK 6.5.18 It is easy to check that in the above theorem Fix(F ) = {x ∈ C : x ∈ F (x)} is compact. If we combine Theorem 6.5.17 with Remarks 6.5.16 and 6.5.18 and if we recall that a locally compact multifunction is usc if and only if it is closed (i.e., its graph is closed; see Proposition 6.1.10), we can have the following multivalued generalization of the Schauder–Tychonov fixed point theorem (see Theorem 3.5.28). The result is known as the Kakutani–Ky Fan fixed point theorem. THEOREM 6.5.19 If X is locally convex space, C ⊆ X is a nonempty, compact, and convex set, and F : C −→ Pf c (C) is closed, then the set Fix(F ) is nonempty and compact. A related topological fixed point theorem for multifunctions is the following. THEOREM 6.5.20 If X is a locally convex space, C ⊆ X is a nonempty, compact, and convex set, and F : C −→ 2C \{∅} is a multifunction with convex values such that for each y ∈ C the set F + ({y}) = {x ∈ C : y ∈ F (x)} is open, then F admits a fixed point.
6.5 Fixed Points and Carath´eodory Multifunctions
511
PROOF: The family F + ({y}) y∈C is an open cover of C. Due to the compactness m of C, we can find a finite subcover F + ({yk }) k=1 and a corresponding continuous partition of unity {pk }m k=1 . We set m u(x) = pk (x)yk for all x ∈ C. k=1
Clearly u : C −→ C is continuous and u(x) ∈ F (x) for all x ∈ C (recall that F is convex-valued). Apply Theorem 3.5.28 to obtain x ∈ C such that u(x) = x ∈ F (x). Next let X, Y be Banach spaces and C ⊆ X, D ⊆ Y are nonempty, closed, convex sets. In what follows by (D, w) we denote D furnished with the relative weak topology of Y . We consider multifunctions G : C −→ 2C \{∅} that admit the following decomposition G = K ◦ N, (6.32) where N : C −→ 2D \ {∅} is usc from C with the relative strong (norm) topology into (D, w) and has weakly compact and convex values and K : (D, w) −→ C is sequentially continuous, namely w
yn −→ y
in D ⇒ K(yn ) −→ K(y)
in C.
Also we assume that G is compact; that is, it maps bounded sets in C into relatively compact sets in C. We emphasize that with the above hypotheses G need not have convex values. We have the following multivalued versions of the nonlinear alternative theorem (see Theorem 3.5.16) and of the Leray–Schauder alternative principle (see Corollary 3.5.18). Both theorems can be proved using degree-theoretic arguments. Details can be found in Bader [49]. THEOREM 6.5.21 If X is a Banach space and G : B R −→ 2X \{∅} is a compact multifunction that admits the decomposition (6.32), then at least one of the following statements holds. (a) There exists x0 ∈ ∂BR and λ ∈ (0, 1) such that x0 ∈ λG(x0 ). (b) Fix(G) = ∅. THEOREM 6.5.22 If X is a Banach space, G : C −→ 2C \ {∅} is a compact multifunction that admits the decomposition (6.32), and 0 ∈ C, then at least one of the following statements holds. (a) G has a fixed point. (b) The set S = {x ∈ C : x ∈ λG(x) for some 0 < λ < 1} is bounded. Let us conclude our discussion of the fixed point theory of multifunctions, with two results on the topological structure of the set Fix(F ) for h-contractions. First we have a definition. DEFINITION 6.5.23 (a) Let A be a subset of the metric space X. Then A is called a retract if there exists a continuous map (retraction) r : X −→ A such that rA = idA .
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6 Multivalued Analysis
(b) A metric space E is said to be an absolute retract if for every metric space X and every nonempty closed set A ⊆ X, each continuous map f : A −→ E admits a continuous extension on all of X. REMARK 6.5.24 By virtue of Theorem 3.1.10 (Dugundji’s extension theorem), every closed convex set of a normed space is a retract. Every homeomorphic image of an absolute retract is also an absolute retract. Moreover, if E is an absolute retract and A is a retract of E, then A is also an absolute retract. The first structural theorem for the set Fix(F ) is due to Ricceri [514]. THEOREM 6.5.25 If X is a Banach space, C ⊆ X is nonempty, closed, and convex and F : C −→ Pf c (X) is an h-contraction, then Fix(F ) is an absolute retract. The second structural theorem, is due to Bressan–Cellina–Fryszkowski [97]. THEOREM 6.5.26 If (Ω, Σ, µ) is a finite nonatomic measure space, X is a separable Banach space, Y = L1 (Ω, X), and F : Y −→ 2Y \{∅} is an h-contraction with bounded, closed, and decomposable values, then Fix(F ) is an absolute retract. We conclude this section with a few facts about Caratheodory multifunctions. First suppose that T is a locally compact, σ-compact metric space equipped with a (regular) measure µ of bounded variation and defined on the σ-field S of a µ-measurable set (i.e., S = B(T )µ ). Also X is a Polish space and Y is a separable metric space. Recall that for Pk (Y )–valued multifunctions continuity and h-continuity coincide (see Corollary 6.1.40) and that Pk (Y ), h is a separable metric space (see Proposition 6.1.32(b)). Therefore invoking Theorem 6.2.9, we obtain the following result. PROPOSITION 6.5.27 If F : T ×X −→ Pk (Y ) is a multifunction such that (i) For every x ∈ X, t −→ F (t, x) is measurable and (ii) For every t ∈ T , x −→ F (t, x) is continuous,
then for every ε > 0 we can find Tε ⊆ T compact with µ(T \ Tε ) < ε such that F T ×X ε is continuous. Another result in the same direction is the following. PROPOSITION 6.5.28 If F : T × X −→ 2Y \{∅} is a multifunction such that (i) (t, x) −→ F (t, x) is measurable and (ii) For every t ∈ T , x −→ F (t, x) is lsc,
then for every ε > 0 we can find Tε ⊆ T compact with µ(T \ Tε ) < ε such that F T ×X ε is lsc.
PROOF: Fix y ∈ Y and consider the function (t, x)−→ϕy (t, x) = d y, F (t, x) . Evidently ϕy is L×B(X)-measurable (see Proposition 6.2.4) and for every t ∈ T ϕy (t, ·) is upper semicontinuous (see Proposition 6.1.15(a)). We can find a sequence {ϕn }n≥1 of Carath´eodory functions on T × X into Y such that ϕn (t, x) ↓ ϕy (t, x) for all
6.5 Fixed Points and Carath´eodory Multifunctions
513
(t, x) ∈ T × X. Invoking Theorem 6.2.9 given ε > 0, we can find Tε ⊆ T com pact with µ(T \ Tε ) < ε such that ϕn T ×X is continuous. Hence ϕy T ×X is lower ε ε semicontinuous. A new application of Proposition 6.1.15(a) implies that F T ×X is ε lsc. REMARK 6.5.29 The result fails if for all t ∈ T, F (t, ·) is usc. Next we present two parametrized versions of Michael’s selection (see Theorem 6.3.6). PROPOSITION 6.5.30 If Y is a separable reflexive Banach space and F : T × X −→ Pf c (Y ) a multifunction such that (i) (t, x) −→ F (t, x) is measurable and (ii) For every t ∈ T , x −→ F (t, x) is lsc, eodory functions such then there exists a sequence fm : T ×X −→ Y, m ≥ 1, of Carath´ that for all (t, x) ∈ T × X we have F (t, x) = {fm (t, x)}m≥1 . PROOF: By virtue of Proposition 6.5.28, for every n ≥ 1 we can find Tn ⊆ T compact such that µ(T \ Tn ) < 1/n and F T ×X is lsc. We apply Theorem 6.3.11 n to obtain continuous functions fn,m : Tn × X −→ Y, n, m ≥ 1, such that F (t, x) = {fnm (t, x)}m≥1 for all (t, x) ∈ Tn × X. Let fm : T ×X −→ Y, m ≥ 1, be defined by ⎧ ⎨fnm (t, x) if t ∈ Tn and t ∈ / Tk for k < n . fm (t, x) = T 0 if t ∈ T \ n ⎩ n≥1
Clearly for every m ≥ 1, fm is a Carath´eodory function and for all (t, x) ∈ T ×X, we have F (t, x) = {fm (t, x)}m≥1 . If (T, S, µ) is replaced by (Ω, Σ) a measurable space (no topological structure on Ω), then the situation is more involved. Nevertheless by adapting the proof of Michael’s selection theorem (see Theorem 6.3.6) to the present parametric situation, we can have the following theorem. For details we refer to Hu–Papageorgiou [313, p. 235]). THEOREM 6.5.31 If (Ω, Σ) is a complete measurable space, X is a Polish space, Y is a separable Banach space, and F : Ω × X −→ Pf c (Y ) is a multifunction such that (i) (ω, x) −→ F (ω, x) is measurable and (ii) For all ω ∈ Ω, x −→ F (ω, x) is lsc, eodory functions such then we can find a sequence fn : Ω× X −→ Y, n ≥ 1, of Carath´ that F (ω, x) = {fn (ω, x)}n≥1 for all (ω, x) ∈ Ω × X. REMARK 6.5.32 If F (ω, x) is measurable in ω ∈ Ω and usc or lsc in x ∈ X, it is not in general jointly measurable.
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6 Multivalued Analysis
6.6 Convergence of Sets We start this section with the definitions of the different modes of set convergence, which we investigate. DEFINITION 6.6.1 Let (X, d) be a metric space and {An , A}n≥1 ⊆ Pf (X). We h
say that the An s converge to A in the Hausdorff sense, denoted by An −→ A or by h- lim An = A if and only if h(An , A) −→ 0 as n → ∞ (recall h is the Hausdorff n→∞
metric on Pf (X); see Definition 6.1.29). h
REMARK 6.6.2 From Proposition 6.1.31(b), we see that An −→ A if and only if d(·, An ) −→ d(·, A) uniformly on X. Also if X is a normed space and {An , A}n≥1 ⊆ ∗
Pbf c (X), then An −→ A if and only if σ(·; An ) −→ σ(·, A) uniformly on B 1 = {x∗ ∈ X ∗ : x∗ ≤ 1} (see Proposition 6.1.31(c)). h
DEFINITION 6.6.3 Let (X, τ ) be a Hausdorff topological space (τ denotes the Hausdorff topology) and let {An }n≥1 ⊆ 2X \{∅}. We define τ- lim inf An = {x ∈ X : x = τ- lim xn , xn ∈ An , n ≥ 1} n→∞
n→∞
and
τ- lim sup An = {x ∈ X : x = τ- lim xnk , xnk ∈ Ank , nk < nk+1 , k ≥ 1}. n→∞
n→∞
The set τ-lim inf An is the τ -Kuratowski limit inferior of the sequence {An }n≥1 n→∞
and τ-lim sup An is the τ -Kuratowski limit superior of the sequence {An }n≥1 . If n→∞
A = τ-lim inf An = τ-lim sup An , then A is the τ -Kuratowski limit of the sequence n→∞
n→∞ K
τ A. {An }n≥1 and we write An −→
REMARK 6.6.4 If (X, d) is a metric space, then d- lim inf An = {x ∈ X : x = lim d(x, An ) = 0} n→∞
n→∞
and
d- lim sup An = {x ∈ X : x = lim inf d(x, An ) = 0}. n→∞
n→∞
Note that we always have τ -lim inf An ⊆ τ -lim sup An and if the space X is first n→∞ n→∞ An and so it is a closed set. If the topology countable, then τ -lim sup An = n→∞
k≥1 n≥k
τ is clearly understood, then we drop the letter τ . The Kuratowki mode of set-convergence turns out to be suitable for locally compact spaces. In order to deal with sets defined in an infinite-dimensional Banach space (which is not locally compact), we need a new mode of set-convergence, which involves all the useful topologies defined on the Banach space. DEFINITION 6.6.5 Let X be a Banach space. By w- (resp., s-) we denote the weak (resp. strong) topology on X. Let {An }n≥1 ⊆ 2X \ {∅}. We say that the An s converge to A in the Mosco sense if and only if w-lim inf An = s-lim inf An = A. n→∞
n→∞
6.6 Convergence of Sets
515
REMARK 6.6.6 Note that we always have s-lim inf An ⊆ w-lim inf An and sn→∞
n→∞
lim sup An ⊆ w-lim sup An . So comparing Definitions 6.6.3 and 6.6.5, we see that n→∞ M
n→∞
K
K
s w A and An −→ A. An −→ A if and only if An −→
DEFINITION 6.6.7 Let (X, d) be a metric space and {An , A} ⊆ Pf (X). We say W
that the An s converge to A in the Wijsman sense, denoted by An −→ A if and only if for all x ∈ X we have d(x, An ) −→ d(x, A). h
REMARK 6.6.8 From Remark 6.6.2 we already know that An −→ A implies W An −→ A. DEFINITION 6.6.9 Let X be a Banach space and {An , A} ⊆ Pf c (X). We say w that the An s converge to A weakly (or scalarly), denoted by An −→ A, if and only ∗ ∗ ∗ ∗ if for all x ∈ X we have σ(x , An ) −→ σ(x , A). REMARK 6.6.10 From Remark 6.6.2, we know that if {An }n≥1 ⊆ Pbf c (X), then h
w
An −→ A implies An −→ A. It is natural to ask what is the relation between these modes of set-convergence. Already in Remarks 6.6.8 and 6.6.10, we saw some first relations. In fact comparing the definitions and using Remark 6.6.6, we can have the following result which provides a first detailed general comparison between these notions. More specialized results are given later in the section. PROPOSITION 6.6.11 If X is a Banach space and {An }n≥1 ⊆ 2X \ {∅}, then (a) (b) (c) (d)
h
K
W
s An −→ A implies An −→ A and An −→ A. h w If {An , A}n≥1 ⊆ Pbf c (X), then An −→ A implies An −→ A. Ks Kw M A and An −→ A. An −→ A if and only if An −→ Ks W An −→ A implies An −→ A.
Without additional hypotheses we cannot say more. EXAMPLE 6.6.12 (a) In general M -convergence does not imply h-convergence: Let X = l2 and let {en }n≥1 be the standard orthonormal basis of the Hilbert space. M
We define An = {λen : 0 ≤ λ ≤ 1} and A = {0}. Evidently An −→ A but we do not have convergence in the Hausdorff metric because h(An , A) = 1 for all n ≥ 1. (b) In general h-convergence does not imply M -convergence: Let X be a reflexive Banach space and let An = A = ∂B1 = {x ∈ X : x = 1} for all k. The trivially h
An −→ A. Recall that ∂B1 convergence.
w
= {x ∈ X : x ≤ 1}. So we cannot have Mosco
(c) In general K-convergence does not imply W -convergence: Let X = l2 and let {en }n≥1 be the standard orthonormal basis of this Hilbert space. We define An = K
{x ∈ X : x = λe1 + (1 − λ)en ,√0 ≤ λ ≤ 1}, n ≥ 1 and A = {e1 }. Evidently An −→ A but d(0, An ) = 12 e1 + en = 2/2 and so we cannot have W -convergence.
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6 Multivalued Analysis
The next proposition explains the situation in Example 6.6.12(b) and also justifies the use of monotone techniques in variational analysis. PROPOSITION 6.6.13 If (X, d) is a metric space and {An }n≥1 ⊆ 2X \ {∅}, then (a) The sets lim inf An , lim sup An are both closed (possibly empty). n→∞ n→∞ An . (b) If {An }n≥1 is increasing, then lim inf An = lim sup An = n→∞ n→∞ n≥1 (c) If {An }n≥1 is decreasing, then lim inf An = lim sup An = An . n→∞
PROOF: (a): Note that lim sup An = n→∞
n≥1
n→∞
An and so lim sup An is closed. Also
n≥1 n≥1
n→∞
let {xk }k≥1 ⊆ lim inf An and assume that xk −→ x as k −→ ∞. We have d(x, An ) ≤ n→∞
x − xk + d(xk , An ). Hence lim sup d(x, An ) ≤ x − xk . Because this is true for n→∞
all k ≥ 1, we conclude that d(x, An ) −→ 0 as n → ∞, hence x ∈ lim inf An (see n→∞
Remark 6.6.4). Therefore lim inf An is closed. n→∞
(a) and (c): Follow at once from Remark 6.6.4 and part (a). REMARK 6.6.14 Note that in a metric space X,
An ⊆ lim inf An .
k≥1 n≥k
n→∞
h
PROPOSITION 6.6.15 If X is a Banach space, {An }n≥1 ⊆ Pf c (X), and An −→ M
A, then An −→ A. K
s PROOF: From Proposition 6.6.11(a) we know that An −→ A. So we need to show that w-lim sup An ⊆ A. To this end let x ∈ w-lim sup An ⊆ A. Then by
n→∞
n→∞
virtue of Definition 6.6.3 we can find a subsequence {nk } on {n} and xnk ∈ Ank w such that xnk −→ x. Note that A ∈ Pf c (X) and so d(·, A) is convex on X. So it is w-lower semicontinuous and we have d(x, A) ≤ lim sup d(xnk , A). On the other k→∞
hand d(xnk , A) ≤ h(Ank , A) −→ 0 as k −→ ∞. Therefore d(x, A) = 0 and so x ∈ A (because A ∈ Pf c (X)). Hence we infer that w-lim sup An ⊆ A and we conclude that n→∞
M
An −→ A.
K
PROPOSITION 6.6.16 If (X, d) is a metric space, {An }n≥1 ⊆Pf (X), An −→ A, h
and there exists a compact set C ⊆ X such that An ⊆ C for all n ≥ 1, then An −→ A as n → ∞. PROOF: First note that A ∈ Pk (X). Let xn ∈ A be such that d(xn , An ) = sup d(x, An ) : x ∈ A] = h∗ (A, An ). Since {xn }n≥1 ⊆ C, we can find a subseK
quence {xnk }k≥1 of {xn }n≥1 such that xnk −→ x ∈ A. Also since An −→ A, we can find an ∈ An , n ≥ 1, such that an −→ x in X. Then we have h∗ (A, Ank ) = d(xnk , Ank ) ≤ d(xnk , ank ) −→ 0 as k −→ ∞.
6.6 Convergence of Sets
517
Next let un ∈ An , n ≥ 1, such that d(un , A) = sup[d(x, a) : x ∈ An ] = h∗ (An , A). As before we can find a subsequence {unk }k≥1 such that unk −→ u. Because K
An −→ A, we have u ∈ A. Then h∗ (Ank , A) = d(unk , A) ≤ d(unk , u) −→ 0
as k −→ ∞.
From the above arguments, we have h∗ (A, An )
and
h∗ (An , A) −→ 0
as n → ∞,
h
⇒ An −→ A. Now suppose that X is a Banach space and {An }n≥1 ⊆ 2X \{∅}. From Definition 6.6.3 it is clear that for every x∗ ∈ X ∗ , we have σ(x∗ , w- lim sup An ) ≤ lim sup σ(x∗ , An ). n→∞
(6.33)
n→∞
In the next proposition, we present a situation where equality holds in (6.33). PROPOSITION 6.6.17 If X is a Banach space, {An }n≥1 ⊆ 2X \{∅}, and for all n ≥ 1, An ⊆ W ∈ Pwk (X), then w-lim sup An = ∅ and for every x∗ ∈ X ∗ we have n→∞
∗
lim sup σ(x , An ) = σ(x∗ , w- lim sup An ). n→∞
n→∞
PROOF: Because An ⊆ W ∈ Pwk (X), n ≥ 1, from the Eberlein–Smulian theorem, we conclude that w-lim sup An = ∅. Let x∗ ∈ X ∗ and choose an ∈ An such that n→∞
σ(x∗ , An ) −
1 ≤ x∗ , an . n
(6.34)
By passing to a subsequence if necessary, we may assume that an −→ a ∈ wlim sup An . Then from (6.34), we have n→∞
lim sup σ(x∗ , An ) ≤ x∗ , a ≤ σ(x∗ , w- lim sup An ). n→∞
(6.35)
n→∞
Comparing (6.33) and (6.35), we conclude that σ(x∗ , w- lim sup An ) = lim sup σ(x∗ , An ). n→∞
n→∞
A related result is the following. PROPOSITION 6.6.18 If X is a Banach space, {An , A}n≥1 ⊆ 2X \ {∅} and for every x∗ ∈ X ∗ we have lim sup σ(x∗ , An ) ≤ σ(x∗ , A), n→∞
then w-lim sup An ⊆ conv A. n→∞
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6 Multivalued Analysis
PROOF: Let a ∈ w-lim sup A. We can find a subsequence {nk } of {n} and ank ∈ n→∞ w
Ank , k ≥ 1, such that ank −→ a in X. For every x∗ ∈ X ∗ we have x∗ , ank −→ x∗ , a and so x∗ , a ≤ lim sup σ(x∗ , Ank ) ≤ σ(x∗ , A). Hence a ∈ conv A and we conclude k→∞
that w-lim sup An ⊆ conv A. n→∞
PROPOSITION 6.6.19 If X is a Banach space, {An }n≥1 ⊆ Pf c (X), for all n ≥ 1 K
w
w A, then An −→ A. we have An ∈ W ⊆ Pwk (X) and An −→
PROOF: Evidently A ∈ Pf c (X). We fix x∗ ∈ X ∗ and then choose an ∈ An such that x∗ , an = σ(x∗ , An ), n ≥ 1.
(6.36) w
By passing to a suitable subsequence if necessary, we may assume that an −→ Kw A). Hence a ∈ A (because An −→ x∗ , a = lim σ(x∗ , An ) ≤ σ(x∗ , A)
(see (6.36))
⇒ lim sup σ(x∗ , An ) ≤ σ(x∗ , A).
(6.37)
n→∞
From (6.33) and (6.37) and since A = w- lim sup An , we conclude that n→∞
σ(x∗ , An ) −→ σ(x∗ , A)
for all x∗ ∈ X ∗ ,
w
⇒ An −→ A. REMARK 6.6.20 The previous proposition fails if we do not have the uniform boundedness of the sequence {An }n≥1 by W ∈ Pwk (X). Moreover, if in Proposition K
w
w A if and only if An −→ A. For details we refer 6.6.19 X is separable, then An −→ to Hu–Papageorgiou [313, p. 676].
PROPOSITION 6.6.21 If X is a Banach space, {An }n≥1 ⊆ 2X \ {∅}, then for every x ∈ X we have lim sup d(x, An ) ≤ d(x, s- lim inf An ). n→∞
n→∞
PROOF: If s- lim inf An = ∅, then d(·, s- lim inf An ) ≡ +∞ and so the proposition n→∞
n→∞
is trivially true. Hence we may assume that s- lim inf An = ∅. Let a ∈ s- lim inf An . n→∞
n→∞
We can find an ∈ An , n ≥ 1, such that an −→ a in X. Then d(x, An ) ≤ x − an and so lim sup d(x, An ) ≤ x − a. Because a ∈ s- lim inf An was arbitrary, we deduce n→∞
n→∞
that lim sup d(x, An ) ≤ d(x, s- lim inf An ). n→∞
n→∞
PROPOSITION 6.6.22 If X is a Banach space, {An , A}n≥1 ⊆ 2X \ {∅} and for every x ∈ X, we have lim sup d(x, An ) ≤ d(x, A), n→∞
then A ⊆ s- lim inf An . n→∞
6.6 Convergence of Sets
519
PROOF: If a ∈ A, then d(a, An ) −→ 0 and so a ∈ s- lim inf An (see Remark 6.6.4). n→∞
Therefore we have A ∈ s- lim inf An . n→∞
If we combine Propositions 6.6.18 and 6.6.22, we obtain the following result. PROPOSITION 6.6.23 If X is a Banach space, {An , A}n≥1 ⊆ Pf c (X) and W
w
M
An −→ A, An −→ A, then An −→ A. In Propositions 6.6.21 and 6.6.22 we obtained results concerning s- lim inf An . n→∞
We want to have similar results for w- lim sup An . For this purpose we introduce n→∞
the following class of subsets of X. DEFINITION 6.6.24 Let X be a Banach space. We define A = {C ⊆ X : C is nonempty, weakly closed and for every r > 0, C ∩ B r ∈ Pwk (X)}, and Ac = {C ∈ A : C is also convex}. REMARK 6.6.25 Clearly the family A is closed under finite unions, arbitrary intersections, and of course contains the family of all weakly closed and locally weakly compact subsets of X. Moreover, if X is reflexive, then A = {C ⊆ X : C is nonempty and weakly closed}; that is, A = Pf (Xw ). PROPOSITION 6.6.26 If X is a Banach space, {An }n≥1 ⊆ 2X \ {∅} and An ⊆ W ∈ A for every n ≥ 1, then for every x ∈ X, we have d(x, w- lim sup An ) ≤ n→∞
lim inf d(x, An ). n→∞
PROOF: Let x ∈ X and r = lim inf d(x, An ). If r = +∞, then the proposition is n→∞
trivially true. So let r < +∞. Suppose that the result is not true. Then for some x ∈ X we have lim inf d(x, An ) < d(x, w- lim sup An ). (6.38) n→∞
n→∞
We can find a subsequence {nk } of {n}, such that lim d(x, Ank ) = lim inf d(x, Ank ) = r < ∞.
k→∞
n→∞
We pick ank ∈ Ank such that x − ank ≤ d(x, Ank ) +
1 k
for all k ≥ 1.
(6.39)
For k ≥ 1 large we have ank ∈ W ∩ B (r+x+1) ∈ Pwk (X). So by passing to a w further subsequence if necessary, we may assume that ank −→ a ∈ w- lim sup An = ∅. n→∞
Then x − a ≤ lim inf x − ank ≤ lim inf d(x, An ) k→∞
n→∞
(see (6.39))
< d(x, w- lim sup An ) n→∞
≤ x − a,
(see (6.38))
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6 Multivalued Analysis
a contradiction.
Hidden in the above proof is also the result that w- lim sup An = ∅. So we can n→∞
state this separately. PROPOSITION 6.6.27 If X is a Banach space, {An }n≥1 ⊆ 2X \ {∅}, and r = lim inf d(x, An ), then n→∞
(a) w- lim sup An = ∅ implies r < +∞. n→∞
(b) If An ⊆ W ∈ A for all n ≥ 1 and r < +∞, then w- lim sup An = ∅. n→∞
REMARK 6.6.28 If X is reflexive, then in part (b), W = X. So in reflexive Banach spaces, for any sequence {An }n≥1 ⊆ 2X\{∅} we always have d(x, w- lim sup An ) ≤ n→∞
lim inf d(x, An ) for all x ∈ X. n→∞
The next result is analogous to Proposition 6.6.22 but for w-lim sup this time. For a proof of this result, we refer to Hu–Papageorgiou [313, pp. 673–674]. PROPOSITION 6.6.29 If X is a reflexive Banach space with X ∗ locally uniformly convex, {An , A}n≥1 ⊆ Pf c (X) and for every x ∈ X we have d(x, A) ≤ lim inf d(x, An ), then w- lim sup An ⊆ A. n→∞
n→∞
Recalling that weakly convergent sequences in a Banach space are bounded, we see that ! w- lim sup(An ∩ kB1 ). w- lim sup An = n→∞
k≥1
n→∞
This observation leads at once to the following result. PROPOSITION 6.6.30 If X is a Banach space, {An }n≥1 ⊆ 2X \ {∅} and either (i) X ∗ is separable or (ii) X is separable and An ⊆ W ∈ A for all n ≥ 1, then w An ∩ kB1 . w- lim sup An = n→∞
k≥1 m≥1 n≥m
REMARK 6.6.31 Clearly in the above proposition An ∩ kB1 can be replaced by τ An ∩ kB 1 or by An ∩ kB1 with τ = s or τ = w. Note that in general w- lim sup An n→∞
is neither strongly nor weakly closed. Finally in a finite-dimensional setting and under a compactness condition, the situation is very pleasant because we have the following. PROPOSITION 6.6.32 If X is a finite-dimensional Banach space and K w h {An , A}n≥1 ⊆ Pf c (X) with A compact, then An −→ A ⇔ An −→ A ⇔ An −→ W
A ⇔ An −→ A.
6.6 Convergence of Sets
521
In the last part of this section we deal with the convergence of the set-valued integrals and of the sets of Lp -selectors of sequences of measurable multifunctions. A major tool in our considerations is the following result on the pointwise behavior of a weakly convergent sequence in Lp (Ω, X), (1 ≤ p < ∞). PROPOSITION 6.6.33 If (Ω, Σ, µ) is a σ-finite measure space, X is a Banach w space, {fn , f }n≥1 ⊆ Lp (Ω, X, ) 1 ≤ p < ∞, fn −→ f in Lp (Ω, X), and for µ-almost all ω ∈ Ω and all n ≥ 1 fn (ω) ∈ G(ω) ∈ Pwk (X), then f (ω) ∈ conv w- lim sup{fn (ω)} n→∞
µ-a.e. on Ω. PROOF: By Mazur’s lemma we have f (ω) ∈ conv
!
fn (ω)
µ-a.e. on Ω.
n≥k
So for every k ≥ 1, x∗ ∈ X ∗ and ω ∈ Ω \ N, µ(N ) = 0, we have !
x∗ , f (ω) ≤ σ x∗ , fn (ω) = sup x∗ , fn (ω) , n≥k
n≥k ∗
∗
⇒ x , f (ω) ≤ lim sup x , fn (ω) . n→∞
Invoking Proposition 6.6.17, we obtain
x∗ , f (ω) ≤ σ x∗ , w- lim sup fn (ω) , n→∞
∗
∗
for all ω ∈ Ω\N and all x ∈ X ,
⇒ f (ω) ∈ conv w- lim sup fn (ω) . n→∞
Our aim is to derive multivalued analogues of Fatou’s lemma. To do this we need the following measurability result whose proof can be found in Hu–Papageorgiou [313, p. 692]. LEMMA 6.6.34 If (Ω, Σ) is a measurable space, X is a separable Banach space, and Fn : Ω −→ 2X \ {∅}, n ≥ 1, are measurable multifunctions such that Fn (ω) ⊆ W (ω) ∈ A for all ω ∈ Ω and all n ≥ 1, then ω −→ lim sup Fn (ω) is measurable. n→∞
REMARK 6.6.35 If X is reflexive, then W (ω) = X for all ω ∈ Ω and so ω −→ lim sup Fn (ω) is always measurable. n→∞
PROPOSITION 6.6.36 If (Ω, Σ, µ) is a nonatomic, σ-finite measure space, X is a separable Banach space and Fn : Ω −→ 2X \ {∅}, n ≥ 1, are graph-measurable multifunctions such that Fn (ω) ⊆ W (ω) µ-a.e. on Ω, for all n ≥ 1, with W : Ω −→ Pwkc (X) integrably bounded, then w- lim sup Ω Fn dµ ⊆ Ω w- lim sup Fn dµ. n→∞
n→∞
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6 Multivalued Analysis
PROOF: Let x ∈ w- lim sup Ω Fn dµ. We can find a subsequence {nk } of {n} and n→∞ w xnk ∈ Ω Fnk dµ such that xnk −→ x in X. Then we have xnk = Ω fnk dµ with fnk ∈ 1 1 SF1 n ⊆ SW . From Theorem 6.4.23 we know that SW ⊆ L1 (Ω, X) is w-compact. So k
w
we may assume that fnk −→ f in L1 (Ω, X). By virtue of Proposition 6.6.36 we have f (ω) ∈ conv w- lim sup Fn (ω) = conv w- lim sup Fn (ω) and for every n ≥ 1 , ω −→ n→∞
n→∞
Fn (ω) is Σµ -measurable. So Lemma 6.6.34 implies that ω −→ w- lim sup Fn (ω) is n→∞
Σµ -measurable. Using Corollary 6.4.42, we have
w- lim sup Fn dµ cl conv w- lim sup Fn dµ = cl n→∞ n→∞
Ω
Ω ⇒ conv w- lim sup Fn dµ = cl w- lim sup Fn dµ (see Proposition 6.4.48)
⇒
Ω
f dµ ∈ Ω
n→∞
Ω
n→∞
w- lim sup Fn dµ. Ω
n→∞
REMARK 6.6.37 The closure on the set-valued integral
Ω
w- lim sup Fn dµ cann→∞
not be dropped generally. For a situation where this can be done, we refer to Hu–Papageorgiou [313, p. 697]. PROPOSITION 6.6.38 If (Ω, Σ, µ) is a σ-finite measure space, X is a separable Banach space, and Fn : Ω −→ 2X \ {∅}, n ≥ 1, are graph measurable multifunctions such that Fn (ω) ⊆ W (ω) µ-a.e. with W (ω) ∈ Pwk (X) and sup[u : u ∈ F (ω)] ≤ 1 ϕ(ω) µ-a.e. on Ω with ϕ ∈ L1 (Ω)+ , then w- lim sup SF1 n ⊆ conv Swlim sup Fn . n→∞
n→∞
∞
∗ 1 ∗ (Ω, Xw ∗ ) = L (Ω, X)
PROOF: From Corollary 6.4.18, we know that for all u ∈ L we have
σ(u, Fn )dµ, σ(u, SF1 n ) = Ω
⇒ lim sup σ(u, SF1 n ) = lim sup σ(u, Fn )dµ, n→∞ n→∞ Ω
≤ lim sup σ(u, Fn )dµ (by Fatou’s lemma) n→∞
Ω = σ(u, w- lim sup Fn )dµ (see Proposition 6.6.17) n→∞ Ω
1 = σ u, Swlim sup Fn , (see Lemma 6.6.34 and Corollary 6.4.18) 1 ⇒ w- lim sup SF1 n ⊆ conv Swlim sup Fn
(see Proposition 6.6.18).
n→∞
REMARK 6.6.39 If µ is finite, then the pointwise boundedness condition on |W (ω)| = sup{u : u ∈ W (ω)} can be replaced by the condition that |Fn (·)| n≥1 is uniformly integrable.
6.6 Convergence of Sets
523
We can have analogous Fatou-type results for s- lim inf. PROPOSITION 6.6.40 If (Ω, Σ, µ) is a σ-finite measure space, X is a separable Banach space, and Fn : Ω −→ 2X \ {∅}, n ≥ 1, are graph-measurable multifunctions such that sup d 0, Fn (·) ∈ L1 (Ω), then Ω s-lim inf Fn dµ ⊆ s-lim inf Ω Fn dµ. n→∞
n≥1
n→∞
1 PROOF: Let x ∈ Ω s- lim inf Fn dµ. Then x = Ω f dµ with f ∈ Sslim inf Fn . Let n→∞
Hn (ω) = x ∈ Fn (ω) : f (ω) − x ≤ d f (ω), Fn (ω) + 1/n . Clearly Gr Hn ∈ Σµ × B(X) and so by Theorem 6.3.20, we can find fn : Ω −→ X a Σ-measurable function such that fn (ω) ∈ Hn (ω) µ-a.e. on Ω. We have fn (ω) − f (ω) −→ 0 µa.e. on Ω. Therefore by the dominated convergence theorem we have fn −→ f in L1 (Ω, X) and so xn = Ω fn dµ −→ x = Ω f dµ. Because fn ∈ SF1 n , we conclude that
s- lim inf Fn dµ ⊆ s- lim inf Fn dµ. Ω
n→∞
n→∞
Ω
PROPOSITION 6.6.41 If (Ω, Σ, µ) is a σ-finite measure space, X is a separable X Banach n ≥ 1, are graph-measurable multifunctions, and
space, Fn 1: Ω −→ 2 \{∅}, 1 1 sup d 0, Fn (·) ∈ L (Ω), then Sslim inf Fn ⊆ s- lim inf SFn . n→∞
n≥1
PROOF: Using Theorem 6.4.16, for every u ∈ L1 (Ω, X) and every n ≥ 1, we have
d(u, SF1 n ) = d u(ω), Fn (ω) dµ, Ω
⇒ lim sup d(u, SF1 n ) ≤ lim sup d u(ω), Fn (ω) dµ (by Fatou’s lemma) n→∞ Ω n→∞
≤ d u(ω), s- lim inf Fn (ω) dµ, (6.40) Ω
n→∞
(see Proposition 6.6.21). Note that
s- lim inf Fn (ω) = s- lim inf Fn (ω) = x ∈ X : lim d x, Fn (ω) = 0 , n→∞
n→∞
n→∞
hence ω −→ s- lim inf Fn (ω) is Σµ -measurable. Therefore by Theorem 6.4.16, n→∞
Ω
⇒ ⇒
1 d u(ω), s- lim inf Fn (ω) dµ = d(u, Sslim inf Fn ) n→∞
n→∞
1 lim sup d(u, SF1 n ) ≤ d(u, Ss(see (6.40)) lim inf Fn ) n→∞ 1 1 (see Proposition 6.6.22). Sslim inf Fn ⊆ s- lim inf SFn n→∞ n→∞
REMARK 6.6.42 If µ is finite, then the hypothesis that sup d 0, Fn (·) ∈L1 (Ω), n≥1
can be replaced by the weaker hypothesis that d 0, Fn (·) n≥1 is uniformly integrable.
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COROLLARY 6.6.43 If (Ω, Σ, µ) is a σ-finite measure space, X is a separable Banach space, Fn : Ω −→ Pwkc (X), n ≥ 1, are graph-measurable multifunctions with Fn (ω) ∈ W (ω) µ-a.e. on Ω for all n ≥ 1 where W : Ω −→ Pwkc (X) is integrably M M M bounded and Fn (ω) −→ F (ω) µ–a.e. on Ω, then Ω Fn dµ −→ Ω F dµ and SF1 n −→ 1 SF . COROLLARY 6.6.44 If (Ω, Σ, µ) is a σ-finite measure space, X is a finitedimensional Banach space, Fn : Ω −→ 2X \ {∅}, n ≥ 1, are graph-measurable K
multifunctions with sup |Fn (·)| ∈ L1 (Ω) and Fn (ω) −→ F (ω) µ-a.e. on Ω, then
K
Ω
Fn dµ −→
n≥1
Ω
F dµ.
We conclude with two interesting facts concerning weakly convergent sequences in L1 (Ω, RN ). A sequence {fn }n≥1 ⊆ L1 (Ω) that converges weakly but not strongly in L1 (Ω), oscillates wildly around its weak limit. The next proposition is well known from the theory of Lebesgue spaces (see, e.g., Gasi´ nski–Papageorgiou [259, p. 170]). PROPOSITION 6.6.45 If (Ω, Σ, µ) is a finite measure space, {fn , f }n≥1 ⊆ L1 (Ω), w fn −→ f in L1 (Ω) and one of the following two conditions holds, (i) f (ω) ≤ lim inf fn (ω) µ-a.e. or n→∞
(ii) lim sup fn (ω) ≤ f (ω) µ-a.e., then fn −→ f in L1 (Ω). If R is replaced by RN , then conditions (i) and (ii) above are replaced by an extremality condition. The result is due to Visintin [596]. PROPOSITION 6.6.46 If (Ω, Σ, µ) is a finite measure space and {fn }n≥1 ⊆ w L1 (Ω, RN ) satisfy fn −→ f in L1 (Ω, RN ) with
µ-a.e. on Ω, f (ω) ∈ ext conv lim sup{fn (ω)} n→∞
w
then fn −→ f in L1 (Ω, RN ). Using this proposition we can have the following result for the extreme points of the set-valued integral in RN . For the proof of this result, we refer to Hu– Papageorgiou [313, p. 736]. PROPOSITION 6.6.47 If (Ω, Σ, µ) is a finite nonatomic measure space, F : Ω −→ Pf (RN ) is measurable, e ∈ ext Ω F dµ, {fn }n≥1 ⊆ SF1 is uniformly inte grable, and Ω fn dµ −→ e in RN , then we can find f ∈ SF1 such that e = Ω f dµ and fn −→ f in L1 (Ω, RN ).
6.7 Remarks 6.1: There are hyperspace topologies corresponding to the various kinds of continuity of multifunctions introduced in this section. For details in this direction we refer to the books of Beer [61] and Kuratowski [368] and to the paper of Michael
6.7 Remarks
525
[425]. Continuity properties of multifunctions are studied at various levels of generality in the books of Aliprantis–Border [10], Aubin–Cellina [38], Aubin–Frankowska [40], Berge [67], Castaing–Valadier [134], Denkowski–Mig´ orski–Papageorgiou [194], Hu–Papageorgiou [313], Kisielewicz [355], and Klein–Thompson [357]. 6.2: The study of the measurability properties of multifunctions was initiated with the works of Castaing [133] and Jacobs [330]. The approach of Castaing [133] is topological because the multifunctions are defined on a locally-compact Hausdorff topological space equipped with a Radon measure and are compact valued, whereas Jacobs [330] drops this requirement. In Debreu [186] and Rockafellar [521, 529], the multifunctions are defined on a measure space (Ω, Σ, µ) (no topological structure is assumed) and have values in RN . In Debreu [186] the multifunctions are compact valued and in Rockafellar [521, 529] only closed valued. Further contributions to the theory were made by Valadier [591], Leese [373], Himmelberg [301], Himmelberg– Parthasarathy–Van Vleck [304]. When dealing with compact-valued multifunctions, the following result is helpful (see Debreu [186, p. 355]). PROPOSITION 6.7.1 If X is a separable metrizable space, then the Borel
+σ-field of the hyperspace P (X), denoted by B P (X) is generated by the family U :U ⊆ k k X open where U + = C ∈ Pk (X) : C ⊆ U ; also it is generated by the family {U − : U ⊆ X open}, where U − = C ∈ Pk (X) : C ∩ U = ∅ . 6.3: Theorem 6.3.6 is due to Michael [426]. Michael [425, 426, 427, 428] made additional important contributions in this direction. Proposition 6.3.16 is due to Cellina [135], who was the first to conduct a systematic study of the existence of approximate continuous selectors and their use in differential inclusions. For Pkc (RN )-valued multifunction we can have a Lipschitz selector using the Steiner point map. DEFINITION 6.7.2 The Steiner point map is a map s : Pkc (RN ) −→ RN that has the following properties. (a) s(C) ∈ C for all C ∈ Pkc (RN ). (b) s(C + D) = s(C) + s(D) for all C, D ∈ Pkc (RN ). (c) s(U C) = U s(C) for all C ∈ Pkc (RN ) and all U ∈ O(RN ) where O(RN ) is the group of orthogonal transformations on RN . (d) s(λC) = λs(C) for all λ ∈ R and C ∈ Pkc (RN ). Using the Steiner point map, we have at once the following result on Lipschitz continuous selectors. PROPOSITION 6.7.3 If X is a metric space and F : X −→ Pkc (RN ) is hLipschitz, then F admits a Lipschitz continuous selector. This result cannot be generalized to infinite-dimensional Banach spaces. More precisely, we have the following result due to Yost [615]. PROPOSITION 6.7.4 Let X be a metric space and Y a Banach space. Every h-Lipschitz multifunction F : X −→ Pbf c (Y ) admits a Lipschitz continuous selector if and only if dim Y < +∞.
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6 Multivalued Analysis
Theorem 6.3.17 is due to Kuratowski–Ryll–Nardzewski [367]. Theorem 6.3.20 in the form stated there is due to Saint-Beuve [535]. However, earlier versions with additional restrictions, primarily on X, were proved by Yankov [611], von Neumann [456], and Aumann [46]. Scalarly measurable multifunctions were first considered by Valadier [591]. For the graph measurability of the extremal multifunction ω −→ ext F (ω) (see Proposition 6.3.35), we refer to Benamara [63] (who followed a different approach to prove the result), Castaing–Valadier [134, Chapter IV], and Hu–Papageorgiou [313, Section II.4]. A final useful observation concerning measurable selectors is included in the next proposition (see Hu–Papageorgiou [313, p. 173]). PROPOSITION 6.7.5 If (Ω, Σ, µ) is a separable finite measure space and G : Ω −→ Pf (X), then there exists F : Ω −→ Pf (X) ∪ {∅}-measurable multifunction such that F (ω) ⊆ G(ω) µ-a.e. and every measurable selector of G is also a selector of F .
6.4: The importance of decomposability was evident from the beginning to people working in control theory and optimization; see Boltyanski–Gamkrelidze–Pontryagin [82] and Neustadt [457]. The name convex with respect to switching is also used in the control theory literature. Theorem 6.4.16 is a very valuable tool in applications and illustrates the power of the notion of decomposability. It was first proved by Rockafellar [529]. The form presented here is due to Hiai–Umegaki [293]. Theorem 6.4.23 is also useful in applications and was proved by Papageorgiou [475]. Systematic studies of decomposability can be found in Hiai–Umegaki [293], Hiai [294], Olech [468], and Hu–Papageorgiou [313, Section 2.3]. Decomposability is used to define conditional expectation of multifunctions and define multivalued martingales and their extensions. We refer to the papers of Hiai–Umegaki [293], Papageorgiou [475, 480], Wang–Xue [600], and Wang [599]. Property U (see Definition 6.4.31), was introduced by Bourgain [94]. Proposition 6.4.33 relating the weak norm (see Definition 6.4.29) and the weak topology is due to Gutman [279]. Theorem 6.4.35 was proved by Tolstonogov [584], using the Baire category method of De Blasi–Pianigiani [180, 181, 182]. The set-valued integral given in Definition 6.4.38 is due to Aumann [44]. The topological variant given in Definition 6.4.59 is due to Cornwall [157]. The approach based on the R˚ adstr¨ om embedding theorem (see Theorem 6.4.56) is due to Debreu [186]. For more details about embedding theorems of hyperspaces of convex sets, we refer to Schmidt [544]. Studies of the set-valued integral can be found in Klein– Thompson [357], and Hu–Papageorgiou [313]. 6.5: Theorem 6.5.2 was proved by Nadler [449] for multivalued contractions with nonempty, bounded, closed values. Soon thereafter Covitz–Nadler [165], extended the result to multivalued contractions whose values need not be bounded. The stability results presented in Proposition 6.5.4 and Corollary 6.5.5, are due to Lim [383]. Theorem 6.5.10 was first proved by Knaster–Kuratowski–Mazurkiewicz [358] in the special case where C is the set of vertices of a simplex in RN . Their approach is combinatorial based on Sprener’s lemma. The infinite-dimensional version presented here is due to Fan [232]. The finite-dimensional forerunner of Theorem 6.5.13 is due to Hartman–Stampacchia [286], and the infinite-dimensional result presented
6.7 Remarks
527
here is due to Browder [121]. Weakly inward maps (see Definition 6.5.15(b)), were introduced by Halpern–Bergman [283]. Theorem 6.5.17 is due to Halpern [284]. Theorem 6.5.19 is due to Kakutani [337] when X = RN and due to Fan [232] when X is a locally convex space. Theorem 6.5.20 is due to Browder and [118] and Theorems 6.5.21 and 6.5.22 are due to Bader [49]. Additional fixed point theorems for multifunctions can be found in the books of Andres–Gorniewicz [27], Border, [86], Gorniewicz [272], and Hu–Papageorgiou [313, 316]. 6.6: The first systematic study of the limits in Definition 6.6.3 can be found in Kuratowski [366]. The Mosco convergence of sets was introduced by Mosco [445, 446]. From Section 1.5 we know that they two notions correspond to the Γ-convergence and Mosco-convergence of functions, respectively. The W -convergence (see Definition 6.6.7) and the weak or scalar convergence (see Definition 6.6.9), were both introduced by Wijsman [605] to deal with problems in mathematical statistics. A detailed study together with comparison results accompanied by examples and counterexamples, can be found in Hu–Papageorgiou [313, Chapters I and VII]. For the Hausdorff and Kuratowski convergences, we have the following two compactness results. THEOREM 6.7.6 If (X, d) is a metric space, {An }n≥1 ⊆ Pf (X), and for every n ≥ 1, An ⊆ K ∈ Pk (X), then we can find a subsequence {Ank }k≥1 ⊆ {An }n≥1 and h
A ∈ Pk (X), such that Ank −→ A. REMARK 6.7.7 This result is known as Blaschke’s theorem. THEOREM 6.7.8 If (X, d) is a separable metric space and {An }n≥1 ⊆ 2X \{∅}, K
then there exists a subsequence {Ank }k≥1 of {An }n≥1 and A ∈ 2X , such that Ank −→ A. For multivalued Fatou’s lemmata, we refer to Papageorgiou [477] and Hu– Papageorgiou [313, Section VII.3]. Proposition 6.6.45 was extended to functions in the Lebesgue–Bochner space L1 (Ω, X), by Balder [53] and Rzezuchowski [533].
7 Economic Equilibrium and Optimal Economic Planning
Summary. *This chapter is devoted to mathematical economics. We start with the static model of an exchange economy with a measure space of agents (an abstract device to model perfect competition). We introduce the notions of core allocation and of Walras allocations. We show that in the context of perfect competition, the two sets coincide and they are nonempty (existence result). Then we pass to dynamic models and deal with discrete-time infinite horizon multisector growth models. We deal with both discounted and undiscounted model. For the latter, we introduce the optimality notion of “weak maximality”. We prove existence theorems and we establish weak and strong “turnpike theorems”. Turnpike programs are important because every optimal program eventually moves close to a turnpike one. Moreover, they are easier to compute and are relatively insensitive to the optimality criterion. Subsequently, we conduct an analogous study for models with uncertainty (nonstationary discounted and stationary undiscounted). Then we consider continuous-time models and finally we investigate the “expected utility hypothesis (EUH), the main hypothesis in the theory of decision making.
Introduction Mathematical economics deals with the analytical and mathematical aspects of modern economic theory. The field has made remarkable progress in the last forty years and was also the source for significant developments in nonlinear analysis, such as nonsmooth analysis and multivalued analysis. Today the field has expanded very much and covers a variety of topics that cannot be surveyed in just one chapter. For this reason, here we focus on two particular topics that hold a central position in economic theory. We study the theory of competitive markets (in particular their equilibrium theory) and the theory of growth. At the end of the chapter we examine a notion, which is central in decision making and which brings us to the doorsteps of game theory, which is the topic of the next chapter. In Section 7.1, we discuss a static model of an exchange economy. We assume that perfect competition prevails. This is modelled by a continuum of agents (a nonatomic measure space of agents). For such a model, we introduce two equilibrium concepts. The core allocation (which is price-independent) and the Walras N.S. Papageorgiou, S.Th. Kyritsi-Yiallourou, Handbook of Applied Analysis, Advances in Mechanics and Mathematics 19, DOI 10.1007/b120946_7, © Springer Science+Business Media, LLC 2009
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7 Economic Equilibrium and Optimal Economic Planning
allocation (which is price-competitive). We show that in the assumed environment of perfect competition, the sets of these allocations coincide (core–Walras equivalence theorem). We also prove the existence of a Walras allocation. In Section 7.2 we turn our attention to growth theory and study infinite horizon discrete-time multisector growth models. First we deal with the discounted model. We show that it admits an optimal program (path) and then we characterize it via a system of supporting prices. Then we pass to the undiscounted model. In this case the intertemporal utility need not converge and so we need to have a new optimality concept. We introduce the notion of weak maximality and we prove that the model admits a weakly maximal program. In Section 7.3, we prove weak and strong turnpike theorems. A turnpike program is a special stationary program which in general is easy to compute and optimal programs eventually move close to a turnpike. So in general terms, turnpike theory deals with the asymptotic properties of optimal programs. In Section 7.4 we study growth under uncertainty. We investigate both nonstationary, discounted and stationary, undiscounted models. We obtain existence and characterization results that are analogous to the ones proved for the deterministic case (see Section 7.2). In Section 7.5 we discuss a continuous-time, discounted growth model. We prove an existence theorem. Now the choice of the topology on the set of feasible programs is crucial in the analysis of the model. Finally, in Section 7.6, we examine a basic hypothesis in the theory of decision making, namely the Expected Utility Hypothesis (EUH). Using a partial order in the space of measures, we characterize choice behavior that is consistent with EUH. The models considered in this chapter are highly idealized versions of real-life situations and to some readers, may appear to be only of theoretical interest. However, we think this is incorrect. First, there are specific economic situations for which these mathematical models are reasonable and realistic approximations. Second, the postulated optimality of a particular allocation or path (mystical as it may be) can be used to find practical solutions. It can serve as the yardstick against which we measure all other real-life trajectories of the particular economic systems and decide how good our choices actually are.
7.1 Perfectly Competitive Economies: Core and Walras Equilibria A perfectly competitive economy is one in which no individual agent can affect the social outcome of the economic activity by his or her individual decisions. To simplify our model we consider only pure exchange economies in which no production is possible. The prices at which the exchange takes place, are independent of the actions of the agents and are taken by the agents as given. With these prices, the agents can trade any amount of commodities, which of course implies that the supply and demand of each individual agent is negligible compared to the total volume of trade and so cannot influence the price. The mathematical device to express this economic situation is to use a continuum of agents (traders). In particular as we show in the sequel, we assume that the space of agents is a finite nonatomic measure space (e.g., the unit interval with the Lebesgue measure). This way we are able to
7.1 Perfectly Competitive Economies: Core and Walras Equilibria
531
model mathematically the situation in which an individual agent cannot influence the outcome of the collective activity. There are two equilibrium concepts for a pure exchange economy. The first one is a core allocation, whose origins can be traced back to Edgeworth [219], under the name contract curve. To illustrate this notion, consider an economic agent who suggests a feasible allocation to the other agents. Suppose now a group of agents, using their initial endowments, can make each member better off than the proposed allocation. Then we say that this group can block the proposed allocation. A feasible allocation, is a core allocation if no group of agents can block it. The set of core allocations forms the core of the economy. The second equilibrium concept goes back to Walras [598] and refers to the noncooperative allocation of resources via a price system. The idea behind the notion of Walras equilibrium is that when the agents are assumed to know only the price system and their own preferences and initial endowments and are allowed to trade freely among them in a decentralized market, then the result will be allocations that maximize the utilities of the agents (subject to their budget constraints) and equate supply and demand. Comparing the two equilibrium concepts introduced above, we see that in contrast to the Walras (competitive) allocation, a core allocation allows for the possibility of cooperation among agents in the economy. Each Walras allocation belongs to the core and the core is in general larger than the set of Walras allocations. A classical conjecture, which dates back to Edgeworth [219] says that the core shrinks to the set of Walras allocations as the number of economic agents increases. The conjecture was made mathematically precise with the introduction of measure space agents. Then in such a setting Aumann [45] proved that the core coincides with the set of Walras allocations. In this section we prove this equivalence result for a perfectly competitive pure exchange (no production) economy. We start with the description of the mathematical model for a perfectly competitive pure exchange economy. So let (Ω, Σ, µ) be a finite measure space, describing the agents of the economy. More precisely, Ω denotes the set of economic agents, the σ-field Σ represents the system of coalitions that can be formed among the agents, and the finite measure µ measures the size of all the feasible coalitions. The commodity space is RN . So there are N -commodities traded in the market. There is N a consumption correspondence F : Ω −→ 2R \ {∅}. For each agent ω ∈ Ω, let ≺ω denote an irreflexive binary relation defined as F (ω). This is the preference relation of ω and describes his consumption tastes. For two commodity vectors v, v ∈ F (ω) we write v ≺ω v (resp., v ⊀ω v), to express the fact that v is preferred (resp., not preferred) to v by the agent ω ∈ Ω. The preference relation ≺ can be derived from an indifference relation , which is usually assumed to be a reflexive, transitive and complete binary relation. To complete the description of the economy, we are also given a function e ∈ L1 (Ω, RN ) which assigns to each agent ω ∈ Ω, the agent’s initial endowment vector e(ω) ∈ F (ω). So we call e the initial endowment allocation. The economy under consideration is the quadruple (Ω, Σ, µ), F, ≺, e . DEFINITION 7.1.1 Let E = (Ω, Σ, µ), F, ≺, e be an exchange economy as described above. (a) An allocation for the economy E is a function u ∈ L1 (Ω, RN ). We say that the allocation u is feasible if Ω udµ = Ω edµ.
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7 Economic Equilibrium and Optimal Economic Planning
(b) A coalition A ∈ Σ can improve upon the allocation u if there is an allocation w for E such that (i) u(ω) ≺ω w(ω) µ-a.e. on A. (ii) µ(A) > 0 and A wdµ = A edµ. REMARK 7.1.2 In Definition 7.1.1(b), the coalition A ∈ Σ can block allocation u, because it can redistribute the initial endowment among its members in such a way so as to make each agent in the coalition better off (with respect to the allocation u). DEFINITION 7.1.3 Given a pure exchange economy E= (Ω, Σ, µ), F, ≺, e , the core C(E) of the economy E, is the set of all feasible allocations of E, upon which no coalition in Σ can improve. REMARK 7.1.4 The core of an economy is a fundamental concept in mathematical economics, because it provides a precise explanation of competitive behavior. Now let p ∈ RN + \{0} be the price prevailing in the market. Given a commodity vector v ∈ RN , the real number (p, v)RN is the value of the vector v. Clearly two commodity vectors v, v ∈ RN can be exchanged at price p, provided their values are equal; that is, (p, v)RN = (p, v)RN . DEFINITION 7.1.5 Let E = (Ω, Σ, µ), F, ≺, e be a pure exchange economy in which prevails the price system p ∈ RN + \ {0}. (a) The budget set of agent ω ∈ Ω is defined by
b(ω, p) = v ∈ F (ω) : (p, v)RN ≤ p, e(ω) RN and the demand set is defined by d(ω, p) = v ∈ b(ω, p) : there is no y ∈ b(ω, p) such that v ≺ω y (i.e., d(ω, p) consists of those elements in the budget set that are maximal in b(ω, p) with respect to ≺ω ). (b) The allocation-price pair (u, p) ∈ L1 (Ω, RN )×(RN\{0}) is a Walras or competitive equilibrium if and only if (i) u(ω) ∈ d(ω, p) µ-a.e. on Ω; (ii) Ω udµ = Ω edµ (i.e., the allocation u is feasible). We denote by W (E) the set of all Walras equilibriafor the economy E. Next we make precise our hypotheses on the pure exchange E= (Ω, Σ, µ), F, ≺, e . H: (i) (Ω, Σ, µ) is a finite nonatomic measure space. (ii) For all ω ∈ Ω, F (ω) = RN +. (iii)
Ω
edµ ∈ int RN +.
7.1 Perfectly Competitive Economies: Core and Walras Equilibria
533
(iv) For every ω ∈ Ω, ≺ω is irreflexive and transitive. N (v) For every v ∈ RN : v ≺ω y is open. + , the set y ∈ R N (vi) For every v ∈ RN : v ≺ω y is + , the multifunction ω −→ Gv (ω) = y ∈ R graph-measurable. N (vii) If v ∈ RN + and x ∈ R+ \ {0}, then v ≺ω v + x for all ω ∈ Ω.
REMARK 7.1.6 Hypothesis H(i) implies that we are in an economic situation of perfect competition. Hypothesis H(iii) implies that there are resources available to be traded among the agents. Hypotheses H(v), (vi) are continuity and measurability hypotheses on the preference relation and finally hypothesis H(vii) is a monotonicity relation. THEOREM 7.1.7 If E = (Ω, Σ, µ), F, ≺, e is a pure exchange economy satisfying hypotheses H, then C(E) = W (E). PROOF: First we show that W (E) ⊆ C(E). To this end let (u, p) ∈ L1 (Ω, RN ) × (RN \ {0}) be a Walras equilibrium for the economy E. Assume that u ∈ / C(E). Then according to Definition 7.1.3 we can find A ∈ Σ and w ∈ L1 (Ω, RN ) such that A wdµ = A edµ and u(ω) ≺ ω w(ω) µ-a.e. on A. Since (u, p) ∈ W (E), from Definition 7.1.5(b) we have that p, e(ω) RN <
p, w(ω) RN µ-a.e. on A. Hence
p, e(ω) RN dµ < p, w(ω) RN dµ, A A ⇒ p, edµ < p, wdµ ,
A
RN
A
RN
which contradicts the fact that A edµ = A wdµ. So we have proved that W (E) ⊆ C(E). Next let u ∈ C(E). We show that for some price system p ∈ RN + \ {0}, (u, p) ∈ W (E). For this purpose we introduce the multifunction. ω −→ G(ω) = y ∈ RN : v ≺ω y ∪ {e(ω)}.
(7.1)
Evidently G is a graph-measurable multifunction. We claim that
Gdµ − edµ ∩ int(−RN + ) = ∅.
Ω
(7.2)
Ω
Here Ω Gdµ is the set-valued integral introduced in Definition 6.4.38. Note that (7.2) is equivalent
Gdµ − edµ ∩ int(−RN (7.3) + ) = ∅. Ω
Ω
We argue indirectly. So suppose that (7.3) is not true. Then we can find y ∈ int RN + such that
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7 Economic Equilibrium and Optimal Economic Planning
edµ −y ∈ Gdµ,
Ω
Ω 1 ⇒ edµ −y = gdµ with g ∈ SG (see Definition 6.4.38). Ω
(7.4)
Ω
We define C = {ω ∈ Ω : g(ω) = e(ω)}. (7.5) 1 , Evidently A, C ∈ Σ (see hypothesis H(vi)). Because Ω edµ = Ω gdµ and g ∈ SG N from (7.1) we see that µ(A) > 0. Let g : A −→ R+ be the Σ-measurable function defined by 1 g(ω) = g(ω) + y. µ(A) A = {ω ∈ Ω : u(ω) ≺ω g(ω)}
and
Because y ∈ int RN + , from hypothesis H(vi) we have for all ω ∈ A.
g(ω) ≺ω g(ω)
(7.6)
Also from the definition of the multifunction G (see (7.1)), we have for all ω ∈ A.
u(ω) ≺ω g(ω)
(7.7)
From (7.6), (7.7), and the transitivity of the binary relation ≺ω , ω ∈ A, we have for all ω ∈ A.
u(ω) ≺ω g(ω)
(7.8)
Moreover, note that
gdµ = gdµ + y = gdµ− gdµ + y A A
Ω
C = edµ− edµ (see (7.4)) C
Ω = edµ (see (7.5)) A
⇒ g ∈ L1 (Ω, RN ) is a feasible allocation.
(7.9)
Combining (7.8) and (7.9) we reach a contradiction to the fact that u ∈ C(E). So (7.3) and equivalently (7.2) hold. 1 Because G is graph-measurable, e ∈ SG , and the measure µ is nonatomic, we can apply Theorem 6.4.44 and infer that Ω Gdµ − Ω edµ is convex. Because of (7.3), we can use the weak separation theorem and obtain p ∈ RN + \ {0} such that
edµ RN ≤ (p, v)RN for all v ∈ Gdµ, p, Ω
Ω
1 ⇒ p, edµ RN ≤ inf (p, g)RN dµ : g ∈ SG , Ω
Ω
⇒ (p, e)RN dµ ≤ inf (p, x)RN : x ∈ G(ω) dµ (see Theorem 6.4.16). Ω
Ω
(7.10) We claim that from (7.10) it follows that
7.1 Perfectly Competitive Economies: Core and Walras Equilibria
p, e(ω)
≤ inf (p, x)RN : x ∈ G(ω)
RN
µ-a.e. on Ω.
535 (7.11)
Suppose that (7.11) is not true. Then we can find D ∈ Σ with µ(D) > 0 such that
inf (p, x)RN : x ∈ G(ω) < p, e(ω) RN for all ω ∈ D. From Theorem 6.3.24, we know that ω −→ m(ω) = inf (p, x)RN : x ∈ G(ω) is Σµ ∩D-measurable on D. Let ε : D −→ RN + \{0} be a Σ∩D-measurable function such
N that m(ω) + ε(ω) < p, e(ω) RN and consider the multifunction H : D −→ 2R \ {∅} defined by H(ω) = x ∈ G(ω) : (p, x)RN ≤ m(ω) + ε(ω) . Because G is graph-measurable, it follows that GrH ∈ (Σµ ∩ D) × B(RN ). So we can apply Theorem 6.3.20 (the Yankov–von Neumann–Aumann selection theorem) and obtain h : D −→ RN + , Σ ∩ D-measurable such that h(ω) ∈ H(ω) µ-a.e. on D. We have
p, h(ω) < p, e(ω) RN µ-a.e. on D (7.12) and
h(ω) ∈ G(ω)
Define h : Ω −→ RN + by
h(ω) =
h(ω) e(ω)
µ-a.e. on D.
(7.13)
if ω ∈ D if ω ∈ Dc .
1 Evidently h ∈ SG (see (7.13)). Moreover, we have
(p, h)RN dµ = (p, h)RN dµ + (p, e)RN dµ c Ω
D
D < (p, e)RN dµ + (p, e)RN dµ (see (7.12)) Dc
D = (p, e)RN dµ, Ω
which contradicts (7.10). This proves that (7.11) is true. Next we show that
(7.14) p, u(ω) RN = p, e(ω) RN µ-a.e. on Ω.
Because of (7.11), we have that p, e(ω) ≤ p, u(ω) µ-a.e. on Ω. If this inequality is strict for all ω ∈ Ω, µ(E) > 0, then
p, udµ RN = p, udµ + p, udµ RN N R c Ω E E
= (p, u)RN dµ + (p, u)RN dµ c
E
E > (p, e)RN dµ + (p, e)RN dµ E Ec = p, edµ , Ω
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7 Economic Equilibrium and Optimal Economic Planning
a contradiction to the fact that Ω udµ = Ω edµ (recall u ∈ C(E)). So we have proved that (7.14) holds and so u(ω) ∈ b(ω, p) µ-a.e. on Ω. It remains to show that u(ω) ∈ d(ω, p) µ-a.e. on
Ω. By virtue of hypothesis H(iii), we have µ(S) > 0 where S = {ω ∈ Ω : p, e(ω) RN > 0}. Fix ω ∈ S. We can find v0 ∈ RN \ {0} such
+ that (p, v0 )RN < p, e(ω) . Let y ∈ RN + be such that (p, y)RN ≤ p, e(ω) RN . We set yt = tv0 + (1 − t)y for t ∈ (0, 1). Note that yt −→ y as t −→ 0. But because of hypothesis H(v), G(ω)c is closed and so y ∈ / G(ω). This proves that u(ω) ∈ d(ω, p). From this and hypothesis H(vii) it follows that (p, v)RN > 0 for all v ∈ RN + \ {0}, N hence p ∈ int RN we have proved that p ∈ int R and u(ω) ∈ d(ω, p) for + . Therefore +
all ω ∈ S. If ω ∈ S c , then p, e(ω) RN = 0. But because p ∈ int RN + , we deduce that b(ω, p) = {0}. Hence u(ω) = 0 a.e. on S c . Finally we can say that u(ω) ∈ d(ω, p) µa.e. on Ω and p ∈ int RN + , which implies that (u, p) ∈ W (E) (i.e., C(E) = W (E)). Thus far we have introduced the notion of markets with a continuum of traders, we have demonstrated their significance as mathematical models for the intuitive concept of perfect competition, and we proved that under very general conditions, the core of such markets equals the set of all Walras (competitive) equilibria. However, we did not establish the existence of a Walras equilibrium allocation. It may well happen that both sets C(E) and W (E) are empty. In the second part of this section we fill-in this gap by showing that W (E) = ∅ (hence C(E) = ∅ because we always have W (E) ⊆ C(E)). To do this we have to strengthen the hypotheses of our model. We still consider markets with a continuum of agents. So (Ω, Σ, µ) is a nonatomic finite measure space. The nonatomicity of µ expresses the fact that we have perfect competition. We can think of atoms as being oligopolistic agents. The commodity space is RN ; that is, there are N -different commodities traded in the market. Each N trader can trade things in RN + ; that is, for all ω ∈ Ω, F (ω) ∈ R+ . For each trader N (agent) ω ∈ Ω, there is defined on R+ (the set of feasible consumption plans of each trader ω ∈ Ω) a binary relation ω , known as the preference-indifference relation, which is assumed to be reflexive, transitive, and complete. From ω we define the relations ≺ω and ∼ω , called the preference and indifference relations, respectively, as follows. x ≺ω y
if and only if x ω y but not y ω x
(7.15)
x ∼ω y
if and only if x ω y and y ω x.
(7.16)
DEFINITION 7.1.8 We say that the commodity vector x ∈ RN + saturates a trader’s ω ∈ Ω desire, if for all y ∈ RN + , we have y ω x. Also there is an initial endowment e ∈ L1 (Ω, RN + ). Recall that e(ω) presents the commodity vector with which trader ω ∈ Ω comes to the market. So we have an
economy E = (Ω, Σ, µ), RN + , , e . The precise hypotheses on the characteristics of this economy, are the following. H : (i) (Ω, Σ, µ) is a finite nonatomic measure space. (ii) For µ-a.a. ω ∈ Ω, the set F (ω) of all feasible consumption vectors is RN +.
7.1 Perfectly Competitive Economies: Core and Walras Equilibria
537
N (iii) There is an initial endowment e ∈ L1 (Ω, RN + ) such that e(ω) ∈ intR+ µ-a.e. on Ω.
(iv) For µ-a.a. ω ∈ Ω, there is a reflexive, transitive, and complete binary relation ω defined on RN + ; this is the preference–indifference relation for trader ω ∈ Ω and from it we can define a preference relation ≺ω and an indifference relation ∼ω as indicated in (7.15) and (7.16), respectively. (v)
N N N (ω, x, y) ∈ Ω × RN + ×R+ : x ≺ω y ∈ Σ×B(R+ )×B(R+ ).
N (vi) For µ-a.a. ω ∈ Ω and all y ∈ RN + , the sets {x ∈ R+ : y ≺ω x} are open.
(vii) For µ-a.a. ω ∈ Ω, unless the commodity bundle y ∈ RN + saturates the agent’s desire, we have that x − y ∈ int RN + implies y ω x. (viii) There is an allocation v ∈ L1 (Ω, RN + ) such that for µ-a.a. ω ∈ Ω, v(ω) saturates the desire of the trader (ix) For µ-a.a. ω ∈ Ω, the commodity bundle x ∈ RN + can not saturate the desire of the agent unless x − e(ω) ∈ int RN +. REMARK 7.1.9 Hypothesis H (vi) is a continuity in the commodity space for the preference relation ≺ω . In fact it can be shown that this hypothesis together with hypothesis H (iv), implies that there is a continuous utility function x −→ u(ω, x) for each trader ω ∈ Ω (see Debreu [185, p. 56]). Then hypothesis H (v) implies that the utility function (ω, x) −→ u(ω, x) is jointly measurable (a Carath´eodory function). Hypothesis H (vii) is a weak desirability hypothesis. The existence of the special allocation in hypothesis H (viii) is intuitively very acceptable. It says that there is an upper bound on the amount of a commodity that can be profitably used by a trader, no matter what other commodities are or are not available. The fact that v is an allocation (i.e., v ∈ L1 (Ω, RN + )), simply means that the economy as a whole can be saturated, namely the commodity vector Ω vdµ can be distributed among the traders in such a way as to saturate each trader’s desire. Under these hypotheses, we can prove the following existence theorem. THEOREM 7.1.10 If the economy E = (Ω, Σ, µ), RN satisfies hypotheses + , , e H , then W (E) = ∅. PROOF: For each trader ω ∈ Ω and each price vector p ∈ RN + , we define the budget set
b(ω, p) = x ∈ RN + : (p, x)RN ≤ p, e(ω) RN and the preferred set for all y ∈ b(ω, p) . ξ(ω, p) = x ∈ RN + : y ω x N and then set We define V (ω) = x ∈ RN + : v(ω) − x ∈ R+ ϑ(ω, p) = V (ω) ∩ ξ(ω, p).
