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0 on (a,+oo), and p'(x) ~ 0 on [a, +00). Then the problem
(1.19)
(1.20)
p(~) (p(x )y')' = f(x, y, y'),
a
< x < 00,
y'(a) = 0,
has a bounded solution E C 1 [a, 00) n C 2(a, 00). Moreover, if f satisfies the hypotheses Dl and D2 uniformly on compact subintervals of (a, 00), then ' is also bounded on (a,oo). This last theorem should be compared to results in (9) . In order to apply it to the existence of bounded radially symmetric solutions of (3) ,(4) in all of RN, just choose p(x) = X N - 1 with a = o. Any of our first four theorems may be combined with the argument in [1) to give an existence (and uniqueness) result with an appropriate boundary condition at 00. We shall need some tools from [2), which we now state. The stronger form of B 1 ((3) obtained by replacing the weak inequality f(x,y,(3) ~ 0 by f(x,y,(3) > 0 will be designated Bt((3)i the analogous strengthening of B 2((3) is called 8:;((3).
Lemma 1 Suppose f satisfies condition Bt ((3d for some (31 E R and the linear function m(x) = Q + (31 (x - a) satisfies m(x) ~ Yl for a ~ x ~ b. If a ~ r < b and if "(x) = f(x, (x), '(x» , (r) ~ m(r),
'(r)
T
~
X
~ b,
> m'(r) = (3..
then (x) > m(x), '(x) > m'(x) for r < x :5 b.
56 BVPs FOR FDEs Lemma 2 Suppose f satisfies C1 eB1 ) for some (31 > 1, and the function m(x)
IS
defined by m(a) = a, log m'(x) = log (3 exp[Ml (b - x)],
for a::::; x ::::; b, where a
~ Y1
and (3
~
(31 . If a::::; r < band
¢/'(x) = f(x, <jJ(x), ¢>,(x)), t/>(r) ~ m(r), then t/>(x) > m(x), t/>'(x) > m'(x), for
r
r::::; x::::; b,
t/>'(r) > m'(r),
< x ::::; b.
Lemma 3 Suppose f satisfies condition D1 and
t/>"(X) = f(x, t/>(x), t/>'(x)), where a ::::;
r
r::::; X ::::; b,
< b. If 712 ::::; t/>(r) ::::; 7J}' t/>' (r)
~
s ~ Sl(7J2,7J1),
s exp[ -Q1 (712,711)( 711 - 712)],
then either t/>'(x) > s for r ::::; x ::::; b or there exists c t/>'(c) > s.
E
[r,b) such that t/>(c) = 711,
Each of these lemmas have rather obvious dual results, involving the other conditions B:;((32), C2((32), and D 2; see [2] for statements. 2. Proof of Theorem 2 Choose and fix fi satisfying fi2 ::::; fi ::::; fi1· For each integer n ~ "'~a' consider the boundary value problem
(2.1) (2.2)
(2.3)
y" = f(x,y,y')
+ ~(y n
fi) , a::::; x ::::; x' + n ,
g(y(a), y'(a)) y(x' + n) + y'(x' + n)
o fi .
Since this problem satisfies the hypotheses of Theorem 1, then it has at least one solution t/>n on [a, x' + a] . We shall show that Ascoli's theorem may be applied to the sequences {t/>n}, {t/>~} on each compact subinterval of [a, (0) and obtain a solution of (1),(2) as the limit of a subsequence of {t/>n} . If f satisfies B 1((3t} (resp. B 2((32)), then the right side of (21) automatically satisfies the stronger condition Bi((31) (resp. B:; ((32)). Our first major goal is to demonstrate the existence of numbers U1 and U2 so that U2 ::::; t/>n(x) ::::; U1 for a ::::; x::::; x' + n and all n. We need one further preliminary result which pinpoints the purpose of the additional hypotheses of Theorem 2.
57 BVPs ON INFINITE INTERVALS
Lemma 4 Suppose that f(x, y,O) > 0 for x' < x ~ band y > ih . If r/>(b) + ¢/(b) ~ ih, if r/>"(x) = f(x,r/>(x),r/>'(x)), x' ~ x ~ b,
and if a local maximum of r/> occurs at c E (x', b], then r/>( c)
~
iit .
Proof. Suppose that r/> attains a local maximum at some point c E (x' , b), where r/>(c) > iiIi c = b is impossible because r/>(b) + r/>'(b) ~ ih. Thus we have r/>'(c) = 0 and r/>"(c) = f(c, r/>(c), 0) > 0, contradicting the local maximum at x = c. The obvious dual of Lemma 4 will be called Lemma 4'. The next lemma plays a central role in the proof of Theorem 2.
Lemma 5 Suppose the hypotheses of Theorem 2 are satisfied and let r/>n be a solution of the boundary value problem (21)-(23). Then there exist constants Ul and U2 , dependent only on the parameters of the hypotheses relative to the interval [a, x'], and not directly on the function f or the endpoint a, such that for each n, U2 ~ r/>n(x) ~ UI , for a ~ x ~ x' + n. Proof. Throughout this proof, we shall be using the values of the parameters which refer to the interval [a, X1i any explicit occurence of such parameters will always refer to the interval [a, x') . Also we assume, by adjusting Yl and Y2 if necessary, that Yl ~ iit, Y2 ~ iilLetting b = x' and fixing n , we choose a = Yl and let ml(x) be the function of Lemma 1 (resp. Lemma 2) according as the right side of (21) satisfies Bi({h) (resp. Cl (11t)). We then let PI = inf{p(-y) : / ~ O} and choose ml ~ PI and m} ~ max{ml(x) : a ~ x ~ x'} and let tl
_) , ml - PI } . = max { m ,l () a , SI (PI , ml x' - a
Then we choose /1 > 0 such that
where Ql = Ql(Pt,ml). Likewise, we choose m2(x) from Lemma l' (or Lemma 2'), where a = Y2. We put P2 = sup{p(-y) : / ~ O} and choose m 2 ~ P2 , m2 ~ min{m2(x) : a ~ x ~ x'}, and let
and pick /2 < 0 such that
q(-y) ~ t 2 exp[Q2(P2 - m2)), where Q2
= Q2(m2,P2).
for / ~ /2,
58
BVPs FOR FDEs Since rPn satisfies the boundary condition (22), then there exists a/ E R such that = p(!), 4>~(a) = q(!). We now show that /2 ~ / ~ /1. If we assume on the contrary that / > /1> we have 4>n(a) ~ PI> 4>~(a) > tl ~ m~(a). If 4>n(a) ~ ml(a), then Lemma 1 (or Lemma 2) implies that 4>n(x ' ) ~ ml(x' ) ~ YI, 4>~(X') > m~(x') ~ f31 > o. If instead, PI ~ 4>n( a) < ml (a), then Lemma 3 implies that either 4>~ (x) > tl for a ~ x ~ x' or there exists c E [a,x1 such that 4>n(c) = ml, 4>~(c) > t l . In the first alternative,
rPn(a)
4>n(x' ) = 4>n(a) + [
' 4>~(x)dx > PI + tl(X' - a) ~
ml > iiI
and rP~(X') > tl ~ O. In the second alternative, we have 4>n(c) = ml ~ ml(c), 4>~(c) > tl ~ m~(c) and Lemma 1 (or Lemma 2) with r = c guarantees that rPn(x ' ) > ml(x') ~ Yb 4>~(X') ~ m~(x') > O. So, in every case, we conclude that rPn(x ' ) > iiI, 4>~(X') > 0, in contradiction of Lemma 4. Thus,.., ::; /1 . A similar argument shows that / ~ /2. Now choose &1 ~ Yl such that &1 ~ p(!), for /2 ::; / ~ /1, and put ml(x) = ml(x) + (&1 - YI). Likewise, choose &2 ~ Y2 such that &2 ~ p(!), for /2 ~ / ~ /1, and put m2(x) = m2(x)+(&2-Y2) . Finally, let Ul = ml +(&I-Yl), U2 = m2+(&2-Y2). We note the important fact that Ul , U2 depend only on the various parameters which appear in the hypotheses relative to the interval [a, x']; there is no direct dependence on either the function f or the interval [a,x ' + n] . With this preamble, we now complete the proof of Lemma 5 for this case. Suppose on the contrary that 4>n attains a maximum at x where rPn(x) > Ul . Since rPn(a) = p(!) for some / E [,2,/1], then 4>n(a) ~ &1 ~ Ul and so a < x ~ x' + n. By Lemma 4, the maximum must occur at x E (a, x']. Since 4>n (a) ~ &1 = ml (a) and 4>n(x) > Ul ~ ml(x), using the mean value theorem, we find r E (a,x) so that 4>n(r) > ml(r) and 4>~(r) > ml(r). Then Lemma 1 (or Lemma 2) implies that 4>~(x) > m~(x) = f31 > 0, contradicting the maximum at x. Thus 4>n(X) ::; UI on [a, x' + n] and a similar argument proves the inequality U2 ~ 4>n(X) on [a , x' + n]. We now turn out attention to the existence of bounds on
Lemma 6 For each b > a I there exists
x E [a, b] and all n for which x' + n
~
Vi
b + l.
=
4>~(x) .
Vi (b) such that
4>~ (x) ~
Vi for all
Proof. Let 51 = SI(U2, UJ), Ql = QI(U2, Ud be the parameters in Dl associated with f on the interval [a, b + 1] . Put 5 = max{sl' UI - U2} and Vi = s exp[-Ql(UI - U2)]. Suppose there exists n with x'+n ~ HI and r E (a, b) for which 4>~(r) > VI. Applying Lemma 3, we conclude that 4>~(x) > s for r ~ x ~ b + 1, since the alternative of Lemma 3 is clearly impossible. Therefore
4>n(b+ 1) - 4>n(r) =
l
r
b+l
4>~(x)dx
a contradiction since U2 ~ 4>n(x) ~ U1 on [r,b+ 1].
>s
~ Ul - U2,
59 BVPs ON INFINITE INTERVALS Lemma 7 Suppose f satisfies D1 uniformly on all compact subintervals of [a , 00) . Then there exists a constant VI. such that 4>~ (x) :::; VI. for all x E [a , x' + n] and all n . Proof. By hypothesis, we can choose parameters Sl = Sl(U2 , U1) and Q1 = Q1(U2 , U1) for which f satisfies D1 on every compact subinterval of [a, 00). Put s = max{ Sl , yU2} and VI = s exp[-Q1(U2-U1)]. Supposing that there exists nand r E [a, x'+n) for which 4>~ (r) > VI. , Lemma 3 gives a contradiction since 4>~ (x' + n) = y - ¢n (x' + n) :::; y - U2 and ¢n(x) :::; U2 for a :::; x :::; x' + n. The dual of Lemma 6 (resp. Lemma 7) will be called Lemma 6' (resp. Lemma
7'). We now complete the proof of Theorem 2. Given a compact subinterval [a, b] of [a, 00), Lemma 5,6 (and their duals) imply that for x' +n :2: b+ 1, the sequences {¢n}, {¢~} are uniformly bounded on [a, b]. Since f(x, y, z) is continuous on [a, b] x R2, the differential equation (21) shows that the sequence {¢~} is also uniformly bounded on each such interval [a , b]. The familiar diagonalization argument using Ascoli's theorem may now be used to obtain a subsequence of {¢n} which converges (together with the first and second derivatives) uniformly on each compact subset of [a, 00) to a solution ¢ of (1),(2) on [a,oo). From Lemma 5, it follows that U2 :::; ¢(x):::; U1 on [a, 00), so ¢ is bounded. If Lemmas 7, 7' apply, then we also have constants VI., li2 for which li2 :::; 4>'( x) :::; VI. on [a, 00) and so ¢' is also bounded .
3. Proof of Theorem 3 Since it is clear from the hypotheses that Y2 < YI, we choose and fix y satisfying Y2 :::; Y :::; Y1. As before we consider the problem (21)-(23), but we change variables, replacing x by a + x' + n - x to obtain
(3.1)
y" = j(a + x' + n - x,y,y')
(3.2) (3.3)
g(y(x'
+ ~(y - y) , n
a:::; x:::; x' + n,
y(a) - y'(a)
y,
+ n) , y'(x' + n))
o.
Since this problem satisfies the hypotheses of Theorem 1, it has at least one solution on [a,x' + n] . Then ¢n(x) = tPn(a + x' + n - x) is a solution of (21)-(23) . We proceed as in the proof of Theorem 2 to see that Ascoli's theorem may be applied. As before our first goal is the analogue of Lemma 5 for this situation. Until we have achieved this goal, any explicit reference to parameters in the hypostheses will refer to the interval [a,x'] . We may assume that Y1 and Y2 in C1(,8 1) (or B 1(3 1)) and C2((32) (or B 2((32)) satisfy Y1 :2: Y1 and Y2 :::; Y2. In the case that j satisfies B 1((3t} and (31 > t1(g), then there exists 111 and Zl < (31 such that g(y, z) > 0 for y :2: 1111 Z :2: Zl- By increasing Y1 if necessary we may assume that Y1 :2: 111. Similar steps
tPn
60
BVPs FOR FDEs should be taken in the case that j satisfies B2(f32) ' Since these parameters are chosen with respect to [a, x'], they are satisfactory on any subinterval. Define q = f31 if j satisfies BI (f3I) and define q by log q = log f31 exp[MI(x' - a)] if j satisfies C I (f3I). We now show that dE (a, x') and tPn(d) 2 YI imply tP~(d) > -q. Replacing x by a + d - x, we observe that tPn (x) = tPn (a + d - x) satisfies
y"=j(a+d-x,y,y'),
a~x~d.
Defining m(x) as in Lemma 1 (or Lemma 2), with a = YI and b = d, we see that tPn(a) = tPn(d) 2 YI implies tP~(a) ~ m'(a) since otherwise Lemma 1 (or Lemma 2) gives tP~(d) > m'(d) = f31 > ZI and thus
g(tPn(a), tP~(a)) = fJ(tPn(d), tP~(d)) > 0, a contradiction. Now let UI = YI + q2/ 0 + q(x' - a) . Note that the value of UI depends only on the parameters of the hypotheses relative to the interval [a, x']; there is no dependence on the function f or the interval [a, x' + n]. We next show that tPn (x) ~ UI for a ~ x ~ x' + n and all n such that n 2 q/o. We argue by contradiction. If some tPn attains a maximum at some point c where tPn(c) > UI , then Lemma 4 implies that c ~ x'. We assert that tPn(x') 2 YI + q2/ 0 (from which also tP~(x') > -q) i for if not, then the mean value theorem implies the existence of a point d E (c, x') at which tPn(d) > YI + q2/ 0 and tP~(d) ~ -q, a contradiction of the result of the previous paragraph. Let [x', x"] be the largest subinterval of [x', x' + q/o] on which tPn(x) 2 YI. We claim that there exists a point in [x', x"] at which tP~ is positive. If we suppose the contrary, then our hypothesis gives tP~(x)
2 f(x, tPn(x), tP~(x)) 2 0,
and thus tP~ is increasing on [x',x"] . Then integration gives 2
tPn(X") > tPn(x') - q(x" - x') 2 YI lt follows that x" = x'
+ q/o.
2
+~ - ~
= YI .
Then a second integration gives
tP~(x") 2 tP~(x')
+ o(x" -
x') = tP~(x')
+ q > 0,
a contradiction. Thus tPn attains a maximum value greater than YI on the interval (x',x' + n], contradicting Lemma 4 and completing the proof that tPn(X) ~ UI . The construction of a lower bound U2 is similar. Having achieved the analogue of Lemma 5, bounds on the derivatives are provided without change by Lemmas 6 and 7, and the proof is then completed just as in the proof of Theorem 2.
61
BVPs ON INFINITE INTERVALS
4. Proofs of Theorems 4 and 5 To prove Theorem 4, we begin with the problem
(4.1)
y"
= f(x,y,y'),
a + lin ~ x < 00
=A
y(a + lin)
(4.2)
Applying, as appropriate, either Theorem 2 or 3, we see that the problem (27),(28) has at least one bounded solution ¢>n on [a + lin, 00). Moreover, since the hypotheses are satisfied uniformly on closed subintervals of (a, x'], there exist constants VI, V2 , independent of n, for which V2 ~ ¢>n(x) ~ VI for a + lin ~ x < 00. Moreover, the bound provided by Lemmas 6 and 6' for ¢>~ on [a + lin, b] is independent of n. Since f is continuous, the differential equation (27) then provides a uniform bound on ¢>~ on any compact subinterval of (a, 00). Ascoli's theorem may then be used to obtain a subsequence which converges uniformly (together with first and second derivatives) on each compact subinterval of (a, 00) to a solution ¢> of (17) on (a, 00). Let un(x) = ¢>~(x) for a + lin ~ x ~ 1; un(x) = 0 for a ~ x < a + lin. Then
¢>n(x)-A= [un(s)ds,
a+l/n<x~a+l.