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7 Economic Equilibrium and Optimal Economic Planning
Claim 1: For µ-a.a. ω ∈ Ω, p −→ ϑ(ω, p) is continuous on RN \{0}. N Fix ω ∈ Ω\N, µ(N ) = 0. Then for every p ∈ RN + , ϑ(ω, p) ⊆ V (ω) ∈ Pk (R ). So in order to prove that p −→ ϑ(ω, p) is usc, itsuffices to show that it has a closed graph (see Proposition 6.1.10). To this end let (pn , xn ) n≥1 ⊆ Gr ϑ(ω, ·) and assume that
/ ϑ(ω, p). pn −→ p = 0, xn −→ x in RN + . Evidently x ∈ V (ω). Suppose that x ∈ Then x ∈ / ξ(ω, p) and this means that we can find y ∈ b(ω, p) such that x ≺ω y. By virtue of hypothesis H (vi) we can find z ∈ RN + close to y such that
x ≺ω z and (p, z)RN < p, e(ω) RN
(recall that because of hypothesis H (iii) p, e(ω) RN > 0). Therefore we can find n0 ≥ 1 large such that
xn ≺ω z and (pn , z)RN ≤ pn , e(ω) RN for all n ≥ n0 , / ξ(ω, pn ) ⇒ xn ∈
for all n ≥ n0 , a contradiction.
This proves that Gr ϑ(ω, ·) is closed and so p −→ ϑ(ω, p) is usc. Next we show that p −→ ϑ(ω, p) is lsc. Assume that pn −→ p = 0 and let x ∈ ϑ(ω, p). If x saturates the desire of trader ω ∈ Ω, then x ∈ ξ(ω, pn ) for all n ≥ 1 and so if xn = x for all n ≥ 1 we have xn −→ x with xn ∈ ξ(ω, pn ) n ≥ 1. So we may assume that x does not saturate the desire of trader ω ∈ Ω. Because ϑ(ω, pn ) ∈ Pk (RN + ) we can find xn ∈ ϑ(ω, pn ) such that
d x, ϑ(ω, pn ) = x − xn . Let ε > 0 and set sε = (ε, . . . , ε) ∈ int RN + , yε = x + sε . Then by hypothesis H (vii), we have x ≺ ω yε .
Then either for all ε > 0 and all n ≥ 1 large, yε ∈ ϑ(ω, pn ) or else for some ε > 0, we can find a subsequence {nk }k≥1 of {n} such that yε ∈ / ϑ(ω, pn ). In the first case, we have √ xn − x ≤ yε − x ≤ ε N . Because ε > 0 was arbitrary, it follows that xn −→ x and because xn ∈ ϑ(ω, pn ), n ≥ 1, we are done. In the second case, we can find znk ∈ ϑ(ω, pn ) such that yε ≺ω znk . Because N N {z }k≥1⊆V (pn k , znk )RN ≤ n k
(ω)∈Pk (R+ ), we may assume that znk−→z in R . Since
pnk , e(ω) RN , in the limit as k −→ ∞ we obtain (p, z)RN ≤ p, e(ω) RN and so z ∈ b(ω, p). Also by virtue of hypothesis H (vi), we have yε ωz and because x ≺ω yε , hypothesis H (iv) implies that x ≺ω z, a contradiction to the fact that x ∈ ϑ(ω, p). This proves that p −→ ϑ(ω, p) is lsc. So we conclude that p −→ ϑ(ω, p) is continuous (and h-continuous; see Corollary 6.1.40). Claim 2: For all p ∈ RN + \{0}, ω −→ ξ(ω, p) and ω −→ ϑ(ω, p) are graph measurable multifunctions. Clearly the budget multifunction ω −→ b(ω, p) is Pf (RN )-valued and measurable. So by virtue of Theorem 6.3.18, we can find hn : Ω −→ RN + , n ≥ 1 Σ–measurable maps such that b(ω, p) = {hn (ω)}n≥1 for all ω ∈ Ω.
7.1 Perfectly Competitive Economies: Core and Walras Equilibria
539
Invoking hypothesis H (vi) we see that # ξ(ω, p) = x ∈ RN : hn (ω) ω x n≥1
⇒ ω −→ ξ(ω, p) is graph-measurable (see hypothesis H (v)), ⇒ ω −→ ϑ(ω, p) = V (ω) ∩ ξ(ω, p) is graph-measurable. Claim 3: p −→
Ω
ϑ(ω, p)dµ is Pkc (RN ) follows from Theorem 6.4.45 and 6.4.23.
Note that
h
ϑ(ω, p)dµ, Ω
ϑ(ω, p )dµ
Ω
≤ h ϑ(ω, p), ϑ(ω, p ) dµ for all p, p ∈ RN + \ {0}, Ω
⇒ p −→ ϑ(ω, p)dµ is continuous by virtue of Claim 1. Ω
Using Claim 3 we can consider the metric projection on the set denoted by proj ·; Ω ϑ(ω, p)dµ . Set d(p) = proj Ω edµ; Ω ϑ(ω, p)dµ .
Ω
ϑ(ω, p)dµ
Claim 4: p −→ d(p) is continuous.
Suppose that pn −→ p and let d(pn ) = proj Ω edµ; Ω ϑ(ω, pn )dµ , n ≥ 1. By virtue of Proposition 6.1.13 and Claim 3, {d(p n }n≥1 is relatively compact. So we can find a subsequence d(pnk ) k≥1 of d(pn ) n≥1 such that d(pnk ) −→ w in RN + . We have
Ω
edµ − d(pnk ) = d edµ, ϑ(ω, pnk )dµ . Ω
From Claim 3 and Proposition 6.6.32, we have
edµ, ϑ(ω, pnk )dµ −→ d edµ, ϑ(ω, p)dµ d Ω Ω Ω
Ω
and edµ − d(pnk ) −→ edµ − w . Ω
(7.17)
Ω
(7.18) (7.19)
Ω
Combining (7.17) through (7.19) we obtain
edµ − w = d edµ, ϑ(ω, p)dµ , Ω Ω
Ω ⇒ w = proj edµ; ϑ(ω, p)dµ = d(p). Ω
Ω
Then by Urysohn’s criterion for convergence of sequences we have d(pn ) −→ w = d(p) and so we conclude that p −→ d(p) is continuous. N Claim 5: For every p ∈ RN + \{0}, d(p)− Ω edµ ∈ R+ . Suppose that the claim is not true. Then without any loss of generality we may as
N 1 N sume that d1 (p) < Ω edµ (here d(p) = dk (p) k=1 ∈ RN and e = (ek )N k=1 ∈ L (Ω, R+ )). 1 By definition d(p) = Ω udµ with u ∈ Sϑ(·,p) . Let y = (v1 , u2 , . . . , uN ), where
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7 Economic Equilibrium and Optimal Economic Planning
N N v = (vk )N k=1 (see hypothesis H (viii)) and u = (uk )k=1 . Then y(ω) − u(ω) ∈ R+ and v(ω)−y(ω) ∈ RN for all ω ∈ Ω and so it follows that y(ω) ∈ ϑ(ω, p) µ-a.e. on Ω. + Hence
y1 dµ, d2 (p), . . . , dN (p) = ydµ ∈ ϑ(ω, p)dµ. Ω Ω Ω Recall that d1 (p) < Ω e1 dµ and by hypothesis H (ix), we have Ω e1 dµ < Ω v1 dµ. So we can find λ ∈ (0, 1) such that Ω e1 dµ = λ Ω v1 dµ + (1 − λ)d1 (p). Set w = λy + (1 − λ)u and z = Ω wdµ. Then z ∈ Ω ϑ(ω, p)dµ (see Claim 3) and z = Ω e1 dµ, d2 (p), . . . , dN (p) . We have
z −
N 2 2
edµ = ek dµ dk (p) −
Ω
Ω
k=2
(y, γ)RN + (y, γ)RN − γ2 . If (y, γ)RN − γ2 ≥ 0, then y2 > (y, γ)RN ≥ γ2 , which contradicts (7.21). Therefore (y, γ)RN < γ2 . (7.22) 1 We know that y = Ω udµ− Ω edµ with u ∈ Sξ(·,p) . If u(ω, x) is the utility function corresponding to ω −→ ω (see hypothesis H (iv)
and Remark 7.1.9). Then u ω, u(ω) on Ω. ∈ ϑ(ω, p) and u ω, u(ω) ≤ v(ω) µ-a.e. Set w(ω) = u ω, u(ω) . We have Ω wdµ ∈ Ω ϑ(ω, p)dµ and Ω wdµ − Ω edµ ≤ y N (i.e., y − Ω wdµ + Ω edµ ∈ RN + ). Because γ ∈ R+ (see Claim 3), it follows that
wdµ − edµ, γ RN ≤ (y, γ)RN ,
Ω
Ω
⇒ wdµ − edµ, γ RN < γ2 (see (7.22)). (7.23) Ω
Ω
7.1 Perfectly Competitive Economies: Core and Walras Equilibria But because
Ω
wdµ −
Ω
edµ ∈
Ω
ϑ(ω, p)dµ −
wdµ −
Ω
edµ, γ
Ω
RN
Ω
541
edµ, from (7.20) we have
≥ γ2 .
(7.24)
Comparing (7.23) and (7.24), we reach a contradiction. This proves the claim. N N pk = Next let P be the standard price simplex; that is, P = p = (pk )N k=1 ∈ R+ :
1 . Let ϕ : P −→ P be defined by
k=1
p + η(p) .
1
ϕ(p) = 1+
N
ηk (p)
k=1
By virtue of Claim 6, ϕ is continuous and by Brouwer’s fixed point theorem (see Theorem 3.5.3), we can find q ∈ P such that ϕ(q) = q. Then 1+
N
ηk (q) q = q + η(q)
k=1
⇒ η(q) = µq
with µ =
N
η(q) ≥ 0
(see Claim 6).
(7.25)
k=1
We show that η(q) = 0. Suppose that this is not true. From the definition of η and the convexity of Ω ξ(ω, p)dµ, we see that the hyperplane through η(p) + Ω edµ which is perpendicular to η(p) supports the closed, convex set Ω ξ(ω, p)dµ. So if p = q, we have
edµ, η(q) RN ≥ η(q)2 for all y ∈ ξ(ω, p)dµ. y− Ω
Ω
Because η(q) = 0, from (7.25) we have γ > 0 and
edµ, η(q) RN ≥ µ2 q2 > 0 for all y ∈ ξ(ω, p)dµ. y− Ω
(7.26)
Ω
Using hypothesis H (v) and the Yankov–von Neumann–Aumann selection theorem (see Theorem 6.3.20), we can find u ∈ L1 (Ω, RN + ) such that u(ω) ∈ b(ω, q) µ-a.e. and is ≺ω -maximal for µ-a.e. ω ∈ Ω. We have
u(ω) − e(ω), q RN ≤ 0 µ-a.e. on Ω (7.27) u(ω) ∈ ξ(ω, p)
and
µ-a.e. on Ω.
(7.28)
Integrating (7.27) and (7.28) we obtain
udµ − edµ, q RN ≤ 0 with udµ ∈ ξ(ω, p)dµ. Ω
Ω
Ω
(7.29)
Ω
Comparing (7.26) and (7.29), we reach a contradiction. Therefore η(q) = 0. 1 From this we infer that there exists s ∈ Sξ(·,q) such that
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7 Economic Equilibrium and Optimal Economic Planning
sdµ = edµ. Ω
(7.30)
Ω
We claim that (s, q) is a Walras equilibrium (see Definition 7.1.5(b)). To estab1 lish this, it suffices to show s ∈ Sb(·,q) . From hypotheses H (viii) and (ix), we see that
q, e(ω) RN ≤ q, s(ω) RN µ-a.e. on Ω. If this inequality is strict on a set of positive µ-measure, we contradict (7.30). Therefore
q, s(ω) RN = q, e(ω) RN µ-a.e. on Ω, ⇒ s(ω) ∈ b(ω, q)
µ-a.e. on Ω.
REMARK 7.1.11 Walras equilibria are significant only in a perfect competition situation; that is, the market consists of a large number of individually insignificant traders. Theorem 7.1.10, illustrates that in a perfect competition situation, convex preferences are not needed to prove existence.
7.2 Infinite Horizon Multisector Growth Models In this section we turn our attention to dynamic economic models and we examine optimal growth models in discrete time. An important feature of such models is the infinite (unbounded) planning horizon. This results from purely economic considerations. For example, one of the main problems in dynamic economic planning is that of determining what amounts of resources should be allocated to the production of consumption goods and what amounts should be allocated to the production of capital equipment that will be used in the production of future consumption goods. The only rational way to address this question is by considering an infinite time horizon. It is true that we usually hear about five or ten year plans, which set targets for the level of capital accumulation. But then it is natural to ask about the criteria that led to these target levels. The only reasonable answer to this question can be some sort of trade-off between the sacrifices required to accumulate the capital during the planning period and the benefits that will result from the future use of the accumulated capital. In other words, a finite planning horizon requires some method of evaluating end-of-period capital stocks or assets and the only proper evaluation is their value in use in the subsequent future. But, if this is so, then the planning decisions have actually been based on considerations about times well beyond the period of the plan and this will be true independent of whether the planning period is short or long. Therefore the proper way to model this mathematically is through an infinite horizon. So the open endedness of the future in the model is important, because it expresses the fundamental economic fact that the consequences of investment are long-lived. If the criterion to be maximized (the intertemporal utility) is discounted at every step, then an optimal program can be obtained easily using the direct method of calculus of variations. For such a situation the interesting question is to
7.2 Infinite Horizon Multisector Growth Models
543
characterize the optimal programs (paths) using decentralizable conditions. If the discount factor δ equals 1 (i.e., we have an undiscounted instantaneous utility), then the intertemporal utility integral need not converge and we face some serious mathematical difficulties. For this reason we introduce some weaker notions of optimality, which we discuss in the second part of this section. In both the discounted and undiscounted cases, the growth model is a multisector model meaning that there is more than one commodity in the model. We start the discussion with the discounted model. In this case the model is N N described by a triplet (G, u, δ), where G ⊆ RN + ×R+ is the technology set, u : R+ −→ R is the utility function, and δ ∈ (0, 1) is the discount factor. So in our model the commodity space is RN , in other words there are N commodities. In the commodity N N space RN we use the l1 -norm x = |xk | for all x = (xk )N k=1 . Also for x, y ∈ R k=1
we use the following notation. y≤x y <x y!x
if and only if yk ≤ xk for all k ∈ {1, . . . , N }, if and only if y ≤ x and y = x, if and only if yk < xk for all k ∈ {1, . . . , N },
(i.e., x − y ∈ intRN + ). A pair (x, y) ∈ G represents a technologically feasible production pair, where x stands for the initial stock of inputs and y stands for the final output, which can be produced using as inputs the vector x. The hypotheses on the model (G, u, δ) are the following. N H 1 : G ⊆ RN + ×R+ is a nonempty closed set such that
(i) There exists M > 0 such that if (x, y) ∈ G and x ≥ M , then y ≤ x. (ii) If (x, y) ∈ G and x ≤ x , 0 ≤ y ≤ y, then (x , y ) ∈ G. (iii) There exists (x, y) ∈ G such that x ! y. REMARK 7.2.1 Note that we do not assume that the technology set is convex. Hypothesis H1 (i) in economic terms says that if the input exists in sufficiently large quantities, then we have losses due to depreciation. From a mathematical viewpoint, this hypothesis guarantees the boundedness of the set of feasible programs (paths). Hypothesis H1 (iv) is the free disposability hypothesis, very common in economic models. It says that we are always open to accept a supplement of goods, even if we do not have to do anything with them. Finally hypothesis H1 (iii) means that there is a feasible production pair in which the output is strictly bigger than the input. H2 : u : R N + −→ R is continuous. REMARK 7.2.2 Note that no concavity assumption is made on u. H3 : 0 < δ < 1.
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DEFINITION 7.2.3 A program (path) starting at y ∈ RN + , is a sequence {(xn , yn )}n≥0 such that y0 = y, 0 ≤ xn ≤ yn , and (xn , yn+1 ) ∈ G for all n ≥ 1. Associated with such a program {(xn , yn )}n≥0 is a consumption sequence {cn }n≥0 defined by cn = yn − xn ≥ 0 for all n ≥ 0. The utility function u at each stage is evaluated at the consumption level (i.e., we consider u(cn ), n ≥ 0) and it is discounted by the factor δ ∈ (0, 1). So the intertemporal utility generated by a consumption sequence {cn }n≥0 , is given by n δ u(cn ). (7.31) n≥0
LEMMA 7.2.4 If hypotheses H1 hold and (x, y) ∈ G, then (a) x ≤ M implies y ≤ M ; (b) y ≤ max{x, M }. PROOF: (a) We argue indirectly. So suppose that x ≤ M and y > M . Set x = x + (M − x)(e/N ) with e = (1, . . . , 1) ∈ int RN + . Then x ≥ x and so by hypothesis H1 (ii) (the free disposability hypothesis), we have (x , y) ∈ G. Note that x = M and so by virtue of hypothesis H1 (i) we have that y ≤ M , a contradiction. (b) If x ≤ M , then from part (a) we have y ≤ M . This combined with hypothesis H1 (iii) implies that y ≤ max{x, M }. LEMMA 7.2.5 If hypotheses H1 hold and {(xn , yn )}n≥0 is a program starting at y ∈ RN + , then xn , yn , cn ≤ M1 = max{y, M } for all n ≥ 0. PROOF: From Definition 7.2.3 we have that 0 ≤ cn = yn − xn ≤ yn
for all n ≥ 0.
Therefore it suffices to show that yn ≤ M1 for all n ≥ 0. Note that y0 = y and so y0 ≤ M1 . We proceed by contradiction. So suppose that yn ≤ M1 for some n ≥ 1. Then by virtue of Lemma 7.2.4(b) and because (xn , yn+1 ) ∈ G (see Definition 7.2.3), we have xn+1 ≤ yn+1 ≤ max{xn , M } ≤ M1
(by the induction hypothesis).
This implies that for every program {(xn , yn )}n≥0 emanating from y ∈ RN + , the intertemporal utility given in (7.31) is absolutely summable. So we can make the following definition. DEFINITION 7.2.6 A program {(x∗n , yn∗ )}n≥0 starting at y ∈ RN + is said to be optimal if n ∗ n δ u(cn ) ≤ δ u(cn ) n≥0
n≥0
for every other program {(xn , yn )}n≥0 starting from y. Here cn = yn − xn , c∗n = yn∗ −x∗n ≥ 0 for all n ≥ 0.
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545
THEOREM 7.2.7 If hypotheses H1 , H2 , and H3 hold and y ∈ RN + , then there exists an optimal program starting at y. PROOF: Let F (y) be the set of all consumption sequences generated by programs ∞ N that start at y ∈ RN + . Note that F (y) ⊆ l (R )+ and from Lemma 7.2.4(b) we know that if (cn )n≥0 ⊆ F (y), then cn ≤ M1 for all n ≥ 0. Also because G is closed it is easy to see that F (y) is w∗ -closed in l∞ (RN ). Therefore by Alaoglu’s theorem F (y) N is w∗ -compact in l∞ (R n ). Similarly the intertemporal utility∗function U : F (y) −→ R defined by U (c) = δ u(cn ) for all c = (cn )n≥0 ∈ F (y) is w -continuous. Hence by n≥0
the Weierstrass theorem we can find c∗ ∈ F (y) such that U (c∗ ) realizes the maximum of U on F (y). Then the program {(x∗n , yn∗ )}n≥0 which generates the consumption sequence {c∗n }n≥0 is an optimal program. There is a close connection between price systems and optimal programs, which in some respects strongly resembles the relation existing between price systems and optimal programs in finite horizon problems. However, as we show there is a fundamental difference, which stems from the infinite horizon nature of the problem. In order to make all these facts precise, we need to introduce some additional concepts. DEFINITION 7.2.8 (a) The value function V : RN + −→ R is defined by n δ u(cn ), V (y) = n≥0
where {cn }n≥0 is the consumption sequence generated by an optimal program {(xn , yn )}n≥0 starting at y ∈ RN + (see Definition 7.2.6). (b) A sequence {(xn , yn , pn )}n≥0 } is a competitive program starting at y ∈ RN + , if {(xn , yn )}n≥0 } is a program starting at y ∈ RN , pn ∈ RN + for all n ≥ 0 and δ n u(c) − (pn , c)RN ≤ δ n u(cn )−(pn , cn )RN
for all c ∈ RN + , all n ≥ 0 (7.32)
and (pn+1 , y)RN −(pn , x)RN ≤ (pn+1 , yn+1 )RN −(pn , xn )RN
(7.33)
for all (x, y) ∈ G and all n ≥ 0. REMARK 7.2.9 Adding (7.32) and (7.33), we see that if {(xn , yn , pn )}n≥0 is a N competitive program starting at y ∈ RN + ; then for all (x, y) ∈ G, all c ∈ R+ , and all n ≥ 0 we have δ n u(c) + (pn+1 , y)RN − (pn , x + c)RN ≤ δ n u(cn ) + (pn+1 , yn+1 )RN − (pn , yn )RN .
(7.34)
In the definition of a competitive program, inequality (7.32) says that at every time period the consumption plan {cn }n≥0 maximizes the total utility among all other possible consumption vectors. The total utility is defined as δ n u(c) minus the present cost of the consumption (pn , c)RN . In inequality (7.33), we interpret the
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quantity (pn+1 , y)RN −(pn , x)RN as the total profit from operating the technological process (x, y) ∈ G, when the price system {pn }n≥0 prevails in the market. Then inequality (7.33) says that a competitive program from y at every time period maximizes the total profit among all other technologically feasible input-output pairs. Finally inequality (7.34) says that a competitive program from y maximizes the sum of total utility and total profit at every time period in the set of all feasible input-output pairs. DEFINITION 7.2.10 A competitive program {(xn , yn , pn )}n≥0 is said to satisfy the transversality condition if lim (pn , xn )RN = 0. n→∞
REMARK 7.2.11 This condition means that the value of the input asymptotically equals zero. As we show in the sequel this is the condition that is characteristic of infinite horizon problems and which from an economic viewpoint is rather undesirable. But first let us state the theorem on the characterization of optimal programs. To do this we need to modify hypotheses H1 and H2 as follows. N H1 : G ⊆ RN + × R+ is a nonempty, closed, convex set that satisfies hypotheses H1 (i),(ii), and
(iii) (0, 0) ∈ G and if (0, y) ∈ G, then y = 0. (iv) There exists a pair (x, y) ∈ G such that y ∈ int RN . REMARK 7.2.12 Note that now the feasibility set G is convex. This is common hypothesis in economic models and simply means that technological processes can be mixed in arbitrary proportions. The two conditions in hypothesis H1 (iii) are very natural and simply mean that inaction is an option (first condition) and that a nonzero output cannot result from zero input (second condition). Finally hypothesis H1 (iv) means that there is an input x ∈ RN + that produces a strictly positive quantity of each commodity. This condition is a weaker version of hypothesis H1 (iii). Moreover, the input x ∈ RN + is often called sufficient. H2 : u : RN + −→ R is continuous, concave, and strictly increasing in the sense that if c ! c in RN + , then u(c ) < u(c). The next theorem characterizes optimal programs by means of competitive programs. Because the same result is proved in the more general context of stochastic models (see Theorem 7.4.10), here we simply state the theorem and postpone its proof until Section 7.4. THEOREM 7.2.13 If hypotheses H1 , H2 , H3 hold and {(xn , yn )}n≥0 is an optimal N program starting at y ∈ RN + , then there exists a price sequence {pn }n≥0 ⊆ R+ such that (a) The sequence {(xn , yn , pn )}n≥0 is competitive (see Definition 7.2.8(b)). (b) δ n V (y)−(pn , y)RN ≤ δ n V (yn )−(pn , yn )RN for any y ∈ RN + and n ≥ 0. (c) lim (pn , xn )RN = 0 (i.e., the transversality condition holds). n→∞
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547
REMARK 7.2.14 According to (a) the optimal program {(xn , yn )}n≥0 satisfies certain support properties for the technology and utility explained in Remark 7.2.9 (see (7.32) and (7.33)). Condition (b) implies that at every stage we have minimization of the cost among all programs producing no less future value. Finally condition (c) is the transversality condition, which distinguishes infinite horizon from finite horizon problems. Conditions (b) and (c) are not independent, as the next proposition shows. PROPOSITION 7.2.15 If hypotheses H1 , H2 , H3 hold and {(xn , yn , pn )}n≥0 is a competitive program starting at y ∈ RN + , then (b) and (c) in Theorem 7.2.13 are equivalent. PROOF: (b)⇒(c): In statement (b) let y = 0. Then we have
0 ≤ (pn , yn )RN ≤ δ n V (yn ) − V (0) .
(7.35)
By hypothesis H2 , u is continuous. This combined with Lemma 7.2.5 implies that {V (yn )}n≥1 ⊆ R is bounded. Moreover, from Theorem 7.2.7, we have that V is defined on all of RN + . So from (7.35) we deduce that lim (pn , xn )RN = 0 (recall that n→∞
0 < δ < 1; see hypothesis H3 ). (c)⇒(b): Choose any k ≥ 1 and y ∈ RN + . Let {(xn , yn )}n≥0 be a program starting at y. Because by hypothesis {(xn , yn , pn )}n≥1 is a competitive program starting at y, from (7.34) we have for all n ≥ k
δ n u(cn−k )−u(cn ) ≤ (pn+1 , yn+1 )RN − (pn , yn )RN − (pn+1 , yn−k+1 )RN −(pn , yn−k )RN .
(7.36) Summing up (7.36) from n = k to n = k + k1 , k1 ≥ 1, we obtain
k+k1
δ n u(cn−k )−u(cn )
n=k
≤ (pk+k1 +1 , yk+k1 +1 )RN −(pk , yk )RN + (pk , y)RN . From this we obtain δk
k1
δ n u(cn )−
n=0
k+k1
δ n−k u(cn )
n=k
≤ (pk+k1 +1 , yk+k1 +1 )RN −(pk , yk )RN + (pk , y)RN .
(7.37)
Note that as k1 → +∞, the two sums in the left-hand side of (7.37) converge. So using the transversality condition (statement (c)), we obtain n−k δ n u(cn ) − δ u(cn ) ≤ (pk , y)RN −(pk , yk )RN . (7.38) δk n≥0
n≥k
But from the principle of optimality, {(xn+k , yn+k )}n≥0 is an optimal program starting at y ∈ RN + and
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7 Economic Equilibrium and Optimal Economic Planning n−k n δ u(cn ) = δ u(cn+k ) = V (yk ). n≥k
(7.39)
n≥0
Using (7.39) in (7.38) and because {(xn , yn )}n≥0 is any program starting at y ∈ RN + , we have
δ k V (y) − V (yk ) ≤ (pk , y)RN −(pk , yk )RN . Now we show that a competitive program that satisfies the transversality condition is in fact optimal. THEOREM 7.2.16 If hypotheses H1 , H2 , H3 hold and {(xn , yn , pn )}n≥0 is a competitive program starting at y ∈ RN + that satisfies the transversality condition, then {(xn , yn )}n≥0 is an optimal program starting at y ∈ RN +. PROOF: From (7.34) we have k
k
δ n u(cn )−u(cn ) ≤ )RN −(pn , yn − yn )RN (pn+1 , yn+1 − yn+1
n=0
n=0
=
k
(pn+1 , xn+1 + cn+1 −xn+1 −cn+1 )RN
n=0
−(pn , xn + cn −xn −cn )RN
≤ (pk+1 , xk+1 )RN + (pk+1 , ck+1 − ck+1 )RN .
(7.40)
But from (7.32) we know that
(pk+1 , ck+1 −ck+1 )RN ≤ δ k u(ck+1 )−u(ck+1 ) .
(7.41)
Using (7.41) in (7.40), we obtain k+1
δ n u(cn )−u(cn ) ≤ (pk+1 , xk+1 )RN .
(7.42)
n=0
Passing to the limit as k−→+∞ in (7.42) and using the transversality condition, we conclude that {(xn , yn )}n≥0 is an optimal program starting at y ∈ RN +. As we already mentioned in Remark 7.2.11, the transversality condition is rather problematic from an economic viewpoint. Indeed, being asymptotic in nature, it is not a myopic behavior rule for an individual decision maker and for this reason it is unclear how this condition can be verified in a decentralized setting. So, we want to have a reformulation of Theorem 7.2.16 with the transversality hypothesis removed. For this purpose, we introduce the reachability condition. DEFINITION 7.2.17 We say that the discounted multisector growth model (G, u, δ) satisfies the reachability condition (R), if given any y ∈ RN + and any program {(xn , yn )}n≥0 starting at y ∈ RN + , there is an integer k0 ≥ 1 and a program {(xn , yn )}n≥0 starting at y such that yk 0 ≥ yk0 .
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549
REMARK 7.2.18 In essence the reachability condition (R) may be rephrased as follows. The technological production possibilities are such that, beginning with a capital stock from which expansion of stocks are feasible (hypothesis H1 (iii)), it is possible (if need be through pure accumulation of capital over a sufficiently long period), to attain the stocks along any feasible program, at some future date. THEOREM 7.2.19 If hypotheses H1 , H2 , H3 hold, the reachability condition (R) is satisfied and {(xn , yn , pn )}n≥0 is a competitive program starting at any y ∈ RN +, then {(xn , yn )}n≥0 is an optimal program starting at y ∈ RN +. PROOF: Let c = 0 and apply (7.34) to (x, y) ∈ G, c = 0. We obtain δ n u(c) + (pn+1 , y)RN −(pn , x + c)RN = δ n u(0) + (pn+1 , y)RN −(pn , x)RN = δ n u(0) + (pn+1 , y − x)RN + (pn+1 − pn , x)RN ≤ δ n u(cn ) + (pn+1 , yn+1 )RN −(pn , yn )RN .
(7.43)
For k ≥ 2, we have k−1
δ n u(0) +
n=0
k−2
(pn+1 , y − x)RN + (pk , y)RN − (p0 , x)RN .
(7.44)
n=0
Consider the sequence {(xn , yn )}n≥0 defined by xn = xn+k , yn = yn+k
for all n ≥ 0.
{(xn , yn )}n≥0
is a program starting at yk . Invoking the reachability conEvidently dition (R) (see Definition 7.2.17), we can find a program {(xn , yn )}n≥0 starting at y and k0 ≥ 0 such that yk 0 ≥ yk0 = yk0 +k . (7.45) ) ∈ G cn = yn − xn , we have for all n ≥ 0. Applying (7.34) to (xn , yn+1 )RN − (pk+n , yn )RN δ k+n u(cn ) + (pk+n+1 , yn+1
≤ δ k+n u(ck+n ) + (pk+n+1 , yn+k+1 )RN − (pk+n , yn+k )RN .
(7.46)
We adopt the convention that if k0 = 0, the summation from 0 to k0 − 1, equals zero. Summing up (7.46) from n = 0 to n = k0 − 1, we have k0 −1
δ k+n u(cn ) + (pk+k0 , yk 0 )RN − (pk , y0 )RN
n=0 k0 −1
≤
δ k+n u(ck+n ) + (pk+k0 , yk+k0 )RN − (pk , yk )RN .
(7.47)
n=0
Recall that y0 = y, pn ≥ 0 for all n ≥ 0, and use (7.35). Then (7.47) yields k0 −1
δ k+n u(cn )−(pk , y)RN
n=0 k0 −1
≤
n=0
δ k+n u(ck+n )−(pk , yk )RN .
(7.48)
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From (7.44) and (7.48), for every k ≥ 2 we have k0 −1
k−1
δ n u(0) +
n=0
δ k+n u(cn ) +
n=0
k−2
(pn+1 , y − x)RN − (p0 , x)RN
n=0
k+k0 −1
≤
δ n u(cn ) − (p0 , y0 )RN .
(7.49)
n=0
From Lemma 7.2.5 we know that 91 = max{y, M } cn ≤ M1 = max{y, M } and cn ≤ M
for all n ≥ 0.
91 } and M3 = max |u(c)| : 0 ≤ c ≤ M2 e where e = Define M2 = max{M1 , M [1, 1, . . . , 1] ∈ RN . Then from (7.49), we have k−2
(pn+1 , y − x)RN
n=0 k+k0 −2
≤
δ n u(cn ) −
n=0
≤
k−1
k0 −1
δ n u(0) −
n=0
δ k+n u(cn ) + (p0 , x − y0 )RN
n=0
3M3 + (p0 , x − y0 )RN < +∞. 1−δ
So for any k ≥ 2
k−2
(pn+1 , y − x)RN < +∞.
(7.50)
n=0
Because y − x % 0 (see hypothesis H1 (iii)) and pn ≥ 0 for all n ≥ 0, from (7.50) we infer that pn < +∞, n≥0
⇒ pn −→ 0
as n → ∞.
Recall that xn ≤ M1 = max{y, M } for all n ≥ 0 (see Lemma 7.2.5). Therefore (pn , xn )RN −→ 0 as n → ∞. Now we can apply Theorem 7.2.16 and conclude that {(xn , yn )}n≥1 is an optimal program starting at y ∈ RN . Next we present an example of a linear model that satisfies the reachability condition (R). EXAMPLE 7.2.20 The dynamic Leontief model: Let yn ∈ RN be a vector whose ith component yni represents the output of the ith good in time period n ≥ 0. Let A = (aij )N i,j=1 be an N × N -matrix, where aij denotes the current input of the ith N good used to produce one unit of the jth good. Then Ay= aij yj is the vector of j=1 N with lk representing the amount inputs (raw materials). Also let l = (lk )N k=1 ∈ R
7.2 Infinite Horizon Multisector Growth Models
551
of labor required to produce one unit of the kth good. As before u : RN −→ R is a continuous function, representing the utility function and 0 < δ < 1 is the discount N factor. Then the technology feasibility set G ⊆ RN + × R+ is defined by: (x, y) ∈ G if and only if x ≥ Ay, x ≥ 0, y ≥ 0
and
(l, y)RN ≤ 1.
(7.51)
We impose the following conditions on the input coefficient matrix. (L1) A ≥ 0 in the sense that aij ≥ 0 for all i, j ∈ {1, . . . , N } and l % 0. (L2) A is productive, meaning that there is y % Ay and (l, y)RN ≤ 1. n From matrix theory n we know that under the above hypothesis A −→ 0 as n → ∞ −1 and (I − A) = A . Now we can verify that this model fits in the framework of n≥0
the previous analysis. Claim 1: If G is defined by (7.51) and A satisfies hypotheses (L1) and (L2), then G satisfies hypotheses H1 . Proof: Clearly we have the free disposability property H1 (ii). Let m = min lk . 1≤k≤N
Because by (L1) l % 0, we have m>0. Set M = 1/m. If (x, y) ∈ G, then (l, y)RN ≤ 1 N lk yk ≤ 1 and so y ≤ 1/m. Therefore we have (see (7.51)). Hence my ≤ k=1
satisfied hypothesis H1 (i). Finally set x = Ay. Then clearly (x, y) ∈ G and x ! y (see hypothesis (L2)). So we have satisfied hypothesis H1 (iii). N Claim 2: If G ⊆ RN + × R+ is defined by (7.51) and (L1) holds, then G satisfies hypotheses H1 if and only if hypothesis (L2) holds.
Proof: Note that if G satisfies H1 (iii), then it satisfies (L2). This combined with Claim 1 produces the desired equivalence. N Claim 3: If G ⊆ RN + × R+ is defined by (7.51), hypotheses (L1), (L2) hold and {(xn , yn )}n≥0 is a program starting at y ∈ RN + , then there is a program {(xn , yn )}n≥0 starting at y and an integer k0 ≥ 1 such that yk0 = yk0 .
Proof: From the proof of Claim 1, we know that yn ≤ M e for all n ≥ 1 with n e = (1, 1, . . . , 1) ∈ RN + . Because y % 0 and A −→ 0 as n → ∞, we can find k0 ≥ 2 such that Ak0 M e ! y. (7.52) We define a new program {(xn , yn )}n≥0 as follows. x0 = y0 = y, xn
= xn ,
yn
xn = Ak0 −n yk0 = yn =
yn
for 1 ≤ n ≤ k0 − 1,
for n ≥ k0 .
(7.53)
We also introduce the corresponding consumption sequence cn = yn − xn xn
yn
for n ≥ 0.
cn
Clearly ≥ 0, ≥ 0 for n ≥ 0, = 0 for 0 ≤ n ≤ k0 − 1, and cn = cn ≥ 0 for n ≥ k0 . Also y0 = y. So in order to have that {(xn , yn )}n≥0 is a program starting at y, we need to verify that Ayn+1 ≤ xn
and
(l, yn )RN
≤1
for all n ≥ 0 for all n ≥ 1.
(7.54) (7.55)
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From (7.53), we see that (7.54) and (7.55) hold for all n ≥ k0 . Because A ≥ 0 (see hypothesis (L1)), we have y1 = Ak0 −1 yk0 ≤ Ak0 −1 M e, ⇒ Ay1 ≤ Ak0 M e ! y = x0
(see (7.52)).
This proves (7.54) when n = 0. Next if 1 ≤ n ≤ k0 − 1, then because of (7.53), we have = AAk0 −(n+1) yk0 = xn . Ayn+1
This proves (7.54) when 1 ≤ n < k0 − 1. Finally, if n = k0 − 1, then Ayn+1 = Ayk 0 = Ayk0 = Ak0 −(k0 −1) yk0 = xk0 −1 = xn .
This proves (7.54) when n = k0 − 1 and so we have verified (7.54) for all n ≥ 0. It remains to verify (7.55) for 1 ≤ n ≤ k0 − 1. Because {(xn , yn )}n≥0 is a program starting at y ∈ RN + , we have Ayn+1 ≤ xn ≤ yn
for all n ≥ 0.
So for any integer i ≥ 1, we have Ai yn+i = Ai−1 yn+i ≤ Ai−1 yn+i−1
for all n ≥ 0.
Continuing this way until zero, we obtain Ai yn+i ≤ yn
for all n ≥ 0 and all i ≥ 1.
(7.56)
So from (7.53) for 1 ≤ n ≤ k0 − 1, after setting i = k0 − n in (7.56), we obtain yn = Ak0 −n yk0 ≤ yn . Because l % 0 (see hypothesis (L1)) and (l, yn )RN ≤ 1 for all n ≥ 1, we have (l, yn )RN ≤ (l, yn )RN ≤ 1
for 1 ≤ n ≤ k0 − 1.
Therefore we have verified (7.55) and we have proved Claim 3. Claim 3 implies that for the Leontief model the reachability condition (R) holds. So for the Leontief growth model Theorem 7.2.19 applies. Next we turn our attention to growth models that are undiscounted (i.e., δ = 1). In this case the sum describing the intertemporal utility along a program, need not converge. So in order to evaluate the performance of a program, we need to use a criterion different from the standard notion of optimality given in Definition 7.2.6. This leads us to the so-called weak maximality criterion. But first let us describe the model. As before the planning horizon is infinite, namely N0 = {0, 1, 2, . . .} (i.e., we deal with a discrete time, infinite horizon growth model). Also the commodity space is RN (there are N commodities in the market). The technological constraints are described by the following mathematical hypotheses.
7.2 Infinite Horizon Multisector Growth Models
553
N H4 : G : RN + −→ Pf c (R+ ) is a multifunction with a closed and convex graph such that
(i) There exists M > 0 such that |G(x)| ≤ sup{y : y ∈ G(x)} ≤ M for all x ∈ RN +. (ii) If x ≤ x , y ≤ y and y ∈ G(x), then y ∈ G(x ). (iii) There exists (x, y) ∈ Gr G such that x ! y. REMARK 7.2.21 Combining hypothesis H4 (i) with the fact that G has a closed graph, we infer that G is usc (see Proposition 6.1.10). Invoking the Kakutani–Ky Fan fixed point theorem (see Theorem 6.5.19), we have that Γ = {x ∈ RN + : x ∈ G(x)} = ∅. Hypothesis H4 (ii) is the free disposability hypothesis (see also hypothesis H1 (ii)). Hypothesis H4 (iii) is the same as hypothesis H1 (iii) and it says that the economy has an expansible commodity vector x ∈ RN +. Also we have a utility function u(x, y) that satisfies the following hypotheses. H5 : u : Gr G −→ R is an upper semicontinuous concave function such that x −→ u(x, y) is nondecreasing and y −→ u(x, y) is nonincreasing. DEFINITION 7.2.22 A feasible program is a sequence x = {xn }n≥0 , xn ∈ RN + for all n ≥ 0 such that xn+1 ∈ G(xn ) for all n ≥ 0. In what follows by F (x0 ) we denote the set of feasible programs starting at x0 ∈ RN +. REMARK 7.2.23 In contrast to the previous multisector growth model, in this one we have supressed the consumption sequence {cn }n≥0 . In the new model a feasible input-output pair (x, y) includes consumption activities as well as production activities. One might object that this way we are obscuring the distinction between the consumption and production activities. Although this may have some minor implications, such as, for example, obscuring the role of consumer satiation, in general it does not create any conceptual problems and it is reasonable from an economic viewpoint. Moreover, it is mathematically convenient. Due to the infinite planning horizon and the absence of a discount factor (in this model δ = 1), we have difficulties in defining an optimal feasible program, because the sum of the intertemporal utility need not converge. As a first attempt to overcome this difficulty, we introduce the following notion. DEFINITION 7.2.24 (a) Let {αn }n≥0 , {bn }n≥0 be two real sequences. We say m (αn − bn ) ≤ 0. that {bn }n≥0 catches up to {αn }n≥0 , if lim sup m→∞ n=0
(b) A feasible program x={xn }n≥0 starting from x0 ∈RN + is said to be strongly maximal (or optimal ) if and only if the sequence {u(xn , xn+1 )}n≥0 catches up to the sequence {u(yn , yn+1 )}n≥0 for any y = (yn )n≥0 ∈ F (x0 ), in other words lim sup m→∞
m
u(yn , yn+1 ) − u(xn , xn+1 )
n=0
for every y = (yn )n≥0 ∈ F (x0 ).
≤0
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7 Economic Equilibrium and Optimal Economic Planning
REMARK 7.2.25 The catching-up criterion introduces an ordering on the real sequences that is reflexive and transitive. In this ordering, two sequences {αn }n≥0 , {bn }n≥0 are equivalent (each of them catching up the other) if and only m m (αn − bn ) = 0. The ordering is strict if lim sup (αn − bn ) < 0 or if lim m→∞ n=0 m
lim sup
(αn − bn ) = 0 and lim inf
m
m→∞ n=0
(αn − bn ) < 0. However, this ordering is
m→∞ n=0
m→∞ n=0
m (αn − bn ) and lim sup (bn − αn ) m→∞ n=0 m→∞ n=0 n may be strictly positive. To see this consider the sequences αn = (−1) n≥0 and bn = 1/(2n+2 ) n≥0 . In this case both lim sup are equal to 12 . For this reason instead of the strong maximality criterion, we prefer a weaker one, which nevertheless induces a total ordering.
not total. It may happen that both lim sup
m
DEFINITION 7.2.26 (a) Let {αn }n≥0 , {bn }n≥0 be two real sequences. We m (bn − an ) > 0 (equivalently say that {bn }n≥0 overtakes {αn }n≥0 if lim inf lim sup
m
m→∞ n=0
(αn −bn ) < 0).
m→∞ n=0
(b) A feasible program x = {xn }n≥0 starting at x0 ∈ RN + is said to be weakly maximal if and only if no element in F (x0 ) overtakes x, in other words lim inf m→∞
m
u(yn , yn+1 ) − u(xn , xn+1 ) ≤ 0
n=0
for every y = (yn )n≥0 ∈ F (x0 ). REMARK 7.2.27 Note that the overtaking criterion is total and another equivalent way to state it is to say that for every ε > 0 and every m0 ≥ 1, we can find m = m(ε, m0 , y) ≥ m0 such that m n=0
u(yn , yn+1 ) − ε ≤
m
u(xn , xn+1 ).
n=0
In a multisector growth model, of special interest are stationary programs, because they permit a balanced growth of the economy. This justifies the next definiN tion. First note that because Γ = Fix G = {x ∈ RN + : x ∈ G(x)} ∈ Pk (R+ ) and u is upper semicontinuous, the optimization problem u = sup[u(x, x) : x ∈ Γ]
(7.57)
has a solution. Moreover, due to hypothesis H4 (ii) (the free disposability) and hypothesis H5 , we can equivalently rewrite (7.57) as follows, u = max[u(x, y) : y ∈ G(x), x ≤ y].
(7.58)
Let S be the solution set of (7.57) (equivalently of (7.58)), (i.e., S = {x ∈ Γ : u = u(x, x)} = ∅).
7.2 Infinite Horizon Multisector Growth Models
555
DEFINITION 7.2.28 A weakly maximal stationary program is a vector x∗ ∈ S such that the constant (stationary) program x ∗ = x∗n = x∗ n≥0 is weakly maximal within F (x∗ ). We are looking for weakly maximal programs. To this end we start by producing a stationary price system that supports the optimization problem (7.58). PROPOSITION 7.2.29 If hypotheses H4 and H5 hold, then there exists p ∈ RN + such that for all (x, y) ∈ GrG we have u(x, y) + (p, y − x)RN ≤ u. PROOF: Let h : RN ×RN −→ RN be the continuous function defined by h(x, y) = y − x. Then we can write (7.58) as follows. u = max{(x, y) : (x, y) ∈ Gr G, h(x, y) ≥ 0}. By hypothesis H4 (iii) h(x, y) % 0. So by the Kuhn–Tucker theorem, we can find p ∈ RN + such that u(x, y) + (p, y − x)RN ≤ u
for all (x, y) ∈ Gr G.
Using this proposition, we can show that there is no feasible program starting + at x0 ∈ B M = {v ∈ RN + : v ≤ M } that is infinitely better than the value u of problem (7.58). PROPOSITION 7.2.30 If hypotheses H4 and H5 hold, then there exists M1 > 0 + such that for any x = (xn )n≥0 , x0 ∈ B M , such that for all m ≥ 1, we have m
u(xn , xn+1 ) − u ≤ M1 .
n=0
PROOF: By virtue of Proposition 7.2.29, we can find p ∈ RN + such that u(x, y) + (p, y − x)RN ≤ u
for all (x, y) ∈ Gr G.
From this, we have u(xn , xn+1 ) − u = (p, xn − xn+1 )RN for all x = (xn )n≥0 ∈ F (x0 ) m
⇒ u(xn , xn+1 ) − u ≤ (p, x0 )RN − (p, xm )RN ≤ p2M = M1 , n=0
(see H4 (i)).
So there is no feasible program that can produce an intertemporal utility infinitely better than the one generated by a stationary program x∗ ∈ S. However, a program can be infinitely worse (e.g., any nonoptimal stationary program). To avoid this situation, we introduce the following definition.
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7 Economic Equilibrium and Optimal Economic Planning
DEFINITION 7.2.31 A feasible program x = {xn }n≥0 is said to be good if there m
u(xn , xn+1 ) − u ≥ γ; in other exists γ ∈ R such that for all m ≥ 1, we have n=0
words lim inf m→∞
m
u(xn , xn+1 ) − u > −∞.
n=0
REMARK 7.2.32 So a feasible program x = (xn )n≥0 is good, if it can sustain a utility level close to u. In the literature a good program is also called eligible (see Takayama [572]). The next proposition links good programs with the set S of solutions of problem (7.58) (equivalently of (7.58)). PROPOSITION 7.2.33 If hypotheses H4 and H5 hold and x = (xn )n≥0 is a n + 1/(n + 1) xk n≥0 good program with x0 ∈ B M , then L = limit points of k=0
is nonempty and a subset of S. n
PROOF: Note that 1/(n + 1)
+
xk ∈ B M for all n ≥ 0. So by the Heine–Borel
k=0
theorem, we have that L = ∅. Let x∗ ∈ L. Then we can find a sequence m −→ nm such that nm 1 xk −→ x∗ as m → ∞. (7.59) nm + 1 k=0
Exploiting the convexity of Gr G (see hypotheses H4 ), we have
nm n m +1 1 1 xk , xk ∈ Gr G. nm + 1 nm + 1 k=0
(7.60)
k=1
Note that n n m +1 m +1 1 1 1 xk = xk + (xnm +1 − x0 ) nm + 1 nm + 1 nm + 1 k=1
k=0
+
and x0 , xnm +1 ∈ B M . Therefore n m +1 1 xk −→ x∗ nm + 1
as m → ∞.
(7.61)
k=1
Because Gr G is closed from (7.59), (7.61) and (7.60), we infer that (x∗ , x∗ ) ∈ Gr G (i.e., x∗ ∈ Γ). Moreover, because of H5 , Jensen’s inequality, and the fact that by hypothesis x = {xn }n≥0 is a good program, we have (see Definition 7.2.31)
7.2 Infinite Horizon Multisector Growth Models
557
nm
1 γ ≤ u(xk , xk+1 ) − u nm + 1 nm + 1 k=0
nm n m +1 1 1 ≤u xk , xk − u. nm + 1 nm + 1 k=0
(7.62)
k=1
Passing to the limit as m −→ ∞ in (7.62), we obtain
u ≤ lim sup u m→∞
≤ u(x∗ , x∗ )
nm n m +1 1 1 xk , xk nm + 1 nm + 1 k=0
k=1
(because u is upper semicontinuous; see H4 ). (7.63)
Because x∗ ∈ Γ, from (7.63) it follows that u = u(x∗ , x∗ ) ⇒ x∗ ∈ S (i.e., L ⊆ S). This proposition suggests that we should look among the elements of S in order to produce a weakly maximal stationary program. THEOREM 7.2.34 If hypotheses H4 and H5 hold, then there exists a weakly maximal stationary program. ∗ ∗ PROOF: Let p ∈ RN + be as in Proposition 7.2.29 and let x ∈ S such that (p, x )RN ≤ (p, y)RN for ally ∈ S. It exists since S is compact. Consider the stationary program x∗ = x∗n = x∗ n≥0 . We claim that this is a weakly stationary program in F (x∗ ). We argue indirectly. So suppose that the claim is not true. Then there exist z ∈ F (x∗ ), ε > 0, and m0 ≥ 1 such that for all m ≥ m0 we have
Um (x∗ ) + ε ≤ Um (z), where Um (x∗ ) =
m
u(x∗n , x∗n+1 ) =
n=0
m
u(x∗ , x∗ ) and Um (z) =
n=0
m
u(zn , zn+1 ) (see
n=0
Remark 7.2.27). Then ε ≤ Um (z) − Um (x∗ )
for all m ≥ m0 .
(7.64)
From (7.64) and Definition 7.2.31 it follows that z is a good program. So by virtue of Proposition 7.2.33, we have (at least for a subsequence), that m 1 zn −→ y ∗ ∈ L ⊆ S m + 1 n=0
as m −→ ∞.
Hence lim
m→∞
m m
1 1 (p, zn )RN = lim p, zn RN ≤ lim sup(p, zm )RN , m→∞ m + 1 n=0 m + 1 n=0 m→∞
⇒ (p, y ∗ )RN ≤ lim sup(p, zm )RN . m→∞
(7.65)
558
7 Economic Equilibrium and Optimal Economic Planning Then from Proposition 7.2.29 and (7.64), we obtain 0 < ε ≤ (p, x∗ )RN − (p, zm+1 )RN
for all m ≥ m0
⇒ 0 < ε ≤ (p, x∗ )RN − lim sup(p, zm+1 )RN ≤ (p, x∗ )RN − (p, y ∗ )RN , m→∞
(see (7.64)), ⇒ (p, y ∗ )RN < (p, x∗ )RN
and
y ∗ ∈ S.
This contradicts the choice of x∗ ∈ S. This proves that x∗ = {x∗n = x∗ }n≥0 is a weakly maximal stationary program. REMARK 7.2.35 The result fails if the weak maximality criterion is replaced by the strong maximality criterion (see Definition 7.2.24(b)).
7.3 Turnpike Theorems In this section we continue the analysis of multisector growth models with an infinite planning horizon. As in the previous section, the model that we consider is stationary, namely the set of technological possibilities and the utility function are constant in time. The stationarity assumption is of course restrictive, but most ren alistic models such as the constant growth models,
that is, Gn = λ G with λ > 0 (the growth rate) and un (x, y) = u (1/λn )x, (1/λn )y , can be transformed into stationary models. In general, any realistic growth model, will involve some kind of stationarity assumption because any probabilistic prediction of the behavior of the system over a long period is almost always based on some kind of stationarity hypothesis. In the second part of Section 7.2, we saw that in stationary models the stationary programs are important (i.e., programs x = {xn }n≥0 , where xn is independent of time, namely xn = x∗ ∈ RN + ). In this section, we isolate among the stationary programs, a special subclass called turnpike programs (or simply turnpikes). A turnpike in a stationary model is a special stationary optimal program, distinguished by at least three properties. (a) An optimal program eventually moves close to the turnpike. (b) A turnpike is easier to compute than an arbitrary optimal program. (c) A turnpike is relatively insensitive to the optimality criterion. In this section we focus on property (a), which leads to the so-called turnpike theorems. These theorems for infinite programs maintain that these programs in a certain sense approach a turnpike. In fact this property motivated the term turnpike. In most books on the subject, the following analogy is given by way of further explanation. Suppose that we want to drive from town A to town B and the main highway (the turnpike) passes close to A and B. The time of an optimal journey is to drive from town A to the turnpike, follow the turnpike until the exit to town B, and then reach town B. Turnpike theorems for finite programs can have weak and strong forms. Weak turnpike theorems assert that for a sufficiently large planning horizon most of the time optimal programs remain close to the turnpike. However, these theorems say nothing about the whereabouts of the planning intervals where the optimal program deviates considerably from the turnpike. In contrast the strong
7.3 Turnpike Theorems
559
turnpike theorems maintain that these distant points are concentrated at the beginning and at the end of the planning interval. So strong turnpike theorems, show that optimal programs of economic growth are analogous to the time of the optimal journey from town A to town B described above. Turnpike theorems are important because they establish the proximity of optimal programs to special programs with simple structure, the turnpikes (maximal stationary programs). Turnpikes are much easier to obtain compared to a general optimal program. As in Section 7.2, the multisector growth model is described by a pair (G, u). N The set G ⊆ RN + × R+ describes the technological possibilities of the economy. A pair (x, y) ∈ G describes a technological process. So the expenditure of the vector x ∈ RN + (the input) at time n−1 (n ≥ 1), yields the production of the (not necessarily unique) output vector y ∈ RN + at time n (n ≥ 1). The function u(x, y) measures the aggregate utility of the technological process (x, y). The hypotheses on the two items of the economy are the following. N H 1 : G ⊆ RN + × R+ ) is a nonempty, compact, and convex set such that
(i) (0, 0) ∈ G. (ii) We can find (x, y) ∈ G such that x ! y. REMARK 7.3.1 As before hypothesis H1 (i) means that inactivity is one option, and hypothesis H1 (ii) means that there is an expansible stock, namely we can find N N N an input x = (xk )N such that k=1 ∈ R+ and a corresponding output y = (yk )k=1 ∈ R xk < yk for all k ∈ {1, . . . , N }. This is a productivity assumption and it is not that restrictive because the vectors x, y ∈ RN + may differ very little from each other and from the zero vector. Finally the convexity of the set G means that it is possible to mix technological processes in arbitrary proportions. H2 : u : G −→ R is a continuous, strictly concave function. REMARK 7.3.2 The concavity of the utility function reflects the nature of the preferences of the society for variety in general (half an apple and half an orange are preferred to either a whole apple or a whole orange). From hypotheses H1 and H2 we see that the model is stationary, because both the technology set G and the utility function u are time-variant. DEFINITION 7.3.3 A finite or infinite sequence {zn }m n=1 , (m ≤ +∞), is said to be a program (path) of the economy if and only if zn = (xn−1 , yn ) ∈ G
and
xn ≤ yn
for all n ≥ 1.
(7.66)
If the vector y0 ∈ RN + is fixed, then we have program originating from y0 . REMARK 7.3.4 In (7.66), the first condition means that all pairs zn = (xn−1 , yn ) are technologically feasible. The second condition is a resource restriction and simply says that at each stage the input (or expenditures) can not exceed the output (available capital stock) from the previous stage. The vector y0 ∈ RN + is called the initial vector for the program {zn }m n=1 , if y ≥ x0 ; that is, the input at stage zero does not exceed the initial resources.
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7 Economic Equilibrium and Optimal Economic Planning
The next definition describes how we measure the performance of a program m {zn }m n=1 , (m ≤ ∞). We distinguish between finite programs {zn }n=1 , (m < ∞) and infinite programs {zn }n≥1 depending on whether we consider a finite or an infinite planning horizon. In the first case there is no issue of convergence of the intertemporal utility and so optimality is defined in the standard way. In the second case, we are faced with the possible nonconvergence of the intertemporal utility (because there is no discount; i.e., δ = 1) and so to define optimality we use the notions introduced in the last part of Section 7.2. N DEFINITION 7.3.5 (a) A finite program {zn }m n=1 , (m 0 independent of m ≥ 1 such that m
(znm , z ∗ ) ≤ β
for all m ≥ 1.
n=1
PROOF: We have ϑm =
m
(znm , z ∗ ) =
n=1
=
m
ξ(z ∗ ) − ξ(znm )
n=1 m
(see (7.69))
m (p∗ , y ∗ − x∗ )RN u(z ∗ ) − u(znm ) +
n=1 m
−
n=1 ∗ m (p∗ , ynm − xm n )RN + (p , x0 − ym )RN .
n=1
(7.70)
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7 Economic Equilibrium and Optimal Economic Planning
∗ Because p∗ ∈ RN + is the stationary price system supporting the turnpike z = (x∗ , y ∗ ), we have (7.71) (p∗ , y ∗ − x∗ )RN = 0. m ∗ N Also we have xm n ≤ yn and because p ∈ R+ , we obtain
(p∗ , ynm − xm n )RN ≥ 0.
(7.72)
Moreover, because G is by hypothesis compact, we have m )RN ≤ η (p∗ , x0 − ym
for some η > 0, all m ≥ 1.
(7.73)
Returning to (7.70) and using (7.71) through (7.73), we infer that ϑm ≤
m
u(z ∗ ) − u(znm ) + η.
(7.74)
n=1
From (7.74) and the fact that {znm }m n=1 , m ≥ 1 is a good sequence (see Definition 7.3.10) we conclude that ϑm ≤ β for some β > 0 (independent of m ≥ 1) and all m ≥ 1. LEMMA 7.3.13 There exists a function λ : R+ −→ R+ nondecreasing such that λ(0) = 0, λ(ε) > 0 for all ε > 0 and p(z ∗ , z) ≥ λ(z ∗ − z) for all z ∈ G. PROOF: We set λ(ε) = inf p(z ∗ , z) : z ∈ G, z ∗ − z ≥ ε . We only need to show that λ(ε) > 0 for ε > 0. If this is not true, then we can find {zn }n≥1 ⊆ G such that z ∗ − zn ≥ ε for all n ≥ 1, but (z ∗ , zn ) −→ 0 as n → ∞. Because G is compact, we may assume that zn −→ z ∈ G. Clearly z ∗ − z ≥ ε. For any z ∈ G, we have (z ∗ , z) = |ξ(z ∗ ) − ξ(z)| = ξ(z ∗ ) − ξ(z) ≥ 0.
(7.75)
On the other hand 0 = lim (z ∗ , zn ) = ξ(z ∗ )− lim ξ(zn ) = ξ(z ∗ )−ξ(z) ∗
n→∞
n→∞ ∗
(see (7.75))
⇒ ξ(z ) = ξ(z) = max{u(z) + (R , y − x)RN : z = (x, y) ∈ G}.
(7.76)
But ξ is strictly monotone because u is strictly concave (see hypothesis H2 ). So from (7.76) it follows that z∗ = z which contradicts the fact that z ∗ − z ≥ ε.
Now we can prove a weak turnpike theorem for good sequences. So we continue to have z ∗ = (x∗ , y ∗ ) a turnpike and {znm }m n=1 , m ≥ 1 a good sequence starting at y0 ∈ RN + . By rm (ε) we designate the number of the periods k ∈ {1, . . . , m} for which we have zkm − z ∗ ≥ ε (as in Section 7.2 on RN × RN we consider the l1 -norm, 2N |zk |). namely z = k=1
7.3 Turnpike Theorems
563
DEFINITION 7.3.14 A vector y0 ∈ RN + is said to be sufficient if a strictly positive vector can be produced in a finite number of steps starting from y0 ∈ RN + ; that is, we can find a finite program {zn = (xn−1 , yn )}m starting from y such that ym % 0. 0 n=1 THEOREM 7.3.15 If z ∗ = (x∗ , y ∗ ) ∈ RN ×RN is a turnpike and {zn }m n=1 , m ≥ 1, is a good sequence starting at a sufficient vector y0 ∈ RN , then {rm (ε)}m≥1 is bounded. PROOF: Using Lemmata 7.3.12, and 7.3.13, we have rm (ε)λ(ε) ≤
m
(z ∗ , znm ) ≤ β,
for all m ≥ 1,
n=1
β for all m ≥ 1, λ(ε) ⇒ {rm }m n=1 is bounded. ⇒ rm (ε) ≤
However, our goal is to have this theorem for a sequence {znm }m n=1 , m ≥ 1, where the finite program is optimal {znm }m n=1 , m ≥ 1 for the finite horizon economy. So we need to check that the sequence {znm }m n=1 , m ≥ 1, of optimal programs forms a good sequence. For this in turn it is enough to construct an infinite program {z}n≥1 starting at y0 ∈ RN + for which we have m
u(z ∗ ) − u(z n ) ≤ β0
for some β0 > 0, all n ≥ 1.
(7.77)
n=1
Indeed, because of the optimality of {znm }m n=1 , we have m
m u(z ∗ ) − u(znm ) ≤ u(z ∗ ) − u(z n )
⇒ ⇒
n=1 m
for all m ≥ 1,
n=1
u(z ∗ ) − u(znm ) ≤ β0
n=1 {znm }m n=1 ,
for all m ≥ 1,
m ≥ 1, is a good sequence (see Definition 7.3.10).
An infinite program {z n }n≥1 satisfying (7.77) is of course a good program (see Definition 7.2.31). In other words, it cannot be infinitely worse than the turnpike. LEMMA 7.3.16 If y0 ∈ RN + is sufficient (see Definition 7.3.14), then a good program exists with initial vector y0 ∈ RN +. PROOF: Without any loss of generality, we may assume that y0 % 0. Indeed, because y0 is sufficient, according to Definition 7.3.14 with a finite program, we can reach a vector ym % 0. Then we continue as described below. N So y0 % 0. Let y ∈ RN + be the output of the expansible stock x ∈ R+ (see hypothesis H3 (ii)). We may assume that y ! y0 . Because (0, 0) ∈ G and G is convex (see hypothesis H1 ), we have that (µx, µy) ∈ G and of course µx ! µy (i.e., µx remains expansible). Let t ∈ (0, 1) and define
564
7 Economic Equilibrium and Optimal Economic Planning z$n = tn z + (1 − tn )z ∗ ,
where z = (x, y), n ≥ 1.
We show that {$ zn }n≥1 is a good sequence starting at y0 . Because G is convex, we see that z$n ∈ G for all n ≥ 1. To establish that {$ zn }n≥1 is a program we need to show that x $0 ≤ y0 and x $n ≤ y$n for all n ≥ 1, (7.78) (see Definition 7.3.3 and Remark 7.3.4). Note that x $0 = x ! y ≤ y0 . So we have verified the first inequality in (7.78). To check the second inequality in (7.78), by direct substitution we have
$n = tn y − tx + (1 − t)x∗ + (1 − tn )(y ∗ − x∗ ). (7.79) y$n − x Recall that x∗ ≤ y ∗ . Also because x ! y (see hypothesis H1 (ii)), by choosing t close to 1, we see that tx + (1 − t)x∗ ≤ y. Therefore from (7.79), we have x $n ≤ y$n for all n ≥ 1. So we have also verified the second inequality in (7.78). This means that {$ zn }n≥1 is a program starting at y0 . Because u is concave (see hypothesis H2 ), we have zn ) for all n ≥ 1, tn u(z) + (1 − tn )u(z ∗ ) ≤ u($
zn ) ≤ tn u(z ∗ ) − u(z) for all n ≥ 1, ⇒ u(z ∗ ) − u($ m zn ) ≤ β0 for some β0 > 0, all n ≥ 1, u(z ∗ ) − u($ ⇒ n=1
⇒ {$ zn }n≥1 is a good program starting at y0 ∈ RN +. Having this lemma, we can state the weak turnpike theorem for finite programs. THEOREM 7.3.17 If {znm }m n=1 , m ≥ 1, is a sequence of optimal finite programs starting at y0 ∈ RN + , then for every ε > 0, the sequence {rm (ε)}m≥1 is bounded. REMARK 7.3.18 This result says that the number of time periods when the optimal program differs from the turnpike by at least ε > 0 is bounded by a constant β(ε) > 0 that is independent of m ≥ 1. So the optimal program stays near the turnpike except possibly for at most β(ε) periods. In particular we
for all time have rm (ε) /m −→ 0 as m −→ ∞. So the proportion of time when the optimal program deviates from the turnpike goes to zero as the planning horizon increases. In fact we can extend the above weak turnpike theorem to infinite programs. THEOREM 7.3.19 If {zn }n≥1 is a good program, then zn −→ z ∗ as n → ∞. PROOF: As in the proof of Lemma 7.3.12, we can show that m n=1
(z ∗ , zn ) ≤
m
u(z ∗ ) − u(zn ) + η,
for some η > 0, all m ≥ 1.
n=1
From (7.80) and because {zn }n≥1 is a good program, we infer that
(7.80)
7.3 Turnpike Theorems (z ∗ , zn ) −→ 0
565
as n → ∞.
Suppose that for some subsequence {nk } of {n}, we have znk − z ∗ ≥ ε
for all k ≥ 1.
Then from Lemma 7.3.13, we have 0 < λ(ε) ≤ (z ∗ , znk ),
for all k ≥ 1,
a contradiction. Therefore we conclude that zn −→ z ∗ as n → ∞.
REMARK 7.3.20 So according to this theorem every good program, in particular every weakly maximal program, is asymptotic to the turnpike. m THEOREM 7.3.21 If y0 ∈ RN + is a sufficient vector, {zn }n≥1 , m ≥ 1, is a sequence of optimal finite programs starting at y0 , and the turnpike z ∗ ∈ int G, then for any ε > 0, we can find ϑ = ϑ(ε) > 0 (independent of m ≥ 1) such that for any m > 2ϑ, we can find a finite program {$ znm }m n=1 starting at y0 for which we have m ∗ (a) z$n = z for all n ∈ {ϑ, . . . , m-ϑ}. m
znm ) < ε. (b) u(znm ) − u($ n=1
PROOF: Because z ∗ ∈ int G, we can find r > 0 such that Br (z ∗ ) ⊆ G. Let η ∈ (0, r). By virtue of Theorem 7.3.17, we can find β = β(η) > 0 such that m ≥ 1 there are at most (β − 1) time periods n ≥ 1 for which we have znm − z ∗ ≥ η. Fix m > 2β. Among the first β periods {1, 2, . . . , β} and among the last β periods {m − β + 1, . . . , m}, there is at least one time period k = k(m) and i = i(m), respectively, such that zkm − z ∗ < η
and
zim − z ∗ < η.
We consider the following finite program: m ∗ ∗ ∗ ∗ ∗ ∗ m m m , (xm σm = z1m , . . . , zk−1 k−1 , y ), (x , y ), . . . , (x , y ), (x , yi ), zi+1 , . . . , zm . (7.81) Note that in this finite sequence σm of length m, the initial and last parts (with lengths k − 1 and m − i − 1, resp.) coincide with the optimal program. The middle part coincides with the turnpike. The turnpike is linked with the other two parts ∗ ∗ m using the pairs (xm k−1 , y ) and (x , yi ). It is straightforward to check that σm is indeed a finite program starting at y0 . So for every m > 2β we have a program σm = {$ znm }m n=1 as above. Then χ= =
m
u($ znm ) − u(znm )
n=1 m
u($ znm ) − u(znm )
(see (7.81))
n=k
=
i
∗ ∗ ∗ m ∗ u(z ∗ ) − u(znm ) + u(xm k−1 , y ) − u(z ) + u(x , yi ) − u(z ) ,
n=k
(7.82)
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7 Economic Equilibrium and Optimal Economic Planning
(see (7.81)). ∗ N Because {znm }m n=1 is optimal, we have χ ≤ 0. Moreover, if p ∈ R+ is the price system supporting the turnpike (see Proposition 7.3.9), we have
∗ ∗ ∗ m ∗ µ(η) ≤ u(xm k−1 , y ) − u(z ) + u(x , yi ) − u(z ) − (p∗ , y ∗ − yim )RN + (p∗ , x∗ − xm i )RN ≤ χ ≤ 0
(7.83)
with µ(η) independent of m ≥ 1 and µ(η) −→ 0 as η −→ 0. Choose η > 0 small enough so that |µ(η)| < ε. Then set ϑ = ϑ(ε) = β(η) + 1. We see that both statements (a) and (b) of the theorem are satisfied. REMARK 7.3.22 Statement (a) of the theorem says that the program {$ znm }m n=1 coincides with the turnpike except perhaps in the initial and final stages. The length of these parts is bounded by a constant independent of m ≥ 1. Moreover, according to statement (b), this new program {$ znm }m n=1 is ε-optimal. m LEMMA 7.3.23 If {znm }m n=1 is an optimal program, {zn }n=1 is any finite program, and 1 ≤ k ≤ i ≤ m, then i
k−1
(z ∗ , znm ) ≤
n
u(znm ) − u(zn ) + u(znm ) − u(zn )
n=1
n=k
n=i+1
i
∗ ∗ m u(z ∗ ) − u(zn ) + (p∗ , x∗ − xm k−1 )RN + (p , y − yi )RN .
+
n=k
(7.84) PROOF: From (7.69) (definition of the pseudometric ), we have i
i
i−1 (p∗ , ynm − xm u(z ∗ ) − u(znm ) − n )RN
(z ∗ , znm ) =
n=k
n=k
n=k
+ (p∗ , y ∗ − yim )RN − (p∗ , x∗ − xm k−1 )RN i
u(z ∗ ) − u(znm ) + (p∗ , y ∗ − yim )RN − (p∗ , x∗ − xm k−1 )RN .
≤
n=k
(7.85) Because {znm }m n=1 is optimal, we have m
u(zn ) ≤
n=1
⇒
k−1
m
u(znm )
n=1 i m i
u(zn ) + u(znm ). u(zn ) − u(znm ) + u(zn ) − u(znm ) ≤
n=1
n=k
n=i+1
n=k
(7.86) Using (7.86) in (7.85), we obtain (7.84).