Since Un --+ ¢>~ and ¢>n --+ ¢> on (a, a + 1], then the uniform bound on ¢>~ and the Lebesgue bounded convergence theorem imply that
¢>(x)=A+ [¢>'(s)ds,
a<x~a+l
and thus ¢>(x) --+ A as x --+ a, completing the proof of Theorem 4. To prove Theorem 5, we begin with the problem
(4.3)
y" = - p'(x) p( x) y, + f( x, y, y') , a + II n < _ x < 00
(4.4)
y'(a + lin) = O.
This problem satisfies the hypotheses of Theorem 3 because the term - ~y does nothing to change the parameters in the hypotheses satisfied by f(x, y, -z}. Therefore (29),(30) has at least one bounded solution on [a + lin, 00), and as before we use Ascoli's theorem to get a solution ¢> of the differential equation (19) on (a, 00) which is bounded and for which ¢>' is bounded on any bounded interval (a, b]. Let u(x) = p(x)¢>'(x). Then u(x) --+ 0 as x --+ a. Furthermore, if we multiply (19) by p(x) and integrate we obtain
lu(x)1
~ [Ip(s)f(s ¢>(s),¢>'(s))lds.
BVPs FORFDEs Since 4, # are bounded on (a, a + 11 and p(x) is monotone nondecreasing, the continuity of f gives us a constant M for which
Hence IQ(x)I 5 M(x - a ) , so #(x) + 0 as x
-t
a <x 5 a
+ 1,
a, and the boundary condition at x = a is satisfied.
References 1. J. V. Baxley, Nonlinear second order boundary value problems on [0, a), Qualitative Properties of Differential Equations, Proc.1984 Edmonton Conference, W. Allegretto and G. J. Butler, eds., University of Alberta Press, Edmonton, 1986, pp. 50-58. 2. J. V. Baxley, Existence theorems for nonlinear second order boundary value problems, J. Differential Equations 85 (1990), 125-150. 3. J. V. Baxley, A singular boundary value problem in the theory of economic growth, Differential Equations, Proc. 1988 Conference in Athens, Ohio, R. Aftabizedah, Ed., Marcel Dekker, 1989, pp. 40-46. 4. J. Bebernes and L. Jackson, Infinite interval boundary value problems for y" = f(x, Y), Duke Math. Journal 34 (1967), 39-47. 5. H. Berestycki, P. L. Lions, and L. A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in RN, Indiana Univ. Math. J. 30 (1981), 141-157. 6. L. Erbe and K. Schmitt, On radial solutions of some semilinear elliptic equations, Differential and Integral Equations 1 (1988), 71-78. 7. J. Gatica, G. Hernandez, and P. Waltman, Radially symmetric solutions of a class of singular elliptic equations, Proc. Edinburgh Math. Soc. 33 (1990), 169-180. 8. A. Granas, R. B. Guenther, J. Lee, D. O'Regan, Boundary value problems on infinite intervals and semiconductor devices, J. Math. Anal. Appl. 116 (1986), 335-348. 9. Dona1 O'Regan, Singular boundary value problems on the semi-infinite interval, Libertas Math. 1 2 (1992), 109-119.
63
EXISTENCE OF STEADY-STATE SOLUTIONS TO SOME CONSTANT-VOLTAGE PROBLEMS
L. E. BOBISUD Department of Mathematics and Statistics, University of Idaho Moscow, Idaho 83844-1103 USA
1. Introduction
Consider a thin resistive wire stretching along the interval 0 ::; x ::; 1. By "thin" in this context it is understood that cooling of the wire by lateral radiation is accomplished with negligible variation of temperature over a cross section of the wire. It is possible to have a large lateral gradient in the temperature, and hence large heat radiation from the wire, with only small lateral change in temperature of the wire if the thickness orthogonal to the surface is small, by the definition of the derivative. Note that the wire need not have circular or even regular section for this to occur to a satisfactory degree of approximation. Correspondingly, we shall also assume that heat radiation from the wire is governed by its temperature (as in Newton's law of cooling) rather than the normal derivative of the temperature. Application of a (constant) voltage differential if> to the ends of the wire will cause a current 1= 4>1 R, where R is the total resistance of the wire in ohms, to flow and generate heat in the wire as a whole at the rate 12 R = 4>21 R. Unless completely insulated, the wire will dissipate heat through its lateral boundary. If the electrical resistance per unit length r 2: 0 of the wire depends on the temperature u and if we allow spatial dependence as well, the total resistance of the wire is given by
R=
l
r(x, u(x)) dx .
Let k(x, u) denote the heat conductivity of the wire and g(x, u) the rate of lateral heat radiation per unit length; both may depend on position and temperature. Because we wish to include in our formulation such physical systems as a heater element and its power leads, we assume that g, k, and r are merely piecewise continuous functions . Under these conditions we have the steady-state heat conduction equation
(k(x,u(x))u'(x))'
= g(x,u(x)) -
I 2 r(x,u(x)).
64
BVPs FOR FDEs But Ohm's law yields at once that
, , 4h(x,u(x)) (k(x , u(x))u (x)) = g(x, u(x)) - (
J~ r(s, u( s )) ds
)2 '
(1)
Although other boundary conditions could be considered by our methods, we shall restrict ourselves to the Dirichlet boundary condition
u(O) =
Q
u(l) = (3.
(2)
The steady-state problem Eqs. (1)-(2) will not in general admit a solution, for thermal runaway in the corresponding time-dependent system may occur if too great heat generation occurs at high temperatures. Thus our goal is to determine conditions on g, k, and r that will guarantee existence of a steady state. This is similar in spirit to [1), where constant-current devices are considered. In these problems I is fixed, so the functional differential equation (1) is replaced with the ordinary differential equation (ku')' = 9 - Pro A different treatment of a class of constant-voltage devices is given in [4) and several subsequent papers. The specific fixed-point technique we use is application of the topological transversality theorem [5-7) . We exploit the idea, introduced in [2) and refined in [3], of a "forbidden value" of the maximum of a solution; so doing results in substantially weaker hypotheses that involve a supremum instead of a limit (d. hypothesis H3ii below). Eqs. (1)-(2) do not take into account possible variation of density with temperature. In the final section we consider a problem that differs from Eqs. (1)-(2) only in allowing the material density p to be a piecewise continuous function of position and temperature. Because of the temperature dependence, the material will change its length (e.g., the location of its right end point) when the voltage is applied. The new length is not known a priori but must be determined as part of the problem. This additional complication is handled by a device (Lagrangian coordinates) due to Quilghini [8] in this context and used in [1) : each point is mapped back to where it would have been under some arbitrary but fixed temperature distribution. Determination of the appropriate map is part of the new problem, which is a fixed-domain problem of the type specified in Eqs. (1)-(2). Thus existence of solutions for the temperature-dependent-density problem is thrown back to the question of existence for a problem with fixed density. In applications the change in length is likely to manifest itself in sag rather than in change in location of endpoints, but the formulation considered here may be regarded as a one-dimensional model of a physical system with sag. 2. Preliminaries Since we wish to consider physical systems with abrupt spatial changes in material properties, we suppose there is a fixed partition 0 = Xo < Xl < . .. < Xn = 1 of [0, 1]
65 STEADy-STATE SOLUTIONS
such that
HI. 9 and r coincide on each subinterval (Xi-I, Xi) with a function that is continuous on [Xi-I, Xi); k coincides on each (Xi-I, Xi) with a function that is positive and Lipschitz continuous on [Xi-I, X;J. Functions with these properties will be called, respectively, piecewise continuous and piecewise Lipschitz (with respect to the given partition); it seems reasonable to seek a solution which is continuous and whose first and second derivatives are piecewise continuous (with respect to the same partition) and such that the differential equation (1) is satisfied on each (Xi-I, Xi) (equivalently, on each [Xi-I, Xi)). The following lemma is proved in [1) .
°
Lemma 1 Let HI be satisfied and let there exist a constant (J" > such that k(x, u) 2: (J" on [0, 1) x (-00,00). Then for any piecewise continuous function h the boundary value problem (k(x,u(x))u'(x))' = h(x)
u(o)
=
u(I)
Q
= (3
has a unique solution u. By a solution of the equation (k(x,u(x))u'(x))' = h(x) we mean a function u E C[O,I), coinciding on each (Xi-I, Xi) with a function continuously differentiable on [Xi-l>Xi), such that k(x,u(x))u'(x) E C[O,I) n C1[Xi_I,Xi) for each i, and such that the differential equation is satisfied on each subinterval (Xi-l, Xi).
3. Constant Density
As usual in fixed-point arguments, we must first obtain suitable estimates, uniform in the parameter>. E [0,1), for solutions u>. of a suitable one-parameter family of problems, namely
(k(x,u>.(x))u~(x))' = >. [g(X,u>.(X)) -
¢} (
r(x,u>.(x))
J~r(s,u>.(s))ds
rl
u>.(I) = (3.
°
(3) (4)
Note that for >. = this nonlinear problem has a unique solution by Lemma l. Integration over an interval (x, y) yields the basic equalities
k(y-,u>.(y))u~(y-) - k(x+,u>.(x))u~(x+) = >. 2
->.(
J~r(s,u>.(s))ds
fY
)2J.
:z;
t
g(s,u>.(s))ds
r(s,u>.(s))ds
(5)
66 BVPs FOR FDEs and
(6) H2. There exists a number, such that g(x, z) :::; 0 on [0,1) x (-00, ,); r(x, u) >
o.
u~ exists end is zero at any interior local extremum of u.!. and, if H2 holds, then u,\(x) ~ min(a,,8,,) on [0,1).
Lemma 2
Proof. First suppose that u.!. has a local extremum at a point x in (0,1). From Eq. (5) in the limit as x and y approach x we have that k(x+,u.!.(x))u~(x+) = k(x-, u,\(x))u~(x-)), implying that u~(x+ )u~(x-) ~ O. But this and the fact that k -:/; 0 shows that both u~ (x+) and u~ (x-) must be zero. Suppose next that for some .x E [0, 1) u,\ satisfies miIlo<x2
-.x (
J~r(s,u,\(s))ds
{X
)2 iT
r(s, u,\(s)) ds :::; O.
i;
Thus u,\ is nonincreasing On [x, 1), contradicting the boundary condition at 1. It is convenient to set Jl. = min(a,,8, ,); J.L is an a priori lower bound on solutions of Eqs. (3)-(4) for .x E [0,1). H3. Either I.
there exists a constant I< such that
g(x,u) ( ¢> r(x,u) > J~minl'2 21TI) = D (min[O,IJX[I',MoJ T)
=
observe
I .
At an interior maximum or minimum of u), we see from Lemma 2 that u~ = OJ if u), is monotone, there must exist a point xo such that lu~(xo)1 S 1 .8 - al. In either event, there exists Xo E [0,1] such that lu~(xo) 1 S 1.8 - al. Therefore, for any x > Xo
k(x-, u),(x))lu~(x-)1 S and thus
1.8 - al [O,IJX[I'.MoJ max k + DI = D2
lu~(x-)I S D2 / [O,IJmin k x [I',Mo]
= MI'
Of course, by (piecewise) continuity the same bound holds for this bound holds on the whole interval (0,1).
lu~(x+ )Ij
by symmetry,
Theorem 1 Assume that H1-H3 hold; then Eqs. (1)-(2) have a solution. Proof. The conclusions of Lemmas 2 and 3 are valid. Let B be the Banach space of functions in C(O,l] that coincide on each (Xi-I,Xi) with a function continuously differentiable on [Xi-I, x;]j the norm on B is
where-
IIYllo = max IYI, [O,lJ
Ilyllt =
max max ly(x)l1 [X __ 1 .1:';) I
By the Ascoli-Arzela theorem, the injection j : B --+ C(0,1] is completely continuous. Let K be the closed convex subset of those functions y in B satisfying the boundary conditions y(O) = a, y(l) =.8. Let PC denote the class of functions defined on Ui=1 (Xi-I, Xi) and coinciding on each (Xi-I, X;) with a function continuous on [Xi-I, Xi]j PC is equipped with the norm II -IllDefine the map Q on C(0,1] by
(Qy)(x)
= g(x,y(x)) -
(
I
fo r(s,y(s))ds
)2 r (S,y(s))j
70 BVPs FOR FDEs it is easily seen that Q is a continuous map into PC. Let k denote k altered outside the rectangle [0,1] X [11, Mo] so that k(x, u) 2': min(t,.)E[O,I]X[I',Mo] k(t, s) > 0 on [0,1] X (-00,00); a solution of Eqs. (3)-(4) with k replaced by k continues to satisfy the a priori estimates of Lemma 3. Let N : PC -> K be defined by N h = y precisely if y E K with (k(x,y(x))y'(x))' = hex) on each (Xi-I, X;). Lemma 1 guarantees that N is well-defined. The continuity of N as a map into B is straightforward to show; see [1) for details. Let
u = {y E K
: JI. - 1 < y(x) < Mo on [0,1] and
lIy'lh < MI + I};
U is a bounded open subset of K. Any solution u>. E Vof Eqs. (3)-(4) with k replaced by k and oX E [0,1] satisfies JI. ::; u>. < Mo and lu~(x)1 ::; MI by Lemma 3 and thus (i) lies in the interior of U and (ii) is a solution of Eqs. (3)-(4) with k instead of k. H>. : U -> K defined by
H>'y= (No(oXQ)oj)(y) is then a compact homotopy whose fixed points in V are solutions of Eqs. (3)-(4) and conversely; H is therefore fixed-point free on au. Moreover, Hoy = N(O) is the unique solution (by Lemma 1) of the problem
(k(x,y(x))y'(x))' = 0,
yeO) = a,
y(l) =,8.
Thus Ho is a constant map, and so essential. It follows from the topological transversality theorem [5-7] that HI has a fixed point, and therefore that Eqs. (1)-(2) have a solution. 4. Temperature-Dependent Density We now consider a wire that, when zero voltage (or any other convenient voltage) is applied across its ends, occupies the region 0 < ~ < 1 and has temperature distribution T(~); it is not necessary to impose the boundary conditions T(O) = a, T(I) = ,8. We suppose the material property functions k(~,z), g(~,z), r(~,z) and the linear density p(~, z) are known. If now the constant voltage is applied across the ends of the wire, a (different) current will flow, causing the wire to expand or contract to some domain 0 < x < a (we assume for convenience that the left end of the wire is fixed) and the temperature distribution to become ultimately some u(x), 0 < x < a. Consider the map x ...... ~ which sends a point back to where it was before the new voltage and the boundary conditions u(O) = a, u(a) = ,8 were applied. Since mass is conserved, we must have
f p(s,u(s))ds
=
t
p(t,T(t))dt
(8)
71 STEADy-STATE SOLUTIONS
where p{x,u{x» is the density at point x when the temperature distribution is u{·). Clearly, p(x,u(x» = p(e(x),u(x». Using this in the derivativeofEq. (8) we get that d~
p(~(x), u(x»
dx
p(~(x), T(~(x)))'
~(O)
=0
(9)
to determine the function ~(x). In terms of the function ~(x), u satisfies the problem [k(~(x), u(x»u'(x)l' = g(~(x), u(x))
2
u(O)
- (J;'r(~(s),u(s))ds)2r(~(x),u(x)) (0 < x < a),
(10)
= Il!,
(11)
u(a)
= (3
where ~(a) = 1 determines a uniquely. Eqs. (9)-(11) plus ~(a) = 1 define the problem with variable densitYj it is this problem that we want to show has a solution under suitable hypotheses. In this problem we now make the change of independent variable x 1--+ ~ and set z(~(x» = u(x)j we get that
f r(~(s),u(s»ds = loa r(~(s),z(e(s)))ds p(v,T(v))_ = 1 r(v,z(v» ( ()) dv I(z) 1
p v, z v
o
=
(12)
and so we get in place of Eqs. (10)-(11) that
.!!.-[k(~'Z)P(~,Z)dZ] =p(~,T(m[ (t d~
)_L
r (t cl~,T(m d~ p(~,z) 9 \"z I(z)2 \"z z(O) = Il!, z(1) = (3.
)j
(0 and T > there exists a nonnegative Lebesgue integrable function 1/Jl,T (.) defined on [0, T] such that
If(t,,u) l :S 1/Jl,T(t) + tfO(t,,u)F(t,O)
(6)
a.e. in t E [O,T], (t,,u) E M.