Using this lemma we can prove the following result, which is crucial in the proof of the strong turnpike theorem.
7.4 Stochastic Growth Models
567
m m LEMMA 7.3.24 If y0 ∈ RN + is a sufficient vector, {zn }n=1 is an optimal finite m−l program starting at y0 , and ϑ > 0, then we find l = l(ϑ) ≥ 1 such that (z ∗ , znm ) < n=l
ϑ. PROOF: Using (7.84) (see Lemma 7.3.23) with {znm }m n=1 = σm , as in the proof of Theorem 7.3.21, we have m−β
n=β
(z ∗ , znm ) ≤
i
(z ∗ , znm )
n=k
≤ u(z ∗ ) − u(zk ) + u(z ∗ ) − u(zi ) + 2ηp∗ ≤ ϕ(η),
where ϕ(η) −→ 0 as η −→ 0 because u is continuous on G and z ∗ −zk , z ∗ −zi < η. Choose η > 0 small enough so that ϕ(η) < ϑ. Then l = β(η) will do the job. This lemma leads to the following strong turnpike theorem. m m THEOREM 7.3.25 If y0 ∈ RN + is sufficient, {zn }n=1 , m ≥ 1, is a sequence of optimal finite programs, and the turnpike z ∗ ∈ int G, then for any ε > 0, we can find l = l(ε) > 0 (independent of m ≥ 1) such that for any m ≥ 2l, we have z ∗ − znm < ε for all h ∈ {l, . . . , m − l}.
PROOF: Given ε > 0, let ϑ = λ(ε) where λ(·) is as in Lemma 7.3.13. Let l = l(ϑ) ≥ 1 be as in Lemma 7.3.24. If z ∗ − znm ≥ ε for some n ∈ {l, . . . , m − l}, then (z ∗ , znm ) ≥ λ(ε) = ϑ (see Lemma 7.3.13) and this contradicts Lemma 7.3.24. So we have proved the strong turnpike theorem for the pseudometric . Then arguing as in the proof of Lemma 7.3.13, we can have it for the norm in RN ×RN . REMARK 7.3.26 This theorem says that an optimal finite program is in the vicinity of the turnpike for all time periods not less than l periods distant from the endpoints. So compared to Theorem 7.3.17, now we identify those time periods when the optimal program deviates from the turnpike.
7.4 Stochastic Growth Models In the previous sections we studied static and dynamic deterministic economic models. In this section, we examine discrete-time economic growth models with uncertainty. The uncertainty is present both in the utility and in the technological constraints. First we deal with the discounted nonstationary model and we prove the existence of an optimal program, which we characterize with a system of supporting prices. In the second half of the section, we deal with the undiscounted, stationary growth model for which we show that it has a weakly maximal program. So first we describe and study the discounted, nonstationary growth growth model. For this purpose let (Ω, Σ, µ) be a complete probability space. Then ω ∈ Ω represents a possible state of the environment, Σ is the collection of all possible events, and µ is the probability distribution of these events. We have a discrete-time,
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7 Economic Equilibrium and Optimal Economic Planning
infinite planning horizon N0 = {0, 1, 2, . . .} (i.e., the model is a discrete-time, infinite horizon model). To describe mathematically the uncertainty of the system, we consider an increasing sequence {Σn }n≥1 of complete sub-σ-fields of Σ and assume that Σ = Vn≥1 Σn . The sub-σ-field Σn of Σ, represents the information about the state of the environment available up until time n ≥ 0. There are N commodities in the economy. So the commodity space is RN (multisector stochastic model). Finally there is a constant discount factor 0 < δ < 1 (discounted model). At time period n ≥ 1, the technological possibilities of the economy are described N N by a multifunction G : Ω −→ 2R+ ×R+ \ {∅} which has a graph belonging in Σn × N N B(RN + ) × B(R+ ) with B(R+ ) being the Borel σ-field. The set Gn (ω) describes all possible transformations of capital stock at time n ≥ 1, when the state of the environment is ω ∈ Ω. More precisely, if (k, y) ∈ Gn (ω), when the state of the environment is ω ∈ Ω, we can transform a capital input k at time n− 1 into a capital output y at time n. The uncertainty in this production process is manifested by the fact that the graph of Gn is Σn -measurable. Note that the technology feasibility set varies with time and so the model is nonstationary. At every stage n ≥ 1, the utility (gain) achieved by operating a particular technoN logical process (x, y) ∈ Gn (ω), is measured by a utility function un : Ω×RN + ×R+ −→ N R which is assumed to be Σn × B(RN ) × B(R )-measurable. Again the uncertainty + + is expressed by the Σn -measurability of the function. Finally there is a constant discount factor δ ∈ (0, 1). A sequence k = {kn }n≥0 with kn ∈ L∞ (Σn , RN ) is a program (or path). We
say that program k = {kn }n≥0 is feasible if kn (ω), kn+1 (ω) ∈ Gn+1 (ω) µ-a.e. on Ω, for all n ≥ 0. We say that k = {kn }n≥0 is a feasible program starting at x0 ∈ L∞ (Σ0 , RN ), if k0 = x0 . The set all feasible programs starting at x0 , is " of ∞ denoted by F (x0 ). Evidently F (x0 ) ⊆ L (Σn , RN ). n≥0
Given an initial capital stock x0 ∈ L∞ (Σ0 , RN )+ (i.e. x0 (ω) ≥ 0 µ-a.e. on Ω), we want to find an element in F (x0 ) (i.e., a feasible program starting at x0 ) that maximizes the intertemporal utility n+1 U (k) = δ Jn+1 (kn , kn+1 ), (7.87) n≥0
where k = {kn }n≥0 and
un+1 ω, v(ω), w(ω) dµ
Jn+1 (v, w) =
(7.88)
Ω
for all (v, w) ∈ L∞ (Σn , RN )+ × L∞ (Σn+1 , RN )+ , n ≥ 0. The mathematical hypotheses on the technological possibilities multifunction Gn are similar to those of the deterministic discounted multisector growth model. N H1 : Gn : Ω −→ Pf c (RN + ×R+ ) is a multifunction such that N (i) Gr Gn ∈ Σn × B(RN + ) × B(R+ ) (i.e., the multifunction Gn is Σn -graph measurable.
(ii) For every ω ∈ Ω, (0, 0) ∈ Gn (ω) and (0, y) ∈ Gn (ω) implies y = 0.
7.4 Stochastic Growth Models
569
(iii) For every ω ∈ Ω, if (k, y) ∈ Gn (ω), k ≤ k , and y ≤ y, then (k , y ) ∈ Gn (ω) (free disposability). (iv) There exists M > 0 such that if ω ∈ Ω and (k, y) ∈ Gn (ω) with k > M , then y ≤ k. The hypotheses on the instantaneous utility function, are the following. N H2 : un : Ω× RN + × R+ −→ R, n ≥ 1 is a function such that N (i) (ω, k, y) −→ un (ω, k, y) is Σn × B(RN + ) × B(R+ )-measurable.
(ii) For every ω ∈ Ω (k, y) −→ un (ω, k, y) is concave and upper semicontinuous. N (iii) For all (ω, k, y) ∈ Ω × RN + × R+ , we have
|un (ω, k, y)| ≤ ϕn (ω, k|, y) with ϕn : Ω × R+ × R+ −→ R+ a Σn × B(R+ ) × B(R+ )-measurable function, ϕn (ω, ·, ·) is nondecreasing, and sup ϕn (·, v, v)L1 (Σn ) < +∞ for all v ∈ R+ . n≥1
By a price system we mean a sequence p = {pn }n≥0 ⊆ L1 (Σn , RN )+ . Normally prices belong to the dual of the space of production vectors. However, L∞ (Σn , RN )∗ is too big and has no satisfactory economic interpretation. For this reason we limit ourselves to L1 (Σn , RN ) ⊆ L∞ (Σn , RN )∗ . The precise description of L1 (Σn , RN ) as a subspace of L∞ (Σn , RN )∗ is given by the Yosida–Hewitt theorem. For easy reference we state it here. First a definition. DEFINITION 7.4.1 Let (Ω, Σ, µ) be a σ-finite measure space. A functional u∈
L∞ (Ω, RN )∗ is said to be absolutely continuous if u(g) = Ω g(ω), f (ω) RN dµ for some f ∈ L1 (Ω, RN ) and all g ∈ L∞ (Ω, RN ). A functional u ∈ L∞ (Ω, RN )∗ is said to be singular if there exists a decreasing sequence {Cn }n≥1 ⊆ Σ such that µ(Cn ∩C) −→ 0 for every C ∈ Σ with µ(C) < +∞ and u is supported by Cn ; that is, u(g) = 0 for each g ∈ L∞ (Ω, RN ) which is identically zero on one of the Cn s (i.e., u(g) = u(χCn g), n ≥ 1, g ∈ L∞ (Ω, RN )). REMARK 7.4.2 Absolutely continuous elements in L∞ (Ω, RN )∗ are identified with L1 (Ω, RN ). Using the notions of absolutely continuous and singular elements of L∞ (Ω, RN )∗ , we can have a complete description of the dual space L∞ (Ω, RN )∗ . The result is known as the Yosida–Hewitt theorem and for a proof of it, we refer to Levin [377]. THEOREM 7.4.3 If (Ω, Σ, µ) is a σ–finite measure space and u∈L∞ (Ω, RN )∗ , then u can be uniquely written as u = ua + us , where ua is absolutely continuous and us is singular. If u ≥ 0, then ua , us ≥ 0. Moreover, we have u = ua + us .
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7 Economic Equilibrium and Optimal Economic Planning
DEFINITION 7.4.4 A feasible program k∗ = {kn∗ }n≥0 ∈ F (x0 ) is said to be optimal if for all k ∈ F (x0 ). U (k) ≤ U (k ∗ ) To prove the existence of an optimal program for our model, we use the direct method of the calculus of variations. This requires that we prove the upper semicontinuity of the objective functional and the compactness of the constraint set in some useful topology and then apply the Weierstrass theorem. THEOREM 7.4.5 If hypotheses H1 , H2 hold and x0 ∈ L∞ (Σ0 , RN ), then there exists an optimal program in F (x0 ). PROOF: Using hypotheses H1 and arguing as in Lemma 7.2.5, we can have that for all k = {kn }n≥0 ∈ F (x0 ). kn ∞ ≤ M1 = max x0 ∞ , M Let Dn = h ∈ L1 (Σn , RN ) : h∞ ≤ M1 , n ≥ 1 (recall L∞ (Σn , RN ) ⊆ L1 (Σn , RN )). From the Eberlein–Smulian theorem, we infer that Dn ⊆ L1 (Σn , RN ) " is weakly sequentially compact. Therefore Dn is weakly sequentially compact in n≥1 " 1 " 1 L (Σn , RN ). Recall that the weak topology on L (Σn , RN ), is the product n≥1
n≥1
topology of the weak topologies on each factor space L1 (Σn , RN ), n ≥ 0. N We define un : Ω×RN + × R+ −→ R = R ∪ {−∞} by un (ω, k, y) if (ω, k, y) ∈ Gr Gn . un (ω, k, y) = −∞ otherwise It is easy to see that un ∈ Σn ×B(RN )×B(RN )-measurable (see hypotheses H1 (i) and H2 (i)) and for every ω ∈ Ω, un (ω, ·, ·) is concave and upper semicontinuous. Moreover, we have |un (ω, k, y)| ≤ ϕn (ω, k, y) Let U :
"
for all (ω, k, y) ∈ GrGn .
L1 (Σn , RN ) −→ R = R ∪ {−∞} be the expected intertemporal utility
n≥0
corresponding to un , n ≥ 1; that is, n+1 δ U (k) = Jn+1 (kn , kn+1 ) n≥0
for all k = {kn }n≥0 and where
un+1 ω, v(ω), w(ω) dµ
Jn+1 (v, w) = Ω
for all (v, w) ∈ L (Σn , R ) × L (Σn+1 , RN ). " 1 We show that U is weakly sequentially upper semicontinuous in L (Σn , RN ). n≥0 " 1 w L (Σn , RN ), with k m = {knm }n≥0 and So suppose that k m −→ k in F (x0 ) ⊆ 1
N
1
n≥0
7.4 Stochastic Growth Models
571
w
k = {kn }n≥0 . We have knm −→ kn in L1 (Σn , RN ) as m −→ ∞, for all n ≥ 0. Using Fatou’s lemma, we have m lim sup Jn+1 (knm , kn+1 ) ≤ Jn+1 (kn , kn+1 ). m→∞
Given v ∈
"
L1 (Σn , RN ) and K ≥ 0, we set
n≥0
UK (v) =
K
δ n+1 Jn+1 (vn , vn+1 ).
m=0
Using this notation we have lim sup UK (k m ) = lim sup m→∞
m→∞
≤
K
K
m δ n+1 Jn+1 (knm , kn+1 )
m=0
m δ n+1 lim sup Jn+1 (knm , kn+1 ) m→∞
m=0
≤
K
δ n+1 Jn+1 (kn , kn+1 ) = UK (k).
m=0
Because k ∈ F (x0 ), we have ∞ ∞ |U (k) − UK (k)| = δ n+1 Jn+1 (kn , kn+1 ) ≤ δ n+1 β, n=K+1
n=K+1
where β = sup ϕn (·, M1 , M1 ) 1 (see hypothesis H2 (iii)). It follows that n≥1
UK (k) −→ U (k)
as K −→ +∞.
Then we can find a map m −→ K(m) increasing to +∞ such that
lim sup UK(m) (k m ) ≤ lim sup lim sup UK (k m ) m→∞
K→+∞
m→∞
≤ lim sup UK (k) = U (k).
(7.89)
K→+∞
So we have U (k m ) − U (k) = U (k m ) − UK(m) (k m ) + UK(m) (k m ) − U (k) ∞ ≤ δ n+1 β + UK(m) (k m ) − U (k) n=K(m)+1 ∞ ⇒ lim sup U (k m ) − U (k) ≤ lim sup δ n+1 β
m→∞
m→∞ n=K(m)+1
+ lim sup UK(m) (k m ) − U (k) ≤ 0 m→∞
⇒ lim sup U (k m ) ≤ U (k). m→∞
(see (7.89))
572
7 Economic Equilibrium and Optimal Economic Planning This proves the weak sequential upper semicontinuity of U on
"
L1 (Σn , RN ).
n≥0
So by the Weierstrass theorem, we can find k ∗ ∈ F (x0 ) such that U (k ∗ ) = U (k ∗ ) = sup U (k) : k ∈ F (x0 ) . Next we characterize an optimal program by generating a price system that supports it. More precisely, we prove that there exists a price system such that: (a) At every time period, we have minimization of the cost among programs producing no less future value. (b) At every time period, we have maximization of the total utility, defined as the sum of the instantaneous utility and the net profit resulting from operating a particular technological process at that period. The net profit is calculated as the value of the produced output minus the cost of the used input. (c) The expected value of the optimal program goes to zero as the planning horizon expands to +∞ (transversality condition). Recall that the deterministic variant of such a result was given in Theorem 7.2.13. In our effort to produce a price characterization of the optimal program, we need some quantities, which we introduce next. So suppose that f ∈ L∞ (Σn , RN ), n ≥ 0. We define
Dn (f ) = h = {hm }m≥n : hn=f and hm (ω), hm+1 (ω) ∈ Gm+1 (ω) µ-a.e. on Ω m+1 and Vn (f ) = sup δ Jm+1 (hm , hm+1 ) : h ∈ Dn (f ) . m≥n
Clearly Vn is the value (Bellman) function of the sequential optimization problem and as it is well-known it satisfies the dynamic programming functional equation, namely Vn (f ) = sup δ m+1 Jm+1 (f, g) + Vn+1 (g) : g ∈ L∞ (Σn+1 , RN ),
f (ω), g(ω) ∈ Gn+1 (ω) µ-a.e. on Ω .
(7.90)
If k ∗ = {kn∗ }n≥0 ∈ F (x0 ) is an optimal program, then ∗ ∗ ∗ , kn+1 ) + Vn+1 (kn+1 ). Vn (kn∗ ) = δ n+1 Jn+1 (kn−1
(7.91)
Also in what follows by Sn+1 we denote the subset of L∞ (Σn , RN ) × L∞ (Σn+1 , RN ) consisting of pairs of functions which pointwise correspond to a technological feasible production process, namely Sn+1 = (f, g) ∈ L∞ (Σn , RN ) × L∞ (Σn+1 , RN ) :
f (ω), g(ω) ∈ Gn+1 (ω) µ-a.e. on Ω .
7.4 Stochastic Growth Models
573
Note that hypotheses H2 imply that the value function Vn is concave (i.e., −Vn is convex). Then for every z ∈ L∞ (Σn , RN ), we can define the subdifferential of Vn by ∂Vn (z) = p ∈ L∞ (Σn , RN )∗ : Vn (y) − Vn (z) ≤ pn (y − z) for all y ∈ L∞ (Σn , RN ) (see Definition 1.2.28). To generate the desired supporting price system, we need to strengthen hypotheses H2 . N H3 : un : Ω × RN + × R+ −→ R is a function such that N (i) For all (k, y) ∈ RN + × R+ , ω −→ un (ω, k, y) is Σn -measurable.
(ii) For every ω ∈ Ω, (k, y) −→ un (ω, k, y) is continuous concave. (iii) For every (ω, y) ∈ Ω × RN + , k −→ un (ω, k, y) is nondecreasing and for every (ω, k) ∈ Ω × RN + y −→ un (ω, k, y) is nonincreasing. N (iv) For every (ω, k, y) ∈ Ω × RN + × R+ , we have
|un (ω, k, y)| ≤ ϕn ω, k, y ,
with ϕn : Σn × B(R+ ) × B(R+ )-measurable, for every ω ∈ Ω ϕn (ω, ·, ·) is nondecreasing and sup ϕn (·, v, v)1 < ∞ for all v ≥ 0. n≥1
We also need a hypothesis concerning the value function Vn . H4 : For every n ≥ 0, Vn is continuous at some point in L∞ (Σn , RN ) and ∂V0 (x0 ) = ∅. REMARK 7.4.6 From Theorem 1.2.3, we know that the first part of hypothesis H4 is equivalent to saying that Vn is bounded below in a neighborhood of a point. Also from Theorem 1.2.34, we know that if V0 is continuous at x0 ∈ L∞ (Σ0 , RN ), then ∂V0 (x0 ) = ∅. Note that if the technology is rich enough to admit a feasible program v = {vn }n≥0 such that vn (ω), vn+1 (ω) + Bεn +1 ⊆ Gn+1 (ω) µ-a.e. on Ω (interior program), then hypothesis H4 is satisfied. To produce the supporting price system, we need to use the Mackey topology on the Lebesgue space L∞ (Σn , RN ). We have already encountered the Mackey topology in Chapter 6 (see Definition 6.3.29). For easy reference we recall this definition here in the context of the space L∞ (Σn , RN ), where it is used in the sequel. DEFINITION 7.4.7 The Mackey topology m∞ on L∞(Σn , RN ) induced by the
∞ N 1 N pair L (Σn , R ), L (Σn , R ) , is the topology of uniform convergence on the 1 N weakly compact and convex subsets
∞ of L N(Σn , 1R ). So, if by ·, · we denote the duality brackets for the pair L (Σn , R ), L (Σn , RN ) (recall L∞ (Σn , RN ) = m∞ k if and only if for every W ⊆ L1 (Σn , RN ) weakly L1 (Σn , RN )∗ ), then ka −→ compact and convex, we have sup | ka − k, p | : p ∈ W −→ 0.
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7 Economic Equilibrium and Optimal Economic Planning
REMARK 7.4.8 Clearly the w∗ -topology on L∞ (Σn , RN ) = L1 (Σn , RN )∗ is weaker than the Mackey topology. We know that the Mackey
topology is ∗the strongest locally convex topology τ on L∞ (Σn , RN ) such that L∞ (Σn , RN )τ = L1 (Σn , RN )). Convergence in the Mackey topology is related to the convergence in measure, µ denoted by −→. PROPOSITION 7.4.9 If {hk , h}k≥1 ⊆ L∞ (Σn , RN ), sup hk ∞ = η < +∞ and k≥1 µ
∞ hk −→ h as k −→ ∞, then hk −→ h in L∞ (Σn , RN ) as k −→ ∞.
m
PROOF: Let W ⊆ L∞ (Σn , RN ) be a weakly compact and convex set. According to Definition 7.4.7, we need to show that sup | hk − h, w | : w ∈ W −→ 0 as k −→ ∞. Replacing hk by hk − h if necessary, we may assume that h = 0. Because W ⊆ L1 (Σn , RN ) is weakly compact, from the Dunford–Pettis theorem, we have that W is uniformly integrable. Hence the set {v = g∞ w1 : g∞ ≤ η, w ∈ W } is uniformly integrable too. So given ε > 0, we can find ϕ ∈ L1 (Σn )+ , ϕ > 0 such that for all g ∈ L∞ (Σn , RN )+ with g∞ ≤ η and all w ∈ W , we have
g(ω) w(ω)dµ ≤ ε. {gw>ϕ}
So to prove the proposition, it suffices to show that
hm (ω) w(ω)dµ = 0. lim sup m→∞ w∈W
{hm w≤ϕ}
We can always assume that |W |1 = sup w1 : w ∈ W ≤ 1. Note that # {ϕ < λ} {ϕ = 0} = λ>0
and because ϕ(ω) > 0 µ-a.e. on Ω, we see that we can find ϑ > 0 small enough such that
ϕ(ω)dµ ≤ ε. (7.92) {ϕ 0 such that, if A ∈ Σn and µ(A) ≤ δ, then
ϕ(ω)dµ ≤ ε.
(7.93)
A
µ Let γ ∈ 0, min{ε, δ} . Since by hypothesis hk −→ h, we can find k0 ≥ 1 such that for all k ≥ k0 , we have
(7.94) µ {ϕ ≥ ϑ, hk ≥ γ} ≤ γ. Then, for all k ≥ k0 and all w ∈ W , we have
7.4 Stochastic Growth Models
575
≤
hk (ω) w(ω)dµ {hk w≤ϕ}∩{ϕ≥ϑ}∩{hk ≥γ} hk (ω) w(ω)dµ {ϕ≥ϑ}∩{hk ≥γ}
≤ε
(see (7.93) and (7.94)).
(7.95)
Also because we have assumed that |W |1 ≤ 1, for all k ≥ k0 and all w ∈ W , we have
hk (ω) w(ω)dµ < γ. (7.96) {hk 0 and all n ≥ 1. Recall that v = J(k∗ , k∗ ). Exploiting the weak upper semicontinuity of the expected utility function J(·, ·), we have lim sup J n→∞
n+1 n 1 1 km ◦ τ −m , km ◦ τ −m ≤ J(k ∗ , k ∗ ). n + 1 m=0 n + 1 m=1
(7.124)
So, if we pass to the limit as n → ∞ in (7.123) and we use (7.124), we obtain J(k∗ , k∗ ) = v ≤ J(k ∗ , k ∗ ). But recall that k ∗ ∈ S. So J(k∗ , k∗ ) = J(k ∗ , k ∗ ) = v.
We introduce the value loss function ξ(k, y) defined by ξ(k, y) = v − J(k, y) + p, y − k
for all (k, y) ∈ Gr R.
PROPOSITION 7.4.20 If hypotheses H5 , H6 hold and {kn }n≥0 , {zn }n≥0 are two good programs that have sequences of arithmetic means {kn }n≥0 , {z n }n≥0 which converge weakly in L1 (Σ0 , RN ) to the same limit k∗ , then lim inf m→∞
m
n=0 m
≤ lim sup m→∞
J(kn ◦ τ −n , kn+1 ◦ τ −n−1 ) − J(zn ◦ τ −n , zn+1 ◦ τ −n−1 )
ξ(zn ◦ τ −n , zn+1 ◦ τ −n−1 ) − ξ(kn ◦ τ −n , kn+1 ◦ τ −n−1 ) .
n=0
(7.125) PROOF: We have m
J(kn ◦ τ −n , kn+1 ◦ τ −n−1 )−J(zn ◦ τ −n , zn+1 ◦ τ −n−1 )
=
n=0 m
J(kn ◦ τ −n , kn+1 ◦ τ −n−1 )− v − p, kn ◦ τ −n −kn+1 ◦ τ −n−1
n=0
−J(zn ◦ τ −n , zn+1 ◦ τ −n−1 ) + v + p, zn ◦ τ −n − zn+1 ◦ τ −n−1 + p, kn ◦ τ −n − kn+1 ◦ τ −n−1 − p, zn ◦ τ −n − zn+1 ◦ τ −n−1 m
≤ ξ(zn ◦ τ −n , zn+1 ◦ τ −n−1 )−ξ(kn ◦ τ −n , kn+1 ◦ τ −n−1 ) n=0
+ p, zm+1 ◦ τ −m−1 − km+1 ◦ τ −m−1 .
Note that because of Proposition 7.4.19, we have
(7.126)
586
7 Economic Equilibrium and Optimal Economic Planning lim sup p, zm+1 ◦ τ −n − km+1 ◦ τ −m−1 m→∞
≤
lim
m→∞
m+1 1 p, zn ◦ τ −n − kn ◦ τ −n = 0. m + 2 n=0
(7.127)
So, if we pass to the limit as m −→ ∞ in (7.126) and use (7.127), we obtain (7.125). REMARK 7.4.21 Note that the lim inf in the left-hand side of (7.125) is similar to the one involved in the definition of weak maximality (see Definition 7.4.18). The above remark suggests that we should consider the following problem, sup[ϑ(k) : k ∈ F (x0 )], where ϑ(k) = lim
(7.128)
−ξ(kn ◦ τ −n , kn+1 ◦ τ −n−1 ) , with k = {kn }n≥0 . Because
m
m→∞ n=0
−ξ ≤ 0, the limit in the definition of ϑ can be −∞. However, this can not happen if k is good. For this reason (in analogy with the deterministic case) we look at good programs to find a weakly maximal one. This requires that the solution set of (7.119) is a singleton. H7 : There exists at least one good program k = {kn }n≥0 ∈ F (x0 ) and the solution set S0 of (7.119) is a singleton. REMARK 7.4.22 If for all ω ∈ Ω, u(ω, ·, ·) is strictly concave, the second part of hypothesis H7 is automatically satisfied. Now we can prove the existence of a weakly maximal program. THEOREM 7.4.23 If hypotheses H5 ,H6 , and H7 hold, then there exists a weakly maximal program k ∗ = {kn∗ }n≥0 ∈ F (x0 ). PROOF: For every m ≥ 0, we set ϑm (k) =
m
−ξ(kn ◦ τ −n , kn+1 ◦ τ −n−1 )
n=0
for every k = {kn }n≥0 ∈ F (x0 ). Evidently ϑm is w∗ -upper semicontinuous and concave. Moreover, we have ϑm ↓ ϑ as m −→ ϑ is w∗ -upper semicontinu"∞∞. Hence n ous and concave. Also note that F" (x0 ) ⊆ n=0 C ◦ τ and by Tychonov’s theorem ∞ N the product set is w∗ -compact in ∞ n=0 L (Σn , R ). Therefore we can apply the ∗ ∗ theorem of Weierstrass and find k = {kn }n≥0 ∈ F (x0 ) such that ϑ(k ∗ ) = sup ϑ(k) : k ∈ F (x0 ) . We claim that the maximizing feasible program k ∗ is a good program. From hypothesis H7 there is a good program k = {kn }n≥0 ∈ F (x0 ). Then given ε > 0, we can find m0 = m0 (ε) ≥ 1 such that for all m ≥ m0 , we have
7.4 Stochastic Growth Models
587
m
J(kn ◦ τ −n , kn+1 ◦ τ −n−1 ) + p, kn+1 ◦ τ −n−1 − kn ◦ τ −n
≤ ⇒
n=0 ∞
∗ ∗ ◦ τ −n−1 ) + p, kn+1 ◦ τ −n−1 − kn∗ ◦ τ −n + ε, J(kn∗ ◦ τ −n , kn+1
n=0 m
∗ ◦ τ −n−1 ) v − J(kn∗ ◦ τ −n , kn+1
n=0
≤
m
v − J(kn ◦ τ −n , kn+1 ◦ τ −n−1 ) +
n=0
∗ ◦ τ −m−1 − km+1 ◦ τ −m−1 + ε p, km+1
≤ γ + 4M p1 + ε ⇒ k ∗ = {kn∗ }n≥0 ∈ F (x0 ) is a good program (see Definition 7.4.18). If k ∗ is not weakly maximal, then we can find ε > 0, m0 ≥ 1, and z ∈ F (x0 ) such that Vm (k ∗ ) + ε ≤ Vm (z), where Vm (k ∗ ) = and
Vm (z) =
m n=0 m
∗ J(kn∗ ◦ τ −n , kn+1 ◦ τ −n−1 )
J(zn ◦ τ −n , zn+1 ◦ τ −n−1 ).
n=0
Then we have
0 < ε ≤ Vm (z) − Vm (k ∗ ).
(7.129)
∗
Recall that k ∈F (x0 ) too is a good program. In addition, because k ∗ maximizes ϑ on F (x0 ), given δ > 0, we can find m0 = m0 (δ) ≥ 1 such that for all m ≥ m0 , we have m
∗ ∗ ◦ τ −n−1 ) − ξ(zn ◦ τ −n , zn+1 ◦ τ −n−1 ) ≤ δ, ξ(kn ◦ τ −n , kn+1 ⇒
n=0 m
∗ ∗ ◦ τ −n−1 ) + p, kn+1 ◦ τ −n−1 − kn∗ ◦ τ −n v − J(kn∗ ◦ τ −n , kn+1
n=0
− v + J(zn ◦ τ −n , zn+1 ◦ τ −n−1 ) − p, zn+1 ◦ τ −n−1 − zn ◦ τ −n ≤ δ, ∗ ⇒ ε ≤ p, zm+1 ◦ τ −m−1 − km+1 ◦ τ −m−1 + δ. Passing to the limit as m−→∞ and using (7.127) (recall that {zn }n≥0 , {kn∗ }n≥0 are both good programs), we obtain 0 < ε ≤ δ. But δ > 0 was arbitrary. So we let δ ↓ 0 to reach a contradiction. This proves that k ∗ ∈ F (x0 ) is weakly maximal. A careful reading of the previous proof reveals that the hypothesis concerning the existence of a good program (see hypothesis H7 ), was crucial. So we need to produce verifiable conditions that imply the existence of a good program in F (x0 ). For this reason we introduce the following hypotheses.
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7 Economic Equilibrium and Optimal Economic Planning
N H8 : u : Ω×RN + ×R+ −→ R is a function such that N (i) (ω, k, y) −→ u(ω, k, y) is Σ1 × B(RN + ) × B(R+ )-measurable.
(ii) For every ω ∈ Ω, (k, y) −→ u(ω, k, y) is continuous, concave. (iii) For every r > 0, there exists ϕr ∈ L1 (Ω)+ such that |u(ω, k, y)| ≤ ϕr (ω) for µ-almost all ω ∈ Ω and all k, y ≤ r.
H9 : There exists w ∈ L∞ (Σ0 , RN ) such that (w ◦ τ ) ∈ L ω, x0 (ω) µ-a.e. on Ω and x0 (ω) ! w(ω) µ-a.e. on Ω. REMARK 7.4.24 This hypothesis means that the initial capital stock x is expansible. THEOREM 7.4.25 If hypotheses H6 , H8 , and H9 hold, then there exists a good program k = {kn }n≥0 ∈ F (x0 ). PROOF: Hypothesis H9 implies that there exists 0 < λ < 1 such that (1 − λ)k∗ + λx0 ≤ w
in the ordered Banach space L∞ (Σ0 , RN ),
where k∗ ∈ S is a solution of problem (7.119). We set kn = (1 − λn )k∗ ◦ τ n + λn x0 ◦ τ n ∈ L∞ (Σn , RN ),
n ≥ 0.
We claim that k = {kn }n≥0 ∈ F (x0 ). If n = 0, then because of τ 0 -identity, we have k0 = x0 . For n ≥ 1, because of the convexity of Gr L(ω, ·) (see hypothesis H5 (iii)), we have (1 − λn )(k∗ ◦ τ n )(ω) + λn (x0 ◦ τ n )(ω)
∈ L τ n (ω), (1 − λn )(k∗ ◦ τ n )(ω) + λ(x0 ◦ τ n )(ω)
n = L τ (ω), (1 − λn )(k∗ ◦ τ n )(ω) + λ(x0 ◦ τ n )(ω)
= Ln+1 ω, x0 (ω) µ-a.e. on Ω. ∗
∞
(7.130)
Because (1 − λ)k + λx0 ≤ w in L (Σ0 , R ), we have N
kn+1 = (1 − λn+1 )k∗ ◦ τ n+1 + λn+1 x0 ◦ τ n+1
≤ (1 − λn+1 )k∗ ◦ τ n + λn w ◦ τ n ◦ τ.
(7.131)
Then from (7.130), (7.131), and the free disposability hypothesis (see hypothesis H5 (iv)), we have
kn+1 (ω) ∈ Ln+1 ω, kn (ω) = L τ n (ω), kn (ω) µ-a.e. on Ω, ⇒ k ∈ F (x0 ). We claim that k is good. From hypotheses H8 we have that J is continuous, concave on L1 (Ω, RN )×L1 (Ω, RN ). Therefore it is Lipschitz continuous on bounded sets, in particular on C. Therefore we have
7.5 Continuous-Time Growth Models
589
v − J(kn ◦ τ −n , kn+1 ◦ τ −n−1 ) = J(k∗ , k∗ ) − J(kn ◦ τ −n , kn+1 ◦ τ −n−1 ) ≤ lM (λn + λn+1 )4M, for some lM > 0, all n ≥ 1. Summing up, we obtain
v − J(kn ◦ τ −n , kn+1 ◦ τ −n−1 ) < +∞, n≥0
⇒ k = {kn }n≥0 is a good program (see Definition 7.4.18).
7.5 Continuous-Time Growth Models In this section, we turn our attention to continuous-time growth models. We consider a general deterministic model of capital accumulation, with a convex technology and a concave utility function. We establish the existence of an optimal growth path using the direct method of the calculus of variations. This requires a careful choice of topology on the space of feasible programs. First let us describe the model. The planning horizon is R+ = [0, +∞} (continuous-time, infinite horizon model). There are N capital goods in the economy and so the commodity space is RN (multisector growth model). The technological possibilities of the economy are described by a multifunction G : R+ × RN + −→ RN + 2 \ {∅}. If y ∈ G(t, x), then this means that if at time t ≥ 0, we are given a capital stock x, then y can be accumulated as additional capital. Hence at every time instant t ≥ 0, G(t, ·) describes the technological capacity of the economy at that moment. For this reason, we call G the technology multifunction. Note that in our model this multifunction is time-varying and so the model (at least in its technology component) is nonstationary. The performance of a feasible technological process (x, y) ∈ Gr G(t, ·) is measured by an instantaneous utility function u(t, x, y) and future utilities are discounted by a factor δ∈(0, 1). So the intertemporal utility generated by a growth path x is given by
∞
J(x) = e−δt u t, x(t), x (t) dt. 0
Also, there is an initial capital stock x0 ∈ RN +. DEFINITION 7.5.1 A capital accumulation program x is an absolutely continuous map x: R+ −→ RN + . From Lebesgue’s theorem we know that such a function is differentiable almost everywhere and for every t ≥ 0 we have
t x (s)ds. x(t) = x(0) + 0
A program x is feasible if 0 ≤ x(0) ≤ x0 and x (t) ∈ G t, x(t) a.e. on R+ . The set of all feasible programs is denoted by F (x0 ). We assume throughout this section that F (x0 ) = ∅. Note that F (x0 ) ⊆ ACloc (R+ , RN ).
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7 Economic Equilibrium and Optimal Economic Planning
REMARK 7.5.2 The hypothesis that F (x0 ) = ∅ is not at all restrictive. Indeed, we already know from the analysis of the discrete models, that it is reasonable to assume that inaction is an option (i.e., 0 ∈ G(t, 0) a.e. on R+ ) and that free disposability holds. Hence for all (t, x) ∈ R×RN + we have 0 ∈ G(t, x) and so we see that F (x0 ) = ∅. We introduce the measure µ on R+ defined by dµ = e−δt dt. Evidently µ is a finite measure on R+ which is absolutely continuous with respect to the Lebesgue measure. The Radon–Nikodym derivative of µ with respect to the Lebesgue measure is of course e−δt . In what follows by L1 (R+ , µ; R) we denote the space of functions from R+ into R whose absolute value is µ-integrable. Now we can introduce our hypotheses on the technology G and the utility u: N H1 : G : R×RN + −→ Pkc (R ) is a multifunction such that
(i) For all x ∈ RN + , t −→ G(t, x) is graph-measurable. (ii) For almost t ∈ T , x −→ G(t, x) has a closed graph. (iii) There exists a function ϕ ∈ L1loc (R+ ) such that for all x ∈ F (x0 ) we have x (t) ≤ ϕ(t)
a.e. on R+ .
REMARK 7.5.3 Hypothesis H1 (iii) is essentially a growth condition on the technology. For example, assume that for almost all t ∈ T and all x ∈ RN + we have |G(t, x)| = sup y : y ∈ G(t, x) ≤ ξ(t)ϕ(x), 1 with ∞ ξ ∈ L loc (R+ ) and ϕ a nonnegative continuous function on R+ such that dr/ϕ(r) = +∞. Then hypothesis H1 (iii) is satisfied. To see this, fix b > 0 and then choose M > x0 + 1 such that
M
b dr ξ(t)dt. (7.132) > x0 +1 ϕ(r) 0
We claim x(t) ≤ M for all t ∈ T = [0, b]. If this is not true, then we can find t1 , t2 ∈ T such that 0 ≤ t1 ≤ t2 ≤ b and x(t1 ) = x0 + 1, x(t2 ) = M
and
x0 + 1 ≤ x(t) ≤ M
for all t ∈ [t1 , t2 ].
We have x (t) ≤ ξ(t)ϕ(x(t)) a.e. on T,
b
M
t2 dr ξ(s)ds ≤ ξ(s)ds ⇒ ≤ t1 x0 +1 ϕ(r) 0 which contradicts (7.132). In fact we can assume a more general majorant; namely |G(t, x)| ≤ g(t, x)
a.e. on R,
where g : R+ ×R+ −→ R+ is a function such that
for all x ≥ 0,
7.5 Continuous-Time Growth Models • • •
591
For all r ≥ 0, g(·, r) ∈ L1loc (R+ ). For all t ≥ 0, g(t, ·) is nondecreasing. There exist t0 ≥ 0 and u0 > 0 such that the scalar differential equation
u (t) = g t, u(t) , u(t0 ) = u0 has a bounded solution on [t0 , +∞). The hypotheses on the instantaneous utility function, are the following:
H2 : u : R+ ×RN ×RN −→ R = R ∪ {−∞} is a function such that (i) (t, x, y) −→ u(t, x, y) is Borel-measurable. (ii) For almost all t ≥ 0, (x, y) −→ u(t, x, y) is upper semicontinuous. (iii) For almost all t ≥ 0 and all x ∈ RN , y −→ u(t, x, y) is concave. (iv) For almost all t ≥ 0 and all x, y ∈ RN , we have |u(t, x, y)| ≤ α(t) + c(x + y)
with α ∈ L1 (R+ , µ; R)+ and c > 0.
REMARK 7.5.4 By allowing the utility function to take the value −∞, we have incorporated in its definition constraints such as u(t, x, y) = −∞ if x ∈ / RN + . Another economically interesting situation accommodated by hypotheses H2 , is when u(t, x, y) −→ −∞ as x, y −→ 0. Indeed, in many situations it is more natural to heavily penalize the inactivity option, more than simply setting u(t, 0, 0) = 0.
∞ If J(x) = 0 e−δt u t, x(t), x (t) dt, x ∈ F (x0 ), our goal is to find a solution of the following optimization problem, V (x0 ) = sup[J(x) : x ∈ F (x0 )].
(7.133)
As we already mentioned, we approach problem (7.133) using the direct method of the calculus of variations. This requires a careful choice of topology that will make the constraint set F (x0 ) compact and the objective functional (the discounted intertemporal utility) upper semicontinuous. The problem that we face is that even if we choose a good topology on the space of capital stocks, this translates into weak convergence of the corresponding investment flows. Weak convergence does not imply, in general, pointwise convergence (even for a subsequence). This lack of pointwise convergence poses several technical difficulties in the existence theory. For this reason, we have included convexity and concavity conditions in hypotheses H1 and H2 . First we focus on the constraint set and seek a topology that makes it compact. For this purpose, we introduce two topologies. The first one concerns the capital accumulation programs. DEFINITION 7.5.5 The c-topology on C(R+ , RN ) (compact-open topology) (hence on ACloc (R+ , RN ) too) is the topology of uniform convergence on compact sets.
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7 Economic Equilibrium and Optimal Economic Planning
REMARK 7.5.6 The above topology is defined by the family of seminorms pn (x) = max{x(t) : t ∈ [0, n]}, where · is any of the equivalent lp -norms on RN . It is well-known that C(R+ , RN ) endowed with this topology becomes a Fr´echet space. This topology makes the constraint set compact. But in order to be able to show this, we need to consider another topology related to the investment flows. The investment flows are the derivatives of the capital accumulations. Because the latter by definition are absolutely continuous functions, by Lebesgue’s theorem the investment flows belong in L1loc (R+ , RN ). This too is a Fr´echet space, for the topology defined by the sequence of seminorms
n y(t)dt, n ≥ 1. qn (y) = 0
Then using this fact we can define a second topology on the space of absolutely continuous functions from R+ into RN , namely the following. DEFINITION 7.5.7 The α-topology on ACloc (R+ , RN ), is the Fr´echet topology generated by the sequence of seminorms rn (x) = pn (x) + qn (x )
for all n ≥ 1. n Recall that pn (x) = max{x(t) : t ∈ [0, n]} and qn (x ) = 0 x (t)dt. REMARK 7.5.8 Evidently on ACloc (R+ , RN ), the a-topology is stronger than the c-topology. Note that the a-topology is equivalent to the one given by the natural metric on the direct sum RN ⊕ L1loc (R+ , RN ). So we can make the identification ACloc (R+ , RN ) = RN ⊕ L1loc (R+ , RN ). Using this identification we can now determine the dual of ACloc (R+ , RN ) and therefore define the weak topology on ACloc (R+ , RN ). The dual of RN with the lp -norm, is of course RN with the lp -norm, (1/p) + (1/p ) = 1. The dual of L1loc (R+ , RN ) is the space ∗ ∞ {y ∈ L (R+ , RN ) : suppy ∗ is compact}. So x∗ ∈ ACloc (R+ , RN ) if and only if
b
∗
∗ ∗ y , x = v , x(0) RN + w (t), x (t) RN dt 0
∗
for some v ∈ R , some b > 0, and some w ∈ L∞ (T, RN ), T = [0, b]. Therefore with this duality we can define a weak topology on ACloc (R+ , RN ). Using the Dunford– w c Pettis theorem, we can check that if xn −→ x in ACloc (R+ , RN ), then xn −→ x. However, this is no longer true if sequences are replaced by nets. Recall that the generalized Lebesgue’s dominated convergence theorem fails for nets. N
∗
Let ξ ∈ L1loc (R+ ) and define D(ξ) = x ∈ ACloc (R+ , RN ) : x (t) ≤ ξ(t) a.e. on T . We show that when restricted to sets such as D(ξ), the weak and c-topologies coincide. This permits us to show the c-compactness of the set F (x0 ). To this end, first we prove the following lemma.
7.5 Continuous-Time Growth Models
593
c
LEMMA 7.5.9 If {xα }α∈S is a net in D(ξ) and xα−→x in ACloc (R+ , RN ), then w x −→ x in ACloc (R+ , RN ) and x ∈ D(ξ). PROOF: Fix b > 0. Given ε > 0, we can find δ > 0 such that if A ⊆ [0, b] is measurable and |A| < δ (by | · | we denote the Lebesgue measure on R), then we have
ξ(t)dt < ε. (7.134) A
We take 0 ≤ s1 ≤ t1 ≤ · · · ≤ sm ≤ tm ≤ b, with
m
(tk − sk ) < δ. Then
k=1 m
≤
k=1 m
x(tk ) − x(sk )
x(tk ) − xa (tk ) + xa (sk ) − x(sk ) +
k=1
≤ 2mx − xa C([0,b],RN ) +
m k=1
tk
xa (τ )dτ
sk tk
xa (τ )dτ
sk
< 2mx − xa C([0,b],RN ) + ε < 2ε
for a ∈ S large (see (7.134)),
which implies that x ∈ ACloc (R+ , RN ). Next given ε > 0, we can find δ > 0 such that
ξdt < ε and kdt < ε C
C
for all C ⊆ [0, b] measurable with 0 < |C| < δ. Let A ⊆ [0, b] be measurable with 0 < |A|. We can find U ⊆ [0, b] open such that |C = A U | < δ. We know that m U= (sk , tk ). We have k=1
x dt A
A
≤ (xa − x )dt + xa dt + x dt
xa dt −
U
≤
C
C
m
xa (tk ) − x(tk ) + xa (sk ) − xa (sk ) + 2ε
k=1
≤ 2mxa − xC([0,b],RN ) + 2ε,
xa dt −→ x dt. ⇒ A
A
Because A ⊆ [0, b]-measurable was arbitrary, we have xa −→ x in L1 ([0, b], RN ). w
Also xa (0) −→ x(0). Because b > 0 was arbitrary, we conclude that w
xa −→ x
in ACloc (R+ , RN ).
594
7 Economic Equilibrium and Optimal Economic Planning Moreover, from Mazur’s lemma, we infer that x (t) ≤ ξ(t)
a.e. on R+ ,
⇒ x ∈ D(ξ). w
c
LEMMA 7.5.10 If {xa }a∈S ⊆ D(ξ) and xa −→ x in ACloc (R+ , RN ), then xa −→ x and x ∈ D(ξ). PROOF: Let ε > 0 and b > 0 be given. We can find δ > 0 such that
t t ξ(τ )dτ < ε and x (τ )dτ < ε s
s w
for 0 ≤ s ≤ t ≤ b and (t − s) < δ. Because xa −→ x in ACloc (R+ , RN ) for a ∈ S large, we have
δn xa (δn) − x(δn) ≤ xa (0) − x(0) + (xa − x )dτ < ε, 0
where n is an integer such that 0 ≤ n ≤ (1/δ)b. Let t ∈ [0, b]. We choose an integer 0 ≤ n ≤ (1/δ)b with |t − δn| < δ. Then for a ∈ S large, we have
t xa (t) − x(t) ≤ xa (δn) − x(δn) + (xa − x )dτ < 3ε, δn c
⇒ xa −→ x and from Lemma 7.5.9, we have x ∈ D(ξ). From Lemmata 7.5.9 and 7.5.10, we infer the following. PROPOSITION 7.5.11 On the sets D(ξ), the c-topology and the weak-α-topology coincide and of course are metrizable. Moreover, D(ξ) is closed. PROPOSITION 7.5.12 Let Dr (ξ) = D(ξ) ∩ x ∈ ACloc (R+ , RN ) : x(0) ≤ r . N Then Dr (ξ) is c-compact in ACloc (R+ , R ). PROOF: From Proposition 7.5.11 we know that Dr (ξ) is c-closed. Also for every b > 0 and every t ∈ [0, b] Dr (ξ)(t) = {x(t) : x ∈ Dr (ξ)} is bounded and closed, hence compact. Moreover, if t, s ∈ [0, b], 0 ≤ s < t ≤ b, and x ∈ Dr (ξ), we have
t
t x (τ )dτ ≤ ξ(τ )dτ. x(t) − x(s) ≤ s
s
1 t Since ξ ∈ Lloc (R+ ), given ε > 0, we can find δ > 0 such that, if t − s < δ, then ξ(τ )dτ < ε. Hence, we have s
x(t) − x(s) < ε
for all x ∈ Dr (ξ) provided t − s < δ,
⇒ Dr (ξ) is equicontinuous.
7.5 Continuous-Time Growth Models Invoking the Arzela–Ascoli theorem, we conclude that Dr (ξ) is c-compact.
595
In the past literature on the subject, a common mistake was to claim that weak convergence in ACloc (R+ , RN ) implies pointwise convergence of at least a subsequence of the derivatives. Of course this is not true in general. EXAMPLE 7.5.13 For 1 ≤ p < ∞, consider the sequence {xn }n≥1 ⊆ Lp [0, 2π] w defined by xn (t) = cos(nt). The Riemann–Lebesgue lemma implies that xn −→ 0 p in L [0, 2π]. However, because xn 2 = π for all n ≥ 0, we cannot have xn −→ 0 in the norm of Lp [0, 2π], or pointwise, or in measure. In order to get pointwise convergence (of at least a subsequence) out of a weakly convergent sequence, we need additional conditions (see Proposition 6.6.45 and Proposition 6.6.46). These conditions are suggested by the following observation, which is an immediate consequence of Proposition 6.6.33. w
PROPOSITION 7.5.14 If {xn }n≥1 ⊆ L1loc (R+ ) and xn −→ x in L1loc (R+ ), then x ∈ L1loc (R+ ) and lim inf xn (t) ≤ x(t) ≤ lim sup xn (t) a.e. on R+ . n→∞
n→∞
Now we are ready to prove the compactness of the constraint set F (x0 ) in the c-topology (equivalently in the weak-α-topology). THEOREM 7.5.15 If hypotheses H1 hold, then F (x0 ) is c-compact. PROOF: Note that F (x0 ) ⊆ Dx0 (ϕ) and from Proposition 7.5.12, we know that Dx0 (ϕ) is c-compact. So it suffices to show that F (x0 ) is c-closed. To this c end let {xn }n≥1 ⊆ F (x0 ) and assume that xn −→ x. Then xn (0) −→ x(0) in N R , 0 ≤ xn (0) ≤ x0 for all n ≥ 1, hence 0 ≤ x(0) ≤ x0 . Also because of Proposition w 7.5.11 we have that xn −→ x in L1loc (R+ , RN ). For every n ≥ 1, we have
xn (t) ∈ G t, xn (t) a.e. on T. We have xn (t) −→ x(t) for every t ≥ 0. Moreover, by virtue of hypotheses H1 (ii), we have
for all t ≥ 0. (7.135) lim sup G t, xn (t) ⊆ G t, x(t) n→∞
Invoking Proposition 6.6.33 and using (7.135) and the fact that G is convexvalued, we have
x (t) ∈ G t, x(t) a.e. on T, ⇒ x ∈ F (x0 ). Therefore F (x0 ) is c-closed, hence it is c-compact.
To establish the desired upper semicontinuity of J(·) in the c-topology, we will need the following general upper semicontinuity result, whose proof can be found in Hu–Papageorgiou [316, p. 31] (see also Theorem 2.1.28).
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7 Economic Equilibrium and Optimal Economic Planning
THEOREM 7.5.16 If (Ω, Σ, µ) is a finite measure space, Y is a separable Banach space, V is a separable reflexive Banach space, and u : Ω × Y × V −→ R = R ∪ {−∞} is a measurable integrand such that (i) For µ-a.e. ω ∈ Ω, (y, v) −→ u(ω, y, v) is upper semicontinuous; (ii) For µ-a.e. ω ∈ Ω and every y ∈ Y, v −→ u(ω, y, v) is concave; (iii) There exist α ∈ L1 (Ω)+ and c > 0 such that u(ω, y, v) ≤ a(ω) + c(yY + vV ) for almost all ω ∈ Ω and all (y, v) ∈ Y × V ,
then (y, v) −→ Iu (y, v) = Ω u ω, y(ω), v(ω) dµ is sequentially upper semicontinuous from L1 (Ω, Y ) × L1 (Ω, V )w into R = R ∪ {−∞} (by L1 (Ω, V )w we denote the Lebesgue–Bochner space equipped with the weak topology). Using this theorem, we can prove the upper semicontinuity of the objective functional. THEOREM 7.5.17 If hypotheses H1 and H2 hold, then the functional J(·) is cupper semicontinuous on F (x0 ).
c PROOF: Note that R+ , B(R+ ), µ is a finite measure space. Suppose xn −→ x w with xn ∈ F (x0 ). Then from Lemma 7.5.9, we have that xn −→ x in ACloc (R+ , RN ). Applying Theorem 7.5.16, we obtain lim sup J(xn ) ≤ J(x) n→∞
⇒ J(·) is c-upper semicontinuous. From Theorems 7.5.15 and 7.5.17, using the theorem of Weierstrass, we conclude that THEOREM 7.5.18 If hypotheses H1 and H2 hold, then there exists x∗ ∈ F (x0 ) such that J(x∗ ) = V (x0 ) (see (7.133)). The open-endedness of the planning horizon, although conceptually and mathematically elegant, is impractical from a computational point of view. To actually solve such a problem, we must usually content ourselves with finite horizon approximates. So we need to know that the values of these finite horizon approximations converge to the value V (x0 ) of the original problem (see (7.133)). Hence we consider the following finite horizon growth problem: b
Vb (x0 ) = sup e−δt u t, x(t), x (t) dt : x ∈ Fb (x0 ) . (7.136) 0
Here the feasibility set Fb (x0 ), is defined by
Fb (x0 ) = x ∈ AC([0, b], RN + ) : x (t) ∈ G t, x(t) 0 ≤ x(0) ≤ x0 .
a.e. on [0, b],
7.5 Continuous-Time Growth Models
597
We want to determine conditions which guarantee that Vb (x0 ) −→ V (x0 )
as b −→ +∞.
For this reason, we strengthen hypotheses H2 as follows. H3 : u : R+ ×RN ×RN −→ R = R ∪ {−∞} is a function that satisfies hypotheses H2 and also (v) There exists ψ ∈ L1 (R+ , µ) such that for almost all t ≥ 0, all x ≥ 0, and all y ∈ RN , we have ψ(t) ≤ u(t, x, y). THEOREM 7.5.19 If hypotheses H1 and H3 hold, then b −→ Vb (x0 ) is continuous on R+ . PROOF: Note that by Theorem 7.5.18, we have V (x0 ) ∈ R. We can find x ∈ F (x0 ) such that
∞
−∞ < V (x0 ) − ε ≤ e−δt u t, x(t), x (t) dt ≤ V (x0 ) < +∞. (7.137) 0
Then because of hypothesis H3 (v) we have
b
e−δt u t, x(t), x (t) dt = lim b→+∞
0
∞
e−δt u t, x(t), x (t) dt.
(7.138)
0
Let b > 0 and set Tb = [0, b]. We have xT ∈ Fb (x0 ) and so b
b
e−δt u t, x(t), x (t) dt ≤ V (x0 ),
0
e−δt u t, x(t), x (t) dt ≤ lim inf Vb (x0 ), b→+∞ 0
∞ −δt
e u t, x(t), x (t) dt ≤ lim inf Vb (x0 ) ⇒ V (x0 ) − ε ≤ ⇒
b
lim
b→+∞
b→+∞
0
(see (7.137)). Because ε > 0 was arbitrary, we let ε ↓ 0 and obtain V (x0 ) ≤ lim inf Vb (x0 ). b→+∞
(7.139)
Next let ξ = lim supb→+∞ Vb (x0 ). We can find bn ↑ +∞ such that ξ−
1 ≤ Vbn (x0 ) n
for all n ≥ 1.
Similarly as for the infinite horizon problem, the finite horizon problem has a solution (see Theorem 7.5.18). So we can find xn ∈ Fbn (x0 ) such that
b
1 e−δt u t, xn (t), xn (t) dt. (7.140) ξ− ≤ n 0 Extend xn on all of R+ by setting xn (t) = 0 for all t > bn . Denote the extended function by xn . Then xn ∈ ACloc (R+ , RN ) and in fact xn ∈ Dx0 (ϕ). From Proposition 7.5.12, we know that Dx0 (ϕ) is c-compact. So we may assume that
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7 Economic Equilibrium and Optimal Economic Planning c
xn −→ y
with y ∈ Dx0 (ϕ).
From this it follows that xn (t) −→ y(t) and
for all t ≥ 0, uniformly on compact sets in R+
xn −→ y in L1loc (R+ , RN ) (see Lemma 7.5.9). w
Fix k ≥ 1. Then for every n ≥ k, we have
xn (t) ∈ G t, xn (t) a.e. on Tbk . By virtue of Proposition 6.6.33, we obtain
y (t) ∈ G t, y(t)
a.e. on R+ .
(7.141)
Moreover, we have 0 ≤ y(0) ≤ x0 and so we conclude that y ∈ F (x0 ). In addition, using Theorem 7.5.17, we have
∞
∞
e−δt u t, xn (t), xn (t) dt ≤ e−δt u t, y(t), y (t) dt. (7.142) lim sup n→∞
0
0
Using hypothesis H3 (v), we have
bn 0
≤
∞
e−δt u t, xn (t), xn (t) dt +
e−δt u t, xn (t), xn (t) dt.
∞
e−δt ψ(t)dt
bn
(7.143)
0
Note that because ψ ∈ L1 (R+ , µ), we have
∞ ψ(t) −→ 0
as n → ∞.
(7.144)
bn
Passing to the limit as n → ∞ in (7.143) and using (7.144), we obtain
bn
lim sup n→∞
e−δt u t, xn (t), xn (t) dt
0 ∞
e−δt u t, xn (t), xn (t) dt ≤ lim sup n→∞ 0
∞
e−δt u t, y(t), y (t) dt (see (7.142)). ≤
(7.145)
0
From (7.140) and (7.145) it follows that
∞
ξ≤ e−δt u t, y(t), y (t) dt ≤ V (x0 ) < +∞, 0
⇒ lim sup Vbn (x0 ) ≤ V (x0 ). n→∞
(7.146)
From (7.139) and (7.146), we conclude that Vbn (x0 ) −→ V (x0 ).
7.6 Expected Utility Hypothesis
599
∗ ∗ Let S and S(x0 ) = x∗ ∈ F (x0 ) : V (x0 ) = b (x0 ) = x ∈ F (x0 ) : Vb (x0 ) = Jb (x ) J(x∗ ) . Here
b
Jb (x) =
e−δt u t, x(t), x (t) dt
for all x ∈ ACloc (R+ , RN ).
0
From the above proof, we deduce the following. PROPOSITION 7.5.20 If hypotheses H1 and H3 hold and bn ↑ +∞, then lim sup Sbn (x0 ) ⊆ Sb (x0 ) in the c-topology. n→+∞
7.6 Expected Utility Hypothesis In this section we deal with an issue that brings us closer to the subject of the next chapter, which is the theory of games. The Expected Utility Hypothesis (EUH for short), is the dominant theory of decision making under risk, employed in economics and finance. In simple terms it says that decision makers choose (or ought to choose) among lotteries, by computing the expected value of the utility of the lottery prizes and finally selecting the lottery with the largest expected utility. In this section we want to characterize the choice behavior, which is consistent with the EUH, in risky situations where there are objective probabilities and a ranking of prizes. This setting covers many situations of interest in economic theory. The items chosen are not commodities, but lotteries. By a lottery we understand a probability distribution on a set of prizes. There is also a set of observations. To each observation corresponds a budget, which is a set of lotteries available to the decision maker and also a choice from each budget, which is a nonempty subset of the budget. If the choices are singletons, then we have a choice function. More generally, for at least some observations the choices may be more than one. This case we refer to as the choice multifunction. The choice function or multifunction determines the choice behavior of the decision maker. There is also a utility function. Since the prizes are monetary (remember we have assumed that there is an a priori ranking of prizes), it is reasonable to assume that the utility function is strictly increasing. Then a choice behavior is EU-rational if it maximizes the expected utility. Let us make precise the mathematical setting for the items described above. There is a set observation T which is a compact Hausdorff topological space and a space of prizes X which is a compact subset of R. A lottery is a probability measure µ on X (i.e., µ ∈ M1+ (X)). We topologize M1+ (X) as follows. DEFINITION 7.6.1 The weak topology on M1+ (X) is the weakest topology on M1+ (X) that makes continuous all functions ξh : M1+ (X) −→ R, h ∈ C(X) defined by
ξh (µ) =
h(x)dµ(x). X
We denote the weak topology on M1+ (X) by w M1+ (X), C(X) or simply by w. REMARK 7.6.2 A subbasic element for the weak topology on M1+ (X) is given by
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7 Economic Equilibrium and Optimal Economic Planning
hdµ − hdµ < ε , Uh,ε (µ) = µ ∈ M1+ (X) : X
X w
where h ∈ C(X), ε > 0. Evidently for a net {µa }a∈J in M1+ (X), we have µa −→ µ if and only if for every h ∈ C(X), we have
hdµa −→ hdµ. X
X
Evidently the weak topology on M1+ (X) is the relative w∗ -topology on M1+ (X), when the latter is viewed as a subspace of the dual Banach space M (X) = C(X)∗ . Hence the weak topology on M1+ (X) is metrizable. + A budget is a set of lotteries. There is a budget multifunction B : T −→ 2M1 (X) \ {∅}. This multifunction provides to the decision maker a set of lotteries for each observation t ∈ T . A choice function is a measurable selector t −→ µt of the multifunction t −→ Bt . For each observation t ∈ T, µt represents the choice of the decision maker. DEFINITION 7.6.3 A choice function µ : T −→ M1+ (X) is said to be EU-rational if there is a strictly increasing utility function u : X −→ R (hence automatically Borel-measurable), such that for every observation t ∈ T , the choice µt maximizes the expected utility over the budget set
u(x)dν(x) ≤ u(x)dµt for every ν ∈ Bt and every t ∈ T. X
X
We say that u rationalizes µ. Given a choice function µ, a compound lottery can be found by first choosing t ∈ T at random according to a prior probability λ on T and then choosing a prize at random in Bt according to µt . This composition process generates a new lottery ν on X as follows. For every Borel set A ⊆ X, we have
µt (A)dλ(t); ν(A) = T
that is, ν = T µt dλ(t), where the vector-valued integral is understood as a Gelfand integral (see Denkowski–Mig` orski–Papageorgiou [194, p. 615]). Recall that M (X) is the Banach space of all finite Borel measures on X. We know that M (X) = C(X)∗ . DEFINITION 7.6.4 (a) If µ, ν ∈ M (X), then we say that ν ≺ µ if and only if
udν ≤ udµ for all u ∈ C(X) nondecreasing. X
X
(b) A choice function µ : T −→ M1+ (X) is ex-ante dominated if there exist another choice function ν : T −→ M1+ (X) and a probability measure λ on T such that
µt dλ(t) ≺ νt dλ(t) (see (a)). T
T
In this case we say that µ is dominated by ν with respect to the prior λ.
7.6 Expected Utility Hypothesis
601
REMARK 7.6.5 In what follows K ={µ∈M (X):0 ≺ µ}, the positive cone of ≺. We characterize EU-rational choice functions. This requires a closer look at the order described by ≺. We start with a result of Nachbin [447, 448]. In what follows M+ (X) denotes the nonnegative elements of M (X) (i.e., µ(A) ≥ 0 for every Borel set A ⊆ X), G = {(x, y) ∈ X × X : y ≤ x}, and D is the diagonal of the Cartesian product X × X. THEOREM 7.6.6 µ ∈ K (i.e., 0 ≺ µ) if and only if there exists ξ ∈ M+ (G) such that
gdµ = g(x) − g(y) dξ for all g ∈ C(X). X
G
This theorem has some remarkable consequences, which are useful in the sequel. COROLLARY 7.6.7 If u∈C(X), it is strictly increasing, µ∈K, and then µ = 0.
X
udµ = 0,
PROOF: According to Theorem 7.6.6, we can find ξ ∈ M+ (G) such that
udµ = u(x) − u(y) dξ. X
G\D
Because u is strictly increasing we have u(x) − u(y) > 0 We have
for all (x, y) ∈ G \ D.
u(x) − u(y) dξ,
udµ =
0= X
G\D
⇒ ξ(G \ D) = 0
(see (7.147)).
Hence for any g ∈ C(X), we have
gdµ = X
(7.147)
g(x) − g(y) dξ = 0
G\D
⇒ µ = 0. From the above corollary, we get the following result at once. COROLLARY 7.6.8 The following statements are equivalent. udν < X udµ. (a) For all u ∈ C(X) strictly increasing, we have X (b) For all u ∈ C(X) nondecreasing, we have X udν ≤ X udµ and µ = ν. COROLLARY 7.6.9 If g ∈ C(X) and for every µ ∈ K \ {0} we have then g is strictly increasing.
X
gdµ > 0,
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7 Economic Equilibrium and Optimal Economic Planning
PROOF: Suppose y < x and let ξ = δ(x,y) ∈ M1+ (G) (by δ(x,y) we denote the Dirac measure concentrated at (x, y)). We choose µ ∈ K \ {0} corresponding to ξ. We have
0< gdµ = g(x) − g(y) dξ = g(x) − g(y), X
G\D
⇒ g is strictly increasing. Finally from Corollaries 7.6.7 and 7.6.8, we obtain the following. COROLLARY 7.6.10 If g0 ∈ C(X) is strictly increasing, then the set C = {µ ∈ K : X g0 dµ = 1} is a w∗ -closed and convex set in M (X) = C(X)∗ and K = {ϑµ : ϑ ≥ 0, µ ∈ C}. Now we can give some alternative descriptions of ≺ restricted on M1+ (X). PROPOSITION 7.6.11 If µ, ν ∈ M1+ (X), then the following statements are equivalent. (a) ν ≺ µ. (b) For every u : X −→ R nondecreasing, we have
X
(c) For every u ∈ C(X) strictly increasing, we have
udν ≤
X
udν
0 was arbitrary. So we let ε ↓ 0 and obtain ν(A) ≤ µ(A). (b)⇒(e): This follows at once by taking u to be the indicator function of the set {y : x ≤ y}. (c)⇒(e): This implication holds because the indicator of {y : x ≤ y} is the poitwise limit of continuous nondecreasing functions.
7.6 Expected Utility Hypothesis
603
(b)⇒(d): Obvious. (e)⇒(d): From Kamae–Krengel–O’Brien [339, Theorem 1], we know that if (e) holds, then there are random variables Sµ and Sν defined on [0, 1] equipped with the Lebesgue measure with values in X and distributions µ and ν, respectively, such that Sν ≤ Sµ and the inequality is strict on a set of positive Lebesgue measure. Therefore if u : X −→ R is strictly increasing, then we have u(Sν ) ≤ u(Sµ ) with strict inequality on a set of positive Lebesgue measure. From this we conclude that (d) holds. REMARK 7.6.12 In the literature, the partial order induced on M1+ (X) by any of the equivalent statements of Proposition 7.6.11, is called first-order stochastic dominance. PROPOSITION 7.6.13 If V is a compact Hausdorff topological space, Y is a ∗ separable Banach space, F : V −→ 2Y \{∅} is a multifunction with w∗ -compact and convex values which is Vietoris continuous (see Definition 6.1.2(c)) from V into Y ∗ endowed with the w∗ -topology (denoted by Yw∗∗ ), and ! ! Rc = conv F (v) and G = F (v)dλ(v) λ∈M1+ (V )
v∈V
V
then Rc = G and G is w∗ -compact and convex. PROOF: From Proposition 6.1.13, we know that F (V ) ⊆ Y ∗ is w∗ -compact. Recall that Yw∗∗ has the convex compact property (i.e., compact sets in Yw∗∗ have a compact closed convex hull). Therefore we deduce that Rc is w∗ -compact, convex. First we show that (7.148) G ⊆ Rc . We argue indirectly. So suppose that we can find y ∗ ∈ G, such that y ∗ ∈ / Rc . Then by the strong separation theorem, we can find y ∈ Y such that σ(y, Rc ) < y ∗ , y
(7.149)
(by ·, · we denote the duality brackets for the pair (Y ∗ , Y )). By definition
y∗ = u∗ (v)dλ(v), V
where Then
λ ∈ M1+ (V
) and u : V −→ X is w∗ -measurable and u∗ (v) ∈ F (v) for all v ∈ V .
y ∗ , y =
∗
u∗ (v), y dλ(v) ≤ V
σ y, F (v) dλ(v) ≤ σ(y, Rc ).
(7.150)
V
Comparing (7.149) and (7.150), we reach a contradiction. Therefore (7.148) holds. Next we show that Rc ⊆ G. (7.151) If y ∗ ∈ G, then by definition we can find λ ∈ M1+ (V ) such that
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7 Economic Equilibrium and Optimal Economic Planning
σ y, F (v) dλ(v) for all y ∈ Y, y ∗ , y ≤
(7.152)
V
if y ∗ ∈ Rc . Note that because Y is separable, the relative w∗ -topology on Rc is w∗
compact and metrizable. So we can find {yn∗ }n≥1 ⊆ conv F (V ) such that yn∗ −→ y ∗ in Y ∗ . There exist {λn }n≥1 ⊆ M1+ (V ), each with finite support, such that yn∗ =
f (v)dλn (v), V
where f : V −→ X ∗ is a w∗ -measurable selector of F . We have
σ y, F (v) dλn (v). yn∗ , y ≤
(7.153)
V w∗
We may assume that λn −→ λ in M (V ) = C(V )∗ and λ ∈ M1+ (V ). Because v −→ σ y, F (v) is continuous (recall F is Vietoris continuous), we have
σ y, F (v) dλn (v) −→
V
σ y, F (v) dλ(v).
(7.154)
V
Therefore, if we pass to the limit as n → ∞ in (7.153) and we use (7.154), we obtain y ∗ , y ≤
σ y, F (v) dλ(v), V
⇒ y∈G
(see (7.148)),
⇒ Rc ⊆ G.
(7.155)
Combining (7.151) and (7.155), we conclude that Rc = G.
Now we can have the main theorem on the rationalization of the choice function. THEOREM 7.6.14 If (i) The set of prizes X is a compact subset of R; (ii) The set of observations T is a compact Hausdorff topological space; (iii) If on M1+ (X), we consider the weak topology (see Definition 7.6.1), the budget + multifunction B : T −→ 2M1 (X)\{∅} has compact convex values and it is Vietoris continuous; and µ : T −→ M1+ (X) is the choice function, then (a) If µ is ex-ante dominated, then it is not EU-rational. (b) If µ is continuous, then one of the following alternatives holds. (b1 ) µ is rationalized by a continuous strictly increasing utility function. (b2 ) µ is ex-ante dominated.
7.6 Expected Utility Hypothesis
605
PROOF: (a) Clear from Definition 7.6.4. (b) Suppose µ is continuous and it is ex-ante dominated. Then we can find a measurable selector ν of the budget multifunction such that for some λ ∈ M1+ (T ), we have
µt dλ(t) ≺ νt dλ(t) (see Definition 7.6.4(b)), T
T ⇒ (νt − µt )dλ(t) & 0. (7.156) T
Set G=
!
(Bt − µt )dλ(t) : λ ∈ M1+ (T ) .