Theorem 1 Assume that A is closed and that the growth condition (-y) is satisfied. Then if rtK denotes the set of all admissible pairs for which J (x, u) :S J( , where J( > is a fixed positive constant, the set
°
HK
= {g(.):
g(t)
= D(t,xt} ,
{x,u} E rt K }
is a set of equiabsolutely continuous functions on each interval [0, T], T > 0.
°
°
Proof: Let T > and € > be given. From the growth condition (-y), for u = ~€(I{ + 1t1 there exists 1/J",TO E L1 ([0, T]j IR+) such that for almost all t E [0, T], (t, , u) E M we have If(t,
, u)1 :S 1/J",T(t) + u f°(t, , u)F(t, 0).
77 HEREDITARY LAGRANGE PROBLEMS
Then for each measurable subset E C [0, T] and {x, u} E OK we have
i 1~g(t)1 dt
i 1~ D(t, Xt)1 dt
=i
If(t, Xt, u(t))1 dt
~ i[?jJU,T(t)+(1fO(t,xt,u(t))F(t,O)] dt
~
i
[?jJU,T(t) + (1.f(t,Xt,u(t))F(t,
l
r(s,xs,u(s))ds)] dt
< i ?jJu,T(t) dt +(1 JE [fO(t, Xt, u(t))F(t,
l
~ i ?jJ",T(t) dt 00
+(11+
[fO(t, xt,u(t))F(t,
r(s, x., u(s)) ds)] dt
l
r(s, x., u(s)) ds)] dt
~ i?jJ",T(t)dt+(1I< f I 0 be fixed and define, for t 2:: 0,
(3(t)p+l
¢ 0 there is a 6 = 6(x, f) > 0 such that lIiP(8, t)x - xII ::; f, for all 8 E 0 whenever 0 ::; t ::; 6.
(3) iP(8, t) is a bounded linear mapping from X into X that satisfies the co cycle identity:
iP(8,s+t)=iP(8.t,s)iP(8,t) 8E0; s,tEJR
(2.2)
(4) For each t E JR the mapping of £: into X given by (x, 8) -+ iP(8, t)x is continuous.
88
BVPs FOR FDEs Properties (2) and (3) imply that for each (x, 0) E £ the solution operator t is continuous for t E JR. Indeed one has II~(O, t
+ h)x -
~(O, t)xll
= II[~(O . t, h) -
-+
~(O, t)x
Il~(O, t)xll
which goes to zero as h -+ O. The cocycle identity (2.1) implies that ~(O,t) is an isomorphism with inverse ~-1(0, t) = ~(O
. t, -t)
Vt
E
JR
Proposition 2.1 Let 7r = (~, (7) be a linear skew-product flow on £ . Then there exist constants M ~ 1, a > 0 such that II~(O, t)11
:::; Mea1tl ,
0 E 0, t E JR.
(2.3)
2.2. Projectors and Subbundles A Banach bundle £ with fiber X over a base space 0 with projection P is denoted by (£, X, 0, P), or £ for short, and is defined as follows:
(1) X is a fixed Banach space and 0 is a compact Hausdorff space. (2) The mapping P : £
--+
0 is a continuous mapping.
(3) For each 0 E 0, P-l(O) = £(0) is a Banach space, which is referred to as the fiber over O. (4) For each 0 E 0, there is an open neighborhood U of 0 in 0 and a homeomorphism T :
and
P-l(U) -+ XxU such that for eachN E U, P-l(N) is a mapped onto XX {N} P- 1 (N) --+ X x {N} is a linear isomorphism.
T :
(5) The norms 11·11
= 11·110 on the fiber P-l(O)
vary continuously in O.
One can use the local coordinate notation (x, 0) to denote d. typical point in a Banach bundle £. By this we mean that (x,O) E £ . This is a shortened way to refer to property (4) above. For any subset F C £ we define the fiber F(O) := {x EX: (x,O) E F} . So £(0) := X x {OJ. Also, we define £0 = {(x,O) E £ : x = O} the zero fiber and for U C 0, we shall defin~ the set F(U) := UoEuF(O). A mapping P : £ -+ £ is said to be a projection if P is continuous and has the form P(x, 0) = (P(O)x,O) where P(O) is a bounded linear projection on the fiber £(0). For any projector P we define the range and the null space by
'R = 'R(P) = {(x, 0) E £: P(O)x = x} and
N = N(P) = {(x, 0) E £: P(O)x = OJ.
89 DYNAMICAL SPECTRUM
Since P is continuous, this means that the fibers R(fJ) and .N(fJ) vary continuously in fJ. This also means that P(fJ) varies continuous in the operator norm. The following result can be found in Sacker- Sell [13]. Lemma 2.1 Let P be a projector on £. Then Rand.N are closed in £, and one has
R(fJ) n.N(fJ) for all fJ E
= {OJ,
R(fJ) +.N(fJ)
=£
e.
Definition 2.2 A subset v is said to be a subbundle of £ if there is a projector P on £ with the property that R(P) = V In this case W Whitney Sum.
= .N(P)
is a Complementary subbundle, i. e.,
£ = V
+W
as a
Lemma 2.2 Let V C £ with the properties:
(A) V is closed. (B) V(fJ) is a linear subspace of £(fJ) for all fJ E (C) codim V(fJ) is finite for all fJ E
e.
e.
(D) codim V(fJ) is locally constant on
e.
Then V is a sub bundle of £ . Proof See [13].
o
2.3. Stable, Unstable and the Initial Bounded Sets Let 7r = (cI>, u) be a given linear skew-product flow defined on £ we define the shifted flow as follows: 7r"
= (cI>>"u),
cI>,,(fJ,t)
= e-"tcI>(fJ,t)
= X x e. For A E JR
for t E JR , fJ E
u" = {(x, fJ) E £ : lIe-"tcI>(fJ, t)xll -+ 0,
t
-+
e.
-oo}
The set U" is the unstable set, S" is the stable set , and B" is the initial bounded set corresponding to 7r". If A = 0 we shall denote B = Bo, U = Uo and S = So . We are interested in knowing when S" and U" are complementary invariant subbundles of £. The answer of this question can be formulated in terms of dichotomies.
90 BVPs FOR FDEs Definition 2.3 A project P on £ is said to be invariant if we have P(fJ.t)~(fJ,
~(fJ,
t) =
t)P(fJ), t E JR., fJ E 0
(2.4)
which is equivalent to:
P(8.t) = ~(fJ, t)p(fJ)~-l(fJ, t), t E JR., fJ E 0
(2.5)
Definition 2.4 We shall say that a linear skew-product flow 'Ir = (~,O') on £ has an exponential dichotomy (ED) over an invariant set 0, were 0 C 0, if there is an invariant Projector P on £ and constants k ~ 1, (3 > 0 such that
(2.6) 11~(fJ,t)[I - P(fJ)l~-I(fJ,s)11
for all fJ E
:S ke{3(t-s J,
s~t
(2.7)
0
Remark 2.1 (1) If 0 = {fJ}, then E.D corresponds to the usual concept of dichotomy (2) If 0
= 0, then E.D over 0 is equivalent to the splitting of £.
(3) P(fJ) varies continuously over (4) k,(3 depend of
0.
0.
Proposition 2.2 If'lr is a linear skew-product flow on £ = X x 0 admits an exponential dichotomy over 0, then one has that the initial bounded set B = £0 and the correspondent Projector P is such that: R(P)
= 5(0),
N(P)
= U(0)
£ = R(P) E9N(P) = 5(0) E9U(0) (The whitney sum of two bundles) 3. The Dynamical Spectrum Let 0 be an invariant subset of 0 under the flow under 'Ir is defined as follows: p(0) :=
P
0'.
Then the resolvent p(0) of
E JR. : 'irA admits an exponential dichotomy over
The spectrum 2:(0) of 0 under
'Ir
is defined as follows
Our main results are the following Theorems:
0}.