(7.157)
T
If µ is not ex-ante dominated, then G ∩ K = ∅ (see (7.156)). Let C ⊆ K be as in Corollary 7.6.10 (i.e., a w∗ -closed convex base of the cone K). Then G ∩ C = ∅. Recall that the weak topology on M1+ (X), is the relative w∗ -topology, that + M1 (X) inherits from the dual Banach space M (X) = C(X)∗ . So t −→ Bt − µt is Vietoris continuous into M (X) with the w∗ -topology (denoted by M (X)w∗ ) and has nonempty, compact, and convex values in M (X) w∗ . Because ∗ G ∩ C = ∅, by the strong separation theorem we can find u ∈ C(X) = M (X)w∗ such that and
ν, u ≤ 0
for all ν ∈ G
ξ, u > 0
for all ξ ∈ C.
(7.158)
(7.159)
Here by ·, · we denote the duality brackets for the pair C(X), M (X) . From (7.159), we have
ξ, u = udξ > 0 for all ξ ∈ C. X
This combined with Corollaries 7.6.10 and 7.6.9, implies that u ∈ C(X)
is strictly increasing.
We claim that it rationalizes µ. To this end let t ∈ T and let ν ∈ Bt . Consider λ ∈ M1+ (T ) defined by λ = δ{t}
(the Dirac measure concentrated on t).
Note that ν − µt ∈ Bt − µt and so
(ν − µt )dλ(t) = ν − µt ∈ G. T
Then because of (7.158), we have u, ν − µt ≤ 0,
udν ≤ udµt , ⇒ X
⇒ u
X
rationalizes µ
(see Definition 7.6.3).
EXAMPLE 7.6.15 The compactness of the space X of prizes cannot be dropped: Indeed consider the situation where T ={t}, X
= {0, 1, 2, . . .} and Bt = {µk }k≥0 with µ0 = δ{t} and for n ≥ 1, µn = (1/n)δ{0} + 1 − (1/n) δ{2} . The choice µ0 is not ≺-dominated, but we can check that it is not EU-rational.
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7 Economic Equilibrium and Optimal Economic Planning
7.7 Remarks 7.1: The first to formulate and study of model of a pure exchange was Edgeworth [219]. He introduced a model with two economic agents and two commodities and using the recontracting adjustment process (i.e., no allocation occurs if it can be blocked – recontracted out – by a group of agents), he defined the well-known (Edgeworth) contract curve. This curve represents the locus of points where the marginal rate of substitution between the pair of goods, is the same for both agents. This curve can be illustrated, using a diagrammatic construction known as the Edgeworth box. The core, introduced in Definition 7.1.3 is a generalization of the contract curve. Note that the notion of core does not involve prices and it is the result of pure trading among the agents. In contrast, the other equilibrium notion, the Walras (or competitive) equilibrium, introduced in Definition 7.1.5(b), assumes that in the market prevail some prices for the commodities, which are accepted by all agents involved. But, intuitively speaking, we know that in reality, prices are a convenience and simply provide a common yardstick to measure all traded quantities. Therefore it is reasonable to claim that the two equilibrium notions should lead to the same allocations. A moment’s reflection can lead to the conclusion that this should be the case in a perfect competition environment (see Theorem 7.1.7). This means that no individual agent or group of agents has enough power alone to influence the outcome of the economic process. The idea to model perfect competition using a continuum of agents (more precisely a nonatomic measure) is due to Aumann [43, 45]. In Aumann [43] we find the core-Walras equivalence theorem (see Theorem 7.1.7), and in Aumann [45], the author establishes the existence of a Walras (competitive) equilibrium (see Theorem 7.1.10). The results of Aumann were extended to economies with production by Hildenbrand [295]. There are some other equilibrium notions such as Pareto equilibrium, quasi-competitive equilibrium, and expenditure minimizimg equilibrium. A comparative study of all these different equilibrium notions can be found in Hildenbrand [296]. There is another approach in the core theory in which the primitive notion in the economic model is the coalition and not the individual agent. This changes the necessary mathematical tools and so multifunctions are replaced by set-valued measures (multimeasures). This approach was initiated by Vind [595]. There is still a third, more complicated pure exchange model, with “big” and “small”traders due to Dreze–Gabsewicz–Schmeidler–Vind [207]. In the last twenty years, a lot of effort was put into extending equilibrium theory to economies with an infinite-dimensional commodity space. In this direction we mention the works of Bewley [74], Aliprantis–Brown [8], Florenzano [250], MasColell [411], Jones [333], Zame [618], and the book of Aliprantis–Brown–Burkinshaw [9], where a more detailed bibliography can be found. The books of Debreu [185], Hildenbrand [297], Hildenbrand–Kirman [298], and Takayama [572] contain additional material on economic equilibrium theory. In particular, Hildenbrand [297] deals with economies described by a continuum of agents. It does historical justice to say that the founding fathers of mathematical economics are Cournot, Walras, Pareto, Edgeworth, and Menger. Antoine Cournot (1801–1877) was French and is considered to have made the initial step in developing econometrics. Leon Walras (1834–1910) was French and the first holder in 1870 of the chair of political economy at the University of Lausanne in Switzer-
7.7 Remarks
607
land. His two main contributions in the theory of mathematical economics are the development of the marginal utility approach to the theory of value (1873) and the development of the theory of general equilibrium (1874–1877). Vifredo Pareto (1848–1923) was Italian and the successor of Walras to the chair of political economy at the University of Lausanne (1892). He is better known for the Pareto optimum, which is described as a position from which it is impossible to improve anyone’s welfare by altering production or exchange without impairing someone else’s welfare. He is also the founder of positive economics, an economic science purged of all ethical elements. Pareto rejected socialism, justified the inequality of income, and viewed with sympathy the coming to power in Italy of the fascists (October 1922). Francis Ysidro Edgeworth (1845–1926) was British and professor of political economy at Oxford University from 1891 to 1922. He is credited with inventing indifference curves and the contract curve. Finally Carl Menger (1840–1921) was Austrian and one of the formulators (together with Walras) of the marginal utility theory of value. He is also the founder of the Austrian school of economic thought. 7.2: The question of price characterization of efficient capital accumulation and allocation of resources in infinite horizon economies (“efficient pricing”) started with the work of Malinvaud [403, 404], and continued by Gale [254] in the context of optimal growth for multisector growth models (“competitive pricing”). Gale [254] introduced the notion of strong maximality (see Definition 7.2.24(b)) and assumed that the utility function is strictly concave. Brock [107] objected to this hypothesis, primarily because it excludes the von Neumann model. Brock [107] replaced the strict concavity of the utility function by the assumption that the optimal stationary program is unique and used the weak maximality criterion (see Definition 7.2.26(b)). Other contributions to the subject were made by Peleg [492], Peleg–Yaari [493], McKenzie [420, 423], Mitra [431, 432, 434], Mitra–Zilcha [433], Dechert–Nishimura [183], Khan–Mitra [350], and Joshi [334]. The books of Makarov–Rubinov [402] and Takayama [572] deal with discrete time growth models and contain additional references on the subject. Starting with the work by Hurwicz [319] on informational decentralization, people raised the question of characterizing the optimality of competitive programs in terms of conditions that can be verified by agents in an informationally decentralized mechanism, which involves two basic restrictions; first that there is initial dispersion of information and each economic unit has only partial knowledge of the environment; second that there is limited communication and so it is impossible to completely centralize dispersed information about the environment. Significant contributions in this direction, were made by Brock–Majumdar [111], Hurwicz– Majumdar [320], Hurwicz–Weinberger [321], and Dasgupta–Mitra [177, 178] who introduced the reachability condition (R) (see Definition 7.2.17) and have additional particular economic models which satisfy this condition. 7.3: Turnpike theory originates with a simple remark in the book of Dorfman– Samuelson–Solow [203]. Radner [509] proved a weak turnpike theorem for the von Neumann–Gale model, and McKenzie [418] did this for a Leontief-type model. A strong turnpike theorem was first proved by Nikaido [461]; see also Tsukui [589] and Winter [607]. Additional results can be found in McKenzie [419, 420, 421, 422, 423], Araujo–Scheinkman [28], Benveniste–Scheinkman [65], Fershtman–Mullar [242], and Read [512], and in the books of Arkin–Evstingeev [29] and Takayama [572].
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7.4: Optimal growth under uncertainty was first investigated in the early seventies. We mention the works of Brock–Mirman [108, 109], Dana [173], Dynkin [215], Evstingeev [230], Jeanjean [331], Radner [510], and Tacsar [570]. Brock–Mirman modelled the uncertainty in their model with a sequence of independent, identically distributed (iid) random variables. Dynkin–Evstingeev, Jeanjean, Radner, and Tascar considered a stationary Markov process modelling the uncertainty in the economy, considered the more general situation of a probability distribution on the set of sequences of states. This approach incorporates the other two via Kolmogorov’s theorem (for the idd case) and the Ionescu–Tulcea disintegration theorem (for the Markovian case). Soon thereafter Zilha [628, 629] considered the discounted model (see Zilha [628]) and the undiscounted model using the strong maximality criterion (see Zilha [629]) and established the existence of supporting prices for optimal programs. More recent works on the subject are those by Evstingeev–Katyshev [231], Majumdar–Radner [400], Majumdar–Zilha [401], Nyarko [466], Zilha [630], and Pantelides–Papageorgiou [469]. In particular, Nyarko [466] extends to a stochastic economic growth model the work of Dasgupta–Mitra [177] on temporal and informational decentralization. Various aspects of the theory of stochastic economic dynamics can be found in the book of Arkin–Evstingeev [29]. 7.5: Continuous-time economic growth problems present more technical difficulties than the discrete ones. So the choice of the topology on the set of feasible programs is crucial. The investigation of the c-topology and the α-topology was conducted by Becker–Boyd–Sung [60], who this way were able to rectify some erroneous claims existing in the literature. Namely, Brock–Haurie [110] and Yano [612] claimed that stock convergence implies, at least for a subsequence, flow convergence almost everywhere and Balder [55] claimed that the c-topology and the weak α-topology coincide. We should mention that the problem studied by Becker–Boyd–Sung [60] involves a recursive utility. Undiscounted continuous-time growth models can be found in the works of Brock–Haurie [110], Papageorgiou [479], and in the book of Carlson–Haurie–Leizarowitz [130]. 7.6: The model used in this section is due to Border [88]; see also Border [87]. Earlier works on the subject by Fishburn [248] and Ledyard [372] assumed the set of observations and the budget set are finite. Definition 7.6.4(a) is due to Nachbin [447]; see also Nachbin [448]. Proposition 7.6.11 is essentially due to Kamae–Krengel– O’Brien [339].
8 Game Theory
Summary. *Game-theoretic models provide a substantial amount of generalization of the basic notions of mathematical economics, such as core, equilibria, saddle points and intertempolar optimum. In this chapter, we deal with different models in game theory. We start with noncooperative games which lead to the notion of “Nash equilibria”. Then we pass to cooperative games, for which we can define the notion of core. After that we consider random games with a continuum of players. For such games, we show the existence of Cournot–Nash equilibria. We also consider Bayesian games stochastic 2-player, and zero-sum games. Finally, using the notion of ε-subdifferential of convex functions, we prove the existence of approximate Nash equilibria for noncooperative games.
Introduction Any discussion of economic models is incomplete if it is not accompanied by a presentation of game theory. This is because the models in game theory provide a substantial amount of generalization of the basic notions of mathematical economics such as core, equilibria, saddle point, and intertemporal optimum. A game is a decision problem for a group of players who may have cooperative or noncooperative (conflicting) objectives to optimize, with different or equal information structures and finite strategies from which to choose. In this chapter we present some basic issues from the theory of games. We deal with both static and dynamic (over an infinite discrete-time horizon) games and with deterministic and stochastic games. In Section 8.1, we examine noncooperative games with n-players. The basic equilibrium concept for such games, is the so-called Nash equilibrium. We establish the existence of such equilibrium. In Section 8.2, we deal with cooperative games with n-players. We consider the characteristic function form of such games and we study both side-payment games and no-side-payment games. For both the central stability concept is the core. We present theorems that guarantee the nonemptiness of the core. In Section 8.3 we deal with random games, with a continuum of players (namely, atomless measure space of players) and, an infinite-dimensional strategy space. For N.S. Papageorgiou, S.Th. Kyritsi-Yiallourou, Handbook of Applied Analysis, Advances in Mechanics and Mathematics 19, DOI 10.1007/b120946_8, © Springer Science+Business Media, LLC 2009
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such games, using notions and results from multivalued analysis, we prove the existence of Cournot–Nash equilibria. Continuing with random games, in Section 8.4 we consider Bayesian games with an infinity of players and an infinite-dimensional strategy space. In such games the players have different information structures that are modelled as sub-σ-fields of the σ-field of all possible outcomes. Again we show that under reasonable hypotheses, Bayesian games admit Cournot–Nash equilibria. In Section 8.5 we investigate stochastic 2-player zero-sum games, using the stochastic dynamic programming model, which we also present. We show the existence of a saddle point. Finally in Section 8.6 we introduce and use the notion of ε-subdifferential of a convex function, in order to show that noncooperative n-player games with noncompact strategy sets admit approximate Nash equilibrium.
8.1 Noncooperative Games–Nash Equilibrium In this section we study n-person games in normal form, in which the players exhibit a noncooperative (individualistic) behavior. The central concept for such games is the notion of Nash equilibrium. We start with a description of the n-person game in normal (strategic) form. So let N = {1, 2, . . . , n}, n ≥ 2, be the set of players. Each player k ∈ N has a strategy set Xk which describes all strategies available to him. We set X=
n =
Xk
(the multistrategies set).
k=1
Also a set S(N ) ⊆ X is given describing the feasible multistrategies. The preference relation of player k ∈ N among the strategies available to him, is described by a utility function uk : S(N ) −→ R, which to each feasible multistrategy x ∈ S(N ) assigns the number uk (x) measuring the utility produced by x. Of course he prefers multistrategies that yield a maximum utility. We can also think of uk as a loss function, in which case −uk can be regarded as a utility function. The interpretation of uk as a utility function, which we have adopted here, is motivated from the economic applications. From the viewpoint of player k ∈ N , the set X of multistrategies, is split as follows X = X × Xk
with X =
=
Xi .
i=k
We think of kc = N \ {k} as a coalition adverse or complementary to player k ∈ N . Also let pk and p be the projections from X onto Xk and X, respectively. The decomposition (x, xk ) ∈ X × Xk of a multistrategy x ∈ X = X × Xk is important for player k ∈ N , because this way he distinguishes the component xk which he can control from the components x = {xi }i=k ∈ X over which he has no control. DEFINITION 8.1.1 A game in normal form is the collection {Xk , uk }k∈N .
8.1 Noncooperative Games–Nash Equilibrium
611
Suppose that no player has any information about the choices made by the other players. Then a cautious way to proceed is for each player to maximize her minimum utility. More precisely, for every player k ∈ N , let vk (xk ) = inf uk (x) : x ∈ S(N ), pk (x) = xk . (8.1) The quantity vk (xk ) in (8.1) represents the least utility for player k among all feasible multistrategies, when she employs strategy xk . Then she will try to maximize vk (·); that is,
vk = sup vk (xk ) : xk ∈ pk S(N ) . (8.2) Note that vk is the outcome of a sup inf process. DEFINITION 8.1.2 The vector v = {vk }n k=1 is the conservative value of the game and a multistrategy x∗ = {x∗k }n k=1 ∈ S(N ) such that vk = vk (x∗k ) is a conservative or maximin multistrategy. We have the following existence theorem for conservative multistrategies. THEOREM 8.1.3 If the strategy sets Xk , k ∈ N , are Hausdorff topological spaces, S(N ) ⊆ X is compact (for the product topology on X), and uk : S(N ) −→ R, k ∈ N , is upper semicontinuous, then the game admits a conservative multistrategy x∗ ∈ S(N ). PROOF: Note that vk (xk ) = inf uk (x) : x ∈ S(N ), pk (x) = xk = inf uk (x, xk ) : (x, xk ) ∈ S(N ) .
So from Theorem 6.1.18(a) is upper we have that vk : pk S(N ) −→ R
semicon tinuous. Because pk S(N ) is compact in Xk , we can find x∗k ∈ pk S(N ) such that vk = vk (x∗k ), k ∈ N. Evidently x∗ = {x∗k }k∈N is a conservative multistrategy for the game.
REMARK 8.1.4 In general note that player k ∈ N will reject any multistrategy x yielding a utility uk (x) smaller than vk . So vk is the minimum that each player can tolerate. In the above definition of conservative strategy, every player chooses his strategy independently of the rest of the players. We want to consider a different kind of behavior, in which every player varies his strategy based on the choice of the complementary coalition. Roughly speaking, what happens is the following: As soon as player k announces her intention to play, the complementary coalition N \ {k} moves first and then player k responds. With such a rule, an equilibrium will be a
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multistrategy x∗ in which no player k can increase her utility by moving away from strategy x∗k , provided that the other players do not alter their choice. So we have uk (x∗ , x∗k ) = sup uk (x∗ , y) : y = pk (x), x ∈ S(N ) . (8.3) In other words, (8.3) says that given the multistrategy x∗ of the complementary coalition N \ {k}, the k-player responds by maximizing the utility y −→ uk (x∗ , y)
y = pk (x),
x ∈ S(N ).
DEFINITION 8.1.5 Given a game {Xk , uk }k∈N in normal form, a Nash equilibrium is a feasible multistrategy x∗ ∈ S(N ), such that for every k ∈ N , we have uk (x∗ , y) ≤ uk (x∗ , x∗k )
for all y = pk (x), x ∈ S(N ).
REMARK 8.1.6 So in the Nash equilibrium there is no incentive for a player k ∈ N to change her strategy x∗k by herself. This is a noncooperative equilibrium concept. To study the notion of Nash equilibrium, we introduce the function ξ : S(N ) × S(N ) −→ R defined by ξ(x, z) =
n
ui (x) − ui (x, zi ) .
(8.4)
i=1
PROPOSITION 8.1.7 If {Xk , uk }k∈N is a game in normal form and for x ∈ S(N ) we have ξ(x, z) ≥ 0 for all z ∈ S(N ), (8.5) then x is a Nash equilibrium for the game. The converse is true provided that S(N ) is a product space. PROOF: Suppose that x ∈ S(N ) satisfies (8.5) and let z ∈ S(N ) such that p(z) = x = {xi }i=k . Using (8.5), we have 0 ≤ ξ(x, z) =
n i=1
=
ui (x) − ui (x, zi )
ui (x) − ui (x, xi ) + uk (x) − uk (x, zk )
i=k
= uk (x) − uk (x, zk ). There x ∈ S(N ) is a Nash equilibrium (see Definition 8.1.5). n n " " Now suppose that S(N ) = Si ⊆ Xi and let x∗ ∈ S(N ) be a Nash equilibi=1
i=1
rium. Then by Definition 8.1.5, we have ui (x∗ ) − ui (x∗ , zi ) ≥ 0
for all zi ∈ Si , i ∈ {1, . . . , n} = N.
Adding these inequalities, we obtain ξ(x∗ , z) ≥ 0 for all z ∈ S(N ).
To prove the existence of a Nash equilibrium, we need the following abstract theorem, known as Ky Fan’s inequality. For a proof, we refer to Aubin–Ekeland [39, p. 327].
8.1 Noncooperative Games–Nash Equilibrium
613
THEOREM 8.1.8 If K ⊆ RN is a nonempty, compact, convex set and ξ : K × K −→ R is a function satisfying (i) For every z ∈ K, x −→ ξ(x, z) is upper semicontinuous; (ii) For every x ∈ K, z −→ ξ(x, z) is quasiconvex (see Definition 2.3.11); (iii) For every z ∈ K, ξ(z, z) ≥ 0, then we can find x∗ ∈ K such that inf ξ(x∗ , z) ≥ 0.
z∈K
Now we can state and prove the main existence theorem for Nash equilibria. N THEOREM 8.1.9 If {Xk , uk }k∈N is a game in normal form, S(N
) ⊆ R is compact convex, for every k ∈ N uk is continuous, and for every x ∈ p S(N ) , z −→ uk (x, z) is quasiconcave, then there exists a Nash equilibrium.
PROOF: The function ξ(x, z) =
n
ui (x) − ui (x, zi ) ,
(x, z) ∈ S(N )×S(N )
i=1
is continuous in x and quasiconvex in z. Moreover, ξ(z, z) ≥ 0
for all z ∈ S(N ).
So we can apply Theorem 8.1.8 and obtain x∗ ∈ S(N ) such that ξ(x∗ , z) ≥ 0
for all z ∈ S(N ).
This then by virtue of Proposition 8.1.7 implies that x∗ ∈ S(N ) is a Nash equilibrium for the game. At this point it is instructive to briefly present a special case, known as a bimatrix game. EXAMPLE 8.1.10 Bimatrix Game: This is a two-player game; that is, N = {1, 2} and the two strategy sets X1 and X2 are finite sets; for example, X1 = {1, . . . , l1 } and X2 = {1, . . . , l2 }. Also we have two utility functions u1 and u2 . We set apq = u1 (p, q)
and
bpq = u2 (p, q)
for all p ∈ X1 , q ∈ X2 .
So apq is the first player’s payoff when the first player chooses a (pure) strategy p ∈ X1 and the second chooses a (pure) strategy. Similarly for the payoff bpq of the 2 second player. Then the game Xk , uk k=1 reduces to the following bimatrix game: 5 6 (a11 , b11 ) · · · (a1l2 , b1l2 ) . (al1 1 , bl1 1 ) · · · (al1 l2 , bl1 l2 ) Evidently a Nash equilibrium in general need not exist because the strategy sets X1 and X2 are not in general convex. The sets X1 and X2 are said to represent the pure strategies available to player 1 and player 2, respectively. When apq + bpq = 0 for all (p, q) ∈ {1, . . . , l1 } × {1, . . . , l2 }, we say that we have a zero-sum game. It
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represents a situation of direct conflict between the two players, because the gain of one is equal to the loss of the other. To be able to produce a Nash equilibrium, we need to consider mixed strategies. Namely, we are led to another game in normal form with strategy sets lk lk l X k = x = (xi )i=1 ∈ R+k : xi = 1 ,
k = 1, 2.
i=1
A point in the simplex X k is a mixed strategy and each component xi represents the probability attached to the pure strategy i. When the two players choose mixed strategies (x, y) ∈ X 1 × X2 , then the payoffs are apq xp yq for player 1 (8.6) p,q
and
bpq xp yq
for player 2.
(8.7)
p,q
In this case all hypotheses of Theorem 8.1.9 are satisfied and so a Nash equilibrium exists. Now let us see a specific bimatrix game with a unique Nash equilibrium. This is the famous prisoner’s dilemma problem. There are two players who are prisoners and have been charged with some joint crime. Each player has a strategy set with two elements (strategies). The first strategy is to plead innocent and the second is to plead guilty. If both prisoners plead guilty they will be punished. If both plead innocent, then there is no solid case against them and their punishment will be light. If one pleads guilty and other pleads innocent, then the first will be rewarded for his honesty and will be released, and the second will be punished severely. So let the bimatrix representing the game be the following one 5 6 (10, 10) (0, 15) . (8.8) (15, 0) (5, 5) The entries in this bimatrix are not the terms of imprisonment, but are the utility levels. Evidently the Nash equilibrium for this bimatrix game exists uniquely and is the pair of strategies that generates (5, 5); that is, both prisoner’s plead innocent. If we pass to the mixed strategies the situation does not change. The Nash equilibrium remains the same (a pure strategy) and is unique. Indeed let (x∗ , y ∗ ) ∈ X 1 ×X 2 be a Nash equilibrium. Then given y ∗ , x∗1 solves max 10.x1 y1∗ + 0.x1 y2∗ + 15.(1 − x1 )y1∗ + 5.(1 − x1 )y2∗ (see (8.6)) 0≤x1 ≤1
⇒ x∗1 = 0. By using (8.7) we obtain y1∗ = 0. So indeed is the strategy producing utility (5, 5) (see (8.8)) as before. Note that in this game the incentive, for a rational player concerned only with his own survival, is to confess and let the other suffer the consequences. But, as both are motivated to act in this way, they end up with an outcome that is worse for both than if they had been able to make a binding agreement between themselves
8.1 Noncooperative Games–Nash Equilibrium
615
that no one confesses. This example illustrates how a rational behavior at a micro level leads to an apparently irrational macro outcome. An interesting interpretation of the bimatrix game is in wage negotiations in a labor market. In this situation, player 1 is the labor union, player 2 is the management. Player 1 wants high wages, whereas management wishes to give as small a wage increase as possible. Hence the game is clearly a zero-sum game. Next we go beyond games in normal form by introducing the notion of feasibility. This notion allows us to deal with situations in which the choices of players cannot be made independently (as was the case thus far). For example, consider several mining companies which extract coal from a field. Each company chooses to extract a certain amount xk and to sell it. The price of the coal depends on the amount sold. So each company (producer) has a partial control of the price and consequently of its n profits. On the other hand xk can not be chosen independently, because xk (n = k=1
number of producers), cannot exceed the total amount of coal in the field. For this reason, for every player k ∈ N , we introduce a correspondence Fk : X −→ Xk , which describes those strategies available to player k ∈ N , when all players have chosen their strategy. The way we have defined Fk it also depends on the strategies of player k. This is done only for reasons of technical convenience. In concrete situations Fk is independent of players k choice. Also now, in contrast to the previous model, we do not assume that the preferences of the players are represented by utility functions. Instead we are more general and assume that each player k ∈ N , has a good reply n " multifunction Uk . So Uk : X = Xi −→ Xk and yk ∈ Uk (x), if yk is a good reply i=1
for player k ∈ N to multistrategy x ∈ X. DEFINITION 8.1.11 A generalized game or abstract economy is a collection {Xk , Fk , Uk }k∈N . REMARK 8.1.12 This is not a game. No player can individually play this game, because he needs to know the strategies of the other players in order to determine his feasibility set. For this reason we call it a generalized game. A generalized game is a good mathematical tool to produce existence theorems in various applied contexts. In this general setting, the notion of Nash equilibrium takes the following form: DEFINITION 8.1.13 A Nash equilibrium for an abstract economy {Xk , Fk , Uk }k∈N , is a multistrategy n " x∗ ∈ X such that x∗ ∈ F (x∗ ) = Fk (x∗ ) and Fk (x∗ ) ∩ Uk (x∗ ) = ∅ for all k ∈ N . k=1
REMARK 8.1.14 So a Nash equilibrium for an abstract economy, is a multistrategy for which no player has a good reply. PROPOSITION 8.1.15 If Xk ⊆ Rlk are nonempty, compact, convex sets and n " Xk −→ 2Xk is a multifunction such that Uk : k=1
(i) For every x ∈ X, Uk (x) is convex and
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8 Game Theory
(ii) For every {xk } ∈ X, Uk {xk } = x ∈ X : xk ∈ U (x) is open, then we can find x = {xk }n k=1 ∈ X such that xk ∈ Uk (x)
or
Uk (x) = ∅
for all k ∈ N = {1, . . . , n}.
PROOF: Let Γk = {x ∈ X : Uk (x) = ∅}. Because of hypothesis (ii), the set Γk is open in X. Let Uk = Uk Γ : X −→ 2Xk \{∅}. Apply Proposition 6.3.4 to obtain a k continuous function uk : Γk −→ Xk such that uk (x) ∈ Uk (x) for all x ∈ Γk . Consider the multifunction Gk : X −→ 2Xk \{∅} such that if x ∈ Γk {uk (x)} . Gk (x) = Xk otherwise Evidently Gk is usc and has nonempty, compact, and convex values. Hence so is n " Gk (x) from X into the nonempty, compact, and the multifunction x −→ G(x) = k=1
convex subsets of X. So we can apply the Kakutani–Ky Fan fixed point theorem (see Theorem 6.5.19) and obtain x∗ ∈ X such that x∗ ∈ G(x∗ ). If Gk (x∗ ) = Xk , then ∗ ∗ ∗ ∗ x∗k ∈ Gk (x∗ ) (x∗ = {x∗k }n k=1 ) implies that xk = uk (x ) ∈ Uk (x ). If Gk (x ) = Xk , ∗ then Uk (x ) = ∅. REMARK 8.1.16 A careful reading of the above proof reveals that if one of the Uk s is single-valued, then the result is automatically true (in this case the continuous selection γk is obtained trivially). COROLLARY 8.1.17 If Xk ⊆ Rlk is nonempty, compact, and convex, and for every k ∈ N = {1, . . . , n} the multifunction Uk : X −→ 2Xk has an open graph and satisfies xk ∈ / conv Uk (x) for all x ∈ X, then we can find x ∈ X such that Uk (x) = ∅ for all k ∈ N . PROOF: The multifunction x −→ conv Uk (x) satisfies hypotheses (i) and (ii) of Proposition 8.1.15. So by virtue of that result, we can find x = {xk }n k=1 ∈ X such that xk ∈ convUk (x) or conv Uk (x) = ∅, k ∈ N . But by hypothesis xk ∈ / conv Uk (x). Therefore Uk (x) = ∅ for all k ∈ N . THEOREM 8.1.18 If {Xk , Fk , Uk }k∈N is an abstract economy and for every k ∈ N, (i) Xk ⊆ Rlk is nonempty, compact, and convex, (ii) Fk : X −→ 2Xk has nonempty, closed, and convex values, is usc, for every x ∈ X we have Fk (x) = cl int Fk (x) and x −→ int Fk (x) has an open graph, (iii) Gr Uk is open in X × Xk , (iv) For every x ∈ X, xk ∈ / conv Uk (x), then the abstract economy admits a Nash equilibrium x∗ ∈ X (see Definition 8.1.13).
8.2 Cooperative Games PROOF: Let V0 = X, Vk = Xk for all k ∈ N and V = V0 × of V is denoted by (x, y) ∈ V with x ∈ V0 and y ∈
n "
n "
617
Vk . A generic element
k=1
Vk . For every k ∈ N , we
k=1
introduce a multifunction Hk : V −→ 2
Vk
defined by
H0 (x, y) = {y}
(8.9)
and for k ∈ N, Hk (x, y) =
int Fk (x) conv Uk (y) ∩ int Fk (x)
if yk ∈ / Fk (x) . if yk ∈ Fk (x)
(8.10)
Clearly H0 is single-valued continuous, whereas for k ∈ N, Hk has convex values (see hypothesis (ii)) and yk ∈ / Hk (x, y) (see hypothesis (iv)). We claim that GrHk ⊆ V × Vk is open. To this end let Ak = (x, y, zk ) : zk ∈ int Fk (x) Bk = (x, y, zk ) : yk ∈ / Fk (x) Ck = (x, y, zk ) : zk ∈ conv Uk (y) . Because by hypothesis (ii), x −→ int Fk (x) has an open graph, we deduce that Ak is open. Suppose that yk ∈ / Fk (x). Then we can find a closed neighborhood D of yk such that Fk (x) ⊆ Dc and because Fk is usc, it follows that Bk is open. Moreover, Ck is open because of hypothesis (iii). Since GrHk = (Ak ∩ Bk ) ∪ (Ak ∩ Ck ), we infer that indeed Gr Hk ⊆ V × Vk is open. Apply Proposition 8.1.15 (see also Remark 8.1.16) to obtain (x∗ , y ∗ ) ∈ V such that x∗ = y ∗
and
Hk (x∗ , y ∗ ) = ∅
for all k ∈ N
(see (8.9)).
(8.11)
From (8.10) and (8.11), we obtain conv Uk (x∗ ) ∩ int Fk (x∗ ) = ∅ ∗
∗
⇒ Uk (x ) ∩ int Fk (x ) = ∅ ∗
for all k ∈ N,
for all k ∈ N.
(8.12)
∗
But by hypothesis (ii), Fk (x ) = cl [int Fk (x )]. So from (8.12), we conclude that Uk (x∗ ) ∩ Fk (x∗ ) = ∅
for all k ∈ N,
⇒ x∗ ∈ X is a Nash equilibrium (see Definition 8.1.13).
8.2 Cooperative Games In the previous section, we examined noncooperative games and we introduced the concept of Nash equilibrium according to which each player will consider only unilateral strategy changes in deciding whether or not she can be made better off. In
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8 Game Theory
this section we consider games in which players cooperate in order to improve their position. So coalitions of players may have more power to make their members better off than they would be by acting individually. So let N = {1, . . . , n} be the set of players and 2N \ {∅} (all nonempty subsets of N ) be the set of all possible coalitions of players. Throughout this section, we use the following notation. If {Xk }k∈N is"a family of sets and B ∈ 2N \ {∅}, " B then X = Xk . Similarly we set RB = R. By pB : X N −→ X B (resp., k∈B
k∈B
pB : RN −→ RB ), we denote the projection map. The simplest cooperative game is a game in characteristic function form with side payments (a side-payment game). This is defined as a function v : 2N \{∅} −→ R, which for every coalition B ∈ 2N \ {∅} assigns the maximal total utility of the coalition that its members can generate by their cooperation regardless of the actions of the players outside the coalition. A side-payment game corresponds to a game in normal form (see Definition 8.1.1). Indeed, for each player k ∈ N , his strategy set is given by the mutually disjoint union ! Xk = Xk,B : B ∈ 2N \ {∅}, k ∈ B . If x ∈ Xk,B , then player k ∈ N cooperates with the players in the coalition B. We set uk (x, xk ) = −∞ (x = pB c (x); see Section 8.1), / Xm,B . if for the coalition B for which xk ∈ Xk,B , there exists m ∈ B such that xm ∈ Also let uk be independent of {xk }k∈N \B where B is the coalition for which xk ∈ Xk,B . Then the required game satisfies for every coalition B ∈ 2N \ {∅}
uk ≤ v(B) = uk (x, xk ) k∈B for all k ∈ B, xk ∈ Xk,B . (uk )k∈B : k∈B
But clearly the associated game in normal form is not easy to deal with and for this reason, we prefer to work directly with the value function ν. The basic solution concept for cooperative games, is analogous to the notion of core of a pure competition economy (see Definition 7.1.3). DEFINITION 8.2.1 The core of a side-payment game v is the set C(v) of all u = (uk )k∈N ∈ RN such that (a) uk ≤ v(N ). k∈N uk < v(B). (b) There is no coalition B ∈ 2N \{∅} such that k∈B
REMARK 8.2.2 In the above definition, the requirement (a) corresponds to the feasibility of the utility vector u = {uk }k∈N , and the requirement (b) implies that no coalition can improve upon u ∈ RN . Note that the core C(v), is the set of solutions of the 2n -linear inequalities, −(χN , x)Rn ≥ −v(N ) and (χB , x)Rn ≥ v(B), ek ,ek = (ek,l )n where for every B ∈ 2N \{∅}, χB = l=1 with k∈B
ek,l = δkl =
1 0
if k = l . if k = l
8.2 Cooperative Games
619
The last observation leads us to consider systems of linear inequalities. So let A be an m × n-matrix and b ∈ Rm . First we consider the following system of linear equalities with a nonnegativity constraint. Ax = b
with x ≥ 0.
(8.13)
DEFINITION 8.2.3 We say that system (8.13) is consistent, if there is x ∈ Rn + (i.e., x ∈ Rn , x ≥ 0) that satisfies Ax = b. Such a vector is called a solution of (8.13). PROPOSITION 8.2.4 System (8.13) is consistent (i.e., it has a solution x ∈ Rn +) if and only if (y, b)Rm ≥ 0
for all y ∈ Rm such that A∗ y ≥ 0.
PROOF: Let Γ(A) = {Ax : x ≥ 0}. Clearly Γ(A) ⊆ Rm is a nonempty, closed, convex cone and 0 ∈ Γ(A). Note that the system (8.13) is consistent if and only if b ∈ Γ(A), then b = Ax for some x ≥ 0 and so if A∗ y ≥ 0, then (A∗ y, x)Rn ≥ 0, hence (y, Ax)Rm = (y, b)Rm ≥ 0. Suppose b ∈ / Γ(A). Then by the strong separation theorem we can find y ∈ Rm \ {0} and ε > 0 such that (y, b)Rm + ε ≤ (y, u)Rm ⇒ (y, b)Rm ≤ −ε
for all u ∈ Γ(A),
(because 0 ∈ Γ(A)).
On the other hand we have (y, u)Rm ≥ 0 for all u ∈ Γ(A). Indeed, if we can find u ∈ Γ(A) such that (y, u)Rm < 0, then because Γ(A) is a convex cone, for ϑ > 0 large enough, we have ϑu ∈ Γ(A)
and
(y, ϑu)Rm < (y, b)Rm + ε,
a contradiction.
REMARK 8.2.5 This result is often called the Minkowski–Farkas lemma. COROLLARY 8.2.6 If A is an m×n-matrix and b ∈ Rm , then the system Ax ≥ b is consistent (i.e., there exists x ∈ Rn such that Ax ≥ b) if and only if for every ∗ y ∈ Rm + for which we have A y = 0, then (y, b)Rm ≤ 0. PROOF: Let Im be the m × m-identity matrix. The system Ax ≥ b is consistent if and only if the system ⎡ ⎤ ? ⎡x ⎤ > x1 1 (8.14) A −A −Im ⎣ x2 ⎦ = b with ⎣ x2 ⎦ ≥ 0 x3 x3 is consistent. Then by virtue of Proposition 8.2.4, (8.14) is consistent if and only if for every y ∈ Rm for which we have
∗ A − A∗ − Im y ≥ 0, then (y, b)Rm ≥ 0. Now we return to the side-payment game and introduce the following notion.
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8 Game Theory
DEFINITION 8.2.7 A collection S of coalitions (i.e., S ⊆ 2N \{∅}) is said to be balanced if there exists a set {λB }B∈S ⊆ R+ such that λB χB = χN . B∈S
Recall that for every B ∈ 2N \{∅} χB =
ek ,
k∈B n where ek = (ek,l )n l=1 ∈ R+ with
ek,l = δkl = Note that
1 0
if k = l . if k = l
λB χB = χN if and only for every k ∈ N ,
λB = 1. The set
B∈S
B∈S
k∈B
{λB }B∈S is called the associated balancing coefficients. The side-payment game v is said to be balanced if for every balanced S ⊆ 2N \ {∅} with associated balancing coefficients {λB }B∈S , we have λB v(B) ≤ v(B). B∈S
The notion of a balanced side-payment game is related to the nonemptiness of the core. THEOREM 8.2.8 If v : 2N \ {∅} −→ R is a side-payment game, then the core C(v) of v is nonempty if and only if for every family {λB }B∈2N \{∅} ⊆ R+ for which λB χB = χN , we have B∈2N \{∅}
λB v(B) ≤ v(N ).
B∈2N \{∅}
PROOF: By virtue of Corollary 8.2.6, we have that C(v) = 0 if and only for every family {λB }B∈2N \{∅} ∪ ξ ⊆ R+ for which λB χB + ξ(−χN ) = 0, B∈2N \{∅}
we have
λB v(B) + ξ −v(N ) ≤ 0.
(8.15)
B∈2N \{∅}
If ξ = 0, then λB = 0 for all B ∈ 2N \ {∅}, satisfies (8.15). If ξ > 0, then we set λB = λB t and we obtain λB v(B) ≤ v(N ). B∈2N \{∅}
8.2 Cooperative Games
621
A shortcoming of the model of a side-payment game is that the notion of total utility v(B) for a coalition B ∈ 2N \{∅} is hard to interpret, unless the players have utility functions. If the players have preferences over outcomes that are not representable by utility functions, then the game must specify the physical outcomes that a coalition can guarantee for its members. The preferences can then be described as binary relations on vectors of physical outcomes and it is not necessary to rely on a utility function. To accommodate this situation, we introduce a more general game. So for every coalition B ∈ 2N \{∅}, we set n for every k ∈ B . RB = x = (xk )n k=1 ∈ R : xk = 0 A game in characteristic function form without side-payments (or simply a nonn side-payment game) is a multifunction V : 2N \{∅} −→ 2R \ {∅} such that V (B) ⊆ RB
for every B ∈ 2N \ {∅}.
We have that u ∈ V (B) if and only if cooperation among the members of coalition B, can bring about the utility allocation (uk )k∈B to the members of B. An equivalent definition of a non-side-payment game is a multifunction V : 2N \ n {∅} −→ 2R \{∅} such that u, u ∈ Rn
and
uk = uk
u ∈ V (B)
if and only if
for all k ∈ B
implies
u ∈ V (B).
Evidently the set V (B) is a cylinder with basis V (B); that is, V (B) = u ∈ Rn : (uk , δkB )k∈B ∈ V (B)
where δkB =
1 0
if k ∈ B . if k ∈ N \B
We adopt this definition of a non-side-payment game. Again we can associate with a game in normal form. However, its formulation is very cumbersome and so it is much easier to work with the value multifunction B −→ V (B). For such a game, the notion of core is defined as follows. DEFINITION 8.2.9 The core of a non-side-payment game V , is the set C(V ) of n all u = (uk )n k=1 ∈ R such that (a) u ∈ V (N ). (b) We can find coalition B ∈ 2N \{∅} and u ∈ V (B) such that uk < uk
for all k ∈ B.
REMARK 8.2.10 In the above definition, requirement (a) is a feasibility condition on the core vector and requirement (b) simply says that no coalition can improve upon u.
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8 Game Theory
DEFINITION 8.2.11 A non-side-payment game V is said to be balanced if for every balanced collection of coalitions S (see Definition 8.2.7) we have # V (S) ⊆ V (N ). B∈S
In what follows, we show that the notion of a balanced game, is sufficient for the nonemptiness of the core. However, in contrast to side-payment games (see Theorem 8.2.8) it is not necessary. But first we need to state a version of the KKM-theorem (see Theorem 6.5.9) due to Shapley [551], which we need in the proof of the theorem for the nonemptiness of the core. For every coalition B, ∆B = conv{ek : k ∈ B}. PROPOSITION 8.2.12 If CB : B ∈ 2N \ {∅} is a family of nonempty closed subsets of ∆N such that for each B ∈ 2N \ {∅}, ∆B ⊆ CD , then there is a balanced collection S such that
D⊆B
#
CB = ∅.
B∈S
Using this proposition we can now establish the nonemptiness of the core C(V ). n THEOREM 8.2.13 If V is a non-side-payment game, b = (bk )n k=1 ∈ R is defined by bk = sup uk : u = (ui )n i=1 ∈ V ({k})
and we have (i) For every B ∈ 2N \ {∅}, V (B) − Rn + = V (B); (ii) There exists M > 0 such that for every B ∈ 2N \ {∅} u ∈ V (B) ∩ {b} + Rn implies uk < M +
for all k ∈ B;
(iii) For every B ∈ 2N \ {∅}, V (B) is closed in Rn ; (iv) V is balanced, then C(V ) is nonempty. PROOF: We may assume without any loss of generality that b = 0. Let M > 0 be as in hypothesis (ii) and for every coalition B define DB = conv{−M nek : k ∈ B}. We consider the function ξ : DN −→ R defined by ! V (B) . ξ(y) = max λ ∈ R : y + λχN ∈ B∈2N \{∅}
The function ξ is continuous (see Theorem 6.1.18(c)). Hence the function y −→ g(y) = y + ξ(y)χN is continuous too. We set
8.2 Cooperative Games
623
CB = y ∈ DN : g(y) ∈ V (B) = g −1 V (B) , ⇒ CB ⊆ DN
is closed.
(8.16)
Claim: If B, B are coalitions and DB ∩ CB = ∅, then B ⊆ B . Clearly the claim is true if B = N . So suppose that |B | < n and let y ∈ DB ∩ CB . We have yk = −M n, k∈B
hence, we can find k0 ∈ B such that Mn < −M, |B | + ξ(y) ≥ 0 (because g(y) ∈ Rn + ),
yk0 ≤ − ⇒ yk0
⇒ ξ(y) > M
(see (8.17)).
(8.17)
(8.18)
On the other hand, we have g(y) ∈ V (B)
(see (8.16)), for all k ∈ B
⇒ yk + ξ(y) < M
for all k ∈ B
⇒ yk < 0
(see hypothesis (ii)),
(see (8.18)),
⇒ B ⊆ B. Thus we have satisfied all the hypotheses of Proposition 8.2.12. According to that result we can find a balanced collection S0 such that #
CB = ∅.
B∈S0
Let y0 ∈
B∈S0
CB . Then we have
g(y0 ) ∈ g
#
CB
⊆
B∈S0
#
g(CB )
B∈S0
⊆
#
V (B)
(see (8.16))
B∈S0
⊆ V (N )
(because S0 is balanced).
(8.19)
definition of ξ, we see that g(y0 ) is on the boundary of From (8.19)Nand the V (B) : B ∈ 2 \{∅} . This together with hypothesis (i) imply that g(y0 ) can not be improved upon by any coalition. Therefore g(y0 ) ∈ C(V ); that is, C(V ) = ∅. REMARK 8.2.14 Hypothesis (i) is usually called the comprehensiveness condition (free disposability in economic models). Also hypothesis (iii) can be replaced by the weaker condition: (iii) V (N ) ⊆ Rn is closed.
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8 Game Theory
8.3 Cournot–Nash Equilibria for Random Games In Section 8.1 we considered noncooperative games and introduced an equilibrium notion for them, due to Nash (see Definition 8.1.5). The game had a finite number of players each of whom is described by a strategy set and a utility (payoff) function defined on the Cartesian product of the strategy sets. It was assumed that the strategy sets were subsets of R and in fact without any additional effort we can also assume that each strategy set is a subset of a finite-dimensional Banach space. In this section we present a twofold generalization of this model. We allow the strategy set of each player to be a subset of an infinite dimensional Banach space. On the other hand, we consider a continuum of players, in analogy to the model of perfect competition investigated in Section 7.1. So the space of players is a finite measure space. Each player has a preference multifunction, which describes his or her preference relation on the set of feasible strategies. This relation need not be transitive and complete and so may not be representable by a utility function. So let (T, T , µ) be a complete finite nonatomic measure space. Here T is the set of players, T is the set of all possible coalitions among the players, and µ is a set function measuring the size of each coalition. The space of all possible strategies (decisions) of the players is a separable Banach space X. More precisely, there is a multifunction F : T −→ 2X \{∅} which for each player t ∈ T describes his strategy set F (t) (i.e., F (t) ⊆ X is the set of all strategies (actions, decisions) available to player t ∈ T ). The randomness of the game is described by a complete probability space (Ω, Σ, µ), with Ω corresponding to the set of all possible states of nature, and Σ to the collection of all possible outcomes. Also given is a multifunction P : Ω × T × SF1 −→ 2X which describes the random preferences of the players. More precisely, for every (ω, t) ∈ Ω × T , P (ω, t, u) ⊆ F (t), and for µ-almost every player t ∈ T , the set P (ω, t, u) consists of all the strategies (actions), that he/she strictly prefers to her own strategy u(t), given that ω ∈ Ω is the state of the environment and given that the decisions of all players (modulo null-coalitions), are represented by u ∈ SF1 . So we see that each player’s preference pattern is influenced by the strategies of the other players (modulo null coalitions) and by the realized state of nature. However, each player must choose his strategy independently, after having observed the realized state ω ∈ Ω of nature. So the players’ actions can be modelled by a map g : Ω −→ SF1 , which prescribes for each possible state of nature ω ∈ Ω, the strategies of all players modulo null coalitions. Note that the players act independently and noncooperatively. So the random game under consideration is described by the quadruple R = (T, T , µ), (Ω, Σ, ν), F, P . DEFINITION 8.3.1 A Cournot–Nash equilibrium for the random game R, is a
map g : Ω −→ SF1 which is Σ, B L1 (T, X) -measurable such that for ν-a.a. ω ∈ Ω, we have
P ω, t, g(ω) = ∅ for µ-a.a. t ∈ T.
REMARK 8.3.2 Every Σ, B L1 (T, X) -measurable map g : Ω −→ SF1 , is called a strategy rule. According to the above definition, no player t ∈ T outside a powerless (due to the nonatomicity of µ) null coalition, can produce a strategy, which he/she strictly prefers to the equilibrium strategy g(ω)(t).
8.3 Cournot–Nash Equilibria for Random Games
625
Now we are ready to introduce the precise mathematical hypotheses on the data of the random game. In what follows (SF1 , w) is the set SF1 with the relative weak L1 (T, X)-topology. H1 : (T, T , µ) is a complete, finite, nonatomic measure space. REMARK 8.3.3 As was the case with the economic model of pure competition considered in Section 7.1, the nonatomicity of the measure space of players expresses the fact that there is no coalition of players which has more influence on the game than the others. H2 : F : T −→ Pwkc (X) is graph-measurable and integrably bounded (see Definition 6.4.22).
H3 : dom P = (ω, t, u) ∈ Ω × T × SF1 : P (ω, t, u) = ∅ ∈ Σ× T ×B L1 (T, X) and dom conv P (ω, t, ·) is w-open.
H4 : There is no Σ, B L1 (T, X) -measurable map f : Ω −→ SF1 (a decision rule),
such that for ν-a.a. ω ∈ Ω, we have f (ω)(t) ∈ conv P ω, t, f (ω) µ-a.e. on T . H5 : There exists a multifunction H : domP −→ Pf c (X) that is graph-measurable and (i) For every (ω, t, u) ∈ Ω × T × SF1 , H(ω, t, u) ⊆ conv P (ω, t, u) ⊆ F (t). (ii) For every (ω, t) ∈ Ω × T, u −→ H(ω, t, u) is usc from (SF1 , w) into Xw . REMARK 8.3.4 If (ω, t, u) −→ conv P (ω, t, u) is graph measurable and for all (ω, t) ∈ Ω × T, u −→ P (ω, t, u) is usc with closed values, then we can take H = convP . Let (Y, d) be a separable metric space and consider a multifunction G : Ω × T × Y −→ Pf c (X) such that for every (ω, t, y) ∈ Ω × T × Y we have G(ω, t, y) ⊆ F (t). 1
Then we define Γ : Ω × Y −→ 2SF by Γ(ω, y) = u ∈ SF1 : u(t) ∈ G(ω, t, y) µ-a.e. on T . PROPOSITION 8.3.5 If hypotheses H1 , H2 hold and for every (ω, t) ∈ Ω × T , G(ω, t, ·) is usc from Y into X furnished with the weak topology (denoted by Xw ), then (a) For every ω ∈ Ω, Γ(ω, ·) is usc from Y into (SF1 , w). (b) If Gr G ∈ Σ × T × T × B(Y ) × B(X), 1 then we can find a multifunction Γ : Ω×Y −→ 2SF such that GrΓ ∈ Σ×B(SF1 ) and for ν-a.a. ω ∈ Ω we have Γ(ω, ·) = Γ (ω, ·).
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8 Game Theory
PROOF: (a) By virtue of hypothesis H1 and Theorem 6.4.23, we have that SF1 is nonempty, convex, and w-compact. Moreover, (SF1 , w) is metrizable. So in order to prove the desired upper semicontinnuity of Γ(ω, ·), it suffices to show that Gr Γ(ω, ·) is sequentially closed in Y × (SF1 , w) (see Proposition 6.1.10). To this end let (yn , un ) ⊆ Y × SF1 and suppose that yn −→ y
in (Y, d)
w
un −→ u
and
in L1 (T, X).
We have un (t) ∈ G(ω, t, yn )
µ-a.e. on T.
Invoking Proposition 6.6.33, we obtain u(t) ∈ conv w- lim sup G(ω, t, yn )
µ-a.e. on T.
But since G(ω, t, ·) is w-usc and has closed convex, values, we obtain u(t) ∈ G(ω, t, y)
µ-a.e. on T,
⇒ u ∈ Γ(ω, y). This proves the upper semicontinuity of y −→ Γ(ω, y) from Y into (SF1 , w). (b) Let
ϑ(ω, t, y, x) = iG(ω,t,y) (x) =
0 +∞
if x ∈ G(ω, t, y) , otherwise
the indicator function of the set G(ω, t, y). By virtue of the graph measurability of G, we have that ϑ is measurable. Moreover, because Gr G(ω, t, ·) is closed in Y ×Xw (due to the upper semicontinuity of y −→ G(ω, t, y) from Y into Xw ), we have that ϑ(ω, t, ·) is lower semicontinuous on Y × Xw and of course it is convex in x ∈ X. Therefore we can find an increasing sequence ϑn : Ω × T × Y × X −→ R, n ≥ 1, of measurable functions such that for every n ≥ 1 and every (ω, t) ∈ Ω × T , ϑn (ω, t, ·, ·) is Lipschitz continuous on Y ×X with Lipschitz constant n ≥ 1 and ϑn ↑ ϑ as n → ∞. Let
Iϑn (ω, y, u) = ϑn ω, t, y, u(t) dµ and Iϑ (ω, y, u) = ϑ ω, t, y, u(t) dµ. T
T
Using Theorem 2.1.28, we can check that for every ω ∈ Ω, Iϑn (ω, ·, ·) is continuous on Y ×L1 (T, X). Also for every n ≥ 1 and every (y, u) ∈ Y ×SF1 , ω −→ Iϑn (ω, y, u) is Σ-measurable. Thus by virtue of Theorem 6.2.6 for every n ≥ 1, (ω, y, u) −→ Iϑn (ω, y, u) is Σ×B(Y )×B(SF1 )-measurable. Note that from the monotone convergence theorem, we have Iϑn ↑ Iϑ
as n → ∞,
⇒ Iϑ is Σ × B(Y ) × B(SF1 )-measurable
recall that B(SF1 ) = B (SF1 , w) . Finally let Γ be defined by GrΓ = (ω, y, u) ∈ Ω×Y ×SF1 : lim Iϑn (ω, y, u) ≤ 0 .
n→∞
Then Γ (ω, y, u) has all the desired properties.
Now we are ready to establish the existence of a Cournot–Nash equilibrium (see Definition 8.3.1 for the random game under consideration.
8.3 Cournot–Nash Equilibria for Random Games
627
THEOREM 8.3.6 If hypotheses H1 , H2 , H3 , H4 , and H5 hold, then the random game R admits a Cournot–Nash equilibrium. PROOF: As we already pointed out (SF1 , w) is compact, and metrizable (see the proof of Proposition 8.3.5). Let G : Ω×T ×SF1 −→ 2X be defined by H(ω, t, y) if y ∈ dom P (ω, t, ·) G(ω, t, y) = . (8.20) F (t) otherwise Evidently G(ω, t, y) ∈ Pwkc (X) and G(ω, t, y) ⊆ F (t) for all (ω, t, y) ∈ Ω×T ×SF1 . Moreover, because of hypothesis H5 (ii), we have that for all (ω, t) ∈ Ω × T , G(ω, t, ·) is usc from (SF1 , w) into Xw . So, if we define Γ(ω, y) = u ∈ SF1 : u(t) ∈ G(ω, t, y) µ-a.e. on T , then we can apply Proposition 8.3.5 with Y = (SF1 , w) and
G, Γ as above,
to deduce that for every ω ∈ Ω, the multifunction y −→ Γ(ω, y) is usc from (SF1 , w) into (SF1 , w). Because Γ(ω, ·) has nonempty, compact, convex values in (SF1 , w), we can apply Theorem 6.5.19 (the Kakutani–Ky Fan fixed point theorem) and deduce that for every ω ∈ Ω, we can find u ∈ SF1 (depending on ω ∈ Ω) such that
u ∈ Γ(ω, u).
(8.21) 1
From Proposition 8.3.5(b), there is a multifunction Γ : Ω×SF1 −→ 2SF such that Γ is graph-measurable and for ν-a.a. ω ∈ Ω, we have Γ(ω, ·) = Γ (ω, ·).
(8.22)
So from (8.21) and (8.22) and the µ-completeness of Σ, we have D = (ω, u) ∈ Ω × SF1 : u ∈ Γ(ω, u) ∈ Σ × B(SF1 ). Invoking Theorem 6.3.20 (the Yankov–von Neumann–Aumann selection theo
rem), we obtain a Σ × B(SF1 ) -measurable map u : Ω −→ SF1 such that
u(ω) ∈ Γ ω, u(ω) ν-a.e. on Ω. This implies that for ν-a.a. ω ∈ Ω, we have
u(ω)(t) ∈ G ω, t, u(ω)
µ-a.e. on T.
Suppose that we can find B ∈ Σ with ν(B) > 0 such that for every ω ∈ B
domP ω, t, u(ω) = ∅ µ-a.e. on T. Then for every ω ∈ B, we have
u(ω)(t) ∈ H ω, t, u(ω) ⊆ conv P ω, t, u(ω)
µ-a.e. on T
(see (8.20)),
which contradicts hypothesis H4 . Therefore we conclude that for ν-a.a. ω ∈ Ω,
P ω, t, u(ω) = ∅ for µ-a.a. t ∈ T, ⇒ u ∈ SF1 is a Cournot-Nash equilibrium for R.
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8 Game Theory
8.4 Bayesian Games In the previous section, we studied the existence of equilibria for random games in which all players shared the same information about the state of the environment. In this section, we consider the situation in which the players have private information (imperfect information in the terminology of Harsanyi [285] or differential information). Motivated by the analogous situation in statistics, we call such games Bayesian games. We consider the situation with a countable number of players and an infinite-dimensional strategy space. The precise mathematical model of the game is the following. Let (Ω, Σ, µ) be a complete probability space. Here Ω denotes the set of all possible states of the environment, the σ-field Σ corresponds to the collection of all possible events and the measure µ describes the distribution of these events. Also T is a countable set and its elements represent the players of the game. The players choose their strategies (actions) from a separable Banach space X. Each player t ∈ T , has a strategy (action) multifunction Ft : Ω −→ 2X \{∅}. If the state of the environment is ω ∈ Ω, the set Ft (ω) ⊆ X represents the set of strategies (actions) available to player t ∈ T when the state of the " environment is ω ∈ Ω. Also each player t ∈ T has a utility function ut (ω, ·) : Fs (ω) −→ R s∈T
that depends on the state of the environment ω ∈ Ω. This utility function describes the preference of the player among the various feasible strategies, when the state is ω ∈ Ω. Moreover, to every player t ∈ T corresponds a complete sub-σ-field Σt of Σ, which represents the private information of that player and a prior pt : Ω −→ R+ , which is a Radon–Nikodym derivative with respect to the probability µ such that p (ω)dµ = 1. t Ω Then the Bayesian game (or game with private information or game with differential information), is the quadruple. B = (Ft , ut , Σt , pt )t∈T . In what follows we use the following notation. SF1 t (Σt ) = vt ∈ L1 (Ω, Σt ; X); vt (ω) ∈ Ft (ω) µ-a.e. on Ω . We set SF1 =
=
SF1 t (Σt )
t∈T
and
SF1t =
=
SF1 s (Σs ).
s∈T
s=t
An element of SF1 t (Σt ) is a strategy for player t ∈ T . For each player t ∈ T , we assume that there exists a finite or countable partition Pt of Ω and Σt = σ(Pt ) (i.e., Pt generates the sub-σ-field Σt containing ω ∈ Ω. We assume that
pt (ω )dµ > 0. At (ω)
8.4 Bayesian Games We set
pt ω At (ω) =
0 pt (ω ) At (ω)pt (ω )dµ
if ω ∈ / At (ω) if ω ∈ At (ω)
.
629
(8.23)
DEFINITION 8.4.1 The conditional expected utility of player t ∈ T , Ut (ω, ·, ·) : SF1 t ×Ft (ω) −→ R, is defined by
ut ω , y(ω ), x pt ω At (ω) dµ
Ut (ω, y, x) =
(see (8.23)).
(8.24)
At (ω)
REMARK 8.4.2 In the above definition we understand Ut (ω, y, x) as the conditional expected utility of player t ∈ T ; when the state of the environment is ω ∈ Ω, the player chooses the strategy x ∈ Ft (ω) and the other players have chosen the strategy profile y ∈ SFt . DEFINITION 8.4.3 A Bayesian Nash equilibrium for the Bayesian game B is a strategy profile y ∗ ∈ SF1 such that for all t ∈ T , we have
Ut ω, y ∗ , yt (ω) = max Ut (ω, y ∗ , v) : v ∈ Ft (ω)
µ-a.e. on Ω.
We introduce some hypotheses on the data of the Bayesian game B. H1 : F : Ω × T −→ Pwkc (X) is a multifunction such that for every t ∈ T, Gr Ft ∈ Σt × B(X), and Ft is integrably bounded. Let Ft (ω) =
"
Fs (ω) and Xt = X for all t ∈ T .
s=t
H2 : u : Ω×
"
Xt , ω −→ R = R ∪ {−∞} is a function such that
t∈T
(i) For every (t, x) ∈ T ×
"
Xt , ω −→ ut (ω, x) is Σ-measurable.
t∈T
(ii) For every (ω, t) ∈ Ω×T , ut (ω, ·, ·) : Ft (ω)×Ft (ω) −→ R and it is continuous " when Ft (ω) is endowed with the relative product weak topology of Xs s=t
and Ft (ω) is endowed with the norm topology. (iii) For every (ω, t, x) ∈ Ω × T ×
"
Fs (ω), the function x −→ ut (ω, x, x) is
s=t
concave on Ft (ω). (iv) There exists h ∈ L1 (Ω, Σ) such that for µ-a.a. ω ∈ Ω and all (t, x) ∈ T × Ft (ω)×Ft (ω) we have |ut (ω, x)| ≤ h(ω). We start with an auxiliary result that establishes the continuity properties of the conditional expected utility (see (8.24)). Let µt (A) = A pt (ω)dµ for every t ∈ T and A ∈ Σ. Evidently this is a probability measure on (Ω, Σ).
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8 Game Theory
PROPOSITION 8.4.4 If hypotheses H1 , H2 hold and for every fixed ω ∈ Ω, t ∈ T and A ∈ Σ, we set
A U t (y, x) = ut ω , y(ω ), x dµt (ω ) with y ∈ SF1t , x ∈ Ft (ω), A
A Ut
is continuous on SF1t ×Ft (ω), when SF1t is equipped with the relative product then weak topology and Ft (ω) is equipped with the norm topology. w
PROOF: Suppose that y n −→ y in SF1t and xn −→ x in Ft (ω). We know that SF1t is weakly compact (see hypothesis H1 and Theorem 6.4.23). We have y n = (ysn )s∈T s=t
w
and ysn −→ ys in SF1 s for all s ∈ T, s = t. w
Claim: For every ω ∈ Ω and every s ∈ T, s = t, we have ysn (ω) −→ ys (ω) in Fs (ω). We fix ω ∈ Ω. Let Ps be the countable Σs -partition of Ω such that σ(Ps ) = Σs . We set Ps = Dsk k≥1 . Hence we have ysn =
zsn,k χDsk
and
ys =
k≥1
zsk χDsk ,
k≥1 k(ω)
where zsn,k , zsk ∈ Fs (ω). For every s ∈ T , we can find a unique Ds
∈ Ps such that
ω ∈ Dsk(ω) . Then for every x∗ ∈ X ∗ , we have x∗ , ysn (ω) =
x∗ , ysn (ω)
χ k(ω) dµ(ω ) k(ω) µ(Ds ) Ds
x∗ , ysn (ω ) = χ k(ω) (ω )dµ(ω ) k(ω) ) Ds Ω µ(Ds Ω
(8.25)
(because ysn (ω ) = ysn (ω) for all ω ∈ Ds ).
k(ω) Evidently x∗ µ(Ds ) ∈ L∞ (Ω, X ∗ ) and ysn χDk(ω) ∈ L1 (Ω, X). Therefore k(ω)
s
w
from (8.25) and because ysn −→ ys in L1 (Ω, X), we obtain x∗ , ysn (ω) −→ x∗ , ys (ω) , w
⇒ ysn (ω) −→ ys (ω)
in Fs (ω) ⊆ X.
This proves the claim. Then using the claim and hypothesis H2 (ii), we have
ut ω, y n (ω), xn −→ ut ω, y(ω), x . From this and the dominated convergence theorem (it can be used here because of hypothesis H2 (iv)), we have A
A
U t (y n , xn ) −→ U t (y, x). Now we can have the equilibrium result for game B.
8.4 Bayesian Games
631
THEOREM 8.4.5 If hypotheses H1 and H2 hold, then the Bayesian game B admits a Bayesian Nash equilibrium. PROOF: By virtue of Proposition 8.4.4, for every ω ∈ Ω Ut (ω, ·, ·) : SF1t × Ft (ω) −→ R is continuous when SF1t is furnished with the relative product weak topology and Ft (ω) is furnished with the norm topology. Moreover, because of hypothesis H2 (iii) for every (ω, y) ∈ Ω × SF1t , Ut (ω, y, ·) is concave. For every t ∈ T , we consider the multifunction Et : Ω × SF1t −→ 2X defined by Et (ω, y) = x ∈ Ft (ω) : Ut (ω, y, x) = max{Ut (ω, y, v) : v ∈ Ft (ω)} . Due to the concavity and continuity of Ut (ω, y, ·), we have that it is weakly upper semicontinuous. Hence because the set Ft (ω) ∈ Pwkc (X), we deduce that Et (ω, y) = ∅. Moreover, the concavity of Ut (ω, y, ·) implies that Et (ω, y) ⊆ Ft (ω) is convex. Theorem 6.1.18(b) implies that for every ω ∈ Ω, y −→ Et (ω, y) is usc from SF1t with the relative product weak topology into the nonempty, closed, and convex subsets of Ft (ω), the latter equipped with the norm topology. From hypothesis H1 , we know that we can find vn : Ω −→ X, n ≥ 1, Σt measurable selectors of Ft (·) such that Ft (ω) = {vn (ω)}n≥1
for all ω ∈ Ω
(see Theorem 6.3.20).
Then exploiting the norm continuity of Ut (ω, y, ·), we have
max Ut (ω, y, v) : v ∈ Ft (ω) = max Ut ω, y, vn (ω) , n≥1 ⇒ ω −→ max Ut (ω, y, v) : v ∈ Ft (ω) = ξt (ω, y) is Σt -measurable, ⇒ GrEt (·, y) = (ω, x) ∈ GrFt : Ut (ω, y, x) = ξt (ω, y) ∈ Σt × B(X) (hypothesis H1 ). So applying Theorem 6.3.20 (the Yankov–von Neumann–Aumann selection theorem), we obtain ft : Ω −→ X a Σt -measurable map such that ft (ω) ∈ Et (ω, y)
for all ω ∈ Ω.
Since Et (ω, y) ⊆ Ft (ω) and by hypothesis H1 , Ft is integrably bounded, it follows 1 that ft ∈ SF1 t (Σt ). So, if we consider the multifunction Γt : SF1t −→ 2SFt (Σt ) defined by Γt (y) = x ∈ SF1 t (Σt ) : x(ω) ∈ Et (ω, y) µ-a.e. on Ω ; then we see that Γt has nonempty values. We claim that it is usc from SF1t with the relative product weak topology into SF1 t (Σt ) with the weak topology. Because SF1 t (Σt ) is weakly compact in L1 (Ω, Σt ; X) (see Theorem 6.4.23), it suffices to show that Gr Γt is closed in SF1t × SF1 t (Σt ) (see Proposition 6.1.10). Because the weak
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8 Game Theory
topologies on SF1t and SF1 t (Σt ) are metrizable, we can use sequences. So suppose w w that yn −→ y in SF1t , xn −→ x in SF1 t (Σt ), and xn ∈ Γt (yn ). Then xn (ω) ∈ Et (ω, yn )
for all n ≥ 1.
µ-a.e. on Ω
Invoking Proposition 6.6.33, we obtain x(ω) ∈ conv w- lim sup Et (ω, yn ) n→∞
⊆ Et (ω, y),
(8.26)
where the inclusion in (8.26) follows from the fact that Et (ω, ·) is usc on SF1t with the relative product weak topology and it has closed and convex values. Therefore it follows that (y, x) ∈ Gr Γt , ⇒ Γt is usc as claimed. Let Γ : SF1 −→ Pwkc (SF1 ) be defined by = Γ(y) = Γt (yt ) t∈T
where y = (yt )t∈T with yt ∈ SF1 t (Σt ) and yt = (ys )s∈T ∈ SF1t . Evidently Γ is usc s=t
when SF1 is furnished with the relative product weak topology. We apply Theorem 6.5.19 (the Kakutani–Ky Fan fixed point theorem), to obtain y ∗ ∈ SF1 such that y ∗ ∈ Γ(y ∗ ). Clearly then
Ut ω, y ∗ , yt∗ (ω) = max Ut (ω, y ∗ , v) : v ∈ Ft (ω)
µ-a.e. on Ω,
⇒ y ∗ is a Bayesian Nash equilibrium for B.
8.5 Stochastic Games In this section, we consider discounted stochastic games in which the state space is a Borel space (i.e., a Borel subset of a Polish space) and the action spaces of the players are compact metric spaces. Under reasonable continuity hypotheses on the reward function and the transition probability describing the law of motion of the system, we show that the discounted stochastic game has a value and both players have optimal stationary policies. To do this we use the so-called Bellman’s principle of optimality for the discounted dynamic programming problem. For this reason our discussion begins with a brief presentation of the dynamic programming model and of the associated Bellman’s principle of optimality. The discounted dynamic programming model is determined by the following objects.
8.5 Stochastic Games
633
•
A state space S, which is assumed to be Borel space (i.e., it is the Borel subset of a Polish space).
•
A space of actions X, which is a Borel space too.
•
A constraint multifunction F : S −→ 2X \ {∅}; this multifunction assigns to each state s ∈ S a nonempty feasible (permissible) set of actions F (s).
•
A law of motion q, which to every pair (s, x) ∈ Gr F assigns a probability measure q ·s, x on the Borel sets of S. This probability measure describes the distribution of the state next visited by the system if the system is in state s ∈ S and action x ∈ X is taken.
•
A bounded reward function r : Gr F −→ R.
•
A discount factor δ ∈ (0, 1). We make the following mathematical hypotheses on the above items:.
H1 : Gr F ∈ B(S ×X) = B(S)×B(X) and contains the graph of a Borel map from S to X.
H2 : q ·s, x is a Borel-measurable transition probability the
on Borel σ-field B(S); that is, for every B ∈ B(S), the function (s, x) −→ q Bs, x is Borel-measurable from S × X into [0, 1]. H3 : The reward function r : Gr F −→ R is bounded and Borel-measurable. Let H1 = S and Hn = Gr F ×Hn−1 for n ≥ 2. These are the sets of all possible histories of the system up to the stage n ≥ 1. Then a policy π is a sequence {πn }n≥1 , where for each n ≥ 1, πn is a conditional probability on B(X) given the past history Hn of the system. We assume that πn satisfies the constraint
πn F (sn )hn = 1 for all histories hn = (s1 , x1 , . . . , sn ). DEFINITION 8.5.1 A history π is said to be stationary if there exists a Borel measurable selector of f of F such that πn = f ; that is, πn f (sn )hn = 1 for all histories hn = (s1 , x1 , . . . , sn ). The stationary policies are identified with SF = set of Borel selectors of F . Any policy π together with the law of motion q, defines a conditional probability pn on the set X × S × X × S × · · · of futures of the system, given the initial state s ∈ S; that is, pπ (·s) = π1 qπ2 q . . . (8.27) The expected total discounted reward is defined by L(π)(s) = Eπ
∞
n=1
δ n−1 r(sn , xn )s ,
s ∈ S.