0
91 DYNAMICAL SPECTRUM
Theorem 3.1 Let 7r = (~, u) be a skew-product flow on £ = X x 0 and 8 a compact connected invariant subset of 0. Then the following statements are valid: (A) There is a > 0 such that 1I~((J,t)1I ~ Me altl , V(J E 0
and
E(8)
=f. 0, E(8,1 C
and t E JR,
[-a,a]
(B) For each set Po < Al < .. . < Am} C p(8) with AO < -a and a < Am such that E(8) n (Ai_I, Ai) =f. ¢ we get that
Vi := Vi (8) = S.d 8)
n U).;_"
i = 1,2,· .. , m
are invariant subbundles of £(8) . (C) Let 7r i be the restriction of 7r to Vi and Ei(8) the spectrum of (Vi, 7r i ) over 8. Then one has that
(D) Ei(8)
= E(8) n (Ai-I, Ai),
i = 1,2, · ··, m
(E) E(8)
= U~I Ei(8)
(F) Vi(8)
n Vj (8)
(G) £(8)
= VI(8) + V~(8) + ... + Vm (8)
= £0(8),
i =f. j (Whitney sum)
In order to get more information about the spectrum we shall put some restriction on the unstable manifold U). for some A E p(8) . Also we will need the following notation: For A E p(8) we shall define: E).(8) := E(8)
n (--00, A)
(3.8)
Theorem 3.2 Assume that dim£(8) = 00, A E p(8), a> 0 is as in part (A) of Theorem 3.1. Then the following statements are valid :
(A) If dimU).
= n(A) < 00,
then A ~ -a
(B) If dimS).
= m(A) < 00,
then A ~ a
(C) If A ~ -a and U). =f. £0(8), then A ~ a
(D) If A ~ a and S). =f. £0(8), then A 2: -a (E) If dimU). --+ n( -a) < 00, as A --+ -a+ then -a E E(8) and dimU). = n(A) < 00 VA E [-a
+ 00) n p(8)
92 BVPs FOR FDEs (F) If dimS" then
(G) If
-+
m(a)
< 00, as A -+ a+
a E ~(0) and dimS" = m(A) VA E (-00, a] n p(0)
1::; dimU"o
= n(Ao) < 00,
then AO E [-a, a]
and
m
~(0) = ~"0(0) u
(U [ai, bi])
(3.9)
1:;:::1
m ::; dimU"o
= n(Ao)
Moreover:
(H) ~"0(0) = [-a,Ao) (I) If
n ~(0)
~(0) C [Ao,a]=> ~(0) = U~l[ai,b;].
(J) &(0)
= S"0(0) + Vl(0) + ... + Vm (0),
where
V;(0)
= U";-l n S,,;>
i
= 1,2, .. . , mj
and with
3.1 . Lemmas
Here we shall derive a number of properties of the spectrum and the resolvent set which will be used in the proof of the main theorems. Lemma 3.1 Let 0 be a compact invariant set in statements are valid:
e
(A) If II~,,(O, t)lI-+ 0 as t -+ +00 for each 0 E and SI"(0) = &(0) for all p. ~ A.
then A E p(0), ~(0) ~ (-00, A),
(B) If II~,,(O, t)11
-+
0 as t
-+
0,
and A E JR. Then the following
-00 for each 0 E 0, then A E p(0), ~(0) ~ (A, +00),
and UA0) = &(0) for all p. ::; A.
Proof We shall prove (A). The proof of (B) is similar. For each 0 E such that II~,,(O, t)1I t ~ T(O).
0, the is T(O) > 0
93 DYNAMICAL SPECTRUM
Consider x E X fixed with IIxll = 1. By the continuity of ~ A (O,t)X with respect to 0 there exist a neighborhood Nx(O) of 0 such that II~A(B,T(O))xll < ~, for all BE Nx(O). Then by the compactness of B we have the following :
e C U~lNx(Oi ) '
T(Ol):::; T(02) :::; . . . :::; T(Om)
We shall put T j = T(Oi), J = 1,2,· · · m. We claim the following:
k
= sup{II~A(O, t)1I
In fact . Assume that K such that
:0E
e,
0:::; t :::; Tm}
< 00 .
= 00 . Then there are sequences {On} C e,
II~A(On) ' tnll
{tn}
C
[0, TmJ
> n , n = 1, 2, 3 . .. .
e
e
Since and [O,TmJ are compact sets we can assume that {On} converges to 00 E and {t n } converge to t* E [0, TmJ. Then by the Banach-Steinhaus Theorem there must be an element Xo E X so that the set:
is unbounded. On the other hand the definition of skew-product flow implies that
which is a contradiction.
e.
Now fix t ~ 0 and let 0 E Then 0 E Nx(Oj,) for some J 1 and II~A(O,Til)xll (!)2 . In the same way O· til E Nx(Oh) for some J 2 and
Now continue this process until one has
T = Til Since
ITl
:::; T :::;
+ .. . + Ti,:::; t < T + TiCIH )
t and 0:::; t -
II~A(O, t)xll
T :::;
and II~A(O, t)xll :::;
1
(2f
Tm , we get the following
= II~A(O.T, t - T)~A(O, T)xll
where a = -,f-In(!) > o. Therefore we have gotten the following
1 :::; k( 2")1
1
:::; k( 2" )t/TI = ke- ext
.(O, t)1I ~ ke-at, 0 E
t ~
13,
o.
From here we get that the skew-product flow 7r>. = (~>.,q) has ED over projections P(O) = I, i.e., A E p(e). On the other hand, if II ~ A, then
II~I'(O , t)1I
= e(>.-I')tll~>.(O, t)1I
~ ke-at, 0 E
13,
t ~
13, with
o.
Therefore,
o Lemma 3.2 Let 13 be a compact invariant set in 0 . Then the resolvent p(e) is open. Moreover if A E p(e), then S>. = SI' and U>. = UI' for all II in a neighborhood of A.
Lemma 3 .3 Let 13 be a compact invariant set in 0 . Then the spectrum E(e) is compact. More specifically, there exists an a > 0 such that, if A > a, then A E p(e), and S>. = E and if A < -a then A E p(e) and U>. = E. Proof Because of Lemma 3.2 we need only to prove that E(e) is bounded. Thanks to Proposition 2.1 we get k ~ 1 and a > 0 such that
II~(O, t)1I ~ ealtl , 0 E
13,
t E JR.
Then if A > a we get 1I~>.(O,t)1I ~ ke{a->.)t -+ 0, as t -+ +00,
for all 0 E
e. Therefore, by Lemma 3.1 one has that (a, 00)
C p(e)
¢:}
E(e)
C
(-oo,a]
Similarly, if A < -a one has II~>.(O, t)1I -+ 0 as t -+ -00 for all 0 E
(-oo, -a) Hence
E(e)
C
C p(e)
¢:}
E(e)
C
e. Consequently
[-a, 00)
[-a, a].
o Lemma 3.4 Let 0 be a nonempty set in 0 and assume that dim E > 1. Then the spectrum E(e) is nonempty.
95 DYNAMICAL SPECTRUM
Proof Pick 80 E
8 and
set Mo
= H(80 )
H(80 )
where
= cl{80 .t : t
E JR}
Then Mo is a compact invariant set and clearly E(Mo) ~ E(8). It will be sufficient to show that E(Mo) is nonempty. From Proposition 2.1, we have k ~ 1 and a > 0 such that 114>(8, t)1I ::; ke'>ltl, 0 E M o, t E JR. By Lemma 3.1 we get the following: (a) If A > a, then A E p(Mo), SA(Mo) = &(Mo) and UA(Mo) = &o(Mo) . (b) If A < -a, then A E p(Mo), UA(Mo) = &(Mo) and SA(Mo) Therefore E(Mo) C [-a,a) .
= &o(Mo) .
Next define
Ao
= infp E p(Mo) : SA(Mo) = &(Mo)}
Then -a ::; A ::; a. For the purpose of contradiction, let us assume that Ao E p(Mo). Then there are two cases to consider:
(i) SAo(Mo)
= &(Mo)
(ii) SAo(Mo)
i: &(Mo) .
For the case (i) the Lemma 3.2 implies that SA(Mo) = &(Mo) for A in a neighborhood of Ao, which contradicts the definition of Ao. For the case (ii) one must have UAo(Mo) i: &o(Mo). From Proposition 2.2 we get that &(8) = UAo (8) + SAo(8), VO E Mo . Then by using Lemma 3.2 once again, we have UA(Mo) i: &o(Mo) in a neighborhood of Ao. This contradicts the fact that UA(Mo) = &o(Mo) for A > Ao close enough to Ao. Therefore Ao E E(Mo).
o Lemma 3.5 Let Al
8
be a compact invariant set in 8. Consider AI, A2 E p(8) with
< A2·
then [AI, A2) C p(8) and SA(8) for all
A E [AI, A2).
= SAl (8),
UA(8)
= UAI (8)
96 BVPs FOR FDEs
Proof In the same way as the proof of Lemma 8 in [12].
o The following Propositions are easy to prove. Proposition 3.1 Let A, Band C be subspaces of X. If C ~ A then
An (B
+ C) =
Proposition 3.2 If.xl>.x 2 E p(0) and
£(0) Lemma 3.6 Let
0
(A n B) + (A n C)
.xl < .x 2, then
= U>.. (0) n S>'2(0) + U>'2(0) + S>.. (0)
be a compact invariant set in 0 and
.xl>.x 2 E p(0)
(3.10) with
.xl < .x2 .
Then the following statements are equivalent: (A) There is a J.l E
(.xl, .x 2 ) n ~(0)
(B) U>.. (0) n S>'2(0)
# £0(0).
Moreover,:F = U>..(0) nU>.,(0) is an invariant subbundle of £
Proof (A) => (B) From Proposition 3.2 we have the following
£ = U>.,(0) + S>.. (0) = u>.. (0) + S>..(0) = U>.,(0) + S>'20) Since U>'2(0) ~ U>.. (0) and S>.. (0) ~ S>'2(0) then
S>..(0) = S>'2(0) and U>..(0) = U>'2(0) Now we can apply Lemma 3.5, it means that
U>.. (0) = U>.(0) and S>.,(0) for
.x E [.xl> .x 2 ]
C p(0) . Therefore
= S>.(0)
(.xl> .x 2 ) n ~(0) = 0 which contradicts (A)
(B) => (A). Define J.l:= infp E p(0) : S>.(0)
Then
.xl < J.l . = U>., nS>'2 nU>. = U>.,
nS>'2
=F
= F.(A) + Fu(A).
Similarly, if A > A2, then
= U>'2 + S>'2
For all A E [AI, A2J we shall use Proposition 3.2 and the fact that £ + S>.,. So we have:
=
U>.,
u>., n (U>'2 + S>'2) u>., n (U>. + S>.) U>. n (U>'2 + S>.,)
U>'2 +F U>. + Fs(A) U>'2 + Fu(A)
Therefore
= U>'2 + F = U>'2 + Fu(A) + F.(A) Fu(A) + F.(A) ~ F, then F = Fu(A) + Fs(A) U>.,
Since
Step 2. p(0) ~ p(0) = 1R \ t(0). In fact, if A E p(0), then there is a projector P : £(8) -+ £(8) and positive constants k and (3 such that
and 1I~>.(e,t)p(e)~~I(e,s)xll:::; kllxlle-.B(t-s),
t 2: s, e E
1I~>.(e,t)(I - p(e))~~l(e,s)xll:::; kllxlle.B(t-s),
0
t:::; s, e E
0
If i> is the restriction of P to F, then
Therefore, by restricting the above inequalities to all (x, e) E F we obtain lI(h(e,t)p(e)ci>~I(e,s)xll :::; kllxlle-.B(t-s),
1lci>>.(O,t)(I - P(O))ci>~I(O,s)xll :::; kllxlle.B(t-s),
t 2: s t:::; s
Thus *>. admits an exponential dichotomy over 0, i.e., A E p(0). So p(0) C p(0) and therefore t(0) C ~(0) . This completes the step 1. Since F ~ S>'2(0), then
1lci>>'2(O,t)xll-+O
as
t-+oo
99 DYNAMICAL SPECTRUM
for all (x, 8) E :F uniformly in x . Hence, applying Lemma 3.1 we obtain that E(0) ~ (-00, A2) and :F.(A) =:F, A ~ A2. In the same way we obtain :F ~ U).., (0) . This implies that II~).., (8, t)1I -+ 0, as t -+ -00 Using Lemma 3.1 again, we get that E(0) ~ (A}, 00) and :Fu =:F. Then
E(0) ~ (AI, A2) n E(0) In order to prove the opposite inclusion, it is sufficient to show that : if (A}, A2), then A E p(0) . In fact, suppose that
A E ,0(0)
n
which is equivalent to :
On the other hand, we already know that E(0) C (A}, A2) ' Therefore,
E(0) n (A}, A2) c E(0) . Step 3. ,0(0) n (A}, A2) C p(0) . In fact , if A E ,0(0) n (AI, A2), then there is a projector Q : :F -+ :F and positive constants k and (3 such that
1I~)..(8,t)(I - Q(8))~:x1(8 , s)xll::; kllxlleP(t-s l ,
t::; s
and for all 8 E 0 Consider the projectors :
and
'R.(P)
= :F = U).., n S)..2 '
N(P)
= U)..2 + S).., .
From the Proposition 3.2 we have
P)..
= PI + QP is a. projector on & such that 'R.(P)..) = S).., + S).., N(P)..) = U).. +U)..2
(3 .12)
100
BVPs FOR FDEs where S~ = R(Q)
U~ = N(Q) .
and
Using (3.11) and (3.12) and the fact that A E (A1> A2), A1> A2 E p(0) one can show that there are positive constants I and a such that
noindent which implies that A E p(0) .
o 3.2. Proof of Main Theorems
Proof of Theorem 3.1 The statements (A) , (B) and (C) follow from Lemma 3.3, 3.4 and 3.6 respectively. The statement (D) follows from Lemma 3.7. (E) From Lemma 3.3 we have that
E(0) c [-a, a] Therefore m
E(0)
m
= E(0) n (U(Ai- 1> Ai)) = U i=l
(F) Consider i
U),;-l
U~; .
~
S),,,
On the other hand we know that
= £0.
(G) From Lemma 3.3 we have that :
= £(0) and U),(0) = £0(0). U~(0) = £(0) and S~(0) = £0(0).
if A > a :::} A E p(0), S),(0) if A
'0 the restriction of 11" to U>'o and ~>'o(S) the spectrum of (U>'o, 11">'0) over S . SO by using Lemma 3.7 we obtain that
Therefore dim Vi ~ 1, which implies that m :=::; dimU>.o ' Now we shall show that the resolvent P>'o(S) of (U>'o, 11">'0) consist of (k + 1) intervals where k :=::; n(Ao) . If P>'o(S) consists of (n(Ao) + 2) intervals: n(>'0)+2
P>'o(S) =
U
i=l
So we get that Then
(e;,d;), di
.o, Ao E [-a, a]
( 4.15) (4.16)
v, = s>.; nU>.._l, i = 1,2, ... ,m
(4.17)
(4.18)
The spectral intervals [a" b,] have been ordered so that b,
:s; ai+l>
i
= 1,2, ... , m-
1.
Let P,: £(0) -+ £(0) denote a projector corresponding to the descomposition (4.4) such that Range (Pi) = n(P i ) = V, and the null space being the sum of the remain Vj and S>'o for j =J i. Then if P>'o : £(0) -+ £(0) is the projector on £ such that
Hence
103 DYNAMICAL SPECTRUM
Given a point (x,O) E £, x (x,O) as follows:
# 0, we shall define the four Lyapunov exponents of
+ A. (x, 0)
= t ....-lim +oo
1 -In lIiP(O, t)xll
(4.19)
At(X,O)
= t ....lim ~InlliP(O,t)xll +oo t
(4.20)
1 A;(X,O)= t .... lim -InlliP(O,t)xll -oo t
(4.21 )
t
(4.22)
Theorem 4.1 (A) If(x,O) E Vi where Vi, is the spectral subbundle associated with Ao and [ai,b i), and x # 0, then the/our Lyapunov exponents (4.7)-(4 .10) lie in [ai,b i]. In particular, if ai = bi then the four Lyapunov exponents agree and the limits lim
t .... +oo
~t In lIiP(O, t)xll = t ....lim ~ In lIiP(8, t)xll -oo t
exist and are equal to ai.