(8.28)
Here Eπ (·s) denotes the conditional expectation with respect to pn (·s) (see (8.27)).
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8 Game Theory
DEFINITION 8.5.2 A policy π ∗ is optimal if L(π)(s) ≤ L(π ∗ )(s) for all policies π and all states s ∈ S (see (8.28)). The next theorem, known as Bellman’s optimality principle, gives a necessary and sufficient condition for optimality of a policy π ∗ . We assume that hypotheses H1 , H2 , and H3 are in effect. THEOREM 8.5.3 A policy π ∗ is optimal if and only if its reward L(π ∗ ) satisfies the optimality equation
u(s )dq(s s, x) : x ∈ F (s) , s ∈ S. (8.29) u(s) = sup r(s, x) + δ S
PROOF: Necessity: Suppose that π ∗ is optimal. Fix a state s ∈ S. For any x ∈ F (s), we define G(x) = {s ∈ S : (s, x) ∈ Gr F }. (8.30) Evidently Gr G ∈ B(X)×B(S) (see hypothesis H1 ). Let f be a Borel-measurable selector of F . It exists by virtue of hypothesis H1 . For x ∈ F (s), we consider the map fx : S −→ X defined by x if s ∈ G(x) . fx (s) = f (s) otherwise Clearly f is Borel-measurable and a selector of F (see (8.30)). If, by (fx , π ∗ ) we denote the policy {fx , π1∗ , π2∗ , . . .}, then by virtue of the optimality of π ∗ , we have
∗ L(fx , π )(s) = r(s, x) + δ L(π ∗ )(s )dq(s s, x) ≤ L(π ∗ )(s). (8.31) S
Because x ∈ F (s) was arbitrary, from (8.31) it follows that
sup r(s, x) + δ L(π ∗ )(s )dq(s s, x) : x ∈ F (s) ≤ L(π ∗ )(s),
s ∈ S.
(8.32)
S
Given any policy π = {πn }n≥1 , we define πn (·|hn ) = πn+1 (·|s, x, hn ), n ≥ 1 and πs,x = {πn }n≥1 . We set (8.33) v(s, x, s ) = L(πs,x )(s ) and we have L(π)(s) =
v(s, x, s )dq(s s, x) dπ1 (x|s),
r(s, x) + δ
F (s)
s ∈ S.
(8.34)
S
From (8.34) it follows that L(π ∗ )(s) ≤ sup r(s, x) + δ
L(π ∗ )(s )dq(s s, x) : x ∈ F (s) .
(8.35)
S
Comparing (8.33) and (8.35), we conclude that
L(π ∗ )(s) = sup r(s, x) + δ L(π ∗ )(s )dq(s s, x) : x ∈ F (s) , S
s ∈ S.
8.5 Stochastic Games
635
Sufficiency: Suppose that L(π ∗ ) satisfies the functional equation (8.29). Let h1n = (s, x1 , . . . , sn , xn ) and u(s) = L(π ∗ )(s), s ∈ S. Then, for any policy π and any state s ∈ S, we have Eπ
δ n u(sn+1 ) − Eπ δ n u(sn+1 )|hn s = 0.
N
(8.36)
n=1
Here Eπ (·|hn ) denotes the conditional expectation given the history hn . Then for pπ (·|s)-almost all futures, we have
Eπ δ n u(sn+1 )|hn
= δn u(s )dq(s |sn , xn ) S
= δ n−1 r(sn , xn ) + δ u(s )dq(s |sn , xn ) − δ n−1 r(sn , xn ) S
≤ δ n−1 u(sn ) − δ n−1 r(sn , xn ).
(8.37)
Using (8.37) in (8.36), we obtain Eπ
δ n u(sn+1 ) − δ n−1 u(sn ) + δ n−1 r(sn , xn ) s
N
n=1
= Eπ δ N u(sN +1 ) − u(s) +
N
δ n−1 r(sn , xn )s ≤ 0.
(8.38)
n=1
We pass to the limit as N −→ ∞ in (8.38). Using the dominated convergence theorem, we have n−1 Eπ δ r(sn , xn )s = L(π)(s) ≤ u(s) = L(π ∗ )(s), s ∈ S, n≥1
⇒ π
∗
is an optimal policy.
Bellman’s optimality principle leads to the existence of optimal stationary policies. H1 : F : S −→ Pk (X) is a measurable multifunction. REMARK 8.5.4 By Theorem 6.3.17 (the Kuratowski–Ryll Nardzewski selection theorem), if F satisfies hypothesis H1 , it admits a Borel-measurable selector. H2 : For every s ∈ S and every bounded Borel measurable function v : S −→ R, the function
x −→ v(s )dq(s |s, x) S
is continuous. H3 : The reward function r : Gr F −→ R is bounded (i.e., |r(s, x)| ≤ M for some M > 0 and all (s, x) ∈ S×X), Borel-measurable and for every s ∈ S, x −→ r(s, x) is continuous.
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8 Game Theory
First let us make a straightforward but useful observation concerning the solution of the optimality functional equation. In what follows by B0 (S) we denote the space of all bounded Borel-measurable functions on S. This is a Banach space for the supremum norm u∞ = sup |u(s)|. s∈S
PROPOSITION 8.5.5 If hypotheses H1 , H2 , and H3 hold, then the optimality functional equation (8.29) has a unique solution u∗ ∈ B0 (S). PROOF: Let u ∈ B0 (S). We consider the optimization problem
u(s )dq(s |s, x) : x ∈ F (s) = ξ(s). sup r(s, x) + δ
(8.39)
S
By virtue of hypothesis H1 we can find a sequence fn : S −→ X, n ≥ 1, of Borel-measurable functions such that F (s) = {fn (s)}n≥1
for all s ∈ S.
Because x −→ S u(s )dq(s |s, x) is continuous (hypothesis H2 ) and x −→ r(s, x) is continuous (see hypothesis H3 ), from (8.39) we have
u(s )dq s |s, fn (s) , ξ(s) = sup r s, fn (s) + δ n≥1
S
⇒ ξ is Borel-measurable. In addition it is clear that ξ is bounded (see hypothesis H3 ). So we can define the operator P : B0 (S) −→ B0 (S) by
u(s )dq(s |s, x) : x ∈ F (s) , s ∈ S. P (u)(s) = sup r(s, x) + δ S
For every u, v ∈ B0 (S) and every s ∈ S, we have |P (u)(s) − P (v)(s)|
u(s ) − v(s ) dq(s |s, x) ≤ sup δ S
≤ δu − v∞ . Therefore P is a contraction map and by the Banach fixed point theorem (see Theorem 3.4.3), P has a unique fixed point, which is the unique solution of the Bellman’s optimality equation (8.29). REMARK 8.5.6 In the above proof, the operator P is known as the dynamic programming operator . Now we can use Theorem 8.5.3 to establish the existence of an optimal stationary policy. THEOREM 8.5.7 If hypotheses H1 , H2 , and H3 hold, then there exists an optimal stationary policy.
8.5 Stochastic Games
637
PROOF: Let u∗ ∈ B(S) be the unique fixed point of the dynamic programming operator. We have
u∗ (s )dq(s |s, x) : x ∈ F (s) = ξ(s). u∗ (s) = sup r(s, x) + δ S
Let Γ(s) = x∗ ∈ F (s) : ξ(s) = r(s, x∗ ) + δ S u(s )dq(s |s, x ∗ ) . Because F (s) ∈ Pk (X) (see hypothesis H1 ) and the function x −→ r(s, x) + δ S u∗ (s )dq(s |s, x) is continuous (see hypotheses H1 and H3 ), from the Weierstrass theorem, we have that Γ(s) = ∅ for all s ∈ S and in fact Γ(s) ∈ Pk (X). Moreover, recalling that a Carath´eodory function is jointly measurable (see Theorem 6.2.6) and because s −→ ξ(s) is Borel-measurable (see the proof of Proposition 8.5.5), we conclude that GrF ∈ B(S) × B(X). Because Γ is compact valued, it follows that Γ is measurable. Therefore by the Kuratowski–Ryll–Nardzewski selection theorem (see Theorem 6.3.17), we can find f ∗ ∈ SF such that f ∗ (s) ∈ Γ(s) for all s ∈ S,
∗ ∗ u∗ (s )dq s |s, f ∗ (s) u (s) = r s, f (s) + δ
S = sup r(s, x) + δ u∗ (s )dq(s |s, x) : x ∈ F (s) S
and so by Theorem 8.5.3 we infer that f ∗ ∈ SF is optimal and L(f ∗ ) = u∗ .
∗ REMARK 8.5.8 In fact we can strengthen the say
above theorem
that ∗f ∈ ∗and ∗ ∗ SF is an optimal policy if and only if u (s) = r s, f (s) + δ S u (s )dq s |s, f (s) for all s ∈ S. Here again u∗ ∈ B(S) is the unique fixed point of the dynamic programming operator P (see Remark 8.5.6). For details we refer to Hernadez– Lerma [290].
Now we pass to discounted stochastic games. We consider a two-player stochastic game. The items determining such a stochastic game are the following. • S is a nonempty Borel subset of a Polish space (a Borel space) and corresponds to the set of all possible states of the system. •
X and Y are compact metric spaces and correspond to the set of actions available to player I and player II, respectively.
•
F1 : S −→ Pf (X) and F2 : S −→ Pf (Y ) are two measurable multifunctions that restrict the actions of the two players; so if the state of the system is s ∈ S, then player I can choose an action from the set F1 (s) and the player II from the set F2 (s).
•
q is the law of motion (transition law) of the system and with every triple (s, x, y) ∈ S ×X ×Y associates probability measure q(·|s, x, y) on the Borel subsets B(S) of S; so if the system is in state s ∈ S and the two players have chosen actions x ∈ X and y ∈ Y , respectively, the system moves to a new state according to the probability distribution q(·|s, x, y).
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8 Game Theory
•
r : S ×X ×Y −→ R is a bounded reward function.
•
δ ∈ (0, 1) is the discount factor; so the unit income today is worth δ n at the nth period in the future.
The game is played as follows. Periodically (say, once a day), players I and II observe the current state s ∈ S of the system and choose actions x ∈ F1 (s) and y ∈ F2 (s), respectively. This choice is made with full knowledge of the history of the system as it has evolved up to the present time. Then as a consequence of the actions chosen by the two players, player II pays player I an amount equal to r(s, x, y). Subsequently the system moves to a new state s according to the distribution q(·|s, x, y). Then the whole process is repeated from the new state s ∈ S. The goal of player I is to maximize the expected discounted reward (income) as the game proceeds over the infinite future and the goal of player II is to minimize the expected discounted loss. For each n ≥ 0, we define the space Hn of all admissible histories of the systems up to time n (the n-histories), by H0 = S
and
Hn = Gr(F1 ×F2 )n ×S = Gr(F1 ×F2 )× Hn−1
for
n ≥ 1.
The generic element hn ∈ Hn , is a vector hn = (s0 , x0 , y0 , . . . , sn−1 , xn−1 , yn−1 , sn ), where (sk , xk , yk ) ∈ Gr(F1 × F2 ) for all k = 0, 1, . . . , n − 1 and sn ∈ S. DEFINITION 8.5.9 (a) A randomized, admissible policy π for player I, is a sequence π = {πn }n≥0 of stochastic kernels (transition probabilities) πn on the Borel σ-field B(X) given Hn and satisfying the constraint
πn F1 (s)|hn = 1 for all hn ∈ Hn and all n ≥ 0. Similarly a (randomized admissible) policy β for player II is a sequence β = {βn }n≥0 of stochastic kernels (transition probabilities) βn on the Borel σ-field B(Y ) given Hn and satisfying the constraint
βn F2 (s)|hn = 1 for all hn ∈ Hn and all n ≥ 0. We denote the set of all policies of player I (resp., of player II) by ∆1 (resp. ∆2 ). (b) A stationary policy for player I (resp., for player II) is a Borel-measurable map 1 1 f : S −→ M+ (X) (resp., a Borel-measurable map g : S −→ M+ (Y )). REMARK 8.5.10 A stationary policy is a particular case of a randomized admissible policy. Indeed, if pn :Hn −→ S is the projection map defined by pn (hn ) = sn (hn = (s0 , x0 , y0 , . . . , sn−1 , xn−1 , yn1 , sn )), then π = {πn = f ◦ pn }n≥0 ∈ ∆1 . So πn is the Dirac measure concentrated at f (sn ). Similarly for player II. A stationary policy is a special case of a Markov policy. Clearly the stationary policies of player I (resp., of player II), can be identified with SF1 (resp., with SF1 ). A pair (π, β) of policies for players I and II associates with every initial state th
s ∈ S, an n =-day expected reward (gain) rn (π, β)(s) for player I (see the dynamic
8.5 Stochastic Games
639
programming model). The total expected discounted reward for player I, is given by n L(π, β)(s) = δ rn (π, β)(s). (8.40) n≥0
The rewards rn (π, β)(·) are Borel-measurable and so from (8.40) it follows that L(π, β)(·) is Borel-measurable. DEFINITION 8.5.11 A policy π ∗ is optimal for player I if inf
sup L(π, β)(s) ≤ L(π ∗ , β)(s)
β∈∆2 π∈∆1
for all β ∈ ∆2 , s ∈ S.
A policy β ∗ is optimal for player II if sup inf L(π, β)(s) ≥ L(π, β ∗ )(s) for all π∈∆1 β∈∆2
π ∈ ∆1 , s ∈ S. The stochastic game is said to have a value if inf
sup L(π, β)(s) = sup inf L(π, β)(s)
β∈∆2 π∈∆1
π∈∆1 β∈∆2
for every s ∈ S.
In this case the function v(s) = inf
sup L(π, β)(s) = sup inf L(π, β)(s),
β∈∆2 π∈∆1
π∈∆1 β∈∆2
s ∈ S,
is called the value function of the stochastic game.
To be able to establish the existence of optimal policies for the two players, we need some mathematical hypotheses on the data {F1 , F2 , q, r} of the stochastic game. H4 : F1 : S −→ Pf (X) and F2 : S −→ Pf (Y ) are measurable multifunctions. REMARK 8.5.12 Because X and Y are compact metric spaces, the measurability
of F1 , F2 is equivalent
to Borel-measurability into the separable metric spaces Pk (X), h and Pk (Y ), h , respectively. Let Cb (S) denote the space of bounded continuous functions on S. This is a Banach space for the supremum norm u∞ = sup |u(s)|, u ∈ Cb (S). Also by s∈S 1 (S) we denote the space of probability measures on S. M+ 1 H5 : q : S ×X ×Y −→ M+ (S) is a map such that
(i) For every (x, y) ∈ X × Y and every B ∈ B(S), s −→ q(B|s, x, y) is Borelmeasurable. (ii) For every s ∈ S, (x, y) −→ q(·|s, x, y) is continuous in the sense that if (xn , yn ) −→ (x, y) in X × Y , then for every u ∈ B0 (S) we have
u(s )dq(s |s, xn , yn ) −→ u(s )dq(s |s, x, y) as n → ∞. S
S
640
8 Game Theory
REMARK 8.5.13 By approximating a function in Cb (S) uniformly by simple functions, we can check that hypothesis H5 (i) is in fact equivalent to saying that 1 for every (x, y) ∈ X × Y, s −→ q(·|s, from S into M+ (S)
x,1 y) is Borel-measurable 1 furnished with the weak topology w M+ (S), Cb (S) . Recall that M+ (S) topologized this way is a Borel space (hence a separable metrizable space). H6 : r : S ×X ×Y −→ R is a function such that (i) For every (x, y) ∈ X × Y, s −→ r(s, x, y) is measurable. (ii) For every s ∈ S, (x, y) −→ r(s, x, y) is continuous. REMARK 8.5.14 The reward function r is a Carath´eodory function, thus Borelmeasurable (see Theorem 6.2.6). 1 1 For u ∈ B(S) and (s, λ, µ) ∈ S × M+ (X) × M+ (Y ), we define
u(s )dq(s |s, λ, µ), Ku (s, λ, µ) = r(s, λ, µ) + δ
(8.41)
S
where
r(s, λ, µ) =
r(s, x, y)dλ(x)dµ(y)
Y X
and
q(B|s, λ, µ) =
q(B|s, x, y)dλ(x)dµ(y) Y
X
for all B ∈ B(S). Because of hypotheses H5 and H6 , the function (s, λ, µ) −→ 1 1 Ku (s, λ, µ) is a Carath´eodory function on S × M+ (X) × M+ (Y ) into R; that is, 1 1 for every (λ, µ) ∈ M+ (X) × M+ (Y ), s −→ Ku (s, λ, µ) is Borel-measurable and 1 1 for every s ∈ S, (λ, µ) −→ Ku (s, λ, µ) is continuous on M+ (X)
1× M+ (Y ), when 1 1 M + (X) and M+ (Y ) are endowed with the weak topologies w M+ (X), C(X) and 1 1 1 w M+ (Y ), C(Y ) , respectively. In what follows M+ (X) and M+ (Y ) are topologized this way.
1
1 Let P1 : S −→ Pk M+ (X) and P2 : S −→ Pk M+ (Y ) be defined by
1 P1 (s) = λ ∈ M+ (X) : λ F1 (s) = 1
1 and P2 (s) = µ ∈ M+ (Y ) : µ F2 (s) = 1 . Note that the function Ku (s, ·, ·) defines a two-person, zero-sum game η(s, u) and P1 (s) and P2 (s) are the spaces of pure strategies in η(s, u) for players I and II, respectively, and Ku (s, ·, ·) is the reward (payoff, utility) function of the game. The next theorem, known as the portmanteau theorem, helps us establish the existence of a pair of optimal stationary strategies for the two players. For a proof of the portmanteau theorem, we refer to Denkowski–Mig` orski–Papageorgiou [194, p. 195] or Parthasarathy [487, p. 40]. Recall that if Z is a separable metric space,
1 1 then M+ (Z) endowed with the weak topology w M+ (Z), Cb (Z) is separable, and metrizable. 1 THEOREM 8.5.15 If Z is a separable metric space and {µn }n≥1 ⊆ M+ (Z), then the following statements are equivalent.
8.5 Stochastic Games w
1 (a) µn −→ µ in M+ (Z) 1 w M+ (Z), Cb (Z) .
(b)
641
w −→ denotes convergence in the weak topology
udµn −→ Z udµ for all u ∈ Ub (Z) = space of bounded, R-valued uniformly continuous functions.
Z
(c) lim sup µn (C) ≤ µ(C) for every closed set C ⊆ Z. n→∞
(d) lim inf µn (U ) ≥ µ(U ) for every open set U ⊆ Z. n→∞
(e) µn (A) −→ µ(A) for every Borel set A ⊆ Z such that µ(∂A) = 0. PROPOSITION 8.5.16 If hypothesis H4 holds, then the multifunctions P1 : 1 1 S −→ 2M+ (X) \ {∅} and P2 : S −→ 2M+ (Y ) \ {∅} have compact values and are Borel-measurable. PROOF: We do the proof for P1 . The proof for P2 is similar. 1 Because X is a compact metric space, then so is M+ (X) furnished with the weak topology. Hence using Theorem 8.5.15 (in particular the equivalence of (a) and (c)), we deduce at once that P 1 has compact values. Also the multifunction F1 is Borel measurable from S into Pk (X), h and the latter is a Polish space. Moreover, for 1 every C ∈ Pk (X), the map λ −→ λ(C) is continuous on M+ (X) (see Theorem 1 8.5.15). Therefore if we consider the function ξ1 : S × M+ (X) −→ R defined by
ξ1 (s, λ) = λ F1 (s) we see that ξ1 is a Carath´eodory function, hence jointly Borel-measurable. But
1 1 GrP1 = (s, λ) ∈ S × M+ (X) : ξ1 (s, λ) = 1 ∈ B(S) × B M+ (X) . Now we can prove the main existence theorem for the stochastic game. THEOREM 8.5.17 If hypotheses H4 , H5 , H6 hold, then the discounted stochastic game has a value, the value function is Borel-measurable, and the two players have optimal stationary policies. PROOF: Recall that for every u ∈ B0 (S), the function (s, λ, µ) −→ Ku (s, λ, µ)
(see (8.41))
1 1 is a Carath´eodory function from S × M+ (X) × M+ (Y ) into R. Hence it is jointly Borel-measurable. Also because of Proposition 8.5.16, we can find λn : S −→ 1 M+ (X), n ≥ 1, Borel-measuarble maps such that
P1 (s) = {λn (s)}n≥1 1 (Y ), we have So for fixed µ ∈ M+
for all s ∈ S.
642
8 Game Theory ϑu1 (s, µ) = max Ku (s, λ, µ) : λ ∈ P1 (s)
= sup Ku s, λn (s), µ , n≥1
⇒ s −→ ϑu1 (s, µ) is Borel-measurable. Moreover, Theorem 6.1.18(c) implies that µ −→ ϑu1 (s, µ)
1 is continuous on M+ (Y ).
1 Because of Proposition 8.5.16, we can find µn : S −→ M+ (Y ), n ≥ 1, Borel-measurable maps such that P2 (s) = {µn (s)}n≥1 for all s ∈ S. Therefore
1 v1 (s) = min ϑu1 (s, µ) : µ ∈ M+ (Y )
= inf ϑu1 s, µn (s) , ⇒ s −→
n≥1 u v1 (s) =
min
max Ku (s, λ, µ) is Borel-measurable.
s −→ v2u (s) = max
min Ku (s, λ, µ) is Borel-measurable.
µ∈P2 (s) λ∈P1 (s)
Similarly we show that λ∈P1 (s) µ∈P2 (s)
Moreover, Theorem 2.3.13, implies that v1u (s) = v2u (s) = v u (s)
for all s ∈ S.
(8.42)
Consider the operator V : B0 (S) −→ B0 (S) defined by V (u)(s) = v u (s). We have that V is a contraction map (see the proof of Proposition 8.5.5). So by the Banach fixed point theorem (see Theorem 3.4.3), there exists unique u∗ ∈ B0 (S) such that V (u∗ ) = u∗ . Then as in the proof of Theorem 8.5.7, we can find Borel maps 1 f ∗ : S −→ M+ (X)
such that
f ∗ (s) ∈ P1 (s)
and
and
1 g ∗ : S −→ M+ (Y ),
g ∗ (s) ∈ P2 (s)
u∗ (s) = min r s, f ∗ (s), µ + δ
for all s ∈ S.
u∗ (s )dq s |s, f ∗ (s), µ : µ ∈ P2 (s)
S
(8.43)
u∗ (s) = max r s, λ, g ∗ (s) + δ
u∗ (s )dq s |s, λ, g ∗ (s) : λ ∈ P1 (s)
S
u∗ (s) = r s, f ∗ (s), g ∗ (s) + δ
(8.44)
u∗ (s )dq s |s, f ∗ (s), g ∗ (s) .
S
(8.45)
8.5 Stochastic Games
643
The system of equations (8.43) through (8.45) may be interpreted as follows. For the game, u∗ (s) is the value of the game and f ∗ (s), g ∗ (s) are, respectively, the optimal strategies for players I and II. We prove that u∗ (·) is the value function of the stochastic game and f ∗ , g ∗ are optimal stationary policies for players I and II, respectively. To this end for every (f , g ) ∈ SP1 × SP2 (recall SP1 (resp., SP2 ) is the set of Borel-measurable selectors of P1 (resp., of P2 )), we set
W (f , g )(u)(s) = r s, f (s), g (s) + δ u(s )dq s |s, f (s), g (s) , s ∈ S. S
We interpret W (f , g )(u)(s) as the expected amount player II pays player I. When the initial state of the system is s ∈ S, player I uses strategy f (s), player II uses strategy g (s), and the game is terminated at the beginning of the second day with player II paying player I the amount u(s ) with s ∈ S being the state of the system on the second day. It is easy to check that W (f , g ) : B0 (S) −→ B0 (S) is a contraction and so it admits a unique fixed point, which is L(f ∗ , g ∗ ). Therefore we have
L(f ∗ , g ∗ )(s) = max r s, λ, g ∗ (s) + δ L(f ∗ , g ∗ )(s )dq s |s, λ, g ∗ (s) : S λ ∈ P1 (s)
= min r s, f ∗ (s), µ + δ L(f ∗ , g ∗ )(s )dq s |s, f ∗ (s), µ : S µ ∈ P2 (s) . Note that if we fix the stationary policy g ∗ of player II in the stochastic game (i.e., player II is allowed to use only this policy), then the stochastic game reduces to a dynamic programming model such as the one considered in the first part of this 1 section. More precisely, the state space is S, the action space M+ (X), the constraint ∗ multifunction is P1 , the law of motion q is given by q (s) , and the reward ·|s, λ, g
function r defined by r (s, λ) = r s, λ, g ∗ (s) . Hypotheses H4 , H5 , H6 imply that these quantities are well-defined and we can use Theorem 8.5.7 to conclude that f ∗ is an optimal stationary policy for the dynamic programming problem. From this it follows easily that L(f ∗ , g ∗ )(s) = sup L(π, g ∗ )(s) : π ∈ ∆1 , s ∈ S. (8.46) In a similar fashion, we show that L(f ∗ , g ∗ )(s) = inf L(f ∗ , β)(s) : β ∈ ∆2 ,
s ∈ S.
(8.47)
Therefore from (8.46) and (8.47) we deduce that L(f ∗ , g ∗ )(s) = inf sup L(π, β)(s) = sup inf L(π, β)(s), ∆2 ∆1
∆1 ∆2
s ∈ S.
(8.48)
From (8.48) it follows that the discounted stochastic game has a value, the value function s −→ L(f ∗ , g ∗ )(s) = u∗ (s) is bounded Borel-measurable (i.e., it belongs in B0 (S)), and f ∗ , g ∗ are optimal stationary policies for players I and II, respectively.
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8 Game Theory
8.6 Approximate Equilibria In this section, we return to the first topic of this chapter, namely noncooperative games. The individual stability in such games was described by the fundamental notion of Nash equilibrium (see Definition 8.1.5). In Theorem 8.1.9 we established the existence of such an equilibrium for cooperative games. However, in order to do that, we had to assume that the strategy set of each player is compact. We want to weaken this compactness condition. Excluding the compactness condition on the strategy set of each player, leads to the notion of ε-equilibrium (approximate equilibrium), because an equilibrium need not exist anymore. We consider a noncooperative n-player game in normal form as in Section 8.1. So let N = {1, . . . , n} be the set of players. Each player has a strategy set Xk which is assumed to be a subset of a Banach space Zk , k ∈ N . In contrast to Section 8.1, in order to make use of the approximate convex subdifferential, we assume that n each player k ∈ N , has a loss function uk : X −→ R. Then u = (uk )n k=1 : X −→ R is the multiloss map. We emphasize that this is done only for reasons of convenience and in what follows if ε = 0 and uk is replaced by −uk , we recover the setting of Section " 8.1. As before, we consider the following splitting of the multistrategy set X= Xk , k∈N = Xi . X = X × Xk with X = i=k
As before, we think of k c = N \ {k} as the coalition adverse (complementary) to player k. By pk : X −→ X and pk : X −→ Xk we denote the projection maps of X onto X and Xk , respectively. Every multistrategy x ∈ X can be written as x = (x, xk ) with x = pk (x) ∈ X and xk = pk (x) ∈ Xk . For each k ∈ N , we set (8.49) βk = inf uk (x) : x ∈ X and throughout this section we assume that βk > −∞ for all k ∈ N . Then the vector β = (βk )n k=1 is called the shadow minimum of the game. Note that u(X) ⊆ β + Rn +. If β ∈ u(X), then β = u(x∗ ) with x∗ ∈ X and so x∗ realizes the infimum in (8.49) for every player k ∈ N . However, this situation is rare and for this reason, we need to introduce approximate equilibria. Recall that in Section 8.1, we introduced the following quantity vk (xk ) = inf uk (x) : x ∈ X, pk (x) = xk . (8.50) This quantity represents the least utility for player k ∈ N among all feasible multistrategies, when she employs strategy xk . DEFINITION 8.6.1 A multistrategy x = (xk )n k=1 ∈ X is said to be an εequilibrium if for some ε ≥ 0 and for all k ∈ N , we have uk (x) ≤ vk (xk ) + ε.
8.6 Approximate Equilibria
645
REMARK 8.6.2 If ε = 0, then we recover the notion of Nash equilibrium (see Definition 8.1.5 with uk there replaced by −uk ). In this case, given the strategy x ∈ X of the adverse coalition kc , the player k ∈ N responds by choosing the strategy xk ∈ Xk that minimizes vk on Xk ; that is, uk (x, xk ) = inf vk (xk ) : x ∈ X, xk ∈ Xk . (8.51) Moreover, if N = {1, 2} and u1 (x) + u2 (x) = 0 (zero-sum game), then the εequilibrium is an ε-saddle point (a saddle point if ε = 0). As in Section 8.1, we introduce the function ξ : X × X −→ R defined by ξ(x, z) =
n
ui (x) − ui (x, zi ) .
i=1
PROPOSITION 8.6.3 If for some ε ≥ 0 and x ∈ X we have sup ξ(x, z) : z ∈ X ≤ ε,
(8.52)
then x is an ε-equilibrium multistrategy. PROOF: Let z = (x, zk ). Then we have ξ(x, z) = uk (x) − uk (z) ≤ ε (see (8.52), ⇒ uk (x) ≤ inf uk (z) : z ∈ X, pk (z) = xk + ε,
for all k ∈ N,
⇒ x ∈ X is an ε-equilibrium. PROPOSITION 8.6.4 If x ∈ X is an ε-equilibrium, then for every multistrategy z ∈ X, we have ξ(x, z) ≤ nε. PROOF: From Definition 8.3.1, we have uk (x) − uk (x, zk ) ≤ ε
for all k ∈ N and all zk ∈ Xk
(see (8.51).
Adding these inequalities over k ∈ N , we obtain ξ(x, z) ≤ nε
for all z ∈ X.
We introduce the conjugate function of zk −→ uk (x, zk ) (see Definition 1.2.15). So
= u∗k (x)(zk∗ ) = sup zk∗ , zk Zk −uk (x, zk ) : z ∈ Z = Zk , pk (z) = x , zk∗ ∈ Zk∗ . k∈N
(8.53) Here by Zk∗ we denote the topological dual of the Banach space Zk and by ·, ·Zk the duality brackets for the pair (Zk∗ , Zk ). Recall that u∗k (x)(·) is convex and lower semicontinuous. In addition to the subdifferential of a convex function introduced in Definition 1.2.28, we can define the approximate subdifferential (ε-subdifferential), which is also a useful tool in convex analysis.
646
8 Game Theory
DEFINITION 8.6.5 Let Y be a Banach space and ϕ : Y −→ R = R ∪ {+∞} be a proper convex function. For each ε ≥ 0 the ε-subdifferential of ϕ at y ∈ dom ϕ, is defined to be the set ∂ε ϕ(y) = y ∗ ∈ Y ∗ : y ∗ , v − y − ε ≤ ϕ(v) − ϕ(y) for all v ∈ domϕ . Here ·, · denotes the duality brackets for the dual pair (Y ∗ , Y ). REMARK 8.6.6 If ε = 0, then the above definition coincides with Definition 1.2.28 (i.e., ∂0 ϕ = ∂ϕ). However there is a basic difference between the subdifferential (ε = 0) and the ε-subdiffrential (ε > 0). Although ∂ϕ = ∂0 ϕ is a local notion, for ε > 0 ∂ε ϕ is a global one, namely the behavior of ϕ on all of X may be relevant for the construction of ∂ε ϕ. This explains why ∂ϕ and ∂ε ϕ have in general different properties, with the most basic difference being that if ϕ ∈ Γ0 (Y ) and y ∈ dom ϕ, for every ε > 0 the set ∂ε ϕ(y) is nonempty, w∗ -closed, and convex. For the function u∗k (x) ∈ Γ0 (Zk∗ ), the ε-subdifferential ∂ε u∗k (x)(zk∗ ) is defined by ∂ε u∗k (x)(zk∗ ) = zk∗∗ ∈ Zk∗∗ : zk∗∗ , v ∗ − zk∗ − ε ≤ u∗k (x)(vk∗ ) − u∗k (x)(zk∗ ) for all vk∗ ∈ Zk∗ . Using the notion of ε-subdifferential, we can produce necessary and, under some additional hypotheses, also sufficient conditions for a multistrategy to be an εequilibrium. THEOREM 8.6.7 If x = (xk )n k=1 is an ε-equilibrium when Xk = Zk for all k ∈ N (i.e., X = Z), then xk ∈ ∂ε u∗k (x)(0). PROOF: From (8.53), we see that u∗k (x)(0) = − inf uk (x, zk ) : z ∈ Z, pk (z) = x ≤ −uk (x) + ε
(because x is an ε-equilibrium)
≤ −vk∗ , xk Zk + u∗k (x)(vk∗ ) + ε ⇒ xk ∈ ∂ε u∗k (x)(0)
(see Definition 8.6.5).
THEOREM 8.6.8 If for every k ∈ N and for some multistrategy x ∈ X, uk (x, ·) is convex and lower semicontinuous on Zk and xk ∈ ∂ε u∗k (x)(0), then x ∈ X is an ε-equilibrium strategy. PROOF: Because by hypothesis xk ∈ ∂u∗k (x)(0), for all k ∈ N and all vk∗ ∈ Zk∗ , we have vk∗ , xk Zk − ε ≤ u∗k (x)(vk∗ ) − u∗k (x)(0), ⇒ vk∗ , xk Zk − u∗k (x)(vk∗ ) ≤ ε − u∗k (x)(0) ⇒ u∗∗ k (x)(xk ) ≤ ε + inf uk (x, zk ) : z ∈ Z, pk (z) = x ⇒ uk (x) ≤ ε + inf uk (x, zk ) : z ∈ Z, pk (z) = x
8.6 Approximate Equilibria
647
(because uk (x, ·) ∈ Γ0 (Zk )). This implies that x ∈ X is an ε-equilibrium.
ateaux differenNow we assume that the loss function uk of each player k, is Gˆ tiable and we examine how the Gˆ ateaux derivative and the ε-equilibria multistrategies are related. THEOREM 8.6.9 If the strategy set Xk ⊆ Zk of each player k ∈ N is closed, n " int Xk = ∅ is an ε-equilibrium with nonempty interior, x = (xk )n k=1 ∈ int X = k=1
with ε > 0 and for every k ∈ N, uk (x, ·) is lower semicontinuous and Gˆ ateaux differentiable on Zk , then there exists a multistrategy y = (yk )n k=1 ∈ X such that √ y − x ≤ n ε ∂uk √ (x, yk ) Z ∗ ≤ ε for every k ∈ N. and ∂zk k PROOF: Recall that in the beginning of the section, we have assumed that the game is bounded below. So we can apply Theorem 2.4.1 (the Ekeland variational principle) and obtain yk ∈ Xk such that √ (8.54) xk − yk ≤ ε √ for all zk ∈ Xk . (8.55) and uk (x, yk ) ≤ uk (x, zk ) + εzk − yk Let hk ∈ Zk and set zk = yk + thk for t > 0 small so that zk ∈ Xk (recall that xk ∈ int Xk ). Then √ 1 − εhk ≤ uk (x, yk + thk ) − uk (x, yk ) . t We let t −→ 0 to obtain √ − εhk ≤
∂uk (x, yk ), hk ∂zk
.
(8.56)
Zk
In (8.56), we take infimum of both sides with respect to all hk ∈ Zk with hk Zk = 1. It follows that ∂uk √ (x) Z ≤ ε, k ∈ N. k ∂zk Also, if we add inequalities (8.56), we conclude that √ x − yX ≤ n ε. THEOREM 8.6.10 If each player k ∈ N has a reflexive strategy space Zk , Xk = Zk , x = (xk )n k=1 ∈ X = Z is an ε-equilibrium multistrategy with ε > 0 and uk (x, ·) ∈ n " ∗ ∗ n ∗ Γ0 (Zk ), then we can find yε = (yε,k )n Zk∗ k=1 ∈ X = Z and yε = (yε,k )k=1 ∈ Z = such that for every player k ∈ N , we have xk − yε,k Zk ≤ √ ∗ Zk∗ ≤ ε; yε,k
k=1
√
ε
(8.57) (8.58)
648
8 Game Theory
moreover, yε = (yε,k )n uk , Xk }k∈N , k=1 ∈ X = Z is a Nash equilibrium of the game {$ where ∗ u $k (x) = uk (x, xk ) − yε,k , xk Z
for every x = (xk )n k=1 ∈ X =
k
n =
Xk .
k=1
PROOF: From Theorem 8.6.7, we have xk ∈ ∂ε u∗k (x)(0) ϑk (x)(vk∗ )
Let (8.59) we have
=
u∗k (x)
−
vk∗ , xk Zk
for all k ∈ N.
(8.59)
, k ∈ N . Then ϑk (x) ∈
Γ0 (Zk∗ )
ϑk (x)(0) ≤ inf ϑk (x)(vk∗ ) : vk∗ ∈ Zk∗ + ε.
and from (8.60)
From (8.60) it follows that ϑk (x)(·) is bounded below on Zk∗ . So we can apply ∗ Theorem 2.4.1 (the Ekeland variational principle) and obtain yε,k ∈ Zk∗ such that for ∗ all vk∗ = yε,k ∗ ϑk (x)(yε,k )−
√
∗ εvk∗ − yε,k Zk∗ < ϑk (x)(vk∗ ) √ ∗ ∗ ϑk (x)(yε,k ) ≤ ϑk (x)(0) − εyε,k Zk∗ .
and
(8.61) (8.62)
∗ From (8.61) it follows that yε,k minimizes the function √ ∗ f (x)(·) = ϑk (x)(·) + ε · −yε,k Zk∗
defined on Zk∗ . So we have
∗ √ ∗ 0 ∈ ∂ ϑk (x)(·) + ε · −yε,k Zk∗ (yε,k ).
Invoking Theorem 1.2.38, we have ∗ ∗ 0 ∈ ∂ϑk (x)(yε,k ) + ∂j(yε,k ),
where j(vk∗ ) =
√
(8.63)
∗ εvk∗ − yε,k Zk∗ . From Example 1.2.41(c), we know that ∗ )= ∂j(yε,k
√
∗∗
εB 1,k
∗∗ B 1,k being the closed unit ball in Zk∗∗ . The space Zk is Zk∗∗ = Zk and so from (8.63), (8.64), and the definition
with have that there exists yε,k ∈ Zk such that
and
(8.64) reflexive, therefore we of ϑk (x)(·), we deduce
∗ yε,k ∈ ∂u∗k (x)(yε,k ) √ yε,k = xk − εvk Zk , vk Zk ≤ 1.
From (8.66), we obtain xk − yε,k ≤
√
ε
which proves inequality (8.57) of the theorem. From (8.60) and (8.62), we have √ ∗ yε,k Zk∗ ≤ ε, which proves inequality (8.58) of the theorem. Next from (8.65), we have
(8.65) (8.66)
8.7 Remarks
∗ , yε,k vk∗ − yε,k
649
∗ ≤ u∗k (x)(vk∗ ) − u∗k (x)(yε,k ) for all vk∗ ∈ Zk∗ , ∗ ∗ ⇒ vk∗ , yε,k Zk − u∗k (x)(vk∗ ) − yε,k , yε,k Z ≤ −u∗k (x)(yε,k ), k ∗ ∗ ∗∗ ⇒ uk (x)(yε,k ) − yε,k , yε,k Z ≤ − sup yε,k , zk Z − uk (x, zk ) : zk ∈ Zk k k ∗ = inf uk (x, zk ) − yε,k , zk Z : zk ∈ Zk k ∗ ∗ ⇒ uk (x, yε,k ) − yε,k , yε,k Z ≤ inf uk (x, zk ) − yε,k , zk Z : zk ∈ Zk Zk
k
k
(8.67) Because uk (x, ·) ∈ Γ0 (Zk ). From (8.67) it follows that u $k (yε ) = min u $k (z) : z ∈ X, pk (z) = yε , $k }k∈N . ⇒ yε = (yε,k )n k=1 ∈ Z is a Nash equilibrium for the game {Xk , u
8.7 Remarks 8.1: Noncooperative games and the associated equilibrium notion (see Definition 8.1.5) were introduced by Nash [451, 452]. Nash proved Theorem 8.1.9 for games where the preferences of the players are described by continuous quasiconcave utility functions and the strategy sets are simplexes. The work of Nash generalized the pioneering work of von Neumann [454], who proved the first minimax theorem (using Brouwer’s fixed point theory) and initiated game theory. The work of Nash achieved the shift of attention from the minimax concept (saddle point) of von Neumann to the new concept of Nash equilibrium (noncooperative equilibrium). It is only with this new notion that the theory of noncooperative games could move with full generality to the case of n-players with n > 2. Theorem 8.1.9 assumes that the utility functions uk (x, yk ) are continuous in both variables. A partial generalization in this direction, can be found in Nikaido–Isoda [460]. Abstract economies (or generalized games; see Definition 8.1.11), were first introduced by Debreu [184], who was influenced by the work of Nash. He introduced and studied a model in which the agents’ (players’) preference relations are described by a utility function (hence they are transitive). Later Mas-Colell [410] and Shafer–Sonnenschein [549] dropped the transitivity requirement and described preferences using a preference multifunction (see Theorem 8.1.18). For further discussion of noncooperative games and abstract economies, with a rich bibliography, we refer to the books of Aubin [36], Border [86], and Ichiishi [324]. 8.2: The first side-payment game was introduced by von Neumann–Morgenstern [455], but had a specific interpretation of the characteristic function v of the game. The notion of core of a side-payment game (see Definition 8.2.1) was introduced by Shapley [550]. The concept of a balanced side-payment game was introduced by Bondavera [84, 85] and for non-side-payment games (see Definition 8.2.9) by Scarf [537]. Theorem 8.2.8 is independently due to Bondavera [84, 85], and Shapley [551]. Side-payment games with infinitely many players have been studied. We mention the works of Schmeidler [541, 542], Kannai [340], and Delbaen [193]. The first to
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8 Game Theory
formulate non-side-payment games, were Aumann–Peleg [41]. Aumann [42] defined the core for such games (see Definition 8.2.9). Theorem 8.2.13, was first proved by Scarf [537]. However, the proof presented here, which is based on the special version of the KKM-Theorem (see Proposition 8.2.12), is due to Shapley [552]. Non-sidepayment games with a continuum of players, can be found in Aumann–Shapley [47], Ichiishi [324], and in the paper of Ichiishi–Weber [323]. 8.3: The first to study the existence of Cournot–Nash equilibria for games with an atomless measure space of players (a continuum game model), was Schmeidler [543]. In Schmeidler’s model the strategy spaces are finite-dimensional. Khan [349] extended this to the infinite-dimensional case. His work and the subsequent extensions by Khan–Papageorgiou [351] and Balder–Yannelis [54] considered models in which each player has a preference multifunction instead of a utility function. So her preference relation need not be transitive or complete. Cournot–Nash equilibrium distributions for games that are viewed as a probability measure on the space of payoff (utility) function were established by Mas Collel [411], and Khan [352]. 8.4: For games with a finite number of players and actions and with private information (imperfect information in the terminology of Harsanyi [285]), equilibrium results were proved by Radner–Rosenthal [511], Milgram–Weber [429], and Mamer–Schilling [405]. Bayesian games with an infinite number of players, were first introduced by Palfrey–Srivastava [473], and Postlewaite–Schmeidler [501], who introduced the information partition approach used here. Bayesian games with an infinite-dimensional strategy space can be found in Balder–Yannelis [54], and Kim– Yannelis [353]. 8.5: The dynamic programming model has its origins in the pioneering work of Bellman [62]. A rigorous foundation of stochastic dynamic programming, was given by Blackwell [77, 78], and Strauch [561]. Advanced treatment of stochastic dynamic programming can be found in the books of Hinderer [305], Striebel [562], Bertsekas– Shreve [72], Dynkin–Yushkevich [216], Arkin–Evstingeev [29], and Maitra–Sudderth [399]. Theorem 8.5.3 is basic in the theory of stochastic dynamic programming and it is due to Blackwell [77]. Discounted stochastic games were investigated by many authors. Indicatively we mention the works of Maitra–Parthasarathy [397, 398], Parthasarathy [488, 489], Himmelberg–Parthasarathy–Raghavan–Van Vleck [302], Himmelberg–Parthasarathy–Van Vleck [303], Nowak [462, 463], and Whitt [604]. Adaptive control problems can be found in the book of Hermandez Lerma [290], and gambling games in the book of Maitra–Sudderth [399]. 8.6: Approximate equilibria for noncooperative games were studied by Tanaka– Yokoyama [575]. The ε-subdifferential of convex functions (see Definition 8.6.5) was studied in a systematic way by Hiriart–Urruty [306, 307].
9 Uncertainty, Information, Decision Making
Summary. *In this chapter, we study how information can be incorporated as a variable in various decision models. In a decision model with uncertainty, it is natural to model information by sub-σ-fields of a probability space which represents the set of all possible states of the “world”. For this reason we topologize the set of sub-σ-fields with two topologies, using tools from probability theory and functional analysis. Then we examine the “ex-post view” and the “ex-ante view”. In both cases, we prove continuous dependence of the model on the information variable. Then we introduce a third mode of convergence of the information variable and we study prediction sequences. Finally we study games with incomplete information or unbounded cost and general state space.
Introduction The goal of this chapter is to study how information can be incorporated as a variable in various decision models. Such a formulation allows us to treat situations with asymmetric information. In Section 9.1 we present a mathematical framework that permits the analytical treatment of the notion of information. We produce two such comparable metric topologies using tools from probability theory and functional analysis. In Section 9.2, we examine the ex-post view, in modelling systems with uncertainty. According to this view, every agent chooses an action after observing his or her information and updating his or her belief. Using the topologies of Section 9.1, we prove the continuity of this model on the information variable. In Section 9.3, we examine an alternative view of uncertain decision systems, known as the ex-ante view. In this alternative approach, an agent formulates a plan of what action to choose at each state before observing her information, subject to the constraint that the plan must be measurable with respect to her information. In Section 9.4, we introduce a third mode of convergence of information, distinct from the ones introduced and studied in Section 9.1. We show that this new mode of convergence is suitable in the analysis of prediction sequences. Section 9.5 formulates and studies a general two-person, zero-sum game with incomplete information. The hypotheses on the data of the model are minimal. N.S. Papageorgiou, S.Th. Kyritsi-Yiallourou, Handbook of Applied Analysis, Advances in Mechanics and Mathematics 19, DOI 10.1007/b120946_9, © Springer Science+Business Media, LLC 2009
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9 Uncertainty, Information, Decision Making
Finally in Section 9.6, we treat games with a general state space and unbounded cost. We prove the existence of equilibria.
9.1 Mathematical Space of Information In many economic models with uncertainty, the information structure is an endogenous variable or an exogenous parameter. A natural and general way to model information is to represent it by sub-σ-fields of a probability space (Ω, Σ, µ), which describes the set of all possible states of the world (Ω), the family (σ-field) of all possible events (Σ), and the distribution of those events (µ). In order to make formal statements on the relation of information to the other variables of the model, we need to define precisely the mathematical space of information and endow it with a suitable topology. This topology should not be too weak, or otherwise various decision variables that depend on information (such as consumer demand) may fail to be continuous. On the other hand the topology should not be too strong (too rich), or otherwise information structures (fields) that are known to lead to similar behavior, may fail to be topologically close to each other. So we need to achieve a rather delicate balance with the topology on the space of information. In this section, we outline a topology that achieves this balance. Let (Ω, Σ, µ) be a probability space. It represents the uncertainty present in the microeconomic model. By S ∗ we denote the set of all sub-σ-fields of Σ. On S ∗ we introduce the relation ∼, defined by for Σ1 , Σ2 ∈ S ∗ , we have Σ1 ∼ Σ2 if and only if they differ only by µ-null sets. So for every A ∈ Σ1 , there is a C ∈ Σ2 such that µ(A C) = 0 and vice versa. The space of information S0 is defined to be the set of ∼-equivalence classes of S ∗ ; that is, S0 = S ∗ / ∼ . In what follows, given Σ ∈ S ∗ and f ∈ L1 (Ω, Σ), by E(f |Σ ) we denote the conditional expectation of f with respect to the sub-σ-field Σ . Using the standard approximation of L1 (Ω, Σ) functions by simple functions, we obtain the following result. PROPOSITION 9.1.1 Σ1 ∼ Σ2 if and only if E(f |Σ1 ) = E(f |Σ2 ) for all f ∈ L1 (Ω, Σ). REMARK 9.1.2 Note that for every Σ ∈ S ∗ and every f ∈ L1 (Ω, Σ), E(f |Σ ) belongs in L1 (Ω, Σ ) ⊆ L1 (Ω, Σ).
Now let L L1 (Ω, Σ) be the Banach space of all bounded linear operators L : L1 (Ω, Σ) −→ L1 (Ω, Σ). Then for Σ ∈ S0 , E(·|Σ ) ∈ L1 (Ω, Σ ) ⊆ L1 (Ω, Σ) (see Remark 9.1.2) and the map Σ −→E(f |Σ ) from S0 into L 1 (Ω, Σ) is injective (see Proposition 9.1.1). So we can embed S0 into L L1 (Ω, Σ) and topologize it
1 with the subspace topology of any topology we consider on the space L L (Ω, Σ) . The
space L L1 (Ω, Σ) is a Banach space with the norm
9.1 Mathematical Space of Information LL = sup
L(f )
1
f 1
653
: f ∈ L1 (Ω, Σ), f = 0 .
The induced norm topology is called operator topology. In this
the uniform topology the map {L, K} −→ L ◦ K ∈ L L1 (Ω, Σ) is jointly continuous. We now introduce another topology on L L1 (Ω, Σ) that is weaker than the uniform operator topology and more suitable for our purposes. DEFINITION 9.1.3 The strong operator topology or topology of pointwise con
vergence is the weakest topology on L L1 (Ω, Σ) such that the maps
Ef : L L1 (Ω, Σ) −→ L1 (Ω, Σ) defined by Ef (L) = L(f ) are continuous for every f ∈ L1 (Ω, Σ). REMARK 9.1.4 A neighborhood basis of the origin is given by the sets
L ∈ L L1 (Ω, Σ) : L(fk )1 < ε, k = 1, . . . , n , 1 this topology where {fk }n k=1 is a finite collection of elements in L (Ω, Σ) and ε > 0. In a net {La }a∈J of operators converges to an operator L ∈ L L1 (Ω, Σ) denoted by s La −→ L if and only if La (f ) − L(f )1 −→ 0 for all f ∈ L1 (Ω, Σ). The map {L, K} −→ L ◦ K is separately but not jointly continuous.
In general the strong operator topology on L L1 (Ω, Σ) is not well-behaved and it is not metrizable. However, the relative topology on S0 ⊆
in particular L L1 (Ω, Σ) is much better behaved because S0 is a uniformly equicontinuous subset. In what follows we assume that the Lebesgue space L1 (Ω, Σ) is separable. This is true when Σ is countably generated and this in turn holds if Σ is the Borel σ-field of a second-countable Hausdorff topological space (e.g., of a separable metric
space). In the sequel by τs we denote the strong operator topology on L L1 (Ω, Σ) . THEOREM 9.1.5 If L1 (Ω, Σ) is separable, then (S0 , τs ) is a Polish space and the metric is given by ds (Σ1 , Σ2 ) =
∞ 1 min E(fk |Σ1 ) − E(fk |Σ2 )1 , 1 , 2k
(9.1)
k=1
where {fk }k≥1 is a countable dense subset of L1 (Ω, Σ). Any two metrics using different countable dense subsets of L1 (Ω, Σ) are uniformly equivalent. PROOF: First we show that ds is a metric. Clearly ds satisfies the triangle inequality. Also from Proposition 9.1.1 we see that Σ1 = Σ2 ⇒ ds (Σ1 , Σ2 ) = 0. On the other hand, if ds (Σ1 , Σ2 ) = 0, then from (9.1) we see that E(fk |Σ1 ) − E(fk |Σ2 )1 = 0
for all k ≥ 1.
(9.2)
Given any h ∈ L1 (Ω, Σ) and ε > 0, we can find k ≥ 1 such that h − fk < ε. Then we have
654
9 Uncertainty, Information, Decision Making E(h|Σ1 ) − E(h|Σ2 )1 ≤ E(h|Σ1 ) − E(fk |Σ1 )1 +E(fk |Σ2 ) − E(h|Σ2 )1
(see (9.2)) ≤ h − fk 1 + fk − h1 < 2ε. Because ε > 0 was arbitrary, we let ε ↓ 0 to obtain E(h|Σ1 ) − E(h|Σ2 )1 = 0 ⇒ Σ1 = Σ2
for all h ∈ L1 (Ω, Σ)
(see Proposition 9.1.1).
Finally it is obvious that ds (Σ1 , Σ2 ) = ds (Σ2 , Σ1 ) for all Σ1 , Σ2 ∈ S0 . Therefore we have shown that ds is a metric on S0 . Note that for all f ∈ L1 (Ω, Σ) and all Σ ∈ S0 , we have E(f |Σ )1 ≤ f 1 , ⇒ S0 is uniformly equicontinuous. So from a result of point-set topology (see, e.g., Kelley [344, p. 238]), we have that the topologies ds and τs on S0 coincide and in fact ds is independent of the particular dense sequence {fk }k≥1 we use. Moreover, the topology is separable. It remains to show that the metric topology is complete. To this end let {Σn }n≥1 be a ds -Cauchy sequence. Then given ε ∈ (0, 1) and k ≥ 1, there exists N such that for m, n ≥ N , we have ε , 2k (see (9.1)), ⇒ E(fk |Σn ) − E(fk |Σm )1 < ε 1 ⇒ E(fk |Σn ) n≥1 ⊆ L (Ω, Σ) is a Cauchy sequence. ds (Σn , Σm )
0, we can find m0 = m0 (ε) ≥ 1 such that E(f |Γm ) − E(f |Λm )1
0 are given, then we can find a finite Σ-partition Σf ∈ S0 of Ω with Σf ⊆ Σ such that ds (Σ, Σf ) < ε. ∞ PROOF: Let {fk }N k=1 ⊆ L (Ω, Σ) and let c> max fk ∞ . Choose r > 0 such that
c < rε/4. For every k = {1, . . . , N } we set
and set
1≤k≤N
ε(i + 1) εi Aki = ω ∈ Ω : ≤ E(fk |Σ)(ω) ≤ 2 2 Σk = σ Ak1 , . . . , Akr , Ak,(−1) , . . . , Ak,(−r) and Σf = σ{Σk }N k=1 . Then, E(fk |Σ)(ω) − E(fk |Σf )(ω) < ε µ-a.e. on Ω.
Next we show how the information topology depends on the probability measure µ. This way, we are able to handle certain asymmetries of the agents’ prior beliefs about uncertainty. Recall that the collection of null-sets affects the information space. For this reason, to be able to proceed further in our discussion we need the following definition. DEFINITION 9.1.11 Let µ, ν be two probability measures on the measurable space (Ω, Σ). We say that µ and ν are equivalent, if they produce the same null sets. REMARK 9.1.12 For equivalent measures there is no ambiguity in the expression almost everywhere. So the space of information S ∗ is the same. Also L∞ (Ω, Σ, µ) = L∞ (Ω, Σ, ν) for µ, ν equivalent. However, it is not true that L1 (Ω, Σ, µ) = L1 (Ω, Σ, ν) (although the common L∞ -space is dense in both). Moreover, the conditional expectations Eµ (·|Σ ), Eν (·|Σ ), Σ ∈ S0 , with respect to µ and ν, are different. The equivalent measures µ, ν are mutually absolutely continuous and so we can define the Radon–Nikodym derivatives (dµ)/(dν), (dν)/(dµ).
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9 Uncertainty, Information, Decision Making
LEMMA 9.1.13 If µ, ν are equivalent probability measures on (Ω, Σ), Σ ∈ S0 , dν and f ∈ L∞ (Ω, Σ), then Eµ (f |Σ ) = Eµ dµ |Σ Eν f dµ |Σ a.e. on Ω. dν
PROOF: For A ∈ Σ , we have
dν dµ Eµ |Σ Eν f |Σ dµ dµ dν
A dν dµ = Eµ Eν f |Σ dµ dµ dν A
dµ = Eν f f dµ, |Σ dν = dν A A
dν dµ ⇒ Eµ (f |Σ ) = Eµ |Σ Eν f |Σ dµ dν
a.e. on Ω.
∞
Taking f = 1 ∈ L (Ω, Σ), we obtain the following corollary. COROLLARY 9.1.14 If µ, ν are equivalent probability measures on (Ω, Σ) and
dν −1 |Σ = Eµ dµ |Σ a.e. on Ω. Σ ∈ S0 , then Eν dµ dν Recall that throughout this section, we assume that L1 (Ω, Σ) is separable. LEMMA 9.1.15 If µ, ν are equivalent probability measures on (Ω, Σ), and ⊆ L∞ (Ω, Σ) is dense {fk }k≥1 ⊆ L∞ (Ω, Σ) is dense in L1 (Ω, Σ, µ), then fk dµ dν k≥1 in L1 (Ω, Σ, ν). PROOF: Let h ∈ L1 (Ω, Σ, ν). We have
dµ dν dµ h − fk dµ 1 = − f = − f h h dν dν k k dν L (ν) dν dµ dν Ω Ω dν = h − fk L1 (µ) dµ dµ ⇒ fk ⊆ L∞ (Ω, Σ) is dense in L1 (Ω, Σ, ν). dν k≥1 Using these auxiliary results, we can now compare the metrics ds (µ) and ds (ν) generating the strong operator topologies (topologies of pointwise convergence) for the two probability measures µ and ν. THEOREM 9.1.16 If µ, ν are equivalent probability measures on (Ω, Σ) and ds (µ), ds (ν) are the metrics generating the corresponding strong operator topologies on S0 , then ds (µ) and ds (ν) are uniformly equivalent. PROOF: Let {fk }k≥1 ⊆ L∞ (Ω, Σ) be dense in L1 (Ω, Σ, µ) with f1 = 1. From Theorem 9.1.5, we know that ds (ν)(Σ1 , Σ2 ) =
∞ 1 min E(fk |Σ1 ) − E(fk |Σ2 )L1 (ν) , 1 k 2
k=1
9.1 Mathematical Space of Information
659
is a metric generating the strong operator topology (topology of pointwise convergence) on S0 for the prior ν. Given ε > 0, we choose k0 ≥ 1 and c > 1 such that 1 0 such that for all A ∈ Σ ν(A) < δ
We have
Eν Ω
So, if we set
implies µ(A) <
dν dµ |Σ1 dν = dµ dν
Eµ Ω
ε . 4c
dν |Σ1 dµ = 1. dµ
dν dµ 1 |Σ1 (ω) (ω) ≤ , A = ω ∈ Ω : Eµ dµ dν δ
we have
ε (by Chebyshev’s inequality). µ(A) ≥ 1 − 4c
k +2 If we set ϑ = δε 1 (2 0 c) , then ds (ν)(Σ1 , Σ2 ) < ϑ implies dµ
dµ δε Eν fk |Σ1 − Eν fk |Σ2 L1 (ν) < dν dν 8c
for all k < k0
and so using Lemma 9.1.15, we obtain Eµ (fk |Σ1 ) − Eµ (fk |Σ2 )L1 (µ)
0 are given, then there exists Ω0 ∈ Σ with µ(Ω0 ) = 1 such that {E(U |Σ0 )(ω) : ω ∈ Ω0 , Σ0 ∈ S0 } is equicontinuous on C and uniformly equicontinuous on every compact K ⊆ C. PROOF: Let x ∈ X and δ > 0 be given. Because for all ω ∈ Ω0 , µ(Ω0 ) = 1, u(ω, ·) is continuous, we can find η = η(x, δ) > 0 such that x ∈ X
and
x − x < η imply |u(ω, x ) − u(ω, x)| < δ.
Then for all ω ∈ Ω0 we have
|E u(·, x)|Σ0 (ω) − E u(·, x )|Σ0 (ω)|
≤ E |u(·, x) − u(·, x )|Σ0 (ω) < δ. This proves the first assertion of the proposition. The second assertion follows from the fact that on compact sets equicontinuity is in fact uniform equicontinuity. Now we can havethe first result on the continuous dependence on the informa
tion. Here L1 (Ω, U )= U:Ω−→U : U is Borel-measurable and E (U, 0) < ∞ .
9.2 The ex-Post View
665
THEOREM 9.2.4 If {Un , U }n≥1 ⊆ D, Un (ω) −→ U (ω) µ-a.e. in U , for every K ⊆ C compact there exists MK > 0 such that for all x ∈ K and all n ≥ 1 we have |un (ω, x)| ≤ MK
and
|u(ω, x)| ≤ MK
µ-a.e. on Ω,
d
s and Σn −→ Σ as n → ∞, then E(Un |Σn ) −→ E(U |Σ) as n → ∞ in L1 (Ω, U ) and in probability.
PROOF: Let K ⊆ X compact and ε > 0 be given. By hypothesis Un (ω) −→ U (ω) µa.e. in U , therefore we have
lim sup |un (ω, x) − u(ω, x)| = 0
n→∞ x∈K
µ-a.e. on Ω.
So we can find n0 (ε) ≥ 1 such that sup un (·, x) − u(·, x)1 < x∈K
ε 2
for all n ≥ n0 .
(9.19)
Also we can find n1 (ε) ≥ 1 such that
ε sup E u(·, x)|Σn − E u(·, x)|Σ 1 < . 3 x∈K
(9.20)
Then for every n ≥ max{n0 (ε), n1 (ε)}, we have
sup E un (·, x)|Σn − E un (·, x)|Σ 1 x∈K
≤ sup E un (·, x)|Σn − E u(·, x)|Σn 1 x∈K
+ sup E u(·, x)|Σn − E u(·, x)|Σ 1 x∈K
+ sup E u(·, x)|Σ − E un (·, x)|Σ 1 x∈K
0 such that, ω ∈ Ω0 , x, x ∈ K with x − x < δn imply
and
|E un (·, x)|Σn (ω) − E un (·, x )|Σn (ω)| < ε
|E un (·, x)|Σ (ω) − E un (·, x )|Σ (ω)| < ε.
(9.22) (9.23)
m Because K is compact it is totally bounded and so we can find {xn k }k=1 ⊆ K m such that K ⊆ Bδn (xn k ). We have k=1
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9 Uncertainty, Information, Decision Making
sup |E un (·, x)|Σn (ω) − E un (·, x)|Σ (ω)|dµ Ω x∈K
≤ sup |E un (·, x)|Σn (ω) − E un (·, xn k )|Σn (ω)|dµ x∈K Ω
n |E un (·, xn + k )|Σn (ω) − E un (·, xk )|Σ (ω)|dµ
Ω
+ sup |E un (·, xn k )|Σ (ω) − E un (·, x)|Σ (ω)|dµ Ω x∈K
≤ 3ε
(see (9.21), (9.22), and (9.23)).
(9.24)
Recall that Un (ω) −→ U (ω) µ-a.e. on Ω in U . So we can find n2 (ε) ≥ 1 such that for all n ≥ n2 (ε), we have
sup |E un (·, x)|Σ (ω) − E u(·, x)|Σ (ω)|dµ x∈K Ω
sup E |un (·, x)| − u(·, x)|Σ (ω)|dµ < ε. (9.25) ≤ Ω x∈K
Finally letting n(ε) = max n0 (ε), n1 (ε), n2 (ε) , for n ≥ n(ε) we have
sup |E un (·, x)|Σn (ω) − E u(·, x)|Σ (ω)|dµ < 3ε Ω x∈K
⇒ E(Un |Σn ) −→ E(U |Σ)
in L1 (Ω, U ) and in probability. d
s COROLLARY 9.2.5 If {Un , U }n≥1 ⊆ D, Un −→ U in L1 (Ω, U ) and Σn −→ Σ, then 1 E(Un |Σn ) −→ E(U |Σ) in L (Ω, U) and in probability.
Next we establish the continuity of the excess demand function on the information variable. THEOREM 9.2.6 The excess demand function de : ∆ × S0 × int C × D −→ L1 (Ω, RN ) is continuous when S0 is endowed with the topology of pointwise convergence. PROOF: Suppose that (pn , Σn , en , Un ) −→ (p, Σ, e, U ) in ∆ × S0 × int C × D as n → ∞, with e ∈ int C. Then by virtue of Corollary 9.2.5, we have that E(Un |Σn )(ω) −→ E(U |Σ)(ω)
in U for µ-a.a. ω ∈ Ω.
(9.26)
Let xn ∈ B(en , pn ) such that
E un (·, xn )|Σn (ω) = max E un (·, x)|Σn (ω) : x ∈ B(en , pn ) . Note that
n≥1
B(en , pn ) ∈ Pk (RN ) and so we may assume that xn −→ x in RN .
From Proposition 9.2.2 and (9.26) we have
E un (·, xn )|Σn (ω) −→ E u(·, x)|Σ (ω)
for µ-a.a. ω ∈ Ω.
9.2 The ex-Post View
667
Because B(en , pn ) −→ B(e, p), is given any x ∈ B(e, p) we can find xn ∈ B(en , pn ), n ≥ 1, such that xn −→ x in C and so
E un (·, xn )|Σn (ω) −→ E u(·, x)|Σ (ω)
for µ-a.a. ω ∈ Ω.
We have
E un (·, xn )|Σn (ω) ≤ E un (·, xn )|Σn (ω) for µ-a.a. ω ∈ Ω,
⇒ E u(·, x)|Σ (ω) ≤ E u(·, x)|Σ (ω) for µ-a.a. ω ∈ Ω. Because x ∈ B(e, p) was arbitrary, it follows that
x = arg max E u(·, x)|Σ (ω) : x ∈ B(e, p) , ⇒ de
is continuous as claimed by the theorem.
DEFINITION 9.2.7 The value of information of the choice problem is the function V : ∆×S0 ×int C ×D −→ R defined by
V (p, Σ0 , e, U ) = U (ω) de (p, Σ0 , e, U )(ω) + e dµ. Ω
Concerning the function V , we have the following continuity result. THEOREM 9.2.8 The value of information map V : ∆ × S0 × int C × D −→ R is continuous when S0 is endowed with the topology of pointwise convergence. PROOF: Suppose (pn , Σn , en , Un ) −→ (p, Σ, e, U ) in ∆ × S0 × int C × D with e ∈ int C. Then from Theorem 9.2.6, we have that de (pn , Σn , en , Un ) −→ de (p, Σ, e, U )
in L1 (Ω, U )
as n → ∞.
(9.27)
Evidently the integral functional
h −→ U (ω) h(ω) + e dµ Ω
is continuous on L1 (Ω, U). Therefore because of (9.27) we have V (pn , Σn , en , Un ) −→ V (p, Σ, e, U ), ⇒ V
is continuous.
REMARK 9.2.9 Of course Theorems 9.2.6 and 9.2.8 are also valid if on S0 we consider the d-metric topology.
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9 Uncertainty, Information, Decision Making
9.3 The ex-ante View In this section we investigate the second way of viewing information in a decision problem and how it affects behavior. This is the ex-ante view or measurability view. According to this approach, the agent formulates a plan of what action to take at each state before observing his information, subject to the constraint that the plan be measurable with respect to his information. So information affects the actions through this measurability constraint. The decision problem has the following structure. There is uncertainty represented by a probability space (Ω, Σ, µ) with Σ countably generated so that L1 (Ω) is separable. The information observed by the agent is represented by a sub-σ-field Σ0 of Σ. We emphasize that this is an ex-ante representation of information and so it does not denote what the agent has observed, but rather what an agent would observe in each state. Before observing his information, the agent formulates a statecontingent plan f : Ω −→ X where X ⊆ RN is a nonempty, compact, and convex set, representing the set of all feasible actions. We must have that f is Σ0 -measurable in order for the plan to be informationally feasible. The agent also faces for the plan f an additional constraint B(p) depending on a variable p ∈ P (p can be, e.g., the price system prevailing in the market). The set B(p) denotes all those plans that are economically feasible when the measurability constraint is not taken into account. The agent has some preferences among the feasible plans, which is described by the complete preorder . Then the decision problem facing the agent, is the following. max : f : Ω −→ X, f is Σ0 -measurable, f ∈ B(p) . (9.28) In problem (9.28), we view p −→ B(p) as the economic constraint multifunction. On the set E of all Σ-measurable functions f : Ω −→ X, we introduce the equivalence relation ∼ of equality µ-a.e. We set E0 = E/ ∼. DEFINITION 9.3.1 A plan is an element of E0 . The measurability constraint multifunction M : S0 −→ 2E0 is defined by M (Σ0 ) = f ∈ E0 : f is Σ0 -measurable . (9.29) Then the overall constraint multifunction ξ : P × S0 −→ 2E0 is defined by ξ(p, Σ0 ) = B(p) ∩ M (Σ0 ).
(9.30)
E0
The solution multifunction S : P × S0 −→ 2 for decision problem (9.28) is defined by S(p, Σ0 ) = f ∈ ξ(p, Σ0 ) : g f for all g ∈ ξ(p, Σ0 ) . (9.31) We want to characterize the continuity of S. DEFINITION 9.3.2 Let Z be a Hausdorff topological space and a complete preorder on Z. Then (a) is lower semicontinuous (lsc for short) if for every z ∈ Z, the set L(z) = {z ∈ Z : z z} is closed.
9.3 The ex-ante View
669
(b) is upper semicontinuous (usc for short) if for every z ∈ Z, the set U (z) = {z ∈ Z : z z } is closed. (c) is continuous if it is both lsc and usc. REMARK 9.3.3 Recall that a general binary relation R on a set X is a preorder (or quasiorder), if it is reflexive (i.e., xRx) and transitive (i.e., xRy, yRz imply xRz). If in addition xRy and yRx imply x = y, then R is called a partial order (or simply an ordering). We should mention that some authors do not require a partial order to be reflexive. If for a preorder R we necessarily have xRy or yRx or both, then we say that R is a complete or total preorder. Evidently a complete preorder is reflexive. A total partial order is called a linear order . Given a preorder on X, we can define the associated strict preorder ≺ by x≺y
if and only if x y
and
y x.
Evidently ≺ is irreflexive and transitive. If the set X is a Hausdorff topological space, then in analogy to Definition 9.3.2 we can say that ≺ is lsc (resp., usc), if for every x ∈ X, the set Ls (x) = {x ∈ X : x ≺ x } (resp., the set Us (x) = {x ∈ X : x ≺ x}) is open. Note that for a total preorder , lower semicontinuity of the corresponding ≺ is equivalent to the lower semicontinuity of (see Definition 9.3.2(a)). Similarly for the upper semicontinuity. The parameter space P is assumed to be Hausdorff topological space. Now we want to topologize E0 . We consider two topologies on E0 . The first is the norm topology of L1 (Ω, RN ), denoted by s-. The second is the weak topology of L1 (Ω, RN ), denoted by w-. PROPOSITION 9.3.4 If u : Ω × X −→ R is a function such that (i) For all x ∈ X, ω −→ u(ω, x) is Σ-measurable, (ii) For µ-a.a. ω ∈ Ω, x −→ u(ω, x) is continuous, (iii) ω −→ max |u(ω, x)| belongs in L1 (Ω), x∈X
then the integral operator U : E0 −→ R defined by
u ω, x(ω) dµ U (x) = Ω
is s-continuous. PROOF: This is an immediate consequence of a dominated convergence theorem and from the fact that from a strongly convergent sequence in L1 (Ω, RN ), we can extract a µ-a.e. convergent subsequence. REMARK 9.3.5 Note, however, that U is not in general w-continuous. Let Ω = [0, 1], Σ = B([0, 1]), and µ = λ = the Lebesgue measure. Also suppose N = 1 and X = [0, 1] ⊆ R. Consider the utility function u(x) = x2 . Then if {xn }n≥1 is the sequence of Rademacher functions; that is,
670
9 Uncertainty, Information, Decision Making 1 if t ∈ 2kn , k+1 k = even 2n xn (t) = , −1 otherwise w
then xn −→ 0 in L1 ([0, 1]) but U (xn ) = 1 U (0) = 0. Our stated goal is to determine the continuity properties of the solution multifunction S(p, Σ0 ) (see (9.31)). A reasonable approach is to use the Berge maximum theorem (see Theorem 6.1.18). Then according to that result if ξ is lower semicontinuous and has a closed graph and the preference relation is continuous (in the sense of Definition 9.3.2(c)), then S has closed graph. This requires that on E0 we employ the w-topology, which makes E0 w-compact and assume that is w-continuous, a rather restrictive requirement as was observed in Remark 9.3.5. So we see that although the weak topology is a good choice for E0 because it gives compactness of that space (see Theorem 6.4.23), however, this choice is too restrictive for the preference relation , where the natural choice is the strong topology. Hence we have conflicting requirements and eventually what we need is a version of Theorem 6.1.18 when two different topologies are involved. Such a result was proved by Horsley–Van Zandt–Wrobel [312, Corollary 8]. PROPOSITION 9.3.6 If (P, τ ) is a Hausdorff topological space, V is a linear space on which we have two linear topologies s, w, whose restrictions on any straight line in V are identical to the usual topology of R, X is a convex and w-compact subset of V , B : P −→ 2X \{∅} has w-compact values and it is τ -to-w usc and τ -to-s lsc, is a total preorder on X which is s-lsc and w-usc, then the optimal action correspondence S(p) = x ∈ B(p) : for all z ∈ B(p) z x , p ∈ P, from P into 2X \ {∅} is τ -to-w usc with w-compact values. Using this proposition in the case of our decision problem, we obtain the following result. PROPOSITION 9.3.7 If (P, τ ) is a Hausdorff topological space, on S0 we have a Hausdorff topology τ0 , ξ is τ × τ0 -to-s lower semicontinuous, and Gr ξ is closed for the product topology τ ×τ0 ×w, is s-lsc and w-usc, then Gr S is closed for the product topology τ × τ0 × w and has nonempty and w-compact values. The above result is missing the concrete topology τ0 on S0 that will make the measurability constraint multifunction M (see (9.29)) s-lsc and w-closed. In Section 9.1, we defined and discussed the ds -metric topology on S0 (the topology of pointwise convergence). PROPOSITION 9.3.8 If on S0 we consider the ds -metric topology, then the map (f, Σ0 ) −→ E(f |Σ0 ) is w × ds -to-w continuous. PROOF: Recall that the weak topology on E0 is metrizable. So we can work with w×d sequences. Suppose (fn , Σn ) −→s (f, Σ). For every h ∈ L∞ (Ω), we have
9.3 The ex-ante View
671
E(fn |Σn )hdµ = Ω
fn E(h|Σn )dµ. Ω
Because L∞ (Ω) ⊆ L1 (Ω), we have E(h|Σn ) −→ E(h|Σ) in L1 (Ω). Also note that w∗
because X is compact and fn −→ f in L1 (Ω), we also have that fn −→ f in L∞ (Ω). Then
fn E(h|Σn )dµ −→ f E(h|Σ)dµ = E(f |Σ)hdµ, Ω
Ω
Ω ⇒ E(fn |Σn )hdµ −→ E(f |Σ)hdµ. w
Ω
Ω
∞
Because h ∈ L (Ω) was arbitrary, we conclude that w
E(fn |Σn ) −→ E(f |Σ)
in L1 (Ω).
PROPOSITION 9.3.9 If on S0 we consider the ds -metric topology, then M : S0 −→ 2E0 has nonempty, w-compact, convex values and it is ds -to-s lower semicontinuous and Gr M is ds ×w-closed. PROOF: First we show the ds -to-s lower semicontinuity. To this end suppose ds Σn −→ Σ. It suffices to show that M (Σ) ⊆ s- lim inf M (Σn ).
(9.32)
So let g ∈ M (Σ). Then gn = E(g|Σn ) ∈ M (Σn ) for all n ≥ 1 and gn −→ g in L1 (Ω). This proves (9.32) and so we have the s-lower semicontinuity of M . Next we show that Gr M is ds × w-closed. To this end let {(Σn , gn )}n≥1 ⊆ Gr M d
w
s and Σn −→ Σ, gn −→ g in L1 (Ω). From Proposition 9.3.8 we have
w
gn = E(gn |Σn ) −→ E(g|Σ)
in L1 (Ω),
⇒ g = E(g|Σ) (i.e., g ∈ M (Σ)). Therefore Gr M is ds × w-closed.
PROPOSITION 9.3.10 If (P, τ ) is a Hausdorff topological space, S0 is endowed with the ds -metric topology, and B : P −→ 2E0 \{∅} has a graph that is τ × w-closed, then ξ : P ×S0 −→ 2E0 \{∅} has a graph that is τ ×ds × w-closed. DEFINITION 9.3.11 We say that the constraint multifunction B : P −→ 2E0 \{∅} is state independent if, for all p ∈ P, all g ∈ B(p), and all Σ0 ∈ S0 , we have E(g|Σ0 ) ∈ B(p). REMARK 9.3.12 One way to interpret the above condition is to say that the economic constraint multifunction B does not reveal information. Another interpretation is to say that information is not necessary in order to satisfy the constraint. There are economic models in which this condition is not satisfied (e.g., the microeconomic rational expectations general equilibrium model).
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9 Uncertainty, Information, Decision Making
PROPOSITION 9.3.13 If (P, τ ) is a Hausdorff topological space, S0 is endowed with the ds -metric topology, and B : P −→ 2E0 \{∅} is τ -to-s lower semicontinuous, then ξ : P ×S0 −→ 2E0 \{∅} is τ ×ds -to-s lower semicontinuous. PROOF: First we show that
ξ(p, Σ0 ) = E B(p)|Σ0
for all (p, Σ0 ) ∈ P × S0 .
(9.33)
Let g ∈ ξ(p, Σ0 ). Then from (9.30) we see that g ∈ B(p) and because g is Σ0 -measurable, we have g = E(g|Σ0 ). Therefore g ∈ E B(p)|Σ0 and we have proved
(9.34) ξ(p, Σ0 ) ⊆ E B(p)|Σ0 .
On the other hand, if g ∈ E B(p)|Σ0 , then we can find h ∈ B(p) such that g = E(h|Σ0 ). Because B is by hypothesis state-independent, we have g ∈ B(p). Because g is Σ0 -measurable we have g ∈ M (Σ0 ), hence g ∈ B(p) ∩ M (Σ0 ) = ξ(p, Σ0 ). Therefore we have proved that
E B(p)|Σ0 ⊆ ξ(p, Σ0 ). (9.35) From (9.34) and (9.35), we conclude that (9.33) holds. Let η1 : P ×S0 −→ E0 ×S0 be defined by η1 (p, Σ0 ) = B(p) × {Σ0 }. Evidently this is τ × ds -to-s × ds lower semicontinuous. Also let η2 : E0 × S0 −→ E0 be defined by η2 (g, Σ0 ) = E(g|Σ0 ). We know that η2 is s × ds -to-s continuous (see Proposition 9.3.8). From (9.33) we see that ξ = η2 ◦ η 1 , ⇒ ξ
is τ × ds -to-s lower semicontinuous.
REMARK 9.3.14 The result remains true if instead we assume that the economic constraint multifunction B : P −→ 2E0 \ {∅} is τ -to-w lower semicontinuous (see Proposition 9.3.8). The above results yield the following theorem on the continuity of the solution multifunction ξ(p, Σ0 ). THEOREM 9.3.15 If (P, τ ) is a Hausdorff topological space, on S0 we consider the ds -metric topology, and (i) B : P −→ 2E0 \{∅} is τ -to-s lower semicontinuous and Gr B is τ × w-closed, (ii) B is state-independent (see Definition 9.3.11),
9.4 Convergence of σ-Fields and Prediction Sequences
673
(iii) is s-lower semicontinuous and w-upper semicontinuous, then the solution multifunction S : P ×S0 −→ 2E0 has nonempty, w-compact values, and Gr S is τ ×ds ×w-closed. PROOF: From Propositions 9.3.10 and 9.3.13 we have that (p, Σ0 ) −→ ξ(p, Σ0 ) is s-lsc and w-closed and clearly has nonempty values. These facts in conjunction with hypothesis (iii) permit the use of Proposition 9.3.6, from which we get the conclusion of the theorem.
9.4 Convergence of σ-Fields and Prediction Sequences In this section motivated from the work in Section 9.1 and in particular from Definition 9.1.8(b), we define still another mode of convergence of sub-σ-fields, which turns out to be suitable for the convergence of prediction sequences. So let (Ω, Σ, µ) be a complete probability space. To simplify our considerations throughout this section we assume that all sub-σ-fields considered are µ-complete. DEFINITION 9.4.1 Let {Σn }n≥1 be a sequence of sub-σ-fields of Σ (a) µ- lim inf Σn is the sub-σ-field Σ0 such that for every f ∈ L∞ (Ω), we have n→∞
E(f |Σ0 )1 ≤ lim inf E(f |Σn )1 .
(9.36)
n→∞
and Σ0 is maximal among all sub-σ-fields satisfying (9.36), namely if Σ0 is a sub-σ-field for which (9.36) is true (with Σ0 replaced by Σ0 ), then Σ0 ⊆ Σ0 . (b) µ- lim sup Σn is the sub-σ-field Σ0 such that for every f ∈ L∞ (Ω), we have n→∞
lim sup E(f |Σn )1 ≤ E(f |Σ0 )1 .
(9.37)
n→∞
and Σ0 is minimal among all sub-σ-fields satisfying (9.37), namely if Σ0 is a sub-σ-field for which (9.37) is true (with Σ0 replaced by Σ0 ), then Σ0 ⊆ Σ0 . Immediately from this definition, we have the following. PROPOSITION 9.4.2 If {Σn }n≥1 ∈ S0 , then we always have µ- lim inf Σn ⊆ µ- lim sup Σn .
n→∞
n→∞
REMARK 9.4.3 The inclusion in the above result can be strict. PROPOSITION 9.4.4 If {Σn }n≥1 ∈ S0 , then µ- lim inf Σn = n→∞ lim inf µ(A C) : C ∈ Σn = 0 .
n→∞
A ∈ Σ :
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9 Uncertainty, Information, Decision Making
PROOF: In what follows Σl = A ∈ Σ : lim inf µ(A C) : C ∈ Σn = 0 . n→∞
For any sets A1 , A2 , C1 , C2 ∈ Σ we have
µ (A1 ∩ A2 ) (C1 ∩ C2 ) ≤ µ(A1 C1 ) + µ(A2 C2 )
and µ (A1 ∪ A2 ) (C1 ∩ C2 ) ≤ µ(A1 C1 ) + µ(A2 C2 ).
(9.38) (9.39)
From (9.38) and (9.39), we see that A1 , A2 ∈ Σl ⇒ A1 ∪ A2 ∈ Σl
and
A1 ∩ A2 ∈ Σl .
(9.40)
Suppose that {Cn }n≥1 ⊆ Σl and set C=
!
Cn ,
Cm =
n≥1
m !
Cn .
n=1
Given ε > 0, we can find m = m(ε) ≥ 1 such that ε µ(C Cm ) < . 2 Because of (9.40), we have that Cm ∈ Σl . So from the definition of Σl , we can find n0 ≥ 1 and Dn ∈ Σn such that for n ≥ n0 we have ε µ(Dn Cm ) < 2 ⇒ inf µ(C Cn ) : Cn ∈ Σn ≤ µ(C Cn ) ≤ (C Cn ) + µ(Cm Cn ) < ε for n ≥ n0 ⇒ Σl
is closed under countable unions.
Clearly Σl is closed under complementation and so Σl is a σ-field. It is easy to see that any set A ∈ Σl satisfies
1 1 − µ(A|Σn )dµ = − µ(A|Σl )dµ lim n→∞ Ω 2 2 Ω
(9.41)
and so from (9.41), it follows that for every f ∈ L∞ (Ω), we have E(f |Σl )1 ≤ lim inf E(f |Σn )1 . n→∞
(9.42)
Next we show that if Σl ∈ S0 also satisfies (9.42), then Σl ⊆ Σl . Let C ∈ Σl and take f = 12 − χC in (9.42). We have
1 1 1 − χC dµ = − µ(C|Σl )dµ = 2 2 2 Ω Ω
1 − µ(C|Σn )dµ ≤ lim inf n→∞ 2
Ω 1 − µ(C|Σn )dµ ≤ 1 ≤ lim sup 2 n→∞ Ω 2
1 1 − µ(C|Σn )dµ = , ⇒ lim n→∞ Ω 2 2 ⇒ lim inf µ(C Cn ) : Cn ∈ Σn = 0, n→∞
⇒ Σl ⊆ Σl .
9.4 Convergence of σ-Fields and Prediction Sequences
675
Therefore we conclude that Σl = µ- lim inf Σn . n→∞
Note that {L2 (Ω, Σn )}n≥1 is a sequence of closed subspaces of L2 (Ω). So we can define s- lim inf L2 (Ω, Σn ) and w- lim inf L2 (Ω, Σn ) in the sense of Definition 6.6.3. n→∞
n→∞
THEOREM 9.4.5 If {Σn }n≥1 ⊆ S0 , then s-lim inf L2 (Ω, Σn )= L2 (Ω, µ- lim inf Σn ). n→∞
n→∞
PROOF: First we show that we can find Σ ∈ S0 such that s- lim inf L2 (Ω, Σn ) = L2 (Ω, Σ). n→∞
To do this, we need to show that it is a lattice and it contains the constant functions (see e.g., Schaefer [539, p. 210]). So let f, g ∈ s- lim inf L2 (Ω, Σn ). By n→∞
virtue of Definition 6.6.3 we can find sequences {fn }n≥1 , {gn }n≥1 ⊆ L2 (Ω, Σn ) such that fn −→ f and gn −→ g in L2 (Ω) as n → ∞. Let ∨ denote the maximum operation and recall that for a, b, c, d ∈ R, we have (a ∨ b − c ∨ d)2 ≤ (a − c)2 + (b − d)2 .
(9.43)
Using (9.43), we obtain fn ∨ gn − f ∨ g22 ≤ fn − f 22 + gn − g22 −→ 0
as n → ∞,
⇒ f ∨ g ∈ s- lim inf L2 (Ω, Σn ).
(9.44)
n→∞
Because fn ∧ gn = fn ∨ gn − |fn − gn |, we see that f ∧ g ∈ s- lim inf L2 (Ω, Σn ). n→∞
(9.45)
From (9.44) and (9.45), it follows that s- lim inf L2 (Ω, Σn ) is indeed a lattice. Clearly n→∞
it contains the constant functions. So as we already said, we can find Σ ∈ S0 such that s- lim inf L2 (Ω, Σn ) = L2 (Ω, Σ). n→∞
Let A ∈ Σ. Then we can find {fn }n≥1 ⊆ L2 (Ω, Σn ) such that fn −→ χA in 2 L (Ω) and of course in probability. Set 1 An = ω ∈ Ω : fn (ω) ≥ . 2 We have
An A = ω ∈ Ω : χA (ω) = 0
1 2
1 or χA (ω) = 1 and fn (ω) < 2 1 for all n ≥ 1, ⊆ ω ∈ Ω : χA (ω) − fn (ω) ≥ 2 ⇒ µ(An A) −→ 0 as n → ∞, ⇒ A ∈ µ- lim inf Σn n→∞
and
fn (ω) ≥
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9 Uncertainty, Information, Decision Making
(because An ∈ Σn for every n ≥ 1, see Proposition 9.4.4). Now suppose that A ∈ µ- lim inf n→∞ Σn . Then by Proposition 9.4.4 we can find An ∈ Σn , n ≥ 1, such that µ(An A) −→ 0. Then we have
χAn − χA 22 = |χAn − χA |2 dµ = χAn A dµ = µ(An A) −→ 0, Ω
Ω
⇒ χA ∈ s- lim inf L2 (Ω, Σn ), n→∞
⇒ A ∈ Σ. To further characterize µ- lim inf Σn we need the following general result about n→∞
the Mosco convergence of closed subspaces of a Hilbert space H. PROPOSITION 9.4.6 If H is a Hilbert space, and {Hn }n≥1 is a sequence of closed subspaces of H, then (a) s- lim inf Hn is the maximum among all closed subspaces H of H such that n→∞
pH (x) ≤ lim inf pHn (x) n→∞
for all x ∈ H.
(9.46)
(by pH we denote the orthogonal projection on a closed subspace H of H). (b) w- lim sup Hn is the minimum subspace among subspaces H of H such that n→∞
lim sup pHn (x) ≤ pH (x) n→∞
for all x ∈ H.
(9.47)
PROOF: (a) Let H = s- lim inf Hn . We have n→∞
pHn (x) = pHn pH (x) + pHn (1 − pH )(x)2 = pHn pH (x), pH (x) + pHn pH (x), (1 − pH )(x) + (1 − pH )(x), pHn pH (x) + pHn (1 − pH )(x)2 ≥ pHn pH (x), pH (x) + pHn pH (x), (1 − pH )(x) + (1 − pH )(x), pHn pH (x) −→ pH (x)2 2
(recall pHn (x) −→ x for all x ∈ H). Therefore s- lim inf Hn satisfies (9.36). On the other hand, let H be any closed subspace of H satisfying (9.36). We have pH (x)2 ≤ lim inf pHn pH (x)2 ≤ lim sup pHn pH (x)2 ≤ pH (x) n→∞
n→∞
and
pHn pH (x) −→ pH (x)
as n → ∞.
Hence we have pH (x) − pHn pH (x)2 = pH (x)2 − pH (x), pHn pH (x) − pHn pH (x), pH (x) + pHn pH (x), pH (x) = pH (x)2 − pHn pH (x)2 −→ 0
as n → ∞.
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677
(b) Let H = w- lim sup Hn . Then we have n→∞
(H )⊥ = s- lim inf Hn⊥ n→∞
and from (a) it follows that (1 − pH )(x)2 ≤ lim inf (1 − pHn )(x)2 n→∞
for any x ∈ H,
⇒ x2 − pH (x)2 ≤ x2 − lim sup pHn (x)2 , n→∞ 2
⇒ lim sup pHn (x) ≤ pH (x) 2
n→∞
for all x ∈ H.
On the other hand, let H be a closed subspace of H satisfying (9.47). By following the previous argument backwards, we obtain w- lim sup Hn ⊆ H . n→∞
Using this proposition we can further characterize µ- lim inf Σn . n→∞
THEOREM 9.4.7 If {Σn }n≥1 ⊆ S0 , then µ- lim inf Σn is the maximum among all n→∞
the sub-σ-fields Σ of Σ such that E(f |Σ )2 ≤ lim inf E(f |Σn )2 n→∞
for all f ∈ L2 (Ω).
(9.48)
PROOF: Recall that the conditional expectation E(·|Σn ) is the orthogonal projection of L2 (Ω) onto L2 (Ω, Σn ). From Theorem 9.4.5, we see that E(·|µ- lim inf Σn ) is n→∞
the orthogonal projection of L2 (Ω) onto s- lim inf L2 (Ω, Σn ). So by virtue of Propon→∞
sition 9.4.6, we have
E(f |s- lim inf Σn )2 ≤ lim inf E(f |Σn )2 . n→∞
n→∞
On the other hand if Σ ∈ S0 satisfies (9.48), then from Proposition 9.4.6 and Theorem 9.4.5, we have L2 (Ω, Σ ) ⊆ s- lim inf L2 (Ω, µ- lim inf Σn ), n→∞
n→∞
⇒ Σ ⊆ s- lim inf Σn . n→∞
THEOREM 9.4.8 If {Σn }n≥1 ⊆ S0 , then for any f ∈ L1 (Ω) the following statements are equivalent. (a) f is µ- lim inf Σn -measurable. n→∞
(b) E(f |Σn ) − f 1 −→ 0.
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PROOF: (a)⇒(b): Given ε > 0 we can find g ∈ L2 (Ω, µ- lim inf Σn ) such that n→∞
ε f − g1 < . 3 We have E(g|Σn ) − g2 −→ 0
as n → ∞ (see Theorem 9.4.5).
So we can find n0 = n0 (ε) ≥ 1 such that E(g|Σn ) − g2
0 we can find a set Dε ∈ Σ with µ(Ω\Dε ) < ε such that {h(ω, ·) : ω ∈ Dε } is equicontinuous. PROOF: Recall that µ is tight. Therefore we can apply Theorem 6.2.9 (the ScorzaDragoni theorem) and obtain Dε ⊆ Ω compact subset such that µ(Ω \ Dε ) < ε and hD ×Z is continuous. Therefore it follows at once that {h(ω, ·) : ω ∈ Dε } is ε equicontinuous. Keeping the setting of the previous proposition, we define 1 1 M+,µ (Ω × Z) = {ϑ ∈ M+ (Ω × Z) : ϑΩ = µ}; 1 (Ω × Z) are all probability measures on Ω × Z, such that Ωthat is, M+,µ
their 1 1 marginal ϑΩ equals µ. Clearly M (Ω×Z) inherits the weak topology w M (Ω× +,µ +,µ 1 Z), Cb (Ω × Z) from the space M+ (Ω × Z). As in Proposition 9.5.5, we consider a 1 Carath´eodory function h : Ω × Z −→ R and then define H : M+,µ (Ω × Z) −→ R by
h(ω, z)dϑ. H(ϑ) = Ω×Z 1 (Ω×Z) −→ R is weakly continuous. PROPOSITION 9.5.6 The function H : M+,µ w
1 PROOF: Let ϑn −→ ϑ in M+,µ (Ω × Z). By virtue of Theorem 6.2.9 we can find Dε ⊆ Ω compact with µ(Ω \ Dε ) < ε such that hD ×Z is continuous and so bounded. ε We have
hdϑn = hdϑn + hdϑn (9.51) Dε ×Z
Ω×Z
and
(Ω\Dε )×Z
hdϑn −→
Dε ×Z
hdϑ Dε ×Z
as n → ∞.
(9.52)
Also, if ξ > 0 is the bound of h (recall that by hypothesis h is bounded), then
hdϑn ≤ |h|dϑn ≤ ξε for all n ≥ 1. (9.53) (Ω\Dε )×Z
(Ω\Dε )×Z
So, if we pass to the limit as n → ∞ in (9.51) and we use (9.52) and (9.53), we obtain
hdϑ − ξε ≤ lim inf hdϑn ≤ lim sup hdϑn ≤ hdϑ + ξε. (9.54) Dε ×Z
n→∞
Ω×Z
n→∞
Ω×Z
Dε ×Z
Because ε > 0 was arbitrary we let ε ↓ 0. From (9.54) we infer that
9.5 Games with Incomplete Information
hdϑn −→ hdϑ,
Ω×Z
683
Ω×Z
⇒ H is weakly continuous. Now we are ready to prove an equilibrium theorem for game Γ. THEOREM 9.5.7 If the game Γ is as above and in addition the payoff function u : Ω×X ×Y −→ R is bounded and (i) For all (ω, y) ∈ Ω × Y, x −→ u(ω, x, y) is continuous; (ii) For all (ω, x) ∈ Ω × X, y −→ u(ω, x, y) is continuous, then the game Γ has an equilibrium. PROOF: First we show that the expected payoff function U (ε, τ ) is weakly continuous in each variable separately. For fixed ω1 ∈ Ω1 and distributional strategy τ ∗ for player II, we define the Borel measure ηω 1 on Ω0 × Ω2 × Y by
ηω 1 (A) = 1τ ∗ (dy|ω2 )µ d(ω, ω2 )|ω1 . A
We define H : Ω1 ×X −→ R by
H(ω1 , x) =
u(ω, x, y)dηω 1 .
Ω0 ×Ω2 ×Y
Claim: For every fixed ω1 ∈ Ω1 , x −→ H(ω1 , x) is continuous on X. By virtue of Proposition 9.5.5, for every n ≥ 1, we can find En ⊆ Ω0 × Ω2 × Y a Borel set with ηω 1 (En ) > 1 − (1/n) such that u(ω, ω1 , ω2 , ·, y) : (ω, ω2 , y) ∈ En is equicontinuous. Define Hn : X −→ R by
Hn (x) = u(ω, ω1 , ω2 , x, y)dηω 1 . En
Evidently Hn is continuous. Because u is bounded and ηω 1 (En ) > 1 − (1/n), it follows that Hn −→ H(ω1 , ·) uniformly, ⇒ H(ω1 , ·) is continuous.
Next let J(σ) =
H(ω1 , x)dσ.
Ω1 ×X
From Proposition 9.5.6 we have that J is weakly continuous. We have
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9 Uncertainty, Information, Decision Making
u(ω, x, y)dηω1 dσ J(σ) = Ω ×X Ω0 ×Ω2 ×X
1 = u(ω, x, y)σ(dx|ω1 )τ ∗ (dy|ω2 )dµ Ω×X×Y
= U (σ, τ ∗ ). Therefore U (σ, τ ) is weakly continuous separately on each variable. Now observe that the sets of distributional strategies are convex and weakly compact. So we can apply Theorem 2.3.13 and produce a saddle point (σ, τ ) of U , namely U (σ, τ ) = inf sup U (σ, τ ) = sup inf U (σ, τ ). τ
σ
σ
τ
This pair (σ, τ ) is the desired equilibrium of game Γ.
REMARK 9.5.8 It is interesting to extend this result to the case of nonzero-sum games.
9.6 Markov Decision Chains with Unbounded Costs In this section we consider a general state space Markov decision chain (MDC), with finite action sets and possibly unbounded cost. We prove the existence of optimal stationary discounted policies. So let (Ω, Σ) be a measurable space for which singletons belong to Σ. Also X denotes the action set that is assumed to be countable. Given a state ω ∈ Ω and an action x ∈ X, the next state is chosen according to a transition probability (stochastic kernel), p(·|ω, x) defined on Σ. So for every (ω, x) ∈ Ω × X, we have: (a) p(·|ω, x) is a probability measure on Σ. (b) For every A ∈ Σ, p(A|·, ·) : Ω × X −→ [0, 1] is (Σ × 2X )-measurable. Let µ be a probability measure on Ω, known as the initial measure. Usually µ is the Dirac probability measure concentrated at the known initial state. A policy π is a rule for choosing actions, given the history of the process, and it may be randomized. So, if the policy is π, the process develops as follows. The initial state ω0 is chosen according to the distribution µ. Subsequently the initial action x0 is chosen according to the rule π. Then the next state ω1 is chosen according to the distribution p(·|ω0 , x0 ) and the process continues indefinitely. Of course the above description of a policy (strategy) π is informal. To rigorously define it, we proceed as follows. Let Ω0 , Ω1 , Ω2 , Ω3 , . . . be copies of the state space Ω and X0 , X1 , X2 , X3 , . . . be copies of the action set X. The history of the process up until time n ≥ 0, is defined inductively as follows. (a) H0 = Ω0 . (b) Given Ωn , we set Hn+1 = Hn Xn Ωn+1 . Clearly Hnrepresents the information available to the decision maker at stage 1 n ≥ 0. Let H= Hn and let M+ (X) be the space of all probability measures on X. n≥0
9.6 Markov Decision Chains with Unbounded Costs
685
DEFINITION 9.6.1 A policy (strategy) is a map π : H −→ M+ (X) such that for each n ≥ 0 and x ∈ X, hn −→ π H (hn )(x) is measurable on Hn with respect to the n
product σ-field Σ × 2X × Σ × · · · × Σ. REMARK 9.6.2 By the disintegration theorem (see, e.g., Ash " [30, p. 109]), there is a probability measure on the set of sample paths S = Ωk × Xk under the k≥0
policy π. Equip S with the σ-field Σ × 2X × Σ × 2X × . . . with initial measure µ on (Ω0 = Ω, Σ). Given (ω0 , x0 , . . . , ωn , xn ) we need a measure on Σ. This is the law of motion p(·|ωn , xn ). Also given (ω0 , x0 , . . . , ωn ) we need a measure on 2X . This measure is induced by the probability distribution π(ω0 , x0 , . . . , ωn ). Note that we do not assume that the state space is a Borel space (see Section 8.5). DEFINITION 9.6.3 A stationary policy is a measurable map g : Ω −→ X. REMARK 9.6.4 So whenever ωn = ω, the stationary policy πs chooses the action f (ω). For this reason the stationary policy πs is identified with f . We assume that a cost function is given, namely a function u : Ω × X −→ R+ = R+ ∪ {+∞}. We make the following hypothesis concerning u. H1 : u : Ω×X −→ R+ = R+ ∪ {+∞} is a measurable map such that for every ω ∈ Ω, the set x ∈ X : u(ω, x) < +∞ (9.55) is finite. REMARK 9.6.5 If an action x is infeasible, then u(ω, x) = +∞. With a slight abuse of notation, when a stationary policy g is employed, the cost function evaluated at the point ω is denoted by u(ω, g). There is a discount factor 0 < β < 1. Using it we can now define the total discounted cost under a policy π. For n ≥ 0, we define un : S −→ R = R+ ∪ {+∞} by un (ω0 , x0 , . . .) = u(ωn , xn ). Evidently these functions are measurable. We consider n β u(ωn , xn ). U (ω0 , x0 , . . .) = n≥0
Then U is measurable on S. Conditioned on ω0 = ω, we denote the conditional expectation by Vβ (π, ω). This quantity may be +∞ for some policies. Let Vβ (ω) = inf Vβ (π, ω) : π = policy . We make the following hypothesis. H2 : Vβ (ω) < +∞ for every (β, ω) ∈ (0, 1) × Ω.
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9 Uncertainty, Information, Decision Making
REMARK 9.6.6 This hypothesis implies that given an initial state ω ∈ Ω, there must be at least one action x ∈ X such that u(ω, x) < +∞. Hence in hypothesis H1 , the set given by (9.55) is nonempty. As we already saw in Section 8.5, the proof of the existence of an optimal stationary policy depends on a measurable selection argument, which permits the realization of a stationary policy that achieves the minimum in the discounted optimality equation. The next proposition paves the way to obtain such a measurable selector. PROPOSITION 9.6.7 If hx : Ω −→ R+ = R measurable + ∪ {+∞}, x ∈ X, are functions such that for each ω ∈ Ω, the set x ∈ X : hx (ω) < +∞ is finite, h = min hx , the elements of the countable set X are ordered as x0 < x1 < x2 < · · · x∈X
and we define g : Ω −→ X by g(ω) =smallest x ∈ X such that hx (ω) = h(ω), then g is measurable. PROOF: Evidently h is measurable. We need to show that for every k ≥ 0, g −1 ({xk }) ∈ Σ. Note that k−1 #
#
i=0
i≥k+1
g −1 ({xk })=
{ω ∈ Ω : hxk (ω) < hxi (ω)} ∩
{ω ∈ Ω : hxk (ω) ≤ hxi (ω)},
⇒ g −1 ({xk }) ∈ Σ. If for some ω ∈ Ω, h(ω) = +∞, then the result also holds.
PROPOSITION 9.6.8 (a) If h : Ω × Ω −→ R+ is jointly measurable, then for every x ∈ X, ω −→ Ω h(ω, s)p(ds|ω, x) is measurable. (b) If h : Ω −→ R+ is Σ-measurable and g : Ω −→ X is a stationary policy, then ω −→ Ω h(s)p(ds|ω, g) is Σ-measurable. PROOF: (a) Let h(ω, s)=
N
k=1
λk χCk (ω)χDk (s) with Ck , Dk ∈Σ, k∈{1, . . . , N }. We
have
h(ω, s)p(ds|ω, x) = Ω
N
λk χCk (ω)p(Dk |ω, x),
k=1
⇒ ω −→
h(ω, s)p(ds|ω, x)
is Σ-measurable.
Ω
Because every measurable function h : Ω × Ω −→ R+ can be approximated by an increasing sequence of simple functions, we conclude that ω −→ Ω h(ω, s)p(ds|ω, x) is Σ-measurable. (b) Exploiting as above the approximation of a measurable function h : Ω×Ω −→ R+ by an increasing sequence of simple functions, it suffices to show that for each A ∈ Σ,
ω −→ p A|ω, g(ω) is Σ-measurable. To this end note that ξ : Ω −→ Ω×X defined by
ξ1 (ω) = ω, g(ω)
is Σ, Σ × B(X) -measurable. Recall that ξ2 : Ω × X −→ [0, 1] defined by
9.6 Markov Decision Chains with Unbounded Costs
687
ξ2 (ω, x) = p(A|ω, x) is Σ × B(X)-measurable. Therefore ξ2 ◦ ξ1 is Σ-measurable. But note that
ξ2 ◦ ξ1 (ω) = p A|ω, f (ω) . DEFINITION 9.6.9 If π is a policy and (ω, x) ∈ Ω × X, then the shifted policy π(ω, x) is defined by π(ω, x)(ω0 , x0 , . . . , xn ) = π(ω, x, ω0 , x0 , . . . , xn ). REMARK 9.6.10 According to this definition the decision made by the shifted policy π(ω, x) at the nth stage is the same as the decision made by π at stage n + 1, given that the initial state and the initial decision are (ω, x) ∈ Ω × X. PROPOSITION 9.6.11 For every policy π, the value function ω −→ Vβ (π, ω) is
Σ-measurable. Moreover, for every fixed x ∈ X, (ω, s) −→ Vβ π(ω, x), s is Σ × Σmeasurable on Ω × Ω. PROOF: To prove the first part of the proposition, it suffices to show that for all k ≥ 0, the function ω −→ Eπ (uk |ω0 = ω) is Σ-measurable. We have Eπ (uk |ω0 = ω) =
π(ω)(x)
π(ω, x, ω1 )(y1 )p(dω1 |ω, x)
Ωy ∈X 1
x∈X
π(ω, x, ω1 , . . . , ωk−1 )(xk−1 )p(dωk−1 |ωk−2 , xk−2 )
...
Ωy k−1 ∈X
u(ωk , xk )π(ω, x, ω1 , . . . , ωk )(xk )p(dωk |ωk−1 , xk−1 ).
Ω y ∈X k
From Definition 9.6.1 and Proposition 9.6.8, we see that the first part of the proposition holds. Similarly we deduce the second part of the proposition, by establishing in a similar fashion the joint measurability of (ω, s) −→ Eπ((ω),x) (uk |s0 = s). Now we can prove a theorem on the existence of a β-discounted optimal stationary policy. THEOREM 9.6.12 If hypotheses H1 and H2 hold, then the finite value function ω −→ Vβ (ω) is the minimal nonnegative measurable solution of the optimality functional equation
Vβ (ω) = min u(ω, x) + β Vβ (s)p(ds|ω, x) , ω ∈ Ω. (9.56) x∈X
Ω
If the minimum in (9.56) is realized at g(ω), then ω −→ g(ω) is a stationary optimal policy.
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9 Uncertainty, Information, Decision Making
PROOF: For each n ≥ 0, we define inductively a sequence of functions Vn : Ω −→ R+ = R+ ∪ {+∞} as follows:
Vn−1 (s)p(ds|ω, x) , ω ∈ Ω, n ≥ 1. V0 = 0 and Vn (ω) = min u(ω, x) + β x∈X
Ω
(9.57)
We show the following. • ω −→ Vn (ω) is Σ-measurable.
(9.58)
• {Vn }n≥0 is increasing.
(9.59)
• Vn ≤ Vβ (π, ·) for all n ≥ 0 and all policies π.
(9.60)
Clearly (9.58) follows from (9.57), Proposition 9.6.8(a), and the fact that X is countable. Also (9.59) follows immediately from the inductive definition in (9.57). To prove (9.60), let Vn (π, ω) be the extended discounted cost if we follow policy π for n steps and then the process is terminated with terminal cost zero. We claim Vn ≤ Vn (π, ·)
for all n ≥ 0.
(9.61)
Evidently (9.61) implies (9.60). So we need to prove (9.61). We have u(ω, x)π(ω)(x) = V1 (π, ω). V1 (ω) ≤ x∈X
Suppose (9.61) is true for n − 1. We can easily check that
π(ω)(x) u(ω, x) + β Vn−1 π(ω, x), s p(ds|ω, x) , Vn (π, ω) = Ω
x∈X
with π(ω, x) being the shifted policy (see Definition 9.6.9). From Proposition 9.6.11
we know that (ω, s) −→ Vn−1 π(ω, x), s is Σ × Σ-measurable on Ω × Ω and so Proposition 9.6.8(a) implies that Vn (π, ·) is Σ-measurable. Then by the induction hypothesis
Vn−1 (ω) ≤ Vn−1 π(ω, x), s
⇒ Vn (ω) ≤ u(ω, x) + β Vn−1 π(ω, x), s p(ds|ω, x), Ω
and this proves (9.61) (hence (9.60) too). Then lim Vn = v exists and is Σ-measurable (see (9.58) and (9.59)). Moreover, n→∞
from the monotone convergence theorem, we see that v satisfies
v(s)p(ds|ω, x) , ω ∈ Ω. v(ω) = min u(ω, x) + β x∈X
(9.62)
Ω
By virtue of Proposition (9.61), there exists a stationary policy g : Ω −→ X realizing the minimum in (9.62). So
v(s)p(ds|ω, x). (9.63) v(ω) = u(ω, g) + β Ω
9.7 Remarks
689
From Proposition 9.6.8(b), the integral in the right-hand side of (9.63) produces a Σ-measurable function. Iterating (9.63), we obtain Vn (g, ω) ≤ v(ω), ⇒ Vβ (g, ω) ≤ v(ω). This combined with (9.60) implies that v = Vβ
and
g = stationary optimal policy.
Finally let w be a nonnegative Σ-measurable solution of (9.56) and let gw be the stationary policy realizing the minimum in (9.56) for w (instead of u). Then following the above argument, we obtain Vβ ≤ Vβ (gw , ·) ≤ w, ⇒ Vβ is indeed the minimal solution of (9.56).
9.7 Remarks 9.1: The first to apply topologies on sub-σ-fields to study the dependence of economic models on information was Allen [13], who employed the d-metric topology (see Definition 9.1.17). The d-metric was first defined by Boylan [96], who used it to study the convergence of certain martingale like sequences. The weaker topology of pointwise convergence (see Definition 9.1.3), was introduced by Cotter [160]. Proposition 9.1.9 is due to Fetter [243]. Other topologies considered in probability theory, can be found in Allen [14]. Stinchcombe [559, 560] introduced some Bayesian topologies, in which two information structures are similar if there is only small probability that the posteriors are far apart. Stinchcombe [559, 560], explored the Bayesian topologies on sub-σ-fields that are generated by different vector space topologies on the space of distributions. 9.2: The ex-post (or Bayesian) view was adopted by Allen [13], Cotter [160, 161] and Stinchcombe [559]. In Allen [13] and Cotter [160, 161] the state-dependent posteriors are not well defined and so instead they show that the map from information to expected utility is continuous and then use the fact that the map from utility to actions is continuous. In Stinchcombe [559], the author imposes assumptions on the probability space so that the state-dependent posteriors (regular conditional probabilities) given a sub-σ-field, are well defined. Then he showed that the mapping from information to state-dependent posteriors is continuous under suitable topologies on the space of information. The mapping from posteriors to actions is also continuous. Hence the composition of the two maps is continuous. In this section we present the first approach and outline the continuous dependence results due to Cotter [160] and Allen [13]. 9.3: The ex-ante view was adopted by Van Zandt [592]. Proposition 9.3.6 is due to Horsley–Van Zandt–Wrobel [312]. This approach can be found also in Cotter
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9 Uncertainty, Information, Decision Making
[162] who proved that the set of type-correlated equilibria (an extension of correlated equilibria to Bayesian games) is upper semicontinuous with respect to the information variable. 9.4: Definition 9.4.1 was first introduced by Kudo [365], who also proved Proposition 9.4.4. If Σ = µ- lim inf Σn = µ- lim sup Σn , then we say that the sequence of n→∞
n→∞
Σn s strongly converges to Σ. The use of these limits of sub-σ-fields, to study the convergence of prediction sequences, is due to Tsukada [588]. 9.5: The first general equilibrium result for games with incomplete information was proved by Milgram–Weber [429]. They made two key hypotheses. The first (called the continuity of payoff), is that for fixed ω ∈ Ω, {uk (ω, ·)}n k=1 are uniformly equicontinuous. The second (called continuity of information), is that the joint distribution of the information available to the players, is absolutely continuous with respect to a product measure on the product of the spaces of types. This last hypothesis is in general difficult to verify, unless the information random variables are either independent or discrete and finite-valued. Here we present the result of Mamer–Schilling [405], who were able to weaken the continuity of payoffs hypothesis to continuity of u(ω, ·) on each action variable separately and they eliminated completely the continuity of information hypothesis. 9.6: The first work of Markov decision chains with an arbitrary state space and unbounded costs, is due to Ritt–Sennott [515]. Previous works had a general state space but a bounded cost, see for example Bertsekas–Shreve [72] and Hernandez Lerma [290].
10 Evolution Equations
Summary. *Evolution equations are the abstract formulation of dynamic partial differential equations. In this chapter, we examine both semilinear and nonlinear evolution equations. We start by developing the mathematical tools which are necessary in this study. Among them, the notion of evolution triple plays central role and always allows us to use different spaces within the analysis of a single evolution equation. First we consider semilinear evolutions, which we analyze using the semigroup method. Subsequently, we pass to nonlinear evolutions. We consider two such classes. Evolutions of the subdifferential type (which incorporate variational inequalities) and evolution formulated in the framework of evolution triples with operators of monotone type. The Galerkin method is crucial here. We conclude with the study of second-order evolutions.
Introduction Evolution equations are the abstract formulation of dynamic partial differential equations (i.e., partial differential equations with time t as one of the independent variables). Such equations arise in many branches of science and engineering and describe mathematically important physical processes and phenomena. The basic theoretical question related to such problems is whether for given initial data the equation has a solution at least locally in time and whether this solution is unique. It is also important to know how this solution depends on the initial data. We investigate these issues in the context of certain broad classes of semilinear and nonlinear evolution equations. In Section 10.1 we present the mathematical tools needed to study evolution equations. Central in this section is the notion of evolution triple of spaces. Evolution triples provide a natural and flexible framework for the study of evolution equations. Evolution triples realize the modern strategy to study pdes, which is to use many different spaces in the same problem. In Section 10.1, we examine some basic function spaces associated with evolution triples. Section 10.2 deals with semilinear evolution equations. We employ the semigroup method and prove existence, uniqueness, and blow-up of solutions for such evolutions. We also apply the abstract results to some parabolic initial-boundary value problems. N.S. Papageorgiou, S.Th. Kyritsi-Yiallourou, Handbook of Applied Analysis, Advances in Mechanics and Mathematics 19, DOI 10.1007/b120946_10, © Springer Science+Business Media, LLC 2009
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10 Evolution Equations
In Section 10.3 we study nonlinear evolution equations. We consider two classes of such equations. The first class involves time-invariant subdifferential operators and so it incorporates certain variational inequalities. The second class involves problems formulated in the framework of evolution triples. The Galerkin method is the basic tool here. Finally in Section 10.4 we deal with second-order evolutions. The same two classes of equations are studied in this section too.
10.1 Lebesgue–Bochner Spaces In this section we introduce some spaces of Banach space–valued functions that are needed in the study of evolution equations. By a Banach space–valued function, we mean any map that takes values in a Banach space X, for example, a map f : T = [0, b] −→ X. Throughout this section (Ω, Σ, µ) is a finite measure space and X a Banach space. Additional hypotheses are introduced as needed. By · we denote the norm of X and by X ∗ its topological dual. DEFINITION 10.1.1 (a) A function f : Ω −→ X is a simple function if there exist n finitely many sets {Ck }n k=1 ⊆ Σ that are mutually disjoint and {xk }k=1 ⊆ X such that n f (ω) = xk χCk (ω) for all ω ∈ Ω. k=1
(b) A function f : Ω −→ X is said to be strongly measurable if there exists a sequence of simple function sn : Ω −→ X, n ≥ 1 such that lim f (ω) − sn (ω) = 0
n→∞
µ-a.e. on Ω.
(c) A function f : Ω −→ X is said to be weakly measurable if for any x∗ ∈ X ∗ the R-valued function ω −→ x∗ , f (ω) is Lebesgue measurable. Hereafter, by ·, · we denote the duality brackets for the pair (X ∗ , X). The next theorem, usually known as the Pettis measurability theorem relates the notions of strong and weak measurability (see Definition 10.1.1(b),(c)) and its proof can be found, for example, in Gasi´ nski–Papageorgiou [259, p. 109]. THEOREM 10.1.2 A function f : Ω −→ X is strongly measurable if and only if (a) f is almost separably valued (i.e., there exists a µ-null set C ⊆ Ω such that f (Ω\C) is separable in X) and (b) f is weakly measurable. In particular, if X is separable, then the notions of strong and weak measurability are equivalent.
10.1 Lebesgue–Bochner Spaces
693
DEFINITION 10.1.3 Let f : Ω −→ X be strongly measurable. We say that f is Bochner integrable if there exists a sequence {sn }n≥1 of simple functions such that
f (ω) − sn (ω)dµ = 0. lim n→∞
In this case
Ω
f (ω)dµ = lim
n→∞ Ω
Ω
sn (ω)dµ and it is called the Bochner integral of f .
REMARK 10.1.4 It is easy to see that the above definition of the Bochner integral is independent of the choice of the sequence {sn }n≥1 of simple functions. For any C ∈ Σ we set C f (ω)dµ = Ω χC (ω)f (ω)dµ. An easy consequence of Definition 10.1.3 is the following convenient criterion for Bochner integrability. PROPOSITION 10.1.5 If f : Ω −→ X is strongly measurable, then f is Bochner integrable if and only if f (·) ∈ L1 (Ω). Moreover, we have
f (ω)dµ ≤ f (ω)dµ. Ω
Ω
The Bochner integral possesses almost the same properties as the Lebesgue integral. In the next theorem we have gathered the basics of those properties. We omit the proofs which can be found in Gasi´ nski–Papageorgiou [259, Section 2.1]. THEOREM 10.1.6 (a) If fn : Ω −→ X, n ≥ 1, is a sequence of Bochner integrable w f (ω) µ-a.e. in X, then f is Bochner integrable and functions and fn (ω) −→ f (ω)dµ ≤ lim inf Ω fn (ω)dµ. Ω n→∞
µ
(b) If fn : Ω −→ X, n ≥ 1, is a sequence of Bochner integrable functions, fn −→ f (i.e., for every ε > 0, µ {ω ∈ Ω : fn (ω) − f (ω) ≥ ε} −→ 0 as n → ∞) and there exists h ∈ L1 (Ω)+ such that fn (ω) ≤ h(ω) µ-a.e. on Ω, then s f is Bochner integrable and C fn dµ −→ C f dµ for all C ∈ Σ. In fact, lim Ω fn − f dµ = 0. n→∞
(c) If m(C) = C f dµ, C ∈ Σ, then m is a vector measure (i.e., m(∅) = 0) and if {Cn }n≥1 ⊆ Σ is a sequence of pairwise disjoint sets and C = Cn , then n≥1 m(Cn ) (the sum is actually absolutely convergent). Moreover, m ! µ m(C) = n≥1 (i.e., lim C f dµ = 0) and |m|(C) = C f dµ for all C ∈ Σ, with |m| being µ(C)→0
the total variation of the vector measure m. COROLLARY 10.1.7 If f, g : Ω −→ X are Bochner integrable and C f dµ = gdµ for all C ∈ Σ, then f (ω) = g(ω) µ-a.e. on Ω. C PROOF: Let m(C) = C (f − g)dµ, C ∈ Σ. Then by hypothesis m(C) = 0 for every C ∈ Σ, hence |m|(C) = 0. But |m|(C) = f − gdµ for all C ∈ Σ. So C f − gdµ = 0 for all C ∈ Σ, hence f (ω) − g(ω) = 0 µ-a.e. on Ω. C
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10 Evolution Equations
The next theorem states a useful property of the Bochner integral, which has no counterpart in the Lebesgue integration. For a proof of this result we refer to Gasi´ nski–Papageorgiou [259, p. 116]. THEOREM 10.1.8 If Y is another Banach space, A : D(A) ⊆ X −→ Y is a closed linear operator,
and f : Ω −→ D(A) ⊆ X and Af : Ω −→ Y are both Bochner integrable, then A Ω f dµ = Ω Af dµ. Given f, g : Ω −→ X strongly measurable, we say that f ∼ g if and only if f (ω) = g(ω)µ-a.e. Evidently this is an equivalence relation. Now we can define the so-called Lebesgue–Bochner spaces. DEFINITION 10.1.9 (a) Let 1 ≤ p < ∞. By Lp (Ω, X) we denote the space of all the equivalence classes for the relation ∼ of the strongly measurable functions f : Ω −→ X such that f (·) ∈ Lp (Ω)+ . We set
1/p f (ω)p dµ . f Lp (Ω,X) = Ω
(b) The space L∞ (Ω, X) is defined to be the space of all equivalence classes for the relation ∼ of the strongly measurable functions f : Ω −→ X such that f (·) ∈ L∞ (Ω)+ . We set f L∞ (Ω,X) = ess supf (·). REMARK 10.1.10 For every 1 ≤ p ≤ ∞, Lp (Ω, X) is a Banach space. Moreover, if Σ is countably generated and X is a separable Banach space, then Lp (Ω, X), 1 ≤ p < ∞ is separable. Recall that, if Ω is an open or a closed subset of RN , then the Borel σ-field B(Ω) is countably generated. For 1 < p < ∞ Lp (Ω, X) is uniformly convex if and only if X is uniformly convex. Simple functions are dense in Lp (Ω, X) for 1 ≤ p < ∞ and countably-valued functions of L∞ (Ω, X) are dense in L∞ (Ω, X). Finally if Ω ⊆ RN is bounded open, then C ∞ (Ω, X) is dense in Lp (Ω, X), 1 ≤ p < ∞. The characterization of the dual of these spaces is not easy. The reason for this is the fact that the Radon–Nikodym theorem is not in general true for vector measures. This leads to the following classification of Banach spaces. DEFINITION 10.1.11 A Banach space X has the Radon–Nikodym property (RNP for short) if for every finite measure space (Ω, Σ, µ) and every vector measure m : Σ −→ X of bounded variation such that m ! µ, we can find f ∈ L1 (Ω, X) such that m(C) = C f (ω)dµ for all C ∈ Σ. Large classes of Banach spaces have the RNP (see Diestel–Uhl [199, pp. 79–82]). THEOREM 10.1.12 (a) Reflexive Banach spaces have the RNP. (b) Separable dual Banach spaces have the RNP. Using this notion we can have a Riesz representation theorem for the Lebesgue– Bochner spaces. For a proof see Diestel–Uhl [199, pp. 89–100].
10.1 Lebesgue–Bochner Spaces
695
THEOREM 10.1.13 If X is a Banach space such that X ∗ has the RNP and 1 ≤ p < ∞, then Lp (Ω, X)∗ = Lp (Ω, X ∗ ), (1/p) + (1/p ) = 1. There is a version of this theorem when X ∗ does not satisfy the RNP. We state here the case p = 1 which appears often in applications. DEFINITION 10.1.14 Given two functions g, h : Ω −→ X ∗ that are w∗ measurable (i.e., for every x ∈ X, ω −→ g(ω), x, ω −→ h(ω), x are Σ-measurable), we have g∼h
if and only if g(ω), x = h(ω), x
µ-a.e. on Ω
for every x ∈ X. (10.1)
Note that in (10.1) the µ-null set in general depends on x ∈ X. Also ∼ is an equiv∗ alence relation. By L∞ (Ω, Xw ∗ ), we denote the linear space of the ∼-equivalence classes of functions g : Ω −→ X ∗ that are w∗ -measurable and there exists c ≥ 0 such that | g(ω), x | ≤ cx µ-a.e. on Ω for every x ∈ X. (10.2) Again the µ-null set in (10.2) may depend on x ∈ X. The infimum of all c ≥ 0 ∗ ) and it is easily seen to be a norm such that (10.2) holds is denoted by gL∞ (Ω,Xw ∗ ∞ ∗ for the space L (Ω, Xw∗ ). ∗ REMARK 10.1.15 If X is separable and g ∈L∞ (Ω, Xw ∗ ), then ω −→ g(ω)X ∗ belongs in L∞ (Ω)+ and ∗ ) = ess supg(·)X ∗ . gL∞ (Ω,Xw ∗
Ω
The next theorem is usually called the Dinculeanu–Foias theorem (Ionescu– Tulcea [328, p. 95]). THEOREM 10.1.16 If X is an arbitrary Banach space, then L1 (Ω, X)∗ = ∗ L∞ (Ω, Xw ∗ ) and the duality pairing between the two spaces is given by
g(ω), f (ω) dµ. g, f 0 = Ω
Using the notion of RNP, we can also extend the fundamental theorem of the Lebesgue calculus to vector-valued functions. DEFINITION 10.1.17 A function f : T = [0, b] −→ X is said to be absolutely continuous if for every ε > 0, we can find δ > 0 such that for every sequence of disjoint (cn − αn ) < δ, we have f (cn ) − f (αn ) < ε. intervals (αn , cn ) ⊆ T verifying n≥1
n≥1
THEOREM 10.1.18 If X has the RNP and f : T −→ X is absolutely continuous, then f is strongly differentiable a.e. on T , f ∈ L1 (T, X), and
t
f (t) = f (0) + 0
f (s)ds
for all t ∈ T.
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10 Evolution Equations
In applications this theorem is used in the context of reflexive Banach spaces (see Theorem 10.1.12(a)). For this reason in the next definition we assume that X is reflexive. DEFINITION 10.1.19 Suppose X is a reflexive Banach space. (a) By AC 1,p (T, X), 1 ≤ p ≤ ∞, we denote the space of all absolutely continuous p functions
f : T −→ X such that f ∈ L (T, X) (see Theorem 10.1.18). (b) By D (0, b), X we denote the space of all continuous
linear operators from D(0, b) = Cc∞ (0, b) into X. An element u ∈ D (0, b), X is an X-valued distri
bution on (0, b). If u ∈ D (0, b), X , then for a positive integer k ≥ 1 dk ϑ
Dk u(ϑ) = (−1)k u
dtk
for all ϑ ∈ D(0, b) = Cc∞ (0, b),
defines another vectorial distribution Dk u ∈ D (0, b), X , known as the k1 vectorial distributional derivative of u. If k = 1, then we write D = D. (c) Let 1 ≤ p ≤ ∞. The vectorial Sobolev space W 1,p (0, b), X is defined to be the space of all functions u ∈ Lp (T, X) such that the vectorial distributional derivative Du also belongs in Lp (T, X).
As in the case for R-valued functions, the spaces AC 1,p (T, X) and W 1,p (0, b), X can be identified (see Barbu [58, p. 19] and Gasi´ nski–Papageorgiou [259, p. 138]). THEOREM 10.1.20 If X is a reflexive Banach space and u ∈ Lp (T, X), 1 ≤ p ≤ ∞, then the following statements are equivalent.
(a) u ∈ W 1,p (0, b), X . (b) There exists u ∈ AC 1,p (T, X) such that u(t) = u(t) a.e. on (0, b). Suppose X and Y are Banach spaces and X is embedded continuously and densely into Y . Then we can easily verify that • Y ∗ is embedded continuously into X ∗ .
(10.3)
• If X is reflexive, then the above embedding is also dense.
(10.4)
With this in mind we make the following definition, which is basic in the study of evolution equations. DEFINITION 10.1.21 By an evolution triple (or Gelfand triple), we understand a triple of spaces X ⊆ H ⊆ X∗ such that (a) X is a separable, reflexive Banach space. (b) H is a separable Hilbert space that is identified with its dual (pivot space). (c) X is embedded continuously and densely into H.
10.1 Lebesgue–Bochner Spaces
697
REMARK 10.1.22 By virtue of (10.3) and (10.4), for an evolution triple (X, H, X ∗ ) we have X ⊆ H and H ∗ = H ⊆ X ∗ with the embedding being continuous and dense. In what follows by ·, · we denote the duality brackets for the pair (X ∗ , X) and by (·, ·) the inner product of H. Also by · (resp., | · |, · ∗ ), we denote the norm of X (resp., of H, X ∗ ). We have ·, · H×X = (·, ·) and x∗ ≤ c|x| for some c > 0 and all x ∈ H. (10.5) Using evolution triples we can define the following Banach space, which is a useful tool in the study of evolution equations. DEFINITION 10.1.23 Wp (0, b) = x ∈ Lp (T, X) : x ∈ Lp (T, X ∗ ) , 1 < p < ∞, (1/p) + (1/p ) = 1. The space Wp (0, b) is equipped with the norm xWp (0,b) = xLp (T,X) + x Lp (T,X ∗ ) which clearly makes Wp (0, b) a Banach space. REMARK 10.1.24 If in the above definition, we interpret the derivative of x as a vectorial distributional derivative into X ∗ (weak derivative), by virtue of Theorem ∗ 10.1.20, we see that it is a strong derivative if
we view∗ x as an X -valued function. 1,p ∗ 1,p Evidently Wp (0, b) ⊆ AC (T, X ) = W (0, b), X . THEOREM 10.1.25 If (X, H, X ∗ ) is an an evolution triple and 1 < p < ∞, then Wp (0, b) is embedded continuously and densely in C(T, H). PROOF: Let c < 0 < b < d and x ∈ Wp (0, b). Because x ∈ AC 1,p (T, X ∗ ), we can extend x on (c, 0) and on (b, d) by setting x(t) = x(0) for t ∈ (c, 0) and x(t)
= x(b) for t ∈ (b, d). We choose ϕ ∈ Cc∞ (c, d) such that ϕT = 1. We set x(t) = x ϕ(t) for all t ∈ (c, d). Then clearly x ∈ Wp (c, d) and xT = x. Also we have xWp (0,b) ≤ xWp (c,d) ≤ c(ϕ)xWp (0,b) and we can find ε > 0 small such that x(c,c+ε) = 0 and
with c(ϕ) > 0
x(d−ε,d) = 0.
1 Let ϑ ∈ Cc∞ (−1, 1) be a mollifier (i.e., ϑ ≥ 0, −1 ϑ(t)dt). We set ϑm (s) = mϑ(ms) and define the corresponding mollification of x; that is,
d
x(s)ϑm (t − s)ds.
xm (t) = c
We know that xm ∈ Cc∞ (c, d), X for m ≥ 1 large and xm −→ x in Wp (c, d) with xm Wp (c,d) ≤ xWp (c,d) . We have
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10 Evolution Equations
1 d |xm (t)|2 = xm (t), xm (t) = xm (t), xm (t) 2 dt
t 1 ⇒ |xm (t)|2 = xm (s), xm (s) ds 2 c
t xm (s)∗ xm (s)ds ≤
(see 10.5)),
c
≤ xm 2Wp (c,d)
(by the Cauchy–Schwartz inequality). (10.6)
From (10.6), we infer that for m, n ≥ 1, we have 1 |xm (t) − xn (t)|2 ≤ xm − xn 2Wp (c,d) 2 ⇒ {xm }m≥1 ⊆ C(T, H) is Cauchy.
for all t ∈ [c, d]
Therefore we have xm −→ x in C(T, H), hence x = x ∈ C(T, H). So we have proved that Wp (0, b) is embedded continuously in C(T, H). The embedding is also dense because the set of all X-valued polynomials is dense in both Wp (0, b) and C(T, H). From the above proof, we deduce the following corollary.
COROLLARY 10.1.26 If x, y ∈ Wp (0, b), then t −→ x(t), y(t) is absolutely continuous and d
x(t), y(t) = x (t), y(t) + y (t), x(t) a.e. on T. dt
(10.7)
REMARK 10.1.27 We call (10.7) the integration by parts formula for the elements in Wp (0, b). The embedding of Wp (0, b) in C(T, H) is not in general compact. We can have compact embedding of Wp (0, b) into Lp (T, H). This is a particular case of Theorem 10.1.29 below. First we need the following interpolation lemma, sometimes called in the literature Ehrling’s inequality. LEMMA 10.1.28 If X, Y, Z are Banach spaces such that X ⊆ Y ⊆ Z, the embedding of X into Y is compact, and the embedding of Y into Z is continuous, then given ε > 0, we can find c(ε) > 0 such that for all x ∈ X we have xY ≤ εxX + c(ε)xZ .
(10.8)
PROOF: Suppose that (10.8) is false. Then we can find ε > 0 and sequence {xn }n≥1 ⊆ X such that xn Y > εxn X + nxn Z
for all n ≥ 1.
We set yn = xn /xn X , n ≥ 1. Evidently yn X = 1 for all n ≥ 1 and we have yn Y ≥ ε + nyn Z
for all n ≥ 1.
(10.9)
10.1 Lebesgue–Bochner Spaces
699
Because {yn }n≥1 ⊆ X is bounded and by hypothesis X is embedded compactly in Y and Y is embedded continuously in Z, by passing to a suitable subsequence if necessary, we may assume that yn −→ y
in Y
and in
as n → ∞.
Z
(10.10)
From (10.9) and (10.10), we obtain yZ = 0
and
yY ≥ ε > 0,
a contradiction.
THEOREM 10.1.29 If X, Y, Z are Banach spaces with X, Z reflexive such that X ⊆ Y ⊆ Z, the embedding of X into Y is compact, the embedding of Y into Z is continuous, and for 1 < p, q < ∞, we define the Banach space Wpq (0, b) = x ∈ Lp (T, X) : x ∈ Lq (T, Z) , then Wpq (0, b) is embedded compactly in Lp (T, Y ). PROOF: Clearly Wpq (0, b) is a reflexive Banach space. Let {xn }n≥1⊆Wpq (0, b) be w a bounded sequence. We may assume that xn −→ x in Wpq (0, b) as n → ∞. We have w
xn −→ x
in Lp (T, X)
and
xn −→ x w
in Lq (T, Z)
as n → ∞.
Without any loss of generality we may assume that x = 0. First we show that xn (t) −→ 0 in Z for every t ∈ T . We show this convergence for t = 0 and for the other points in T the proof is similar. Because xn ∈ AC 1,r (T, X), r = min{p, q}, we have
t xn (0) = xn (t) − xn (s)ds, t ∈ T. 0
Integrating this inequality we obtain
s t
1 s xn (t)dt − xn (τ )dτ dt xn (0) = s 0 0 0 = an + cn , with an =
1 s
s
xn (t)dt
and
cn = −
0
1 s
s
(10.11)
(s − τ )xn (τ )dτ, n ≥ 1.
0
Given ε > 0, we choose s > 0 small so that
s ε xn (τ )Z dτ ≤ . cn Z ≤ 2 0 w
(10.12)
For this fixed s > 0 and because xn −→ 0 in Lp (T, X) (recall x = 0), we see that w
an −→ 0
in X
as n → ∞,
⇒ an −→ 0
in Z
as n → ∞.
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10 Evolution Equations
So we can find n0 = n0 (ε) ≥ 1 such that for all n ≥ n0 , we have an Z ≤
ε . 2
(10.13)
Returning to (10.11) and using (10.12) and (10.13), we have xn (0)Z ≤ ε ⇒ xn (0) −→ 0
for all n ≥ n0 , in Z
as n → ∞.
(10.14)
Note that Wpq (0, b) is embedded continuously in C(T, Z) (see Theorem 10.1.20). So it follows that {xn }n≥1 ⊆ C(T, Z) is bounded. Therefore by virtue of the dominated convergence theorem (see (10.14)), we have xn −→ 0
in Lp (T, Z)
as n → ∞.
(10.15)
Invoking Lemma 10.1.28, for any δ > 0 we have xn Lp (T,Y ) ≤ δxn Lp (T,X) + c(δ)xn Lp (T,Z) ≤ δM + c(δ)xn Lp (T,Z) ⇒ xn Lp (T,Y ) −→ 0
(with M = sup xn Lp (T,X) < ∞) n≥1
as n → ∞ (see (10.15).
This proves that Wpq (0, b) is embedded compactly in Lp (T, Y ). Another useful result in this direction is included in the next proposition.
PROPOSITION 10.1.30 If X, Y are Banach spaces, X is reflexive and is embedded continuously in Y , and x ∈ L∞ (T, X) ∩ C(T, Yw ), then x ∈ C(T, Xw ) (here Xw , resp., Yw , denotes the space X, resp., Y , endowed with the weak topology). ·
PROOF: By replacing Y with X Y if necessary, we may assume that the embedding of X into Y is also dense. Then Y ∗ is embedded continuously and densely in X ∗ (see (10.3) and (10.4)). If tn −→ t in T , then because by hypothesis x ∈ C(T, Yw ), we have y ∗ , x(tn )Y −→ y ∗ , x(t)Y
as n → ∞,
for all y ∗ ∈ Y ∗ .
(10.16)
Here by ·, ·Y we denote the duality brackets for the pair (Y ∗ , Y ). First we show that x(t) ∈ X for all t ∈ T and x(t)X ≤ xL∞ (T,X)
for all t ∈ T.
(10.17)
To see this, extend x by zero outside T , denote this extension by x and then regularize x. So we can find a sequence {xn }n≥1 ⊆ C 1 (T, X) such that xn (t)X ≤ xL∞ (T,X) and
∗
∗
y , xn (t)Y −→ y , x(t)Y
for all t ∈ T, all n ≥ 1 as n → ∞,
for all y ∗ ∈ Y.
We have | y ∗ , xn (t)Y | = | y ∗ , xn (t)X | ≤ y ∗ X ∗ xL∞ (T,X) .
(10.18)
10.2 Semilinear Evolution Equations
701
Here by ·, ·X we denote the duality brackets for the pair (X ∗ , X). Passing to the limit as n → ∞ in (10.18), we obtain | y ∗ , x(t)Y | = | y ∗ , x(t)X | ≤ y ∗ X ∗ xL∞ (T,X)
for all t ∈ T and all y ∗ ∈ Y ∗ . (10.19)
Because Y ∗ is dense in X ∗ , from (10.19) it follows that x(t) ∈ X
for all t ∈ T
and
(10.17) holds.
(10.20)
∗ ∗ Let x∗ ∈ X ∗ . Because Y ∗ is dense in X ∗ , we can find {ym }m≥1 ⊆ Y ∗ , ym −→ x∗ in X ∗ as m −→ ∞. We have ∗ ∗ ym , x(tn )X −→ ym , x(t)X
Also
as n → ∞
∗ ym , x(t)X −→ x∗ , x(t)X
for all m ≥ 1
(see (10.20)). (10.21)
as m −→ ∞.
(10.22)
Because of (10.21) and (10.22) we can find a sequence n −→ m(n) not necessarily strictly increasing, such that m(n) −→ ∞ as n → ∞ and ∗ ym(n) , x(tn ) X −→ x∗ , x(t)X as n → ∞. (10.23) Then we have | x∗ , x(tn )X − x∗ , x(t)X | ∗ ∗ , x(tn ) X + ym(n) , x(tn ) X − x∗ , x(t)X ≤ x∗ , x(tn )X − ym(n)
⇒ x∗ , x(tn )X −→ x∗ , x(t)X
as n → ∞
(see (10.23)) (10.24)
Because in (10.24) x∗ ∈ X ∗ was arbitrary, we conclude that x ∈ C(T, Xw ).
10.2 Semilinear Evolution Equations In this section we examine semilinear evolution equations using semigroups of linear operators (see Section 3.2). So let X be a Banach space, A : D(A) ⊆ X −→ X be the infinitesimal generator of a C0 -semigroup {S(t)}t≥0 and f : [0, b] × X −→ X be a function (the nonlinear perturbation). We consider the following semilinear Cauchy problem:
) * x (t) = Ax(t) + f t, x(t) , t ∈ T = [0, b], . (10.25) x(0) = x0 ∈ X We introduce the following hypotheses on the nonlinearity f (t, x). H(f ): f : T ×X −→ X is a function such that (i) For every x ∈ X, t −→ f (t, x) is strongly measurable.
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10 Evolution Equations (ii) There exists k ∈ L1 (T )+ such that for almost all t ∈ T and all x, u ∈ X, we have f (t, x) − f (t, u) ≤ k(t)x − u and
f (t, 0) ≤ k(t).
DEFINITION 10.2.1 A function x ∈ C(T, X) is said to be a mild solution of (10.25) if it is a solution of the following integral equation
x(t) = S(t)x0 +
t
S(t − τ )f τ, x(τ ) dτ,
t ∈ T = [0, b].
(10.26)
0
REMARK 10.2.2 Hereafter we do not distinguish between the mild solutions of (10.25) and the solutions of (10.26). PROPOSITION 10.2.3 If A is the infinitesimal generator of a contraction semigroup {S(t)}t≥0 ; that is, S(t)L ≤ 1
for all t ≥ 0,
f (t, x) satisfies hypotheses H(f ), and x0 ∈ X, then problem (10.25) admits a unique mild solution x(·; x0 ) ∈ C(T, X) satisfying
and
k(τ )dτ x0 0
t k(τ )dτ x0 − x0 . x(t; x0 ) − x(t; x0 ) ≤ exp x(t; x0 ) ≤ exp
t
(10.27) (10.28)
0
PROOF: On C(T, X) we introduce the following equivalent norm
|u| = max exp −L
t
k(τ )dτ x(t) : t ∈ T
(10.29)
0
with L > 1. We consider the nonlinear operator ξ : C(T, X) −→ C(T, X) defined by
ξ(x)(t) = S(t)x0 +
t
S(t − τ )f τ, x(τ ) dτ.
0
We show that ξ is a | · |-contraction. To this end, for x, u ∈ C(T, X), we have ξ(x)(t) − ξ(u)(t)
t
S(t − τ ) f τ, x(τ ) − f τ, u(τ ) dτ = 0
t k(τ )x(τ ) − u(τ )dτ. ≤ 0
t Multiplying (10.30) with exp −L 0 k(τ )dτ , we obtain
(10.30)
10.2 Semilinear Evolution Equations
703
t
k(τ )dτ ξ(x)(t) − ξ(u)(t) exp −L 0
t
τ
t
exp −L k(s)ds exp −L k(s)ds k(τ )x(τ ) − u(τ )dτ ≤ 0 τ 0
t
t
exp −L k(s)ds k(τ )dτ (see (10.29)) ≤ |x − u| 0
τ
1 ≤ |x − u| L 1 |x − u|. L Because L > 1, it follows that ξ is a |·|-contraction and so by Banach’s contraction principle, we deduce that there exists a unique x(·; x0 ) ∈ C(T, X) such that
x(·; x0 ) = ξ x(·; x0 ) . ⇒ |ξ(x) − ξ(u)| ≤
Clearly this is the unique mild solution of (10.25). We have
t
f τ, x(τ ) dτ x(t; x0 ) ≤ x0 + 0
t
f (τ, 0) + k(τ )x(τ ) dτ ≤ x0 + 0
t
k(τ ) 1 + x(τ ) dτ. ≤ x0 + 0
Invoking Gronwall’s inequality, we infer that
t k(τ )dτ (1 + x0 ). x(t; x0 ) ≤ exp 0
Moreover, we have
t
x(t; x0 ) − x(t; x0 ) ≤ S(t)(x0 − x0 )+ S(t − τ ) x(τ ; x0 ) − x(τ ; x0 ) dτ 0
t k(τ )x(t; x0 ) − x(t; x0 )dτ. ≤ x0 − x0 + 0
Once again Gronwall’s inequality implies that
t k(τ )dτ x0 − x0 . x(t; x0 ) − x(t; x0 ) ≤ exp 0
REMARK 10.2.4 Of course the result is still true if A is the infinitesimal generator of a C0 -semigroup {S(t)}t≥0 satisfying S(t)L ≤ M eωt
for all t ≥ 0
with M ≥ 1, ω > 0.
In this case the estimates (10.27) and (10.28) have the following form,
t
k(τ )dτ (1 + x0 ) x(t; x0 ) ≤ M exp ωt + M 0
t
k(τ )dτ x0 − x0 for all t ∈ T. and x(t; x0 ) − x(t; x0 ) ≤ M exp ωt + M 0
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10 Evolution Equations
In Proposition 10.2.3 we assumed that x0 ∈ X. For any x0 ∈ X that does not belong in D(A), in general t −→ S(t)x0 is not necessarily differentiable and neither belongs in D(A) for t > 0. Therefore, if x0 ∈ D(A) (smooth initial condition), we expect to deduce some regularity for the mild solution. PROPOSITION 10.2.5 If A is the infinitesimal generator of a C0 -semigroup and f (t, x) satisfies hypotheses H(f ), with k ∈ L∞ (T )+ and x0 ∈ D(A), then the unique mild solution x(·; x0 ) is Lipschitz continuous. PROOF: Without any loss of generality we assume that {S(t)}t≥0 is a contraction semigroup. For any r > 0, we set x(t) = x(t + r). Evidently x is a mild solution of (10.25) with initial condition x(r). So from (10.28) it follows that
t x(t + r) − x(t) ≤ exp k(τ )dτ x(r) − x0 . (10.31) 0
Also note that
r
S(t − τ )f τ, x(τ ) dτ x(r) − x0 = S(r)x0 − x0 + 0
r
r
r S(τ )Ax0 dτ + f (τ, x0 )dτ + k(τ )x(τ ) − x0 dτ ≤ 0
0
(because x0 ∈ D(A))
0
r k(τ )(1 + x0 )dτ + k(τ )x(τ ) − x0 dτ 0 0
t
k(τ )x(τ ) − x0 dτ. ≤ Ax0 + (1 + x0 )k∞ r + r
≤ Ax0 r +
0
(10.32) From (10.32) and Gronwall’s inequality, we obtain x(r) − x0 ≤ cr
for some c > 0 (independent of r ∈ T ).
(10.33)
Using (10.33) in (10.31), we conclude that x(·; x0 ) is Lipschitz continuous.
COROLLARY 10.2.6 If X is a reflexive Banach space, A is the infinitesimal generator of a C0 -semigroup, f satisfies hypotheses H(f ) with k ∈ L∞ (T )+ , and x0 ∈ D(A), then the mild solution x(·; x0 ) is a strong solution; that is, it satisfies (10.25) pointwise almost everywhere. Moreover, if f is t-independent, then
the mild solution is a classical solution; that is, x(·; x0 ) ∈ C(T, X) ∩ C 1 (0, b), X . Proposition 10.2.3 is still valid if T = [0, b] is replaced by R+ . So we have the following. PROPOSITION 10.2.7 If A is the infinitesimal generator of C0 -semigroup, f (t, x) satisfies hypotheses H(f ), and x0 ∈ X, then problem (10.25) admits a unique global mild solution x(·; x0 ) ∈ C(R+ , X) and
t x(t; x0 ) − x(t; x0 ) ≤ exp k(τ )dτ x0 − x0 . 0
10.2 Semilinear Evolution Equations
705
PROOF: It is similar to the proof of Proposition 10.2.3, so we only sketch it here. Again we introduce the map ξ : C(R+ , X) −→ C(R+ , X), defined by
t
S(t − τ )f τ, x(τ ) dτ, t ≥ 0. ξ(x)(t) = S(t)x0 + 0
t Let V = x ∈ C(R+ , X) : supt≥0 exp −L 0 k(τ )dτ x(t) < ∞ , with L > 1. In V we introduce the norm
t
k(τ )dτ x(t). |x| = sup exp −L t≥0
0
Clearly V equipped with this norm is a Banach space. Moreover, ξ(V ) ⊆ V . Indeed, for x ∈ V , we have
t
ξ(x)(t) ≤ S(t)x0 + S(t − τ )f τ, x(τ ) dτ 0
t
f τ, x(τ ) dτ ≤ x0 + 0
t k(τ )(1 + x(τ ))dτ ≤ x0 + 0
t
t
t
k(τ )dτ ξ(x)(t) ≤ c1 + exp −L k(τ )dτ k(τ )x(τ dτ, ⇒ exp −L 0
0
0
c1 > 0, ⇒ |ξ(x)| < ∞ (i.e., ξ(x) ∈ V ). As in the proof of Proposition 10.2.3, we can check that ξ is a contraction. So by the Banach contraction principle it has a unique fixed point in V , which is a mild solution of (10.25). Moreover, the uniqueness also holds in C(R+ , X). Indeed if x, u ∈ C(R+ , X) are mild solutions of (10.25) and y = x − u, then
t
S(t − τ ) f τ, x(τ ) − f τ, u(τ ) dτ, y(t) = 0
t k(τ )y(τ )dτ, t ≥ 0, ⇒ y(t) ≤ 0
⇒ y(t) = 0
for all t ≥ 0
(by Gronwall’s inequality),
⇒ x = u. The Lipschitz dependence on the initial condition follows via Gronwall’s inequality, as in the proof of Proposition 10.2.3. As before, for simplicity we assume that A is the infinitesimal generator of a contraction semigroup. Then from the Hille–Yosida theorem (see Theorem 3.2.87), we know that for every λ > 0, the operator I − λA is one-to-one from D(A) ⊆ X onto X and so by Banach’s theorem we know that Jλ = (I −λA)−1 ∈ L(X). This operator is called the resolvent of A. Using Jλ we also define
(10.34)
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10 Evolution Equations 1 (I − Jλ ) ∈ L(X) λ
Aλ =
(10.35)
which is the so-called Yosida approximation of A (see also Definition 3.2.42, where these notions were defined in the context of nonlinear monotone operators defined on a pivot Hilbert space). As in Proposition 3.2.44, we can show that the following are true for the operators Jλ and Aλ defined in (10.34) and (10.35), respectively. PROPOSITION 10.2.8 (a) For any λ>0 and x ∈ X, we have Aλ (x) = A(Jλ x). (b) For any λ > 0 and x ∈ D(A), we have Aλ x = Jλ (Ax). (c) For any λ > 0 and x ∈ D(A), we have Aλ x ≤ Ax. (d) For any x ∈ X, we have lim Jλ x = x. λ→0
(e) For any x ∈ D(A), we have lim Aλ x = Ax. λ→0
A is unbounded, therefore a mild solution need not be strongly differentiable. This may be an inconvenience in many applications. The following approximation result helps us to overcome such an inconvenience. PROPOSITION 10.2.9 If A is the infinitesimal generator of a contraction semigroup, f (t, x) satisfies hypotheses H(f ), x0 ∈ X, x ∈ C(T, X) is a mild solution of problem (10.25), and xλ ∈ C(T, X) is the solution of
t
xλ (t) = Sλ (t)x0 + Sλ (t − τ )f τ, xλ (τ ) dτ, t ∈ T, 0
where Sλ (t) = eAλ t =
(Aλ t) k!
k≥0
k
(the semigroup generated by Aλ ∈ L(X)) then
lim max xλ (t) − x(t) = 0.
λ→0 t∈T
PROOF: As before using hypothesis H(f )(ii) and Gronwall’s inequality, we obtain
b
sup Sλ (r) − S(r) f τ, x(τ ) dτ. (10.36) xλ (t) − x(t) ≤ 0 r∈T
But we know that for any x ∈ X. Sλ (t)x −→ S(t)x
as λ ↓ 0
uniformly in t ∈ T.
So from (10.36) and the dominated convergence theorem, we conclude that lim max xλ (t) − x(t) = 0.
λ→0 t∈T
In Section 3.2, we saw the Hille–Yosida characterization of the infinitesimal generator of a C0 -semigroup. In the next theorem we produce a different characterization for the infinitesimal generators of contraction semigroups. The result is known as the Lumer–Phillips theorem and its proof can be found in Yosida [614, p. 250].
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707
THEOREM 10.2.10 If A is a closed and densely defined linear operator in a Banach space X, then A is the infinitesimal generator of a contraction semigroup if and only if the operator −A is m-accretive. Recall that if A is a closed operator in a Banach space X with domain D(A), then we can consider D(A) with the graph norm xD(A) = x + Ax.
(10.37)
Due to the closedness of A, (D(A), · D(A) ) is a Banach space that is embedded continuously in X. We consider the following Cauchy problem: ) * x (t) = Ax(t) + g(t), t ≥ 0, . (10.38) x(0) = x0 We assume that −A is closed, densely defined, and m-accretive, hence it is the infinitesimal generator of a contraction semigroup {S(t)}t≥0 . We have the following existence result for problem (10.38). PROPOSITION 10.2.11 If A is closed, densely defined, −A is m-accretive, g ∈ C 1 (R+ , X), and x0 ∈ D(A), then problem (10.38) admits a unique classical solution x such that
x ∈ C 1 (R+ , X) ∩ C R+ , D(A) which can be expressed as
t
S(t − τ )g(τ )dτ.
x(t) = S(t)x0 + 0
PROOF: We know that t −→ S(t)x 0 (the solution of the homogeneous equation) belongs in C 1 (R+ , X) ∩ C R+ , D(A) . So we need to check that
t
S(t − τ )g(τ )dτ
u(t) = 0
belongs in C 1 (R+ , X) ∩ C R+ , D(A) and satisfies (10.38). To this end we consider
t
u(t + r) − u(t) 1 t+r S(t + r − τ )g(τ )dτ − S(t − τ )g(τ )dτ = r r 0 0
t
1 t+r = S(t + r−τ )g(τ )dτ + S(t + r−τ )−S(t −τ ) g(τ )dτ r t 0 (10.39)
1 t
1 t+r S(z)g(t + h−z)dz + S(z) g(t + h−z)−g(t−z) dz. = r t r 0 (10.40) We let r −→ 0 and in the right side of (10.40), we obtain
t S(t)g(0) + S(z)g (t − z)dz ∈ C(R+ , X). 0
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10 Evolution Equations
So it follows that u ∈ C 1 (R+ , X). Moreover, the right-hand side in (10.39) gives S(0)g(t) − Au(t) = g(t) − Au(t). Hence u ∈ C R+ , D(A) and we have proved the proposition.
COROLLARY 10.2.12 If A is closed, densely defined, −A is m-accretive, g ∈
C R+ , D(A) , and x0 ∈ D(A), then problem (10.38) has a classical solution x given by
t
S(t − τ )g(τ )dτ,
x(t) = S(t)x0 +
t ≥ 0.
0
COROLLARY 10.2.13 If A is closed, densely defined, −A is m-accretive, g ∈ C(R+ , X), g is strongly differentiable with g ∈ L1loc (R+ , X), and x0 ∈ D(A), then problem (10.38) has a classical solution x given by
t x(t) = S(t)x0 + S(t − τ )g(τ )dτ, t ≥ 0. 0
PROOF: Let b > 0, T = [0, b]. By hypothesis g ∈ L1 (T, X). We set
t u(t) = S(t − τ )g(τ )dτ, t ∈ T. 0
We can easily check that u ∈ C(T, X). Also from (10.40) we see that u is strongly differentiable almost everywhere on T and at points of strong differentiability we have
t u (t) = S(t)g(0) + S(τ )g (t − τ )dτ 0
t S(t − τ )g (τ )dτ ∈ C(T, X). = S(t)g(0) + 0
Therefore for almost all t ∈ R+ , we have u (t) = Au(t) + f (t),
⇒ u ∈ C 1 (T, X) ∩ C T, D(A) . REMARK 10.2.14 The assumption g ∈ C(R+ , X) alone is not enough to guarantee the existence of a classical solution. COROLLARY 10.2.15 If X is a reflexive Banach space, A is closed, and densely defined, −A is m-accretive, g : R+ −→ X is Lipschitz continuous, and x0 ∈ D(A), then problem (10.38) has a classical solution x given by
t x(t) = S(t)x0 + S(t − τ )g(τ )dτ, t ≥ 0. 0
Now we use Proposition 10.2.11 to obtain an existence result for classical solutions for problem (10.25).
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709
PROPOSITION 10.2.16 If A is closed, densely defined, −A is m-accretive, f ∈ C 1 (X, X), and there exists k > 0 such that f (x) − f (u) ≤ kx − u
for all x, u ∈ X,
then for every x0 ∈ D(A) problem (10.25) admits a unique global classical solution. PROOF: By virtue of Proposition 10.2.7, problem (10.25) has a unique global mild solution x ∈ C(R+ , X). Because of the hypotheses on f and Proposition 10.2.11, it suffices to show that x ∈ C 1 (R+ , X). To this end, we consider the following auxiliary problem,
) * u (t) = Au(t) + f x(t) u(t), t ≥ 0, . (10.41) u(0) = f (x0 ) + Ax0 ∈ X Problem (10.41) has a unique global mild solution u ∈ C(R+ , X) given by
u(t) = S(t) f (x0 ) − Ax0 ) +
t
S(t − τ )f x(τ ) u(τ )dτ,
t ≥ 0.
0
We have x(t + r) − x(t) − u(t) r
t+r 1
= S(t) S(r) − I x0 + S(t + r − τ )f x(τ ) dτ r 0
t
t
S(t − τ )f x(τ ) dτ−S(t) f (x0 )+Ax0 − S(t−τ )f x(τ ) u(τ )dτ − 0
0
t
S(r)x0 − x0
f x(τ + r) −f x(τ ) − Ax0 + − f x(τ ) u(τ ) dτ ≤ r r 0
1 r
S(t + r − τ )f x(τ ) dτ − S(t)f (x0 ) + r 0 = ξ1 + ξ2 + ξ3 . (10.42) Because by hypothesis x0 ∈ D(A), we have ξ1 −→ 0
as r ↓ 0.
(10.43)
Also due to the continuity of x and f , we have ξ3 −→ 0
as r ↓ 0.
Next note that because f is strongly differentiable, we have
f x(τ + r) − f x(τ )
x(τ + r) − x(τ ) − f x(τ ) r r x(τ + r) − x(τ )
ϑ x(τ + r) − x(τ ) ≤ r (with ϑ(s) −→ 0 as s −→ 0). So we have
(10.44)
(10.45)
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10 Evolution Equations
f x(τ + r) − f x(τ )
− f x(τ ) u(τ ) r
f x(τ + r) − f x(τ )
x(τ + r) − x(τ ) − f x(τ ) ≤ r r
x(τ + r) − x(τ ) + f x(τ ) − u(τ ) r x(τ + r) − x(τ )
ϑ x(τ + r)−x(τ ) +k x(τ + r) − x(τ ) − u(t) ≤ r r (10.46)
using (10.46) and recalling that f x(τ ) L ≤ k for all τ ≥ 0. Because of Proposition 10.2.5, we see that x(τ + r) − x(τ )
ϑ x(τ + r) − x(τ ) −→ 0 r
as r −→ 0.
(10.47)
Using (10.43), (10.44), (10.46), and (10.47), we see that given ε > 0, we can find r > 0 small such that
t x(t + r)−x(t) x(τ +r)−x(τ ) −u(t) ≤ ε+k −u(τ ) dτ. r r 0 Invoking Gronwall’s inequality and because ε > 0 was arbitrary, we conclude that x(t+r)−x(t) − u(t) = 0, r for all t ≥ 0, ⇒ x (t) = u(t) lim
r→0
⇒ x ∈ C 1 (R+ , X). Thus far all the existence results that we proved required that the nonlinearity f (x) be globally Lipschitz. This is a rather restrictive hypothesis. In particular it implies that f (x) ≤ f (0) + kx. This means that f has at most linear growth and so we rule out several interesting nonlinearities (such as quadratic). For this reason we would like to be able to relax this global Lipschitzness hypothesis. In this direction we have the following existence result. PROPOSITION 10.2.17 If A is a closed, densely defined operator, −A is maccretive, f : X −→ X is Lipschitz continuous on bounded sets, and x0 ∈ X, then we can find b > 0 depending on x0 such that problem (10.25) admits a unique local solution x ∈ C(T, X). Moreover, if x0 ∈ D(A), then x is Lipschitz continuous on T . Finally if X is reflexive and x0 ∈ D(A), then the mild solution x is in fact classical. PROOF: Let r = 1 + x0 and let kr > 0 be the Lipschitz constant of f on B r = {x ∈ X : x ≤ r}. Let 0 < b < 1/kr and define Cb = x ∈ C(T, X) : x(t) ≤ r, t ∈ T = [0, b] .
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711
Clearly Cb is a closed and convex subset of C(T, X). We consider the nonlinear map ξ : Cb −→ C(T, X) defined by
t
S(t − τ )f x(τ ) dτ for all t ∈ T. ξ(x)(t) = S(t)x0 + 0
We have
ξ(x)(t) ≤ x0 + b f (0) + kr r
for all t ∈ T.
If we choose b > 0 such that b < min
1 1 , , kr f (0) + kr r
(10.48)
then we deduce that ξ(x)(t) ≤ x0 + 1 = r. Therefore, ξ : Cb −→ Cb . Moreover, for every x, u ∈ Cb , we have
t
S(t − τ ) f x(τ ) − f u(τ ) dτ ξ(x) − ξ(u)C(T,X) = sup t∈T
0
≤ b kr x − uC(T,X) . Because bkr < 1 (see (10.48)), we see that ξ is a contraction on Cb . So by Banach’s fixed point theorem, we can find a unique fixed point of ξ. It is easily seen that this is the unique mild solution of (10.25). If x0 ∈ D(A), Proposition 10.2.5 implies that x is Lipschitz continuous. Finally when X is reflexive and x0 ∈ D(A), Corollary 10.2.6 implies that the unique mild solution is in fact a classical one. As is the case with ordinary differential equations in RN , we can always produce a maximal solution; that is, a solution defined on a maximal time interval. PROPOSITION 10.2.18 If A is a closed, densely defined operator, −A is maccretive, f : X −→ X is Lipschitz continuous on bounded sets, and x0 ∈ X, then problem (10.25) has a unique mild solution on a maximal time interval [0, bmax ) such that either (a) bmax = +∞; that is, the unique mild solution is global, or (b) bmax < +∞ and lim x(t) = +∞ (i.e., we have a blow-up of the solution in t→b− max
finite time). PROOF: Suppose x1 , x2 are two mild solutions defined on T1 = [0, b1 ] and T2 = [0, b2 ], respectively, and assume that b1 < b2 . Then T1 ⊆ T2 and due to the uniqueness of the mild solution on T1 we have x2 T = x1 . 1
Invoking Zorn’s lemma, we can find a maximal interval Tmax = [0, bmax ) on which the unique mild solution exists. We need to show that if bmax < +∞, then
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10 Evolution Equations lim x(t) = +∞.
(10.49)
t→b− max
We proceed indirectly. So suppose that (10.49) is not true. We show that we can continue x beyond bmax , a contradiction to the maximality of the interval Tmax = [0, bmax ). So suppose we can find {tn }n≥1 ⊆ Tmax such that tn −→ b− max
as n → ∞
and
x(tn ) ≤ c for some c > 0, all n ≥ 1.
We consider the following Cauchy problem
) * u (t) = Au(t) + f u(t) , t ≥ 0, . u(0) = x(tn ), n ≥ 1
(10.50)
By virtue of Proposition 10.2.17, problem (10.50) has a unique mild solution on some interval [0, b], with b > 0 depending on c > 0. Let n ≥ 1 be large so that bmax < b + tn . We set x(t) if t ∈ [0, tn ] . (10.51) y(t) = if t ∈ [tn , tn + b] u(t − tn ) We show that y is a mild solution of (10.25) on the interval [0, b + tn ]. To this end note that
t
S(t − τ )f y(τ ) dτ for t ∈ [0, tn ] (see (10.51)) y(t) = S(t)x0 + 0
and
u(t) = S(t)x(tn ) +
(10.52) t
S(t − τ )f u(τ ) dτ
for t ∈ [0, b].
(10.53)
0
From (10.52) we see that y ∈ C([0, b + tn ], X) is a mild solution of (10.25) on the time interval [0, tn ]. Now, if t ∈ [0, b], we have
t
y(t + tn ) = u(t) = S(t)x(tn )+ S(t−τ )f u(τ ) dτ (see (10.51) and (10.53)) 0
tn
S(t − τ )f x(τ ) dτ = S(t) S(tn )x0 + 0
t
+ S(t−τ )f u(τ ) dτ 0
tn
S(t + tn − τ )f x(τ ) dτ = S(t+tn )x0 +
0
tn+t
+
S(t+tn − τ )f u(τ − tn ) dτ
tn
= S(t + tn )x0 +
t+tn
S(t+tn − τ )f y(τ ) dτ
(see (10.51)).
0
(10.54) From (10.54) we conclude that y is a mild solution of (10.25) on [0, b + tn ] which strictly contains Tmax , a contradiction to the maximality of Tmax .
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713
Using Propositions 10.2.17 and 10.2.18, we can have an existence and uniqueness result for a global mild solution, under conditions that are slightly weaker than the global Lipschitz condition considered in the first part of this section. COROLLARY 10.2.19 If A is a closed, densely defined operator, −A is maccretive, and f : X −→ X is Lipschitz continuous on bounded sets, there exist c1 , c2 > 0 such that f (x) ≤ c1 + c2 x
for all x ∈ X
(10.55)
and x0 ∈ X, then problem (10.25) admits a unique maximal mild solution x ∈ C(R+ , X). PROOF: From Proposition 10.2.18 we know that problem (10.25) admits a unique maximal mild solution x ∈ C(Tmax , X), Tmax = [0, bmax ). We have
t
x(t) = S(t)x0 + 0
t
⇒ x(t) ≤ x0 + 0
≤ x0 +
S(t − τ )f x(τ ) dτ
for t ∈ Tmax = [0, bmax ),
f x(τ ) dτ c1 + c2 x(τ ) dτ
t
(see (10.55)).
0
If bmax < +∞, via Gronwall’s inequality, we obtain x(t) ≤ M
for some M > 0
and all t ∈ Tmax .
(10.56)
But from Proposition 10.2.18, we know that we must have lim x(t) = +∞,
t→b− max
a contradiction to (10.56).
COROLLARY 10.2.20 If X = H = Hilbert space, −A is maximal monotone, f : H −→ H is Lipschitz continuous on bounded sets, and there exist c1 , c2 > 0 such that (f (x), x)H ≤ c1 + c2 x2
for all x ∈ H,
(10.57)
then for every x0 ∈ X problem (10.25) admits a unique global mild solution x ∈ C(R+ , X). Moreover, if x0 ∈ D(A), then this unique global mild solution is in fact a classical one. PROOF: First suppose that x0 ∈ D(A). Then from Propositions 10.2.17 and 10.2.18, problem (10.25) admits a maximal classical solution x ∈ C(Tmax , H) with Tmax = [0, bmax ). We have
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10 Evolution Equations
x (t) = Ax(t) + f x(t) for all t ∈ Tmax ,
⇒ x (t), x(t) H = Ax(t), x(t) H + f x(t) , x(t)
H
1 d ⇒ x(t)2 ≤ c1 + c2 x(t)2 2 dt (because − A is monotone and using (10.57))
t x(τ )2 dτ ⇒ x(t)2 ≤ c3 + c2
for all t ∈ Tmax .
0
(10.58) As in the proof of Corollary 10.2.19, (10.58) implies that bmax = +∞. Now suppose that x0 ∈ X.Then we can find {xn 0 }n≥1 ⊆ D(A) such that xn 0 −→ x0
in X
as n → ∞.
Let xn ∈ C(R+ , X) be the global classical solution emanating from xn 0 (see the first part of the proof). As above, using (10.57), we show that for every b > 0, we have xn (t) ≤ M
for all n ≥ 1, all t ∈ T = [0, b],
with M > 0 independent of n ≥ 1. Also if n, m ≥ 1 are fixed and we set u(t) = xn (t) − xm (t), then we have
)
t ∈ T,
* u (t) = Au(t) + f xn (t) − f xm (t) , t ≥ 0, . m u(0) = xn 0 − x0 , n ≥ 1
(10.59)
From (10.59) as above, using (10.57) and the monotonicity of −A, we obtain 1 d u(t)2 ≤ kM u(t)2 2 dt
(10.60)
with kM > 0 being the Lipschitz constant of f on B M . From (10.60), via Gronwall’s inequality, we obtain m 2 u(t)2 = xn (t) − xm (t)2 ≤ cxn 0 − x0 ,
for some c > 0 independent of n, m ≥ 1. Therefore {xn }n≥1 ⊆ C(T, X) is Cauchy and so xn −→ x in C(T, X). Evidently x is mild solution of (10.25) on T = [0, b]. Because b > 0 is arbitrary, it follows that x is a global mild solution. As before we can easily check that x is unique. We conclude this section by applying the abstract results to some concrete parabolic problems. So let Z ⊆ RN be a bounded domain with a C ∞ -boundary. First we consider the following initial-boundary value problem:
10.2 Semilinear Evolution Equations ⎧ ∂x ⎫ ⎨ ∂t + x(t, z) = g(t, z) on R+ × Z, ⎬ . ⎩ x ⎭ = 0, x(0, z) = x (z), z ∈ Z 0 R ×Z
715
(10.61)
+
We make the following hypotheses on g. H(g): g : T ×Z −→ R is a function such that g(0, ·) ∈ L2 (Z) and (i) For all t ∈ T , z −→ g(t, z) is measurable. (ii) For almost all z ∈ Z and all t, t ∈ R+ we have |g(t, z) − g(t , z)| ≤ k(z)|t − t |
with k ∈ L2 (Z).
Then we have the following result concerning problem (10.61). PROPOSITION 10.2.21 If g(t, z) satisfies hypotheses H(g) and x0 ∈ H01 (Z) ∩ H 2 (Z), then problem (10.61) admits a unique global classical solution
x ∈ C R+ , L2 (Z) ∩ C 1 (0, ∞), H01 (Z) ∩ H 2 (Z) . PROOF: Let X = H = L2 (Z) and A : D(A) ⊆ H −→ H defined by Ax = − x ∩ H (Z). Then we can rewrite (10.61) as the following with x ∈ D(A) = equivalent semilinear evolution equation ) * x (t) = Ax(t) + g(t) , (10.62) x(0) = x0 H01 (Z)
2
where g(t) = g(t, ·) ∈ H for all t ≥ 0. We claim that −A is maximal monotone. To this end let h ∈ H = L2 (Z) and consider the following stationary problem * ) u(z) − u(z) = h(z) a.e. on Z, . (10.63) u∂Z = 0
Let V ∈ L H01 (Z), H −1 (Z) be defined by
(Dx, Dy)RN dz. V (x), y H = Z
Evidently V is monotone (hence maximal monotone) and coercive. Therefore it is surjective (see Corollary 3.2.28) and we can find x ∈ H01 (Z) such that V (x) = h ∈ H. Then x is a solution of (10.63) and from the linear regularity theory, we have x ∈ H 2 (Z), ⇒ x ∈ D(A), ⇒ R(I − A) = L2 (Z) = H.
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10 Evolution Equations
Because −A is clearly monotone, it follows that −A is maximal monotone (see Theorem 3.2.30). Note that by virtue of hypothesis H(g)(ii), g : R+ −→ H is Lipschitz continuous. So we can apply Corollary 10.2.15 and obtain a classical solution x such that
x ∈ C R+ , L2 (Z) ∩ C 1 (0, ∞), H01 (Z) ∩ H 2 (Z) . Next we consider the semilinear heat equation:
⎧ ∂x ⎫ ⎨ ∂t + x(t, z) = f0 x(t, z) on R+ ×Z, ⎬ . ⎩ x ⎭ = 0, x(0, z) = x0 (z), z ∈ Z R ×Z
(10.64)
+
PROPOSITION 10.2.22 If f0 ∈ C 1 (R) and f0 is uniformly bounded, then for every x0 ∈ L2 (Z), problem (10.64) admits a unique global mild solution
x ∈ C R, L2 (Z) . Moreover, if x0 ∈ H01 (Z) ∩ H 2 (Z), then the above solution is classical and
x ∈ C 1 R+ , L2 (Z) ∩ C R+ , H01 (Z) ∩ H 2 (Z) . PROOF: As before, let X = H = L2 (Z), A : D(A) ⊆ H −→ H is defined by Ax = − x for all x ∈ D(A) = H01 (Z) ∩ H 2 (Z), and f : L2 (Z) −→ L2 (Z) is defined by
f (x)(·) = f0 x(·) .
Evidently f ∈ C 1 L2 (Z), L2 (Z) and also it is Lipschitz continuous. Then problem (10.64) can be equivalently rewritten as the following abstract evolution equation
* ) x (t) = Ax(t) + f x(t) . (10.65) x(0) = x0 Invoking Proposition 10.2.16 and Corollary 10.2.15, we obtain the conclusions of the proposition. PROPOSITION 10.2.23 If N ≤ 3, f0 (x) = −x3 + x, and x0 ∈ H01 (Z) ∩ H 2 (Z), then problem (10.64) admits a unique classical solution
x ∈ C 1 R+ , L2 (Z) ∩ C R+ , H01 (Z) ∩ H 2 (Z) . PROOF: Again X = H = L2 (Z), A : D(A) ⊆ H −→ H is defined by Ax = − x for all x ∈ D(A) = H01 (Z) ∩ H 2 (Z) and f : L2 (Z) −→ L2 (Z) is defined by f (x)(·) = f0 x(·) . Then problem (10.64) is equivalent to the semilinear evolution equation (10.65).We claim that f : D(A) −→ D(A). Indeed, because N ≤ 3, by the Sobolev embedding theorem, we have that H01 (Z) is embedded continuously in L6 (Z) and H 2 (Z) is embedded continuously into C(Z). Because f0 ∈ C ∞ (R), we see that for all u ∈ D(A), f (u) ∈ H01 (Z) and from the chain rule we have
10.2 Semilinear Evolution Equations
and
717
Df0 u(z) = f0 u(z) Du(z)
D2 f0 u(z) = f0 u(z) D2 u(z) + f0 u(z) Du(z), Du(z) RN .
Therefore we see that f (u) ∈ H 2 (Z) and so we have f : D(A) −→ D(A). Now let H1 = D(A) (with the graph norm) and A1 : D(A1 ) ⊆ H1 −→ H1 is defined by A1 x = Ax with x ∈ D(A1 ) = D(A2 ). Then applying Proposition 10.2.18 on H1 with operator A1 , we obtain a unique maximal classical solution
x ∈ C 1 Tmax , L2 (Z) ∩ C Tmax , H01 (Z) ∩ H 2 (Z) with Tmax = [0, bmax ). We show that bmax = +∞. To this end, it suffices to show that x(t)H 2 (Z) is bounded uniformly in t ≥ 0. So we multiply (10.64) with xt and then integrate over Z. We obtain
∂x 2 1 1 1 d (10.66) Dx2 + x4 − x2 dz + L2 (Z) = 0. dt Z 2 4 2 ∂t Note that x(z)2 ≤
1 x(z)4 + 1. 4
Using this in (10.66) we obtain
∂x 2 1 d (Dx2 + x2 )dz + L2 (Z) = 0, dt Z 2 ∂t
t ∂x 2 dt ≤ M ⇒ xH 1 (Z) ≤ M and 0 ∂t L (Z) 0
for some M > 0.
For r > 0, let u(t) = x(t + r). Evidently we have
u ∈ C 1 [0, bmax − r), L2 (Z) ∩ C [0, bmax − r), H01 (Z) ∩ H 2 (Z) ⎧ ∂u ⎫ 3 ⎨ ∂t + u(t, z) = −u(t, z) + u(t, z) ⎬ and
⎩
u(t, ·)∂Z = 0, u(0, z) = x(r, z)
⎭
.
(10.67)
We set y(t, z) = u(t, z) − x(t, z) = x(t + r, z) − x(t, z). From (10.64) and (10.67), we have ⎧ ∂y ⎫
⎨ ∂t + y(t, z) + y(t, z) u(t, z)2 + u(t, z)x(t, z) + x(t, z)2 − y(t, z) = 0 ⎬ . ⎩ ⎭ y(t, ·)∂Z = 0, y(0, z) = x(r, z) − x0 (z) (10.68) We multiply (10.68) with y(t, z) and then integrate over Z. We obtain
1 d y(t, ·)2L2 (Z) +Dy(t, ·)2L2 (Z) + y 2 (x2 + xu + u2 )dz = y(t, ·)2L2 (Z) . (10.69) 2 dt Z
718
10 Evolution Equations Note that x2 + xu + u2 ≥ 0,
⇒ y 2 (x2 + xu + u2 )dz ≥ 0.
(10.70)
Z
Using (10.70) in (10.69), we have 1 d y(t, ·)2L2 (Z) ≤ y(t, ·)2L2 (Z) 2 dt ⇒ y(t, ·)2L2 (Z) = x(t + r, ·) − x(t, ·)2L2 (Z) ≤ M1 x(r, ·) − x0 2L2 (Z) (10.71) for some M1 > 0 (by Gronwall’s inequality). The constant M1 > 0 depends only on bmax . From (10.71) it follows that ∂x (t, ·) 2 2 ≤ M1 − u0 − u30 + u0 2L2 (Z) < +∞ L (Z) ∂t
(10.72)
(recall u0 ∈ H01 (Z) ∩ H 2 (Z)). We have u(t, ·)H 1 (Z) ≤ M u(t, ·)L2 (Z) 0 ∂x 2 ≤ M1 (t, ·) L2 (Z) + x(t, ·)3 L2 (Z) + x(t, ·)L2 (Z) ∂t ≤ M2 (see (10.72)) (10.73) with M2 > 0 depending only on bmax > 0. So, if bmax < +∞, from (10.73) it follows that lim x(t, ·)L2 (Z) < +∞, t→b− max
a contradiction. Therefore bmax = +∞.
REMARK 10.2.24 A careful reading of the above proof reveals that the result is more generally true if N ≤ 3 and f0 ∈ C 3 (R) with f0 (0) = 0.
10.3 Nonlinear Evolution Equations In this section we deal with nonlinear evolution equations. We focus on two broad classes of nonlinear evolution equations that arise often in applications. The first class concerns dynamic problems driven by subdifferential operators. It contains gradient flows and certain variational inequalities and it is related to the nonlinear semigroup theory briefly mentioned in Section 3.2. The second class considers evolutions monitored by general nonlinear operators of monotone type that are defined within the framework of an evolution triple of spaces (see Section 10.1). We start with the evolution inclusions of the subdifferential type. To prove the main existence and regularity results for such systems, we need some preparatory work.We start with a Gronwall-type lemma, which is a basic tool in deriving a priori estimates.
10.3 Nonlinear Evolution Equations
719
LEMMA 10.3.1 If h ∈ L1 (T )+ , T = [0, b], c ≥ 0 and x ∈ C(T ) satisfies
t 1 2 1 2 h(τ )x(τ )dτ for all t ∈ T, x (t) ≤ c + 2 2 0 t then |x(t)| ≤ c + 0 h(τ )dτ for all t ∈ T . PROOF: For any ε ≥ 0, let ϑε (t) =
1 (c + ε)2 + 2
t
h(τ )x(τ )dτ. 0
We have ϑε (t) = h(t)x(t) for a.a. t ∈ T 1 for all t ∈ T. and x2 (t) ≤ ϑ0 (t) ≤ ϑε (t) 2 From (10.74) and (10.75) it follows that √ ϑε (t) ≤ h(t) 2 ϑε (t)
(10.74) (10.75)
for a.a. t ∈ T.
(10.76) √ The function t −→ ϑε (t) is absolutely continuous and the function r −→ r is locally Lipschitz. So from the Serrin–Vall´ee Poussin chain rule, we have d 1 ϑε (t) = ϑε (t) for a.a. t ∈ T, dt 2 ϑε (t) 1 d ϑε (t) ≤ √ h(t) a.e. on t ∈ T (see (10.75)), ⇒ dt 2
t 1 ⇒ ϑε (t) ≤ ϑε (0) + √ h(τ )dτ for all t ∈ T. (10.77) 2 0 Combining (10.75) and (10.77) we obtain |x(t)| ≤
t √ 2 ϑε (t) ≤ 2ϑε (0) + h(τ )dτ 0
t h(τ )dτ for all t ∈ T. = c+ε+ 0
Because ε > 0 was arbitrary, we let ε ↓ 0, to obtain
t |x(t)| ≤ c + h(τ )dτ for all t ∈ T. 0
Let H be a Hilbert space with inner product denoted by (·, ·)H . The next lemma extends the Serrin–Vall´ee Poussin chain rule.
LEMMA 10.3.2 If ϕ ∈ Γ0 (H), x ∈ W 1,2 (0, b), H , x(t) ∈ D(∂ϕ) a.e. on (0, b),
2 and we can find h ∈ L (T, H) such that h(t) ∈ ∂ϕ x(t) a.e. on T , then the function t −→ ϕ x(t) is absolutely continuous on T and
d
ϕ x(t) = u, x (t) H dt
a.e. on T for all u ∈ ∂ϕ x(t) .
720
10 Evolution Equations
PROOF:
From Corollary 3.2.51(a), we know that for every λ > 0 the function t −→ ϕλ x(t) is differentiable a.e. on T and we have
d ϕλ x(t) = ϕλ x(t) , x (t) dt H
= (∂ϕ)λ x(t) , x (t) a.e. on (0, b), H
t
(∂ϕ)λ x(τ ) , x (τ ) dτ ⇒ ϕλ x(t) − ϕλ x(s) = H
s
for all s, t ∈ T. (10.78)
From Proposition 3.2.44(d)(e) we know that
(∂ϕ)λ x(t) ≤ (∂ϕ)0 x(t) ≤ h(t) a.e. on T
and (∂ϕ)λ x(t) −→ (∂ϕ)0 x(t) a.e. on T as λ ↓ 0. So, from the dominated convergence theorem, we have
(∂ϕ)λ x(·) = ∂ϕλ x(·) −→ (∂ϕ)0 x(·) in L2 (T, H) as λ ↓ 0. Then passing to the limit as λ ↓ 0 in (10.78), we obtain
t
(∂ϕ)0 x(τ ) , x (τ ) dτ ϕ x(t) − ϕ x(s) = s
H
for all s, t ∈ T
(see Corollary 3.2.51(c)). Therefore the function t −→ ϕ x(t) is absolutely
continuous on T . Let t0 ∈ T be a differentiability point for both x(·) and ϕ x(·) and
x(t0 ) ∈ D(∂ϕ). Then for every u ∈ ∂ϕ x(t0 ) , we have
for all y ∈ H. ϕ x(t0 ) − ϕ(y) ≤ u, x(t0 ) − y H Let y = x(t0 ± ε), ε > 0 and divide by ε. We have
1
1
≤ ϕ x(t0 ) − ϕ x(t0 ± ε) u, x(t0 ) − x(t0 ± ε) H . ε ε H Passing to the limit as ε −→ 0, we obtain
ϕ x(t0 ) = u, x (t0 ) H . Now let A : D(A) ⊆ H −→ 2H \{∅} be a possibly multivalued nonlinear operator, x0 ∈ H, and g ∈ L1 (T, H). We consider the following Cauchy problem,
* ) −x (t) ∈ A x(t) + g(t), t ∈ T, . (10.79) x(0) = x0 DEFINITION 10.3.3 (a) A function x ∈ C(T, H) is said to be a strong solution (or solution) for the Cauchy problem (10.79), if t −→ x(t) is absolutely continuous on every compact subinterval of (0, b), x(t) ∈ D(A) a.e. on T , x(0) = x0 and x(·) satisfies the differential equation in (10.79) for almost all t ∈ T .
10.3 Nonlinear Evolution Equations
721
(b) A function x ∈ C(T, H) is said to be a weak solution
for the Cauchy problem (10.79), if there exist sequences {xn }n≥1 ⊆ W 1,1 (0, b), H and {gn }n≥1 ⊆ L1 (T, H) such that for every n ≥ 1, −xn (t) ∈ A xn (t) +gn (t) a.e. on T , gn −→ g in L1 (T, H), and xn −→ x in C(T, H) as n → ∞ with x(0) = x0 . The following existence theorem for problem (10.79) is proved in Barbu [58, p. 124] and in Brezis [102, p. 54]. THEOREM 10.3.4 If A : D(A) ⊆ H −→ 2H is maximal monotone, x0 ∈ D(A), and
g ∈ W 1,1 (0, b), H , then problem (10.79) has a unique (strong) solution x ∈ C(T, H), which is Lipschitz continuous (i.e., x ∈ W 1,∞ (0, b), H ), it is right differentiable at every point t ∈ [0, b), and
0 d+ x for all t ∈ [0, b) (t) = A x(t) + g(t) dt 0 t d+ x
g (τ )dτ (t) ≤ A x(t) + g(t) + dt 0
−
(10.80) for all t ∈ [0, b). (10.81)
1,∞
Moreover, (0, b), H are solutions for the data (x0 , g1 ), (y0 , g2 ) ∈
if x, y∈W D(A)×W 1,1 (0, b), H , respectively, we have
t
x(t) − y(t) ≤ x0 − y0 +
g1 (τ ) − g2 (τ )dτ
for all t ∈ T.
(10.82)
0
REMARK 10.3.5 The result is still true if T =[0, b] is replaced by R+ . Note that (10.80) means that for the multivalued system, among all choices the system tends to minimize its velocity. We can also slightly generalize Theorem 10.3.4 and assume that for some ϑ > 0, x −→ A(x) + ϑx is maximal monotone. Then the existence theorem is still valid, only the estimates in (10.81) and (10.82) need to be modified accordingly. The next theorem concerns weak solutions. THEOREM 10.3.6 If A : D(A) ⊆ H −→ 2H is a maximal monotone operator, x0 ∈ D(A), and g ∈ L1 (T, H), then there exists a unique weak solution x ∈ C(T, H) of (10.79) such that
t
1 1 (10.83) g(τ ) − w, x(τ ) − v H dτ x(t) − v2 ≤ x(s) − v2 + 2 2 s for all 0 ≤ s ≤ t ≤ b and all (v, w) ∈ Gr A. Moreover, if x, y ∈ C(T, X) are weak solutions corresponding to the data (x0 , g1 ),(y0 , g2 ) ∈ D(A)×L1 (T, H), respectively, then we have
t
1 1 g1 (τ ) − g2 (τ ), x(τ ) − y(τ ) H dτ, x(t) − y(t)2 ≤ x(s) − y(s)2 + 2 2 s (10.84) for all 0 ≤ s ≤ t ≤ b.
722
10 Evolution Equations
PROOF: Fix (x0 , g0 ) ∈ D(A)×L1 (T, H). We can find {xn 0 }n≥1 ⊆ D(A) and {gn }n≥1 ⊆
W 1,1 (0, b), H such that xn 0 −→ x0
in H
and
gn −→ g
in L1 (T, H)
as n → ∞.
For every n ≥ 1, we consider the following Cauchy problem
* ) −xn (t) ∈ A xn (t) + gn (t) a.e. on T, . n xn (0) = x0
(10.85)
By virtue of Theorem 10.3.4 problem (10.85) has a unique strong solution xn ∈ W 1,∞ (0, b), H . Moreover, (10.82) implies that for all n, m ≥ 1, we have
b
m xn (t) − xm (t) ≤ xn 0 − x0 +
gn (τ ) − gm (τ )dτ, 0
⇒ {xn }n≥1 ⊆ C(T, H) is Cauchy. So there exists x ∈ C(T, H) such that xn −→ x
in C(T, H).
Evidently x ∈ C(T, H) is a weak solution for problem (10.79). Moreover, because for every n ≥ 1 xn (·) satisfies (10.83) and (10.84), passing to the limit as n → ∞, we deduce that x(·) satisfies them too. Finally the uniqueness of the weak solution follows at once from (10.84). From Theorems 10.3.4 and 10.3.6 we see that in the general case, in order to have a strong solution we need to restrict
the data of the problem, namely we need to assume that x0 ∈ D(A) and g ∈ W 1,1 (0, b), H ; that is, we need a “smooth” initial condition and “smooth” forcing term. We show that in the case of subdifferential evolution inclusions these restrictive requirements can be removed. THEOREM 10.3.7 If A = ∂ϕ : D(∂ϕ) ⊆ H −→ 2H with ϕ ∈ Γ0 (H), x0 ∈ D(∂ϕ), and g ∈ L2 (T, H), then problem (10.79) admits a unique strong solution x ∈ C(T, H) such that √ t −→ t x (t) belongs in L2 (T ),
t −→ ϕ x(t) belongs in L1 (T, H), and it is absolutely continuous on [δ, b], ∀δ > 0. Moreover, if x0 ∈ dom ϕ, then x ∈ L2 (T, H) and t −→ ϕ x(t) is absolutely continuous on T . PROOF: First note that if (v0 , w0 ) ∈ Gr ∂ϕ and we introduce the function ϕ0 (x) = ϕ(x) − ϕ(v0 ) − (w0 , x − v0 )H ≥ 0, x ∈ H,
then the subdifferential inclusion −x (t) ∈ ∂ϕ x(t) + g(t) is equivalent to
−x (t) ∈ ∂ϕ0 x(t) + g(t) − w0 . So, without any loss of generality, we may assume that ϕ(v0 ) = inf ϕ = 0. H
(10.86)
10.3 Nonlinear Evolution Equations
723
Now suppose that x0 ∈ D(∂ϕ) and g ∈ W 1,2 (0, b), H . Then Theorem 10.3.4 implies that there exists a unique strong solution x ∈ W 1,∞ (0, b), H . We multiply the equation with tx (t) and use Lemma 10.3.2. We obtain
d
tx (t)2 + t ϕ x(t) = t g(t), x (t) H a.e. on T, dt
b
b
b
2 ⇒ tx (t) dt + b ϕ x(b) = t g(t), x (t) H + ϕ x(t) dt. 0
0
0
(10.87) Note that
1 1 t g(t), x (t) H ≤ tg(t) x (t) ≤ t g(t)2+ t x (t)2 2 2
for all t ∈ T. (10.88)
Using (10.88) and the fact that ϕ ≥ 0 (see (10.86)) in (10.87), we obtain
b
b
b
tx (t)2 dt ≤ tg(t)2 dt + 2 ϕ x(t) dt. (10.89) 0
0
0
Also from the definition of the convex subdifferential, we have
ϕ x(t) ≤ −x (t) − g(t), x(t) − v0 H (recall that φ(v0 ) = 0, see (10.86))
b
b
ϕ x(t) dt ≤ ⇒ −x (t), x(t)−v0 H dt+ 0
0
b
g(t) x(t) − v0 dt 0
b 1 d g(t) x(t) − v0 dt x(t) − v0 2 dt+ 0 2 dt 0
b 1 ≤ x0 − v0 2 + g(t) x(t) − v0 dt. (10.90) 2 0 b
=−
Next take the inner product of the differential equation with x(t) − v0 and then integrate over [0, s]. We obtain
s
s
−x (t), x(t) − v0 H dt ∈ ∂ϕ x(t) , x(t) − v0 dt H 0 0
s
(10.91) g(t), x(t) − v0 H dt. + 0
We have
s
−x (t), x(t) − v0
0
s
g(t), x(t) − v0
0
dt =
1 1 x0 − v0 2 − x(s) − v0 2 2 2
H
s
and
H
(10.92)
∂ϕ x(t) , x(t) − v0 dt ≥ 0 (because 0 ∈ ∂ϕ(v0 ) and ∂ϕ is
0
H
dt ≥ −
monotone) (10.93) s
g(t) x(t) − v0 dt. 0
Using (10.92) through (10.94) in (10.91), we obtain
(10.94)
724
10 Evolution Equations
s 1 1 g(t) x(t) − v0 dt, x(s) − v0 2 ≤ x0 − v0 2 + 2 2 0
b ⇒ x(s) − v0 ≤ x0 − v0 + g(t)dt for all s ∈ T.
(10.95)
0
Using (10.95) in (10.90), we obtain
b
ϕ x(t) dt ≤ x0 − v0 + 0
b
2 g(t)dt .
We use (10.96) in (10.89) and have
b
b
tx (t)2 dt ≤ tg(t)2 dt + 2 x0 − v0 + 0
(10.96)
0
0
b
2 g(t)dt .
(10.97)
0
Now suppose that x0 ∈ D(∂ϕ) = dom ϕ and g ∈ L2 (T, H). Then we can find {xn 0 }n≥1 ⊆ D(∂ϕ)
and
{gn }n≥1 ⊆ L2 (T, H)
such that xn in H and gn −→ g in L2 (T, H) as n → ∞. 0 −→ x0
Let xn ∈ W 1,∞ (0, b), H , n ≥ 1, be the unique strong solution of the Cauchy n problem with data (x0 , gn ). From (10.82), we see that {xn }n≥1 ⊆ C(T, H) is Cauchy. So we can find x ∈ C(T, H) such that xn −→ x
in C(T, H) as n → ∞.
So we have that xn −→ x in the sense of H-valued distributions on (0, b). Because 2 (10.97) is true for {xn , xn 0 , gn }n≥1 , we deduce that t −→ tx (t) belongs in L (T, H) and √ w √ t xn −→ t x in L2 (T, H) as n → ∞. In particular then, for any 0 < δ < b, we have xn −→ x
in L2 ([δ, b], H) as n → ∞.
If A is the realization of A in L2 ([δ, b], H), then A is maximal monotone and (xn , −xn − gn ) ∈ Gr A
for all n ≥ 1.
From Proposition 3.2.7 it follows that (x, −x − g) ∈ Gr A, )
−x (t) ∈ ∂ϕ x(t) + g(t) x(0) = x0
a.e. on T (because δ ≥ 0 is arbitrary),
* .
Next, let Jϕ : L1 (T, H) −→ R = R ∪ {+∞} be the integral functional defined by b
ϕ u(t) dt if ϕ u(·) ∈ L1 (T, H) 0 Jϕ (u) = . +∞ otherwise
10.3 Nonlinear Evolution Equations
725
We have that Jϕ ∈ Γ0 L1 (T, H) . Also because of (10.96) we see that we can find c0 > 0 such that for n ≥ 1 ⇒ Jϕ (xn ) ≤ c0 , ⇒ Jϕ (x) ≤ lim inf Jϕ (xn ) ≤ c0 , n→∞
⇒ x ∈ dom Jϕ .
So using Lemma 10.3.2, we have that t −→ ϕ x(t) belongs in L1 (T, H) and it is absolutely continuous on [δ, b] for all 0 < δ < b. Finally suppose that x0 ∈ dom ϕ. We have d
ϕ x(t) ≤ g(t) x (t) a.e. on T, dt d
1 1 ⇒ x (t)2 + ϕ x(t) ≤ g(t)2 a.e. on T, 2 dt 2
1 t ⇒ t −→ ϕ x(t) − g(τ )2 dτ is decreasing on T. 2 0 x (t)2 +
Because x0 ∈ dom ϕ, we obtain
1 t ϕ x(t) ≤ ϕ(x0 ) + g(τ )2 dτ 2 0
for all t ∈ [0, b).
(10.98) (10.99)
(10.100)
For δ > 0, from (10.98), we obtain
1 b 1 b x (t)2 dt ≤ ϕ x(δ) + g(t)2 dt 2 δ 2 δ
1 b ≤ ϕ(x0 ) + g(t)2 dt (see (10.98)), 2 0 ⇒ x ∈ L2 (T ).
∞ Finally from (10.100),
we have ϕ x(·) ∈ L (T ) and from Lemma 10.3.2 we conclude that t −→ ϕ x(t) is absolutely continuous.
REMARK 10.3.8 If g ∈ W 1,1 (0, b), H , then x is right differentiable at every
0 + point in (0, b), x(t) ∈ D(A) for every t ∈ (0, b] and − ddtx (t) ∈ ∂ϕ x(t) + g(t) for all t ∈ (0, b].
Subdifferential evolution inclusions exhibit a remarkable smoothing effect. In what follows by {S(t)}t≥0 we denote the semigroup of nonlinear contractions S(t) : D(A) −→ D(A), t ≥ 0, generated by A (see Theorem 3.2.93). THEOREM 10.3.9 If A = ∂ϕ : D(∂ϕ) ⊆ H −→ 2H , ϕ ∈ Γ0 (H), and x0 ∈ D(A), then (a) Sx0 ∈ D(A) for all t > 0 (smoothing effect). +
(b) ddt S(t)x0 = A0 S(t)x0 ≤ A0 (x0 ) + 1t x0 − v0 for all t > 0 and all v0 ∈ D(∂ϕ).
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10 Evolution Equations
PROOF: (a) As in the proof of Theorem 10.3.7, let (v0 , w0 ) ∈ Gr ∂ϕ and define the function ϕ0 (x) = ϕ(x) − ϕ(v0 ) − (w0 , x − v0 )H , x ∈ H. We have ϕ0 ≥ 0 (recall the definition of the subdifferential) and 0 = ϕ0 (v0 ) = min ϕ0 . H
We set x(t) = S(t)x0 , t ≥ 0. Then x ∈ C(T, H) is a strong solution of the Cauchy problem
* ) −x (t) ∈ ∂ϕ0 x(t) + w0 , . (10.101) x(0) = x0 Then by virtue of Remark 10.3.8, we have x(t) = S(t)x0 ∈ D(A) for all t > 0. (b) Also, again from Remark 10.3.8, we have 0
d+ x for all t > 0 (10.102) (t) ∈ −∂ϕ x(t) − w0 dt
t
t t d+ x 2
d+ x ⇒ s ϕ0 x(s) ds − s w0 , (s) ds ≤ (s) ds dt dt H 0 0 0
t
t
ϕ0 x(s) ds − t w0 , x(t) H + w0 , x(s) ds = 0
0
(10.103) (integration by parts).
Because −w0 − x (t) ∈ ∂ϕ0 x(t) a.e on R+ , we have
ϕ0 x(t) ≤ (−w0 − x (t), x(t) − v0 )H a.e. on T,
t
t
ϕ0 x(s) ds ≤ (−w0 − x (s), x(s) − v0 )H ds ⇒ 0
0
=
1 1 x0 − v0 2 − x(t) − v0 2 + t(v0 , w0 )H 2 2
t
− (10.104) w0 , x(s) H ds, t ≥ 0. 0
Using (10.104) in (10.103), we obtain
t d+ x 2 1 1 s (s) ds ≤ x0 − v0 2 + t(w0 , v0 − x(t))H − v0 − x(t)2 dt 2 2 0 1
2 2 2 ≤ x0 − v0 + t w0 , t ≥ 0. (10.105) 2 Note that d+ for all t > 0 and all h > 0, x(t + h) − x(t)2 ≤ 0 dt x(t + h) − x(t) x(s + h) − x(s) ⇒ ≤ for all 0 < s ≤ t, h > 0, h h + d x ⇒ t −→ (t) is decreasing on (0, +∞). dt
10.3 Nonlinear Evolution Equations
727
From this fact and (10.105), we obtain d+ x 2 1 1 2 2 x − v + w , t > 0, (t) ≤ 0 0 0 dt 2 t2 + d 1 ⇒ S(t)x0 ≤ x0 − v0 + w0 , t > 0. dt t Let us present an application of Theorem 10.3.7. So let β : D(β) ⊆ R −→ 2R \{∅} be a maximal monotone function. We know that β = ∂j with j ∈ Γ0 (R). Also let Z ⊆ RN be a bounded domain with a C 2 -boundary ∂Z. We consider the following heat equation with Neumann boundary condition. ⎧ ∂x ⎫ ⎨ − ∂t − x(t, z) = g(t, z) a.e. on (0, b) × Z, ⎬ . (10.106)
⎩ − ∂x ∈ β x(t, z) , x(0, z) = x0 (z) ⎭ ∂n (0,b)×∂Z 2 PROPOSITION 10.3.10 If g ∈ L2 (T ×Z) and x0 ∈ L (Z), then problem (10.106) has a unique solution x ∈ C T, L2 (Z) such that
√
tx ∈ L2 T, H 2 (Z) ,
√ ∂x t ∈ L2 (T × Z). ∂t
Moreover, if x0 ∈ H 1 (Z) and j x0 (·) ∈ L1 (Z), then
x ∈ L2 T, H 2 (Z) ,
∂x ∈ L2 (T × Z). ∂t
PROOF: Let H = L2 (Z) and let ϕ : H = L2 (Z) −→ R = R ∪ {+∞} be defined by ⎧
⎨ 12 Dx(z)2 dz+ j x(z) dσ if x ∈ H 1 (Z) and j x(·) ∈ L1 (∂Z) Z ∂Z ϕ(x) = ⎩ +∞ otherwise. Let A : L2 (Z) −→ L2 (Z) be the operator defined by Ax = − x
∂x for all x ∈ D(A) = x ∈ H 2 (Z) : − (z) ∈ β x(z) ∂n a.e. on ∂Z .
Evidently this is a nonlinear operator and the definition of D(A) makes sense because x ∈ H 2 (Z) implies that x∂Z ∈ H 3/2 (∂Z) and (∂x/∂n) ∈ H 1/2 (∂Z) ⊆ L2 (∂Z). Using Green’s identity, we obtain
∂x − x(y − x)dz = (Dx, Dy − Dx)RN dz − (y − x)dσ Z Z ∂Z ∂n ≤ ϕ(y) − ϕ(x)
for all x ∈ D(A), y ∈ L2 (Z),
⇒ A ⊆ ∂ϕ. To show that A = ∂ϕ, it suffices to show that A is maximal monotone in L2 (Z) and this follows if we show that R(I + A) = L2 (Z). To this end, we consider the following elliptic (stationary) problem.
728
10 Evolution Equations ⎧ ⎨ − xλ (z) + xλ (z) = h(z)
⎫ a.e. on Z, ⎬
, (10.107)
⎭ λ − ∂x (z) = β (z) a.e. on ∂Z x λ λ ∂n
2 with h ∈ L (Z) and βλ = (1/λ) 1 − (1 + λβ)−1 (the Yosida approximation of β). Consider the operator K : L2 (∂Z) −→ L2 (∂Z), where K(x) = y ∂Z with y being the unique solution of ⎫ ⎧ ⎬ ⎨ − y(z) + y(z) = h(z) a.e. on Z, . (10.108) ⎭ ⎩ ∂y (z) = (1 + λβ)−1 x(z) a.e. on ∂Z y(z) + λ ∂n ⎩
Problem (10.108) has a unique solution y ∈ H 2 (Z) and using Green’s identity we can have K(x) − K(u)L2 (∂Z) ≤ ϑx − uL2 (∂Z) 0 < ϑ < 1,
for all x, u ∈ L2 (Z).
So by Banach’s fixed point theorem we can find y ∈ L2 (∂Z) such that K(y) = y. Then the corresponding solution xλ ∈ H 2 (Z) of (10.108) also solves (10.107). Moreover, from Brezis [100], we know that
xλ H 2 (Z) ≤ c 1 + hL2 (Z) for some c > 0 and all λ > 0.
∂x Recalling that the trace map x −→ x∂Z , ∂n is continuous from H 2 ((Z)) ∂Z ∂xλ 3/2 1/2 into H (∂Z) × H (∂Z), we conclude that xλ , ∂n λ>0 ⊆ L2 (∂Z) × L2 (∂Z) is bounded. Also we know that H 2 (Z) is embedded compactly in L2 (Z). Hence we may assume that w
xλ −→ x
in H 2 (Z), xλ −→ x
Moreover, because βλ xλ (·)
in L2 (Z), λ>0
∂xλ w ∂x −→ ∂n ∂n
in L2 (∂Z)
as λ ↓ 0.
⊆ L2 (Z) is bounded, we have
(1 + λβ)−1 xλ −→ x
in L2 (Z) as λ ↓ 0.
(10.109)
Therefore, in the limit as λ ↓ 0, we obtain − x(z) + x(z) = h(z)
a.e. on Z.
2
Also, if β : D(β) ⊆ L2 (∂Z) −→ 2L (∂Z) is defined by
β(x) = y ∈ L2 (∂Z) : y(z) ∈ β x(z) a.e. on ∂Z ,
x ∈ L2 (∂Z)
(the realization (lifting) of β on L2 (∂Z)), then β is maximal monotone and
βλ (x)(z) = βλ x(z) a.e. on ∂Z.
Because βλ (x) ∈ β (I + λβ)−1 x , we obtain w
βλ (xλ ) −→
∂x ∂n
in L2 (∂Z)
as λ ↓ 0
(see (10.109)).
10.3 Nonlinear Evolution Equations
729
Because β is maximal monotone, we deduce that ∂x ∈ β(x). ∂n Hence we conclude that A is maximal monotone and so A = ∂ϕ. Now we equivalently rewrite (10.106) as the following nonlinear evolution inclusion
* ) −x (t) ∈ ∂ϕ x(t) + g(t) a.e. on T, . (10.110) x(0) = x0 with g(t) = g(t, ·) ∈ L2 (Z) for all t ∈ T . Then the conclusion of Proposition 10.3.10 follows from Theorem 10.3.7 applied to the Cauchy problem (10.110). Now we turn our attention to evolution inclusions driven by operators of monotone type defined in the framework of an evolution triple. So let T = [0, b] and let (X, H, X ∗ ) be an evolution triple of spaces (see Definition 10.1.21). We consider the following Cauchy problem,
) * −x (t) + A t, x(t) = f t, x(t) a.e. on T, . (10.111) x(0) = x0 The hypotheses on the data of (10.111) are the following. H(A): A : T ×X −→ X ∗ is a map such that (i) For every x ∈ X, t −→ A(t, x) is measurable. (ii) For almost all t ∈ T , x −→ A(t, x) is hemicontinuous monotone. (iii) For almost all t ∈ T and all x ∈ X, we have A(t, x)∗ ≤ α(t) + cxp−1
with 2 ≤ p < ∞, α ∈ Lp (T ),
1 p
+
1 p
= 1, c > 0.
(iv) For almost all t ∈ T and all x ∈ X, we have A(t, x), x ≥ c1 xp
with c1 > 0.
REMARK 10.3.11 Hypothesis H(A)(ii) implies that for almost all t ∈ T, A(t, ·) is maximal monotone. H(f ): f : T ×H −→ H is a function such that (i) For all x ∈ H, t −→ f (t, x) is measurable. (ii) For almost all t ∈ T , x −→ f (t, x) is sequentially continuous from H into Hw (by Hw we denote the pivot Hilbert space H furnished with the weak topology).
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10 Evolution Equations (iii) For almost all t ∈ T and all x ∈ H, we have f (t, x) ≤ α2 (t) + c2 |x|2/p with α2 , c2 ∈ Lp (T ), p1 + p1 = 1 .
By a solution of (10.111), we understand a function x ∈ Wp (0, b) (see Definition 10.1.23) that satisfies (10.111). In what follows by S(x0 ) we denote the solution set of (10.111). Then S(x0 ) ⊆ Wp (0, b) ⊆ C(T, H) (see Theorem 10.1.25). THEOREM 10.3.12 If hypotheses H(A), H(f ) hold, x0 ∈ H, and X is embedded compactly into H, then S(x0 ) is nonempty, weakly compact in Wp (0, b) and compact in C(T, H). PROOF: The proof of the existence is based on the Galerkin method. Because X is separable reflexive, we can find a basis {vk }k≥1 for X (hence for H and X ∗ too). We set Hn = span{vk }n k=1 ,
n≥1
and endow Hn with the inner product inherited from H. By pn : H −→ Hn we denote the orthogonal projection on Hn . Also let Xn be the finite dimensional space Hn equipped with the X-norm (i.e., we view Hn as a subspace of X rather than of H). Then the dual Xn∗ of Xn is the space Hn furnished with the X ∗ -norm. For every n ≥ 1, we consider An : T × Xn −→ Xn∗ to be restriction (Galerkin approximation) of A(t, ·) on Xn . So we have An (t, x) = y
for x ∈ Xn
with y ∈ Xn∗ satisfying A(t, x), u = y, u ,
for all u ∈ Xn ; that is, An (t, x), uXn = A(t, x), u for all x, u ∈ Xn . Here by ·, ·Xn , we denote the duality brackets for the pair (Xn∗ , Xn ). Clearly for every x ∈ Xn , t −→ An (t, x) is measurable and for almost all t ∈ T, x −→ An (t, x) is continuous. Also we set fn (t, x) = pn f (t, x) for (t, x) ∈ T × Hn . We see that fn is a Carath´eodory function and |fn (t, x)| = |pn f (t, x)| ≤ pn L |f (t, x)| ≤ α2 (t) + c2 (t)|x|2/q
for a.a. t ∈ T, all x ∈ H.
We consider the following finite-dimensional approximation (Galerkin approximation), of problem (10.111).
) * xn (t) + An t, xn (t) = f t, xn (t) a.e. on T, . (10.112) n xn (0) = pn x0 = x0 ∈ Hn From Carath´eodory’s existence theorem, we know that problem (10.112) has at least one solution xn ∈ Wp (0, b). Next we derive some a priori bounds for the
10.3 Nonlinear Evolution Equations
731
sequence {xn }n≥1 . For this purpose we take the duality brackets with xn (t) and obtain
xn (t), xn (t) + An t, xn (t) , xn (t) = pn f t, xn (t) , xn (t) a.e. on T = [0, b]
d ⇒ |xn (t)|2 + 2c1 xn (t)2 ≤ 2|pn f t, xn (t) ||xn (t)|, dt (see Corollary 10.1.26 and H(A)(iv)) εp
p 1 p ≤ 2ξ |p f t, x (t) | + (t) x n n n p εp p (by Young’s inequality) with ε > 0 and ξ > 0 such that | · | ≤ ξ · . We choose 1/p ε = c1εp >0 and obtain
p d a.e. on T with c3 > 0 |xn (t)|2 ≤ c3 pn f t, xn (t) dt p ≤ c3 α2 (t) + c2 |xn (t)|2/p ≤ 2p
−1
c3 α2 (t)p + 2p
−1
c3 cp2 |xn (t)|2
a.e. on T,
(see H(f )(iii)). We integrate this inequality over [0, t] and because |xn 0 | = |pn x0 | ≤ pn L |x0 | = |x0 |, we obtain
t |xn (t)|2 ≤ |x0 |2 + 2p −1 cp3 α2 pp + 2p −1 cp3 c2 (s)p |xn (s)|2 ds. (10.113) 0
From (10.113) using Gronwall’s inequality, we obtain |xn (t)| ≤ M1
for all n ≥ 1
and all t ∈ T = [0, b].
(10.114)
Recall that
d |xn (t)|2 + 2c1 xn (t)2 ≤ 2pn f t, xn (t) |xn (t)| a.e. on T, dt
≤ 2M1 pn f t, xn (t) a.e. on T, (see (10.114)),
b
xn (t)dt ≤ |x0 |2 + 2M1 ⇒ 2c1 0
b
|f t, xn (t) |dt
0
b
≤ |x0 | + 2M1 2
0
⇒ xn L2 (T,X) ≤ M2
for some M2 > 0
Also we know that
xn (t) = −An t, xn (t) + pn f t, xn (t)
2/p
α2 (t) + c2 (t)M1
dt
and all n ≥ 1.
a.e. on T
for all n ≥ 1.
(10.115)
(10.116)
From (10.116) and hypotheses H(A)(iii) and H(f )(iii), we obtain that xn L2 (T,X ∗ ) ≤ M3
for some M3 > 0 and all n ≥ 1.
(10.117)
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10 Evolution Equations
Motivated from (10.115) and (10.117), we introduce the set C = x ∈ Wp (0, b) : xLp (T,X) ≤ M2 and x Lp (T,X ∗ ) ≤ M3 . Evidently C is bounded, closed, and convex (hence w-compact, convex) in Wp (0, b). Invoking Theorem 10.1.29, we deduce that C is compact in Lp (T, H). Let Sn be the solution set of the Galerkin approximation problem (10.113). Note that Sn ⊆ C for all n ≥ 1. By passing to a subsequence if necessary, we may assume that w
xn −→ x
in Wp (0, b), xn −→ x
in Lp (T, H)
and xn (t) −→ x(t) a.e. on T in H.
Also, if Nn (xn )(·) = fn ·, xn (t) = pn fn ·, xn (t) (the Nemitsky operator), then w
Nn (xn ) −→ g
in Lp (T, H)
as n → ∞.
For every D ⊆ T measurable and every h ∈ H, we have
pn f t, xn (t) , h dt Nn (xn )(t), h dt = D D
= f t, xn (t) , pn h dt, D
⇒ g(t), h dt = f t, x(t) , h dt (because pn h −→ h in H). D
D
Because D ⊆ T and h ∈ H, were arbitrary, we infer that
g(t) = f t, x(t) a.e. on T.
(10.118)
Let An : Lp (T, Xn ) −→ Lp (T, Xn ), n ≥ 1, be the Nemitsky operator correspond
ing to An (t, x); that is, An (x)(·) = An ·, x(·) for all x ∈ Lp (T, Xn ). Also let (·, ·)
p denote the duality brackets for the pair L (T, X ∗ ), Lp (T, X) ; that is, (u∗ , x) =
b
u∗ (t), x(t) dt
for all x ∈ Lp (T, X), u∗ ∈ Lp (T, X ∗ ).
0
Then for every y ∈ Lp (T, X), we have
(xn , y) + (An (xn ), y) = (Nn (xn ), y) .
(10.119)
Because of hypothesis H(A)(iii) and (10.115), we may assume that w
An (xn ) −→ v
in Lp (T, X ∗ )
as n → ∞.
So, if we pass to the limit as n → ∞ in (10.119), we obtain
for all y ∈ Lp (T, X), (x , y) + (v, y) = (N (x), y)
with N (x)(·) = f ·, x(·) , (see (10.118)),
⇒ x (t) + v(t) = f t, x(t) a.e. on T.
(10.120)
10.3 Nonlinear Evolution Equations
733
From Theorem 10.1.25, we know that Wp (0, b) is embedded continuously in w w C(T, H). So xn (0) = xn 0 −→ x(0) = x0 and xn (b) −→ x(b) in H. Also
(Nn , (xn ), xn ) − (xn , xn ) = (An (xn ), xn )
1
1 2 ⇒ (Nn (xn ), xn ) − |xn (b)|2 + |xn 0 | = (An (xn ), xn ) 2 2 (see Corollary 10.1.26),
1 1 ⇒ (N (x), x) − |xn (b)|2 + |x0 |2 ≥ lim sup (An (xn ), xn ) . 2 2 n→∞ (10.121) From (10.120), with y = x and (10.121), we have
lim sup (An (xn ), xn ) ≤ (v, x) , n→∞
⇒ lim sup (An (xn ), xn − x) ≤ 0. n→∞
(10.122)
But it is easy to see that A is hemicontinuous and monotone, hence maximal monotone. So it is generalized pseudomonotone and from (10.122) we have A(x) = v; w that is, An (xn ) −→ A(x) in Lp (T, X ∗ ). Thus finally x + A(x) = N (x), x(0) = x0 ,
⇒ x (t) + A t, x(t) = f t, x(t) a.e. on T, x(0) = x0 , ⇒ x ∈ S(x0 ) = ∅. Next we show the topological properties of S(x0 ) ⊆ Wp (0, b). From (10.114), (10.115), and (10.117), we know that we can find M = max{M1 , M2 , M3 } > 0 such that xLp (T,X) , x Lp (T,X ∗ ) , xC(T,X) ≤ M for all x ∈ S(x0 ).
If we set ϑ(t) = α2 (t) + c2 (t)M 2/p , then ϑ ∈ Lp (T ) and we may assume that |f (t, x)| ≤ ϑ(t) for a.a. t ∈ T, all x ∈ H.
Otherwise we replace f (t, x) by f t, rM (x) , with rM : H −→ H being the M -radical retraction; that is, x if |x| ≤ M rM (x) = for all x ∈ H. Mx if |x| > M |x| Then we introduce the set K = h ∈ Lp (T, H) : |h(t)| ≤ ϑ(t)
a.e. on T .
In what follows we consider K equipped with the relatively weak Lp (T, H)topology. Then K becomes a compact metrizable space. For every h ∈ K we consider the auxiliary Cauchy problem
) * −x (t) + A t, x(t) = h(t) a.e. on T, . x(0) = x0
734
10 Evolution Equations
This problem has a unique solution x = η(h) ∈ Wp (0, b). We consider the solution map η : Lp (T, H) −→ C(T, H). Claim: η(K) is compact in C(T, H). Let {xn }n≥1 ⊆ η(K). Then xn = η(hn ) with hn ∈ K, n ≥ 1. From the a priori estimation of the first part of the proof, we know that {xn }n≥1 ⊆ Wp (0, b) is bounded. So we may assume that w
xn −→ x
in Wp (0, b),
xn (t) −→ x(t)
xn −→ x
in Lp (T, H)
for all t ∈ T \ N0 , |N0 | = 0
(see Theorem 10.1.29) in H
p
w
hn −→ h in L (T, H) as n → ∞. The sequence xn (·), xn (·) − x(·)}n≥1 is uniformly integrable. Therefore given ε > 0, we can find t ∈ T \ N0 such that and
b
| xn (s), xn (s) − x(s) |ds < ε
for all n ≥ 1.
(10.123)
t
For t ∈ T , let (·, ·) t denote the duality brackets for the pair Lp ([0, t], X ∗ ), Lp ([0, t], X) . Using Corollary 10.1.26, we have
1 (xn , xn − x) t = |xn (t) − x(t)|2 + (x , xn − x) t . 2
(10.124)
w
2 If
t ∈ T \N0 , then |xn (t) − x(t)| −→ 0. Also because xn −→ x in Wp (0, b), we have (x , xn − x) t −→ 0 as n → ∞. So from (10.124), we infer that
(xn , xn − x)
t
−→ 0
as n → ∞.
We have
b
(xn , xn − x) = (xn , xn − x) t + xn (s), xn (s) − x(s) ds t
≥ (xn , xn − x) t − ε (see (10.123)),
(10.125) ⇒ lim inf (xn , xn − x) ≥ 0 (because ε > 0 was arbitrary).
n→∞
In a similar fashion, we show that
lim sup (xn , xn − x) ≤ 0.
(10.126)
Therefore, from (10.125) and (10.126) we conclude that
(xn , xn − x) −→ 0 as n → ∞.
(10.127)
n→∞
Note that
(xn , xn − x) + (A(xn ), xn − x) =
⇒ lim (A(xn ), xn − x) = 0. n→∞
hn (t), xn (t) − x(t) dt,
b
0
(10.128)
10.3 Nonlinear Evolution Equations
735
As before from (10.128), it follows that
in Lp (T, X ∗ ).
w
A(xn ) −→ A(x) So, in the limit as n → ∞, we have
x + A(x) = h, h ∈ K, x(0) = x0 , ⇒ x ∈ η(K). We show that xn −→ x in C(T, H). We have
1 |xn (t) − x(t)|2 = (hn − h, xn − x) − (A(xn ) − A(x), xn − x) t , 2
b
1 ⇒ |xn (t) − x(t)|2 ≤ | hn (s) − h(s), xn (s) − x(s) |ds 2 0
b
+ | A(s), xn (s) − x(s) |dt + (A(x), xn − x) t . 0
(10.129) Note that
b
| hn (s) − h(s), xn (s) − x(s) |ds −→ 0
as n → ∞.
(10.130)
0
Let βn (t) = A t, xn (t), xn (t) − x(t) . If t ∈ T \N 0 , N0 ⊆ N 0 , |N 0 | = 0, then (10.131) βn (t) ≥ c1 xn (t)p α(t) + cxn (t)p−1 x(t). Let D = t ∈ T : lim inf βn (t) < 0 . This is a Lebesgue measurable subset of T . n→∞
Suppose |D| > 0. For all t ∈ C ∩ (T \ N 0 ) = ∅ we have that {xn (t)}n≥1 ⊆ X is w bounded (see (10.131)). Because xn −→ x in Wp (0, b) and Wp (0, b) is embedded w w continuously in C(T, H), we have xn −→ x in C(T, H) and so xn (t) −→ x(t) in H for all t ∈ T . Therefore, exploiting the reflexivity and separability of X, we have w xn (t) −→ x(t) in X as n → ∞ for all t ∈ T . Then because of the monotonicity of A(t, ·), we have
lim inf A t, xn (t) , xn (t) − x(t) ≥ lim A t, x(t) , xn (t) − x(t) = 0 n→∞
n→0
for all t ∈ D ∩ (T \ N0 ), a contradiction to the definition of D. Therefore |D| = 0 and we have 0 ≤ lim inf βn (t) a.e. on T. n→∞
So from Fatou’s lemma, we obtain
b lim inf βn (t)dt ≤ lim inf 0≤ 0
n→∞
n→∞
b
βn (t)dt = lim (A(xn ), xn − x) = 0,
0
(see (10.128)),
b βn (t)dt −→ 0. ⇒ 0
From (10.131), we see that there exists {γn }n≥1 uniformly integrable sequence such that
736
10 Evolution Equations γn (t) ≤ βn (t)
a.e. on T,
for all n ≥ 1,
⇒ 0 ≤ βn− (t) ≤ γn− (t) a.e. on T,
b βn− (t)dt −→ 0 as n → ∞. ⇒
for all n ≥ 1
0
Therefore, we have
b
b
|βn (t)|dt =
b
βn (t)dt + 2
0
0 1
βn− (t)dt −→ 0,
0
⇒ βn −→ 0 in L (T ),
b
A t, xn (t), xn (t) , xn (t) − x(t) −→ 0. ⇒
(10.132)
0
Finally let ξn (t) = (A(x), xn − x) t . Then ξn ∈ C(T ) and let tn ∈ T such that
tn
ξn (tn ) = max ξn = T
A s, x(s) , xn (s) − x(s) ds
0
= 0
b
χ[0,tn ] A s, x(s) , xn (s) − x(s) ds
= (χ[0,tn ] A(x), xn − x) −→ 0,
⇒ sup (A(x), xn − x) t −→ 0.
(10.133)
T
Returning to (10.129) and using (10.130), (10.132), and (10.133), we obtain xn −→ x
in C(T, H)
as n → ∞
and
x ∈ η(K)
⇒ η(K) is compact in C(T, H). This proves the claim. Note that S(x0 ) ⊆ η(K). Also from the previous part of the proof, we have that S(x0 ) ⊆ Wp (0, b) is weakly closed. So S(x0 ) ⊆ Wp (0, b) is weakly compact in Wp (0, b) and compact in C(T, H).
Let η : Lp (T, H) × H −→ C(T, H) be the map that for each pair (h, x0 ) ∈ p L (T, H)×H assigns the unique solution of
* ) −x (t) + A t, x(t) = h(t) a.e. on T, . (10.134) x(0) = x0
From the proof of Theorem 10.3.12, we have as a byproduct the following continuous dependence result. PROPOSITION 10.3.13 If hypotheses H(A) hold and X is embedded compactly into H, then η : Lp (T, H)w ×H −→ C(T, H) is sequentially continuous. Recall that Sn ⊆ Wp (0, b) is the solution set of the Galerkin approximation (10.109). Another byproduct of the proof of Theorem 10.3.12, is the following result.
10.3 Nonlinear Evolution Equations
737
PROPOSITION 10.3.14 If hypotheses H(A), H(f ) hold, x0 ∈ H, and X is embedded compactly in H, then lim sup Sn ⊆ S(x0 ) and h∗ Sn , S(x0 ) −→ 0 in C(T, H). n→∞
In particular, if S(x0 ) is a singleton (this is the case if f (t, ·) is Lipschitz continuous) and xn ∈ Wp (0, b) is the unique solution of the Galerkin approximation (10.109), then xn −→ x in C(T, H) with x ∈ Wp (0, b) the unique solution of (10.108). We can generalize hypotheses H(A) and still have an existence theorem for problem (10.74). In concrete parabolic problems, the new hypotheses on A(t, x) permit the differential operator to have lower-order terms of nonmonotone nature. In this extended existence theorem, we need the following notion. DEFINITION 10.3.15 Let X be a reflexive Banach space, L : D(L) ⊆ X −→ X ∗ ∗ is a linear maximal monotone operator, and A : X −→ 2X . We say that A is L-pseudomonotone if the following conditions hold. (a) For every x ∈ X, A(x) ⊆ X ∗ is nonempty, weakly compact, and convex. (b) A is usc from every finite-dimensional subspace of X into X ∗ furnished with the weak topology. w w (c) If {xn }n≥1 ⊆ D(L), xn −→ x ∈ D(L) in X, Lxn −→ Lx in X ∗ , x∗n ∈ A(xn ) for w all n ≥ 1, x∗n −→ x∗ in X ∗ and lim sup x∗n , xn − x ≤ 0, then x∗ ∈ A(x) and n→∞
x∗n , xn −→ x∗ , x.
The next result is the analogue of Theorem 3.2.60 for L-pseudomonotone operators. Its proof can be found in Papageorgiou–Papalini–Renzacci [482]. THEOREM 10.3.16 If X is a reflexive Banach space that is strictly convex, L : ∗ D(L) ⊆ X −→ X ∗ is a linear maximal monotone operator, and A : X −→ 2X is bounded, L-pseudomonotone and coercive, then L + A is surjective; that is, R(L + A) = X ∗ . For the extended version of Theorem 10.3.12, we employ the following conditions on the map A(t, x). Here X is a separable reflexive Banach space and X ∗ its topological dual. H(A) : A : T ×X −→ X ∗ is a map such that (i) For all x ∈ X, t −→ A(t, x) is measurable. (ii) For almost all t ∈ T , x −→ A(t, x) is pseudomonotone. (iii) For almost all t ∈ T and all x ∈ X, we have A(t, x)∗ ≤ α(t) + cxp−1 , with 2 ≤ p < ∞, α ∈ Lp (T )+ , p1 + p1 = 1 and c > 0. (iv) For almost all t ∈ T and all x ∈ X, we have A(t, x), x ≥ c1 xp − ϑ(t), with c1 > 0, ϑ ∈ L1 (T )+ .
738
10 Evolution Equations
REMARK 10.3.17 Note that the pseudomonotonicity of A(t, ·) for almost all t ∈ T (see H(A)(ii)) and the p-growth condition (see H(A)(iii)), imply that for almost all t ∈ T, A(t, ·) is demicontinuous. Using an argument similar to that in the last part of the proof of Theorem 10.3.12 with the functions {βn }n≥1 and the set D, we can show the following “lifting” theorem (see also Hu–Papageorgiou [316, p. 41]).
THEOREM 10.3.18 If hypotheses H(A) hold and A : Lp (T, X) −→ Lp (T, X ∗ ) is defined by
A(x)(·) = A ·, x(·) w
(the Nemitsky operator corresponding to A), then A is demicontinuous and if xn −→ x in Lp (T, X) and
lim sup (A(xn ), xn − x) ≤ 0, n→∞
we have A(xn ) −→ A(x) in Lp (T, X ∗ ) and (A(xn ), xn − x) −→ (A(x), x) as n → ∞. w
H(f ) : f : T ×H −→ H is a function such that (i) For all x ∈ H, t −→ f (t, x) is measurable. (ii) For almost all t ∈ T , x −→ f (t, x) is sequentially continuous from H into Hw . (iii) For almost all t ∈ T and all x ∈ H, we have |f (t, x)| ≤ α2 (t) + c2 |x|2/q
with α2 ∈ Lp (T )+ , c2 > 0 if p > 2 and α2 ∈ Lp (T )+ , c2 > 0, c2 β 2 < c1 /2, where β > 0 is such that | · | ≤ β · when p = 2. Using Theorems 10.3.16 and 10.3.18, we can have the following extension of Theorem 10.3.12. For the proof see Hu–Papageorgiou [316, p. 42]. THEOREM 10.3.19 If in the evolution triple (X, H, X ∗ ) the embedding of X into H is also compact, hypotheses H(A) and H(f ) hold, and x0 ∈ H, then the solution set S(x0 ) of problem (10.74) is nonempty, weakly compact in Wp (0, b), and compact in C(T, H). Let us see an application to a nonlinear parabolic problem. So let T = [0, b] and let Z ⊆ RN be a bounded domain with a C 2 -boundary ∂Z. We consider the following nonlinear parabolic initial boundary value problem.
⎧ ∂x ⎫ p−2 Dx(t, x) = f0 t, z, x(t, z) in T × Z, ⎬ ⎨ ∂t − div α(t, z)Dx(t, z) x(0, z) = x0 (z) a.e. on Z, . ⎩ ⎭ x T ×∂Z = 0, 2 ≤ p < ∞ (10.135) We impose the following conditions on the data of (10.135):
10.3 Nonlinear Evolution Equations
739
H(α): α : T ×Z −→ R is a measurable function such that 0 < γ1 ≤ α(t, z) ≤ γ2
for almost all (t, z) ∈ T × Z.
H(f0 ): f0 : T ×Z ×R −→ R is a function such that (i) For all x ∈ R, (t, z) −→ f0 (t, z, x) is measurable. (ii) For almost all (t, z) ∈ T × Z, x −→ f0 (t, z, x) is continuous.
(iii) |f0 (t, z, x)| ≤ η1 (t, z) + η2 |x|2/p for a.a.
Lp T, L2 (Z) and η2 ∈ Lp T, L∞ (Z) .
(t, z) ∈ T × Z, with η1 ∈
2 PROPOSITION 10.3.20 If hypotheses H(α), H(f 0 ) hold2 and x0 ∈ L (Z), then
1,p 2 problem (10.135) has a solution x ∈ L T, W0 (Z) ∩C T, L (Z) such that ∂x/∂t ∈
L2 T, W −1,p (Z) and the solution set of the problem is compact in C T, L2 (Z) .
PROOF: We consider the evolution triple (X, H, X ∗ ), consisting of the following spaces. 1 1 X = W01,p (Z), H = L2 (Z), X ∗ = W −1,p (Z) + =1 . p p From the Sobolev embedding theorem, we know that X is embedded compactly into H (recall p ≥ 2). Consider the time-dependent Dirichlet form α : T × X × X −→ R defined by
α(t, z)Dxp−2 (Dx, Dy)RN dz. α(t, x, y) = Z
Using hypotheses H(α), we can easily check that |α(t, x, y)| ≤ c3 xp−1 y
for some c3 > 0.
So we can introduce a map A : T ×X −→ X ∗ defined by A(t, x), y = α(t, x, y). Then clearly we have. •
For all x ∈ X, t −→ A(t, x) is measurable (see Theorem 10.1.2).
•
For almost all t ∈ T , x −→ A(t, x) is demicontinuous, monotone.
•
For almost all t ∈ T and all x ∈ X, we have A(t, x)∗ ≤ cxp−1 .
•
For almost all t ∈ T and all x ∈ X, we have A(t, x), x ≥ c1 xp .
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10 Evolution Equations
Also let f : T ×H −→ H be the Nemitsky operator corresponding to the function f0 (t, ·, ·); that is,
f (t, x)(·) = f0 t, ·, x(·) . It is straightforward to check that f satisfies hypotheses H(f ). We equivalently rewrite (10.135) as the following nonlinear evolution equation.
) * x (t) + A t, x(t) = f t, x(t) a.e. on T, . x(0) = x0 ∈ H Invoking Theorem 10.3.12, we obtain all the conclusions of the proposition.
10.4 Second-Order Nonlinear Evolution Equations In this section we deal with second-order nonlinear evolution equations. We consider two classes of problems, both defined in the framework of evolution triples. In the first class, the evolution triple consists of Hilbert spaces, the evolution equation involves multivalued terms, and the analysis is conducted using the material from the first half of Section 10.3. As a particular case of this class, we obtain hyperbolic variational inequalities. The second class of problems is formulated on a general evolution triple, involves maximal monotone coercive operators, and the method of proof is based on the results of the second half of Section 10.3. So let T = [0, b] and let (X, H, X ∗ ) be an evolution triple consisting of Hilbert spaces. We are interested in the following second-order nonlinear evolution equation.
* ) x (t) + Ax(t) + B x (t) * g(t) a.e. on T, . (10.136) x(0) = x0 , x (0) = y0 The hypotheses on the data of (10.136) are the following. H(A): A ∈ L(X, X ∗ ) and Ax, x+c1 |x|2 ≥ c2 x2 for all x ∈ X with c1 ∈ R, c2 > 0. ∗
H(B): B : D(B) ⊆ X −→ 2X is maximal monotone.
THEOREM 10.4.1 If hypotheses H(A), H(B) hold, g ∈ W 1,1 (0, b), H , x0 ∈ X, y0 ∈ D(B), and
Ax0 + B(y0 ) ∩ H = ∅, then problem (10.136) admits a unique strong solution x(·) such that
x ∈ W 1,∞ (0, b), X ∩ W 2,∞ (0, b), H d+ x(t) dt
exists in X,
d+ x (t) dt
exists in H, both for all t ∈ [0, b) and
d+ x (t) + Ax(t) + B x (t) * g(t) dt
for all t ∈ [0, b).
PROOF: Let H0 = X ×H. This is a Hilbert space with inner product (u1 , u2 )0 = Ax1 , x2 + c1 (x1 , x2 ) + (y1 , y2 ) for all u1 = [x1 , y1 ], u2 = [x2 , y2 ] ∈ X ×H.
10.4 Second-Order Nonlinear Evolution Equations
741
Let E : D(E) ⊆ H0 −→ 2H0 be the nonlinear operator defined by
E(u) = E([x, y]) = − y, Ax + B(y) ∩ H + ξ[x, y]
for all [x, y] ∈ D(E) = [x, y] ∈ X × H : Ax + B(y) ∩ H = ∅ and with ξ = sup
c1 (x, y) : x ∈ X, y ∈ H, x + |y| = 0 . Ax, y + c1 |x|2 + |y|2
(10.137)
Then we can equivalently rewrite problem (10.136), as the following first-order evolution equation in the Hilbert space H0 = X ×H:
) * u (t) ∈ −E u(t) + ξu(t) + g(t) on T, , (10.138) u(0) = u0 where u(t) = [x(t), x (t)], g(t) = 0, g(t) and u0 = [x0 , y0 ]. It is easy to check that E is monotone in H0 . We claim that in fact E is maximal monotone. It suffices to show that R(I + E) = H0 . Let (v, h) ∈ X × H be arbitrary. Then the conclusion u + E(u) * (v, h) is actually equivalent to the following system ) * x + (−y + ξx) = v, , y + Ax + B(y) + ξy * h ) * (1 + ξ)x − y = v, ⇒ . (1 + ξ)y + Ax + B(y) * h
(10.139)
From the first equation in (10.139) we have x=
1 (v + y). 1+ξ
Using this in the second inclusion of system (10.139), we obtain (1 + ξ)y +
1 1 Ay + B(y) * − Av. 1+ξ 1+ξ
(10.140)
Note that y −→ (1+ξ)y+1/(1+ξ)Ay is continuous, monotone, and coercive from X into X ∗ . Then y(1 + ξ)y + 1/(1 + ξ)Ay + B(y) is maximal monotone, and coercive (see Theorem 3.2.33). Therefore by virtue of Corollary 3.2.28 it is also surjective. So the inclusion (10.140) has a unique solution y ∈ D(E). Hence [v, h] ∈ R(I + E). Therefore, E is maximal monotone and we can apply Theorem 10.3.6 to finish the proof of the theorem. If B = ∂ϕ with ϕ ∈ Γ0 (X), then we have an existence result for nonlinear hyperbolic variational inequalities (see Theorem 10.3.7). COROLLARY
10.4.2 If hypothesis H(A) holds, B = ∂ϕ with ϕ ∈ Γ0 (X), g ∈ W 1,1 (0, b), H , x0 ∈ X, y0 ∈ D(∂ϕ), and
Ax0 + B(y0 ) ∩ H = ∅,
742
10 Evolution Equations
then there exists a unique function x ∈ C(T, X) such that x ∈ C(T, H) ∩ L∞ (T, X), x (t) ∈ dom ϕ
for all t ∈ (0, b)
∞
x ∈ L (T, H)
−x (t) − Ax(t) + g(t), y − x (t) ≤ ϕ(y) − ϕ x(t) ,
for a.a. t ∈ T, all y ∈ X x(0) = x0 , x (0) = y0 . As an application consider the following hyperbolic initial-boundary value problem. Let T = [0, b] and Z ⊆ RN is a bounded domain with a C 2 -boundary ∂Z. ⎧ 2 ⎫ ∂ x ∂x ⎪ ⎨ ∂t2 − x(t, z) + β ∂t * g(t, z) a.e. on T × Z, ⎪ ⎬ ∂x . (10.141) (z), (z, 0) = y (z) a.e. on Z, x(z, 0) = x 0 0 ∂t ⎪ ⎪ ⎩ ⎭ xT ×∂Z = 0 function (hence PROPOSITION 10.4.3 If β : R −→ 2R is a maximal monotone β = ∂j with j ∈ Γ0 (R)) with 0 ∈ D(β), g ∈ W 1,2 (0, b), L2 (Z) , x0 ∈ H01 (Z) ∩ H 2 (Z), and y0 ∈ H01 (Z) with y0 (z) ∈ D(β) a.e. on Z, then problem (10.141) has a unique solution x(t, z) that satisfies
x ∈ C T, H01 (Z) ∩ L∞ T, H 2 (Z)
∂x ∈ C T, L2 (Z) ∩ L∞ T, H01 (Z) ∂t √ ∂2x
∂2x t 2 ∈ L2 (T × Z), t 2 ∈ L∞ T, L2 (Z) . ∂t ∂t
Moreover, if j y0 (·) ∈ L1 (Z), then ∂ 2 x ∂t2 ∈ L2 (T × Z).
PROOF: Let X = H01 (Z), H = L2 (Z) and X ∗ = H −1 (Z). Let A ∈ L H01 (Z), H −1 (Z) be defined by
A(x), y = (Dx, Dy)RN dz for all x, y ∈ H01 (Z). Z
Hence Ax = − x (by integration by parts). Also let ϕ ∈ Γ0 H01 (Z) be defined by
ϕ(x) = j x(z) dz for all x ∈ H01 (Z) (recall β = ∂j with j ∈ Γ0 (R)). Z
Note that {x ∈ L2 (Z) : A ∈ L2 (Z)} = H01 (Z) ∩ H 2 (Z). Therefore, we have
Ax0 + ∂ϕ(y0 ) ∩ H = ∅. Thus we can apply Corollary 10.4.2 and establish the conclusions of the proposition. Next we consider a second-order evolution equation defined in the framework of a general evolution triple. So let T = [0, b] and (X, H, X ∗ ) a general evolution triple of spaces (see Definition 10.1.21). We examine the following second-order Cauchy problem:
)
10.4 Second-Order Nonlinear Evolution Equations 743
* x (t) + A t, x (t) + Bx(t) = f t, x(t), x (t) a.e. on T, . (10.142) x(0) = x0 , x (0) = y0
In problem (10.142), A : T × X −→ X ∗ is a nonlinear in general operator, B ∈ L(X, X ∗ ), f : T ×H ×H −→ H, and (x0 , y0 ) ∈ X ×H. DEFINITION 10.4.4 By a solution of problem (10.142), we mean a function x ∈ C(T, X) such that x ∈ Wp (0, b), (1 < p < ∞), it satisfies the equation in problem (10.142) for almost all t ∈ T and x(0) = x0 , x (0) = y0 . We denote the set of solutions of problem (10.142) by S(x0 , y0 ). REMARK 10.4.5 Because Wp (0, b) ⊆ C(T, H) (see Theorem 10.1.25), it follows that S(x0 , y0 ) ⊆ C 1 (T, H). We impose the following hypotheses on the maps A(t, x), B, and f (t, x, y). H(A) : A : T ×X −→ X ∗ is a map such that (i) For all x ∈ X, t −→ A(t, x) is measurable. (ii) For almost all t ∈ T , x −→ A(t, x) is demicontinuous monotone. (iii) For almost all t ∈ T and all x ∈ X, we have A(t, x)∗ ≤ α(t) + cxp−1 ,
with 2 ≤ p < ∞, α ∈ Lp (T )+ , (1/p) + (1/p ) = 1 and c > 0. (iv) For almost all t ∈ T and all x ∈ X, we have A(t, x), x ≥ c1 xp − α1 (t), with c1 > 0 and α1 ∈ L1 (T )+ . H(B) : B ∈ L(X, X ∗ ), Bx, x ≥ 0 for all x ∈ X and Bx, u = x, Bu for all x, u ∈ X. H(f ): f : T ×H ×H −→ H is a function such that (i) For all x, y ∈ H, t −→ f (t, x, y) is measurable. (ii) For almost all t ∈ T , (x, y) −→ f (t, x, y) is sequentially continuous from H × H into Hw (here Hw denotes the Hilbert space H furnished with the weak topology). (iii) For almost all t ∈ T and all x, y ∈ H, we have |f (t, x, y)| ≤ α2 (t) + c2 |x|2/p + |y|2/p ,
with α2 ∈ Lp (T )+ , c2 > 0.
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10 Evolution Equations
THEOREM 10.4.6 If hypotheses H(A) , H(B) , H(f ) hold and (x 0 , y0 ) ∈ X ×H, then the solution set S(x0 , y0 ) is nonempty, weakly compact in W 1,p (0, b), X , and compact in C 1 (T, H). PROOF: First we derive some a priori bounds for the elements of S(x0 , y0 ). So let x ∈ S(x0 , y0 ). We know that x ∈ Wp (0, b) ⊆ C(T, H). We have
x (t), x (t) + A t, x (t) , x (t) + B x(t) , x (t)
= f t, x(t), x (t) , x (t) a.e. on T. (10.143) Using Corollary 10.1.26, we have
1 d x (t), x (t) = |x (t)|2 2 dt
a.e. on T.
Also because of hypotheses H(A) (iv) and H(B) , we have A t, x (t) , x (t) ≥ c1 x (t)p − α1 (t)
1 d
B x(t) , x(t) and B x(t) , x (t) = 2 dt
(10.144)
a.e. on T
(10.145)
a.e. on T.
(10.146)
So, if we return to (10.143) and use (10.144) through (10.146), we obtain 1 d
1 d |x (t)|2 + c1 x (t)p + B x(t) , x(t) 2 dt 2 dt
≤ f t, x(t), x (t) , x (t) + α1 (t) a.e. on T. Integrating, we have
t 1
1 x (s)p ds + |x (t)|2 + c1 B x(t) , x(t) 2 2 0
t
≤ f s, x(s), x (s) , x (s) ds + c3 for some c3 > 0, 0
t 1 x (s)p ds ⇒ |x (t)|2 + c1 2 0
t
α2 (s) + c2 |x(s)|2/p + |x (s)|2/p |x (s)|ds + c3 ≤
(10.147)
0
(see hypotheses H(B) and H(f )(iii)). Because x ∈ Wp (0, b), we have
t x (s)ds in X for all t ∈ T, x(t) = x0 + 0
t |x (s)|2 ds. ⇒ |x(t)|2 ≤ 2|x0 |2 + 2
(10.148)
0
So, if on the integral in the right-hand side of (10.147) we use Young’s inequality with ε > 0 and (10.148), after some calculations, we obtain
t 1 x (s)p ds |x (t)|2 + c1 2 0
t
t s
εp β p t |x (τ )|2 dτ ds + x (s)p ds ≤ c4 (ε) + c5 (ε) |x (s)|2 ds + c6 (ε) p 0 0 0 0 (10.149)
10.4 Second-Order Nonlinear Evolution Equations
745
with c4 (ε), c5 (ε), c6 (ε) > 0 and β > 0 is such that | · | ≤ β · . We choose ε > 0 such that (εp β p )/p < c1 . We have
t
t s 1 |x (τ )|2 dτ ds. (10.150) |x (t)|2 ≤ c4 (ε) + c5 (ε) |x (s)|2 ds + c6 (ε) 2 0 0 0 From (10.150) and using the generalized Gronwall inequality (see Denkowski– Mig´ orski–Papageorgiou [195, p. 128]), we obtain M1>0 such that |x (t)| ≤ M1
for all t ∈ T
and all x ∈ S(x0 , y0 ).
(10.151)
From (10.148) and (10.151), it follows that we can find M2 > 0 such that |x(t)| ≤ M2
for all t ∈ T
and all x ∈ S(x0 , y0 ).
(10.152)
So, if in (10.149), we use (10.151) and (10.152) and recalling the choice of ε > 0, we see that there exists M3 > 0 such that x Lp (T,X) ≤ M3
Recall that x(t) = x0 +
t
for all x ∈ S(x0 , y0 ).
x (s)ds
(10.153)
in X, for all t ∈ T.
0
Hence there exists M4 > 0 such that x(t) ≤ M4
for all t ∈ T and all x ∈ S(x0 , y0 ).
(10.154)
Using (10.152) through (10.154), directly from the equation in (10.142), we can find M5 > 0 such that x Lp (T,X ∗ ) ≤ M5
for all x ∈ S(x0 , y0 ).
(10.155)
As in the proof of Theorem 10.3.12, because of (10.151) and (10.152), we can replace f (t, x, y) by f1 (t, x, y) defined by
f1 (t, x, y) = f t, rM1 (x), rM2 (y) (rMk is the Mk -radial retraction in H). Note that for almost all t ∈ T and all x, y ∈ H we have |f1 (t, x, y)| ≤ ψ(t) with ψ ∈ Lp (T )+ . Let K : Lp (T, X) −→ C(T, X) be the integral operator defined by
t (Ky)(t) = x0 + y(s)ds for all t ∈ T. 0
With the use of K, we can equivalently recast (10.142) as the following first order evolution equation.
) * y (t) + A t, y(t) + B (Ky)(t) = f1 t, (Ky)(t), y(t) a.e. on T, . (10.156) y(0) = y0 If y solves (10.156), then x(t) = x0 + use the material of Section 10.3.
t 0
y(s)ds solves (10.142). To solve (10.156) we
746
10 Evolution Equations
First assume that y0 ∈ X and set A1 (t, x) = A(t, x + y0 ). We can easily verify that A1 still satisfies hypotheses such as those in H(A) .
Also let B1 ∈ L Lp (T, X), Lp (T, X ∗ ) be defined by
B1 (y)(·) = B K(y + y0 )(·)
and let F1 : Lp (T, X) −→ Lp (T, X ∗ ) be defined by
F1 (y)(·) = f1 ·, K(y + y0 )(·), y(·) + y0 . We consider the following Cauchy problem )
y (t) + A1 t, y(t) + B1 (y)(t) = F1 (y)(t) y(0) = y0
a.e. on T = [0, b],
* .
(10.157)
Note that y ∈ Wp (0, b) solves problem (10.156) if and only if y(·) − y0 solves problem (10.157). Let L : D(L) ⊆ Lp (T, X) −→ Lp (T, X ∗ ) be the linear operator defined by Ly = y for all y ∈ D(L) = y ∈ Wp (0, b) : y(0) = 0 . It is easy to see that L is densely defined, maximal monotone. Also let A1 : Lp (T, X) −→ Lp (T, X ∗ ) be defined by
A1 (y)(·) = A1 ·, y(·) . Clearly A1 is demicontinuous, monotone, hence it is maximal monotone. We set V (y) = A1 (y) + B1 (y) − F1 (y), V : Lp (T, X) −→ Lp (T, X ∗ ). Claim 1: V is L-pseudomonotone (see Definition 10.3.15). w
Clearly V is bounded and demicontinuous. Suppose yn −→ y in Wp (0, b) and assume that
lim sup (V (yn ), yn − y) ≤ 0. (10.158) n→∞
From the definition of V , we have
lim sup (A1 (yn ), yn − y) = lim sup (V (yn ) − B1 (yn ), yn − y) . n→∞
n→∞
(10.159)
Also, if by (·, ·)p p we denote the duality brackets for the pair Lp (T, H), Lp (T, H) , then we have
(10.160) (F1 (yn ), yn − y) = F1 (yn ), yn − y p p −→ 0 as n → ∞. Moreover, because of H(B) , we have
0 = lim (B1 (y), yn − y) ≤ lim inf (B1 (yn ), yn − y) . n→∞
n→∞
If in (10.159), we use (10.158), (10.160), and (10.161), we obtain
lim sup (A1 (yn ), yn − y) ≤ 0. n→∞
(10.161)
(10.162)
10.4 Second-Order Nonlinear Evolution Equations
747
By virtue of the maximal monotonicity of A1 , from (10.162), we have
w (A1 (yn ), yn ) −→ (A1 (y), y) and A1 (yn ) −→ A1 (y) in Lp (T, X ∗ ), w
⇒ V (yn ) −→ V (y)
in Lp (T, X ∗ ).
(10.163)
Also because of (10.158), we have
lim sup (B1 (yn ), yn − y) ≤ 0. n→∞
As before from (10.164) and because B1 is maximal monotone, we have
(B1 (yn ), yn ) −→ (B1 (y), y) w
⇒ (V (yn ), yn ) −→ (V (y), y) .
(10.164)
(10.165)
From (10.163) and (10.165), we conclude that V is L-pseudomonotone. Claim 2: V is coercive. Because of hypothesis H(A) (iv), we have
(A1 (y), y) ≥ c7 ypLp (T,X) − c8
for some c7 , c8 > 0.
(10.166)
Also hypothesis H(B) implies that
(B1 (y), y) ≥ 0,
(10.167)
whereas for F1 , we have
| (F1 (y), y) | = | F1 (y), y p p | ≥ −ψp yLp (T,H) .
(10.168)
Combining (10.166) through (10.168) and because 2 ≤ p < ∞, we conclude that V is coercive. Because of Claims 1 and 2 and using Theorem 10.3.16, we can find y ∈ D(L) ⊆ Wp (0, b) such that L(y) + V (y) = 0 when y0 ∈ X. Then y = y + y0 solves (10.157). Now we remove the restriction that y0 ∈ X. So suppose y0 ∈ H. We can find {y0n }n≥1 ⊆ X such that y0n −→ y0 in H as n → ∞. Let yn ∈ Wp (0, b) be the solution of (10.157) for y0n obtained above. An a priori estimation as in the proof of Theorem 10.3.12 (note that {y0n }n≥1 ⊆ H is bounded), implies that {yn }n≥1 ⊆ Wp (0, b) w is bounded. Therefore we may assume that yn −→ y in Wp (0, b). Then as in the proof of Theorem 10.3.12 (with the Galerkin solutions), we check that y is a solution of (10.157) with y0 ∈ H. This proves the solvability of (10.157), hence the solvability of (10.142) too. Next we prove the compactness properties of the solution set S(x0 , y0 ). The proof follows the steps of the corresponding part of the proof of Theorem 10.3.12. So let Σ = h ∈ Lp (T, H) : |h(t)| ≤ ψ(t) a.e. on T .
Endowed with the relative weak Lp (T, H)-topology, Σ becomes a compact metrizable space. Let Γ : Σ −→ 2C(T,H) be the multifunction which to each h ∈ Lp (T, H) assigns the set of solutions of
748
10 Evolution Equations
) y (t) + A t, y(t) + B (Ky)(t) = h(t) y(0) = y0
a.e. on T,
* .
We saw that Γ(h) = ∅ and in fact Γ(h) is a singleton. Claim 3: Γ(Σ) is weakly in Wp (0, b) and compact in C(T, H). To show the weak compactness in Wp (0, b), it suffices to show that Γ(Σ) is weakly sew quentially closed. So suppose {yn }n≥1 ⊆ Γ(Σ) and assume that yn −→ y in Wp (0, b). Then as in proof of Theorem 10.3.12, we can show that
(xn , xn − x) −→ 0. Also we have
and
0 = lim (B(Ky), yn − y) ≤ lim inf (B(Kyn ), yn − y) n→∞ n→∞
lim (hn , yn − y) = lim (hn , yn − y)p p = 0.
n→∞
n→∞
Therefore finally we have
lim sup (A(yn ), yn − y) ≤ 0. n→∞
Because of the maximal monotonicity of A, we infer that w
A(yn ) −→ A(y)
in Lp (T, X ∗ ).
Because yn + A(yn ) + B(Kyn ) = hn for all n ≥ 1, passing to the limit as n → ∞, we have y + A(y) + B(Ky) = h, y(0) = y0 ,
w
(where hn −→ H in Lp (T, H))
⇒ y ∈ Γ(Σ) (i.e., Γ(Σ) is weakly compact in Wp (0, b)). Next we show that Γ(Σ) is compact in C(T, H). So suppose {yn }n≥1 ⊆ Γ(Σ). w For every n ≥ 1, yn = Γ(hn ), hn ∈ Σ and we may assume that hn −→ h in Lp (T, H). For every t ∈ T and if y = Γ(h), we have
1 |yn (t) − y(t)|2 = (hn − h, yn − y) t − (A(yn ) − A(y), yn − y) t 2
− (B(Kyn ) − B(Ky), yn − y) t . (10.169) Note that
(B(Kyn ) − B(Ky), yn − y) t ≥ 0 (see hypothesis H(B) ).
(10.170)
Using (10.170) in (10.169), we obtain 1 |yn (t) − y(t)|2 ≤ 2
b
0
| hn (t) − h(t), yn (t) − y(t) |
+ 0
b
| A t, yn (t) , yn (t) − y(t) |dt + (A(y), yn − y) t .
10.4 Second-Order Nonlinear Evolution Equations
749
w
Because yn −→ y in Wp (0, b), we have yn −→ y in Lp (T, H) (see Theorem 10.1.29). Hence
b
| hn (t) − h(t), yn (t) − y(t) | −→ 0. 0
Also from the proof of Theorem 10.3.12 we have
b
| A t, yn (t) , yn (t) − y(t) |dt −→ 0 0
and
sup (A(y), yn − y) t −→ 0. t∈T
So finally sup |yn (t) − y(t)|2 −→ 0 (i.e., yn −→ y in C(T, H)), t∈T
⇒ Γ(Σ) is compact in C(T, H). This proves Claim 3. Let S (x0 , y0 ) = x : x ∈ S(x0 , y0 ) . Evidently S (x0 , y0 ) ⊆ Γ(Σ) and it is weakly closed
in Wp (0, b) and strongly closed in C(T, H). Because S(x0 , y0 ) = K S (x0 , y0 ) , we conclude that S(x0 , y0 ) is weakly compact in Wp (0, b) and strongly compact in C 1 (T, H). REMARK 10.4.7 The above theorem remains valid if instead of H(A)(ii), we assume the following more general condition (ii) for almost all t ∈ T, x −→ A(t, x) is pseudomonotone. Moreover, if for almost all t ∈ T, f (t, ·, ·) is Lipschitz continuous, then the solution set S(x0 , y0 ) is a singleton. We conclude this section with an example of a nonlinear hyperbolic initial boundary value problem. So let T = [0, b] and Z ⊆ RN be a bounded domain with a C 2 -boundary ∂Z. We consider the following hyperbolic problem. ⎫ ⎧ ∂2x
⎨ ∂t2 − x(t, z) − div(Dxt p−2 Dxt ) = f0 t, z, x(t, z), xt (t, z) in T × Z, ⎬ . x(z, 0) = x0 (z), xt (0, z) = y0 (z) a.e. on Z, ⎭ ⎩ x T ×∂Z = 0 (10.171) Here xt = ∂x/∂t. The hypotheses on the function f0 (t, z, x, y) are the following. H(f0 ): f0 : T ×Z ×R ×R −→ R is a function such that (i) For all x, y ∈ R, (t, z) −→ f0 (t, z, x, y) is measurable. (ii) For almost all (t, z) ∈ T × Z, (x, y) −→ f0 (t, z, x, y) continuous . (iii) For almost all (t, z) ∈ T ×Z and all x, y ∈ R, we have
|f0 (t, z, x, y)| ≤ α2 (t, z) + c2 |x| + |y| ,
with α2 ∈ Lp T, L2 (Z) , c2 > 0, 2 ≤ p < ∞,
1 p
+
1 p
= 1.
750
10 Evolution Equations
DEFINITION 10.4.8 By a solution of problem (10.171), we mean a function
x ∈ C T, W01,p (Z) such that
and
∂2x
∂x ∈ Lp T, W −1,p (Z) ∈ Lp T, W01,p (Z) , 2 ∂t ∂t
d2 p−2 x(t, z)u(z)dz + Dx (Dx , Du) (Dx, Du)RN dz N dz + t t R dt2 Z Z Z
= f (t, z, x, xt )u(z)dz Z
for all u ∈
W01,p (Z)
(weak solution).
PROPOSITION 10.4.9 If hypotheses H(f0 ) hold, x0 ∈ W01,p (Z), and y0 ∈ L2 (Z), then
the set of (weak) solutions of problem (10.171) is nonempty and compact in C 1 T, L2 (Z) .
PROOF: The evolution triple is X = W01,p (Z),H = L2 (Z), X ∗ = W −1,p (Z). Also A : X −→ X ∗ is defined by
Dxp−2 (Dx, Dy)RN dz for all x, y ∈ W01,p (Z). A(x), y = Z
Clearly it satisfies hypotheses H(A). Also B ∈ L(X, X ∗ ) is given by
Bx, y = (Dx, Dy)RN dz for all x, y ∈ W01,p (Z). Z
It satisfies hypothesis H(B). Finally we set
f (t, x, y)(·) = f0 t, ·, x(·), y(·) . Then f : T ×H ×H −→ H satisfies hypotheses H(f ). Using A, B, and f , we can equivalently rewrite (10.171) in the form of (10.136). We apply Theorem 10.4.6 to conclude the proof.
10.5 Remarks 10.1: The origins of the Bochner integral can be traced in the works of Bochner [81] and Dunford [213]. Some people call it Dunford’s second integral. A more detailed study of the Bochner integral can be found in Diestel–Uhl [199], Denkowski– Mig´ orski–Papageorgiou [194], and Gasi´ nski–Papageorgiou [259]. There is also a weak vector-valued integral known as the Pettis integral. A detailed investigation of it can be found in Talagrand [573]. The RNP (see Definition 10.1.11), is investigated in a systematic way in the books of Diestel–Uhl [199] and Bourgin [95]. Theorem 10.1.16 is due to Dinculeanu–Foias [201]. For X = H = a Hilbert space, Theorem 10.1.18 was proved by Komura [360]. Evolution triples (see Definition 10.1.21) are also known as Gelfand triples , because it was Gelfand who first made systematic use of them (see Gelfand–Shilov [261]). Lemma 10.1.28 appears in the literature under the names Lions lemma or Ehrling’s inequality (see Ehrling [220]). Theorem 10.1.29
10.5 Remarks
751
is due to Aubin [35]. More on evolution triples and the related function spaces, can be found in the paper of Simon [555] and in the books of Gasi´ nski–Papageorgiou [259], Showalter [554], Wloka [608], and Zeidler [621, 622]. 10.2: Semilinear evolution equations using the semigroup method can be found in the books of Amann [19], Henry [291], Pazy [491], Showalter [553, 554], Tanabe [574], and Zheng [625]. The case of a time-dependent operator A, in which case we are dealing with the so-called evolution operators, can be found in Amann [19] and Tanabe [574]. For the applications to parabolic problems, we follow Zheng [625]. 10.3: Evolution equations associated with maximal monotone operators (Hilbert space case) or accretive operators (Banach space case), were the starting point for the introduction of nonlinear semigroups: see Komura [360] (Hilbert spaces) and Crandall–Liggett [166]) (Banach spaces) (see also Segal [546]). Subdifferential evolution inclusion, with time-invariant ϕ, was first investigated by Brezis [99, 100]. Brezis [99] proved Theorem 10.3.7 and also established the regularizing effect on the initial condition. A comprehensive treatment of such evolution equations can be found in the books of Barbu [58] and Brezis [102]. The case of time-dependent subdifferentials, is studied by Attouch–Damlamian [32], Hu–Papageorgiou [314, 315], Kenmochi [347, 348], Yamada [609], and Yotsutani [616]. For the periodic problem, we refer to Bader–Papageorgiou [51]. Further results can be found in Chapter 2 of the book of Hu–Papageorgiou [316]. The Galerkin method together with the monotonicity method was used extensively in the framework of evolution triples by Lions [387]. Extensions can be found in Papageorgiou–Papalini–Yannakakis [483]. We also refer to the books of Showalter [554] and Zeidler [622]. 10.4: Theorem 10.4.1 was originally by Lions–Strauss [386], however the proof given here is due to Brezis (see also Barbu [58, pp. 268–269]). Similar results can be found in Brezis [100]. Second-order evolution equations defined in the framework of a general evolution triple can be found in Zeidler [622, Chapter 33]. The work of Zeidler was generalized by Papageorgiou–Yannakakis [485, 486]. Our presentation here is based on these papers.
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List of Symbols
Symbol
Meaning
L(X, Y )
space of linear operators from X to Y
Γ0 (X)
proper, convex, lower semicontinuous
∗
ϕ
conjugate function
∂ϕ
convex subdifferential
ϕ
0
generalized directional derivative
TC
tangent cone of C
NC
normal cone of C
TC
Clarke tangent cone of C
NC
Clarke normal cone of C
Γ- lim
Γ-limit
B(X)
Borel σ-field of X
Γseq (X1β1 , X2β2 )
multiple Γ-operator
K(C, Y )
compact maps from C into Y
Kf (C, Y )
linear space of finite rank mappings from C to Y
σ(A)
spectrum of A
Lc (X, Y )
compact linear operators from X to Y
Lf (X, Y )
finite rank operators from X to Y
Φ(X, Y )
Fredholm operators from X to Y
Φ+ (X, Y )
semi-Fredholm operators from X to Y
d
Leray–Schauder degree map
782
List of Symbols α
Kuratowski measure of noncompactness
β
ball measure of noncompactness
d0
Nussbaum–Sadovskii degree
d(S)+ = deg(S)+
degree for (S)+ maps
catY A
Ljusternik–Schnirelmann category of A in Y
γ
genus
C01 (Z)
1,2
Wper
(0, b), RN
space of C 1 -functions on Z s.t. on ∂Z are zero
x ∈ W 1,2 (0, b), RN : x(0) = x(b)
Pf (X), Pk (X)
spaces of sets
Pf c (X), Pwkc (X)
spaces of sets
Pkc (X), Pbf c (X)
spaces of sets
m(X ∗ , X)
Mackey topology on X ∗ for (X, X ∗ )
F + (C), F − (C)
inverse images of multifunctions
SFp
Lp -selectors of F
Jλ
resolvent operator
Aλ
Yosida approximation of A
Index
L-pseudomonotone operator, 737 ε-minimizer, 56 m-accretive operator, 184 absolute neighborhood retract, 291 absolute retract, 291, 512 absolutely continuous function, 569, 695 abstract economy, 615 accretive operator, 183 adjoint state, 144 admissible map, 98 admissible pair, 112 admissible policy, 638 allocation core, 531 feasible, 532 initial endowment, 531 Altman’s condition, 242 Ambrosetti–Rabinowitz condition, 422 antimaximum principle, 453 asymptotic center, 236 cone, 43 derivative, 244 asymptotically linear problems, 353 balanced game, 622 non-side payment game, 622 balanced coalition, 620 balanced side-payment game, 620 Banach contraction principle, 225 Banach fixed point theorem, 225 barycenter, 134
Bayesian game, 628 Bayesian Nash equilibrium, 629 Bayesian view, 662 Bellman’s optimality principle, 634 Berge maximum theorem, 462, 670 bilinear map, 105 bimatrix game, 613 Blaschke’s theorem, 527 Bochner integrable function, 693 Bochner integral, 693 Borsuk’s fixed point theorem, 240 Borsuk’s theorem, 205 bounded operator, 163 Boylan metric, 659 Brouwer’s fixed point theorem, 238 budget multifunction, 600 budget set, 532 cancellation law lemma, 468 Carath´eodory function, 470 Caristi’s fixed point theorem, 93 center asymptotic, 236 Cerami condition, 270 Chain Markov Decision, 684 chain rule, 5 Serrin–Vall´ee Poussin, 719 Chebyshev center, 234 radius, 234 Choquet function, 487 Clarke normal cone, 41 Clarke tangent cone, 41 coalition balanced, 620
784
Index
coercive function, 53 coercive map, 166 compact operator, 148 competitive program, 545 completely continuous, 149 comprehensiveness condition, 623 condition Altman’s, 242 Ambrosetti–Rabinowitz, 422, 441 Cerami, 270 comprehensiveness, 623 growth Bernstein–Nagumo–Wintner, 391 maximum, 144 nonuniform nonresonance, 412 Palais–Smale(P S), 92, 270 Palais–Smale(P S)-generalized, 287 R¨ othe’s, 242 reachability, 548 transversality, 144, 546 conditional expected utility, 629 cone τ -closed, 44 τ -recession, 44 asymptotic, 43 Clarke tangent, 41 contingent, 61 convex, 44 dual, 251 fully regular, 245 minihedral, 245 normal, 36, 245 order, 245 positive, 601 recession, 43 regular, 245 solid, 245 strongly minihedral, 245 tangent, 35 total, 245 conjugate function, 18 conservative multistrategy, 611 conservative value, 611 consistent system, 619 consumption sequence, 544 contingent cone, 61 continuity of information, 690 continuity of payoff, 690 continuously Fr´echet differentiable, 6
continuum of agents, 606 contractible set, 290 contraction semigroup, 188 convergence Γτ -, 45 epigraphical, 48 Hausdorff, 514 Mosco, 59, 514 scalar, 515 weak, 515 Wijsman, 515 convex cone, 44 function, 12 strictly, 12 core of a non-side-payment game, 621 of a side-payment game, 618 core allocation, 531 cost function, 685 Courant minimax principle, 289 Courant’s minimax principles, 303 Courant–Fischer–Weyl minimax principle, 289 Cournot–Nash equilibrium, 624, 627 crease, 196 criterion catching, 553 overtaking, 554 critical point, 73, 196, 269 free, 78 nondegenerate, 285 critical value, 269 cyclically monotone map, 175 decomposable set, 487 deformation, 269 invariant, 269 method, 269 theorem, 270, 274 degree, 196 Leray–Schauder, 210, 216 map, 219 Nussbaum–Sadovskii, 217 demand set, 532 demicontinuous map, 165 derivative asymptotic, 244 directional, 16
Index Fr´echet, 3 Gˆ ateaux, 2 generalized directional, 27 partial, 6 diametrical point, 233 diffeomorphism, 11 differentiable twice, 105 Dinculeanu–Foias theorem, 695 direct method, 116 directional derivative, 16 dissipative operator, 184 distance Hausdorff, 465 distance function, 41 distributional strategy σ, 681 distributional strategy τ , 681 domain effective, 12 star-shaped, 429 double sequence lemma, 53 drop set, 95 drop theorem, 94 Du Bois–Reymond’s lemma, 102 dual cone, 251 duality map, 26, 166 with gauge, 166 Dubovickii–Milyutin theorem, 79 Dunford’s second integral, 750 dynamic programming functional equation, 572 dynamic programming operator, 636 economy abstract, 615 effective domain, 12 Ehrling’s inequality, 698 eigenelement, 297 eigenfunction, 297 eigenspace, 153 eigenvalue, 153, 297 eigenvector, 153 Ekeland variational principle, 89 element independent, 661 positive, 251 embedding method, 503 envelope
785
τ -lower semicontinuous, 67 epigraph, 12, 44 equation dynamic programming, 572 Euler, 104 equi-τ -lower semicontinuity, 49 equicoercivity, 53 equilibrium ε, 644 Bayesian Nash, 629 competitive, 532 Cournot–Nash, 624, 627 of a game, 681 Walras, 531 equivalence of probability measures, 657 evolution triple, 696 ex-ante dominated choice function, 600 ex-ante view, 668 ex-post view, 662 expected payoff, 681 extremals, 104 feasible pair, 112 feasible program, 553, 581 Filippov’s implicit function theorem, 482 Finsler manifold, 297 Finsler metric, 297 first partial function, 6 first-order stochastic dominance, 603 fixed point index, 225, 256 fixed point property, 238 formula H¨ ormander’s, 465 Fr´echet derivative, 3 differentiable, 3 Fredholm operator, 160 fully regular cone, 245 function τ -coercive, 66 τ -lower semicontinuous, 64 τ -lower semicontinuous at x, 64 τ -recession, 44 τ -upper semicontinuous, 64 τ -upper semicontinuous at x, 64 conjugate, 18 quasiconcave, 82
786
Index
absolutely continuous, 569 asbolutely continuous, 695 asymptotic, 44 auxiliary, 343 Bochner integrable, 693 Carath´eodory, 470 choice, 599 Choquet, 487 closed, 12 coercive, 53 complementing, 343 convex, 12 convex of compact type, 195 convex–concave, 84 cost, 685 costate, 144 EU-rational choice, 600 ex-ante dominated choice, 600 first partial, 6 gauge, 166 graph-measurable, 471 Hamiltonian, 110 indicator, 12 invariant, 286 Ize’s, 343 Lagrangian, 76, 86 locally Lipschitz, 27 lower semicontinuous, 12 penalty, 392 proper, 12 quasiconvex, 82 recession, 44 regular, 31 saddle, 84 second partial, 6 sequentially coercive, 53 simple, 692 strongly measurable, 692 superpositionally measurable, 471 support, 21, 460 value, 545 value of stochastic game, 639 weakly measurable, 692 functional singular, 569 Gˆ ateaux derivative, 2 differentiable, 2
Galerkin method, 751 game a side-payment, 618 balanced, 622 balanced non-side payment, 622 bimatrix, 613 generalized, 615 in normal form, 610 non-side payment, 621 side-payment balanced, 620 zero-sum, 613 gauge function, 166 Gelfand triple, 750 Gelfand integral, 600 Gelfand triple, 696 generalized game, 615 generalized mountain pass theorem, 279 generalized pseudomonotone operator, 179 generalized subdifferential, 29 generating cone, 245 genus, 293 Krasnoselkii, 293 global extremum, 101 maximum, 101 minimum, 101 glossary, 781 gradient, 270 gradient-like flow, 270 graph, 162 norm, 707 Green’s second identity, 330 H¨ ormander’s formula, 465 Hamiltonian function, 110 system, 110 Hausdorff convergence, 514 distance, 465 measure of noncompactness, 214 metric, 465 hemicontinuous map, 165 Hilbert cube, 475 Hille–Yosida theorem, 191 homotopy pseudomonotone, 223
Index homotopy of class (S)+ , 220 Hopf’s lemma, 304 identity Picone’s, 328 Pohozaev’s, 421, 429 second Green’s, 330 independent element, 661 subset, 661 indicator function, 12 inequality Diaz–Saa, 328 Ehrling’s, 698 Harnack’s, 328 Ky Fan’s, 612 Young–Fenchel, 18 infimal convolution, 20 infinitesimal generator, 188 initial endowment allocation, 531 input sufficient, 546 int dom ϕ, 14 integral Bochner, 693 Gelfand, 600 Pettis, 750 second Dunford’s, 750 integration by parts formula, 698 invariance of domain theorem, 212 inward set, 509 isomorphism, 11 Kakutani–Ky Fan theorem, 114 Knaster–Kuratowski–Mazurkiewicz map, 80 Krasnoselskii genus, 293 Krein’s theorem, 252 Kuratowski limit, 514 interior, 514 superior, 514 Kuratowski measure of noncompactness, 214 Kuratowski–Ryll Nardzewski selection theorem, 480 Ky Fan’s inequality, 612 Lagrange lemma, 102
787
Lagrange multiplier, 76 Lagrange multiplier rule, 35 Lagrangian, 98 function, 76, 86 hyperregular, 109 regular, 109 Lebesgue–Bochner space, 694 Legendre transform, 109 Legendre–Fenchel transform, 18 lemma cancellation law, 468 Du Bois–Reymond’s, 102 Hopf’s, 304 Lagrange, 102 Lions, 750 Minkowski–Farkas, 619 Morse, 285 Leontief dynamic model, 550 Leray–Schauder alternative principle, 242 Leray–Schauder degree, 210 limit lower Γ, 45 lower Kτ , 46 upper Γ, 45 upper Kτ -, 47 linear order, 669 linking set, 276 List of Symbols, 781 Ljusternik’s theorem, 74 Ljusternik–Schnirelmann category, 207, 290 Ljusternik–Schnirelmann–Borsuk theorem, 207 local extremum, 34 linking, 280 maximum, 101 minimum, 101 locally bounded operator, 163 locally Lipschitz function, 27 lower Γ-limit, 45 limit-Kτ , 46 lower semicontinuous function, 12 lower semicontinuous regularization, 48 Lumer–Phillips theorem, 706 Lyapunov’s convexity theorem, 491
788
Index
Mackey topology, 484, 573 manifold Finsler, 297 spherelike, 338 map γ-Lipschitz, 215 γ-condensing, 215 γ-contraction, 215 k-contraction, 225 admissible, 98 asymptotically linear, 244 bilinear, 105 coercive, 166 contractive, 225 cyclically monotone, 175 degree, 219 demicontinuous, 165 duality, 26, 166 duality with gauge, 166 hemicontinuous, 165 inward, 236 Knaster–Kuratowski–Mazurkiewicz, 80 maximal cyclically monotone, 175 minimal section, 187 nonexpansive, 225 normalized duality, 166 policy, 685 proper, 151 quasibounded, 183, 244 regular, 183 smooth, 183 strategy, 685 traverse, 264 truncation, 392 type (S)+ , 183 weakly coercive, 166 mapping finite rank, 150 marginal probability, 681 Markov Decision Chain, 684 maximal accretive operator, 184 maximal cyclically monotone map, 175 maximal monotone operator, 163 maximin multistrategy, 611 maximum condition, 144 principle, 433 measurability view, 668
measurable strongly function, 692 weakly function, 692 measure probability initial, 684 test, 61 Young, 118 method auxiliary variable, 127 deformation, 269 direct, 116, 542 embedding, 503 Galerkin, 751 monotonicity, 751 of Lagrange multipliers, 75 reduction, 116 relaxation, 118 upper-lower solutions, 376, 380 metric Boylan, 659 Finsler, 297 Hausdorff, 465 projection, 39 Michael’s selection theorem, 476 minihedral cone, 245 minimal section map, 187 minimax equality, 79 minimizer-ε, 56 minimum shadow, 644 Minkowski–Farkas lemma, 619 mixed strategies, 614 model Leontief dynamic, 550 mollifier, 197 monotone operator, 163 Moreau–Yosida approximation, 178 Morse index, 285 Morse lemma, 285 Mosco convergence, 514 Mosco sense convergence, 514 mountain pass theorem, 278 multifunction h-continuous, 466 h-contraction, 504 h-lower semicontinuous, 466 h-upper semicontinuous, 466 p-integrably bounded, 495 almost lower semicontinuous, 469 budget, 600
Index choice, 599 closed, 459 constraint, 671 continuous, 457 good reply, 615 graph, 458 graph measurable, 470 integrably bounded, 495 Knaster–Kuratowski– Mazurkiewicz, 506 locally selectionable, 475 lower semicontinuous, 457 measurable, 470 nonexpansive, 506 scalarly measurable, 470 sequentially closed, 459 state independent constraint, 671 strongly inward, 509 upper semicontinuous, 457 Vietoris continuous, 457 weakly inward, 509 weakly lower semicontinuous, 469 multiplication formula, 208 multistrategy, 611 ε-equilibrium, 644 multivalued mapping domain, 162 inverse, 162 Nash equilibrium, 612 neighborhood special, 343 non-side-payment game, 621 nondivergence form, 304 norm graph, 707 weak, 498 normal cone, 36, 245 normal integrand, 119 normal structure, 233 number crossing, 345 Nussbaum–Sadovskii degree, 217 operator L-pseudomonotone, 737 m-accretive, 184 accretive, 183 bounded, 163 compact, 148
789
diagonalizable, 157 dissipative, 184 dynamic programming, 636 evolution, 751 Fredholm, 160 generalized pseudomonotone, 179 index, 160 inf, 126 locally bounded, 163 maximal accretive, 184 maximal monotone, 163 monotone, 163 pseudomonotone, 179 resolvent, 176, 705 semi-Fredholm, 160 strictly monotone, 163 sup, 126 Yosida approximation, 706 optimal admissible pair, 112 control, 112 trajectory, 112 optimal policy, 634 order cone, 245 ordering, 669 strict, 554 pair admissible, 112 feasible, 112 optimal admissible, 112 Palais–Smale condition, 92, 270 Palais–Smale generalized condition, 287 partial order relation, 669 path, 544 path of the economy, 559 payoff expected, 681 penalization technique, 392 perfect competition, 536 Perron–Frobenius theorem, 239 perturbation spike, 136 Pettis integral, 750 plan, 668 Poincar´e half-plane, 111 point bifurcation, 341 critical, 73, 196, 269, 287
790
Index
diametral, 233 invariant, 286 regular, 73 saddle, 78, 79 singular, 343 spectrum, 153 policy admissible, 638 map, 685 randomized, 638 shifted, 687 stationary, 685 Polish space, 471 Pontryagin maximum principle, 136 portmanteau theorem, 640 positive element, 251 preorder continuous, 669 lower semicontinuous, 668 upper semicontinuous, 669 preorder relation, 669 principle antimaximum, 453 Banach contraction, 225 Bellman’s optimality, 634 Courant’s minimax, 303 Ekeland, 89 Leray–Schauder, 242 maximum, 433 Pontryagin maximum, 136 strong maximum, 303, 305, 433 symmetric criticality, 286 Takahashi, 94 uniform antimaximum, 453 weak comparison, 433 prisoner’s dilemma problem, 614 probability initial measure, 684 marginal, 681 transition, 118 problem asymptotically linear, 405 prisoner’s dilemma, 614 value of information, 667 program, 544 capital accumulation, 589 competitive, 545 eligible, 556 feasible, 553, 589
feasible good, 556, 584 finite good, 561 finite optimal, 560 good feasible, 556, 584 infinite optimal, 560 optimal, 544, 553 stationary, 560 strongly maximal, 553 turnpike, 560 weakly maximal, 554, 581 weakly maximal stationary, 555 program of the economy, 559 proper function, 12 property U , 498 convex compact, 603 finite-dimensional, 506 fixed point(ffp), 238 Kadec–Klee, 415 Radon–Nikodym, 694 unique continuation, 333 pseudogradient vector field, 272 pseudometric, 561 pseudomonotone homotopy, 223 pseudomonotone operator, 179 quasibounded map, 244 quasiconcave function, 82 quasiconvex function, 82 quasinorm, 244 quasiorder relation, 669 R¨ othe’s condition, 242 R˚ adstr¨ om embedding theorem, 503 Rademacher’s theorem, 30 radius, 234 Chebyshev, 234 Radon–Nikodym property, 491 randomized policy, 638 Rayleigh quotient, 302 reachability condition, 548 recession function, 44 reduction method, 116 reduction property, 212 regular cone, 245 regular function, 31 regular point, 73 regular value, 196, 269
Index regularization τ -lower semicontinuous , 67 relation indifference, 536 partial order, 669 preference, 536 preference-indifference, 536 preorder, 669 quasiorder, 669 relaxability, 118 relaxation admissible, 118 relaxation method, 118 relaxed control, 118 representation isometric, 286 topological group, 286 resolvent operator, 705 resolvent set, 153 retract, 511 absolute, 512 retraction, 238 rule chain, 5 saddle point, 78 on product space, 79 saddle point theorem, 279 saddle value, 79 Sadovskii fixed point theorem, 242 Sard’s theorem, 197 saturation by a commodity vector, 536 Schauder’s fixed point theorem, 241 Schauder’s theorem, 159 Scorza–Dragoni theorem, 471 second partial function, 6 semigroup C0 , 188 compact, 193 contraction, 188 equicontinuous, 193 of nonexpansive maps, 192 separable probability space, 659 sequence consumption, 544 sequentially τ -lower semicontinuous, 66 τ -lower semicontinuous at x ∈ X, 66 Serrin–Vall´ee Poussin chain rule, 719
set τ -closed, 65 analytic, 473 budget, 532 contractible, 290 decomposable, 487 demand, 532 drop, 95 finitely closed, 506 Haar-null, 60 invariant, 286 inward, 236, 509 linking, 276 resolvent, 153 set-valued integral, 499 sgn, 345 shadow minimum, 644 shifted policy, 687 simple function, 692 solution extremal, 401 lower, 381, 433 mild, 702 strong, 721 upper, 381, 432 vector, 619 weak, 432, 721 Souslin space, 473 space Lebesgue–Bochner, 694 Polish, 471 separable probability, 659 Souslin, 473 tangent, 73 spectral resolution theorem, 157 spectrum, 153 spherelike constraint, 339 spike perturbation, 136 star-shaped domain, 429 state independent, 671 stationary history, 633 policy, 638, 685 Steiner point map, 525 strategy distributional σ, 681 distributional τ , 681 map, 685 rule, 624
791
792
Index
strictly convex, 12 strictly differentiable, 29 strictly monotone operator, 163 strong maximum principle, 303, 433 strong operator topology, 653 strong solution, 720 strong turnpike theorem, 567 strongly minihedral cone, 245 sub-σ-field µ- lim inf Σn , 673 n→∞
µ- lim sup Σn , 673 n→∞
subdifferential, 22 -ε, 646 generalized, 29 subset independent, 661 sufficient input, 546 sufficient vector, 563 support function, 21, 460 system adjoint, 144 consistent, 619 Hamiltonian, 144 Takahashi variational principle, 94 tangent cone, 35 space, 73 vector, 73 theorem Amann’s three fixed points, 262 Banach fixed point, 8, 225 Berge maximum, 462, 670 Birkoff–Kellogg, 212 Blaschke’s, 527 Borsuk’s, 205 Borsuk’s fixed point, 240 Brouwer’s fixed point, 238 Caristi’s fixed point, 93 Courant’s nodal set, 308 deformation, 270, 274, 288 Dinculeanu–Foias, 695 drop, 94 Dubovickii–Milyutin, 79 embedding R˚ adstr¨ om, 503 Filippov’s implicit function, 482 fixed point Kakutani–Ky Fan, 510 Fredholm alternative, 159 generalized mountain pass, 279, 288
Hille–Yosida, 191 implicit function, 8 infinite-dimensional Kuhn–Tucker multiplier, 583 invariance domain, 212 invariance of domain, 207 inverse function, 8, 11 Kakutani–Ky Fan, 114 Krein’s, 252 Kuratowski–Ryll Nardzewski, 480 Ljusternik’s, 74 Ljusternik–Schnirelmann–Borsuk, 207 Lumer–Phillips, 706 Lyapunov convexity, 491 Michael’s selection, 476 mountain pass, 278, 288 Nikaido’s, 37 no retraction, 239 nonlinear alternative, 241 Perron–Frobenius, 239 Pettis measurability, 692 portmanteau, 640 projection Yankov–von Neumann– Aumann, 473 Rademacher’s, 30 saddle point, 279, 288 Sadovskii fixed point, 242 Sard’s, 197 Sch¨ afer’s fixed point, 242 Schauder’s, 159 Schauder’s fixed point, 241 Scorza–Dragoni, 471 selection Yankov–von Neumann– Aumann, 114, 482 spectral resolution, 157 strong turnpike, 567 symmetric mountain pass, 285 turnpike, 558 Tychonov’s fixed point, 244 Vitali’s, 655 weak turnpike, 564 Weierstrass, 66 Yosida–Hewitt, 569 topological complement, 73 topological indices, 290 topology α, 592 c, 591
Index Mackey, 484, 573 narrow, 118 of pointwise convergence, 653 strong operator, 653 uniform operator, 653 weak, 599 total cone, 245 transform Legendre, 109 Legendre–Fenchel, 18 transition probability, 118 transversality condition, 144, 546 truncation technique, 392 turnpike, 560 programs, 558 theorem, 558 turnpikes, 558 twice differentiable, 105 Tychonov’s fixed point theorem, 244 uniform operator topology, 653 upper limit Kτ -, 47 limit-Γ, 45 utility conditional expected, 629 value conservative, 611 critical, 196, 269 of information, 667 regular, 269 value function, 545 value function of stochastic game, 639
793
variational convergence, 61 variational principle Ekeland, 89 vector solution, 619 sufficient, 563 tangent, 73 view Bayesian, 662 ex-ante, 668 ex-post, 662 Walras equilibrium, 531, 532 weak comparison principle, 433 weak maximality criterion, 552 weak maximum principle, 304 weak norm, 498 weak solution, 721 weak turnpike theorem, 564 weakly coercive map, 166 weakly inward map, 236 weakly maximal program, 554 weakly maximal stationary program, 555 Weierstrass theorem, 66 Wijsman convergence, 515 Yankov–von Neumann–Aumann projection theorem, 473 selection theorem, 114, 482 Yosida approximation, 176, 706 Yosida–Hewitt theorem, 569 zero-sum game, 613