(B) If (x, 0) E S>'o, x # 0 then the two Lyapunov exponents (4.7) - (4.8) agree and the limits lim ~ In lIiP(O, t)xll = lim ~ In lIiP(O, t)xll = Ao. t .... +oo t t ....+oo t Proof The proof of (A) is similar to the prove of Theorem 3 in [12]. In order to prove (B) let us consider (x,O) E S>'o with x # O. Then lim II iP>'o (0, t)xll
t~oo
= t ......lim lIe->'otiP(8, t)xll = 0 +oo
Therefore, there is a constant M > 0 such that
Then
So
o = lim~lnlle->.otiP(8,t)xll = tlim [-Ao + ~InlliP(8,t)xll] t ....oo t . 1 hm -In lIiP(O, t)xll
t .... oo
t
= Ao o
104
BVPs FOR FDEs Definition 4.1 For all 0 E 0 we define the upper Lyapunov exponent >';-(0) and the lower Lyapunov exponent >.t (0) as follows: >':;(0) := sup{>.;(x, 0) : x E X,
x =1= O}
>.t(O) := inf{>.t(x,0) : x E X,
x =1= O}
Theorem 4.2 The upper Lyapunov exponent >';-(0) and the lower Lyapunov exponent >.t (0) associated to 0 E 0 are given by : (4.23)
-1 >.t(O) = - 1_00 lim -In II<J>(O.( -t), t)1I t
(4.24)
References 1. S.N.Chow, and H.Leiva, Dynamical spectrum for time dependent linear systems
in banach spaces, Japan J. of Industrial and Applied Math. 11 (1994), 379-415. 2. S.N.Chow, K.Lu, and J.Maliet-Paret, Floquet bundles for scalar parabolic equations, to appear. 3. J.K.Hale, Asymtotic Behavior of Dissipative Systems, Math.Surveys and Monographs, Vo1.25, Amer.Soc., Providence, R.I, 1988. 4. D.Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, New York, 1981. 5. Y.D.Latushkin, and A.M.Stepin, Linear skew-product flows and semigroups of weighted composition operators, to appear. 6. Y.D .Latushkin, and A.M.Stepin, Weighted Translation Operators and Linear Extension of Dynamical Systems, Russian Math . Surveys 46 (1991) . 7. L.T.Magalhaes, The Spectrum of Invariant Sets for Dissipative Semiflows in Dynamics of Infinite Dimensional Systems, NATO AS! series, No.F-37, Springer Verlag, New York, 1987. 8. A.Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vo1.44, Springer Verlag, New York, 1983. 9. R.T.Rau, Hyperbolic evolution groups and dichotomic evolution families, J. Dynamics Differential Equations, to appear.
105 DYNAMICAL SPECTRUM
10. D.Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Annals of Math. 115 (1982), 243-290. 11. R.J.Sacker and G.R.Sell, Existence of dichotomies and invariant splittings for linear differential systems III, J.Differential Equations 22 (1976 B), 497-552 . 12. R.J.Sacker and G.R.Sell, A spectral theory for linear differential systems, J. Differential Equations 27 (1978), 320-358. 13. R.J.Sacker and G.R.Sell, Dichotomies for linear evolutionary equations in Banach spaces, J. Dynamics Differential Equations, to appear.
107
ON BOUNDARY VALUE PROBLEMS FOR FIRST ORDER IMPULSE FUNCTIONAL DIFFERENTIAL EQUATIONS'
ALEXANDER DOMOSHNITSKY Department of Youth Activities Technion, Israel Institute of Technology Haifa 32000, Israel MICHAEL DRAKHLIN The Research Institute The College of Judea and Samaria Kedumim - Ariel, D. N. Efraim 44820, Israel
1. Introduction
In the present paper we investigate the following scalar equation with impulses m+l
(.ex )(t) == x'(t) +
L
i=l
l' x(s )d.ri(t, s) = f(t), t.
(3j
where to
t
E
[O,b],
(1.1)
ti-l
> 0,
j = 1,2, ... ,m.
(1.2)
= O,tm+l = b,f
: [O,b] -+ Rl, is a locally summable function, ri(·,s) : are measurable, ri(t,·): [ti_bti) -+ R\ has bounded variation R;(t) = Var!~t;_lri(t,s), where R; are locally summable functions, Var!~t;_lri(ti's) = 0, ri(t, s) = ri(t, t) for t > s. W.l.o.g. set ri(t, ti-d = 0 for i = 1, .. . , m + 1. The following equation with delayed argument [O,b]-+
R\
(Mx)(t) == x'(t) + p(t)x(t - T(t)) = f(t), x(~) =
0
for
~
(1.3)
< 0,
(1.4)
is a particular case of (1.1) for
ri(t,s)
={
p(t) 0
for for
t - T(t) ~ s, t - T(t) > S,
ti-l ~ s ti-l ~ s
< ti, < ti·
The research is supported in part by the Israeli Ministry of Science and by the Ministry of Absorption, Center for Abeorption in Science.
108
BVPs FOR FDEs Ordinary differential equations with impulses are intensively studied by many authors, see for example the recent monographs by V. Lakshmikantham, D. D. Bainov and P. S. Simeonov [1] and the bibliography there. Various comparison theorems for solutions of the Cauchy and periodic problems for ordinary differential impulse equations have been obtained in [2] . An extension of the results of such type to the impulse equations with delay meets essential difficulties, implied by the following reasons. In the case of ordinary impulse equations (T = 0) the graph of the solution consists of the the graphs of nonimpulsive ordinary differential equation on the intervals (ti' t i +1 ). Evidently, the solution x(t) of the homogeneous equation in this case preserves its sign. This fact leads to the conclusion about invariance of sign of the Green's function G(t,s) for different problems, since the cross-sections of G(t, s) (for fixed s) are just the solutions of the homogeneous equation. The solutions of the equations with deflecting argument in contrast to the ordinary ones may change their sign. This makes their properties essentially different. The oscillation properties of such equations have received the attention of many researchers. Let us remind in this connection the recent monographs by G. S. Ladde, V. Lakshmikanthan, B. G. Zhang [6], and I. Gyori, G. Ladas [7] . The "addition" of impulses in those equations makes the properties of the solutions even more complicated. The problem of existence of a nonoscillatory solution for the impulse equation with delay x'(t) + ax(t - T) = 0, t E [0, +(Xl ), (1.5)
x(tj) = f3jx(tj - 0), x(O = cp(O
= 1,2, ... ,
j
for
e< 0,
(1.6)
where a, T are positive constants, was studied in the well known paper by K. Gopalsamy and B. G. Zhang [3]. They have obtained the following result: there exists a nonoscillatory solution for (1.5) if the two conditions 1
aT
00
(Xl, i.e. "vanishing" of impulses. One of our results makes it possible to eliminate this essential restriction. Moreover, the proposed approach does not assume that the cofficients p, T are positive constants. Let us point out that in [3], as well as in many other investigations, the equation (1.5), (1.6), (1.2) is considered. It does not lead to any additional generality in comparison with the equation (1.3), (1.4), (1.2), since the first can be presented in
109 FIRST ORDER IMPULSE EQUATIONS
our form, where the right hand side
ret)
= J(t) -
where rp(t)
r is defined as follows
p(t)rp(t), rp(t - ret)),
={
t - ret) < 0,
0
t - ret)
~
O.
The idea of considering just the condition (1.4) is that the space of the solutions of the homogeneous equation (1.8), (104), (1.2), where
(Mx)(t)
= 0,
t E [0,+(0),
(1.8)
is one-dimensional. Note that the general solution of (1.1) , (1.2) has the following representation [4] t
x(t)
= X(t)x(O) +
JC(t, s)J(s) ds, o
where X(t) is the solution of the homogeneous equations £x = 0, (1.2) , satisfying the initial condition x(O) = 1, and C(t,s) is the Cauchy function of the equation (1.1), (1.2). In section 2 we propose a theorem on equivalence of several statements in the case that ri(t, .), i = 1, ... , m + 1, are non decreasing functions for t E [0, b] . Some of the statements are nonoscillation of solutions of the homogeneous equation, positivity of the Cauchy function C(t,s), positivity of the Green's function of the periodic problem, existence of a positive function v, satisfying inequality £v ~ 0 and conditions (1.2). Choosing a function v, we get effective tests for nonoscillation and positivity of different Green's functions. In conclusion let us point out that the positivity of the Green's function could be a base for construction of special monotonic techniques for obtaining comparison results for nonlinear impulse equations with delay. 2. On Sign Properties of Green's Functions Consider the equation (1.1), (1.2) . Its general solution has the following representation t
x(t)
=
C(t,O)x(O)
+
JC(t,s)J(s),
(2.1)
o
where C(s, t) is the Cauchy function of (1.1),(1.2). For a fixed s E [0, b] the function C(·,s) is a solution to the homogeneous "s-truncated" equation
(C"x)(t) == x'(t)
t.
m+l
1
t.
+ • x(r)drk(t,r) + 2~1 ti~l x(r)dr;(t,r) =
J.
0,
t
~
s,
(2.2)
110
BVPs FOR FDEs
x(tj) = (3jx(tj - 0),
where k is an integer such that tk-l ~ s < tk, while C(s,s) = l. Denote by D = D[O,bj(tl,"" t m ) the space of functions x : (0, b] continuous on each interval (ti-I, ti) having a finite left limit at t i. If the boundary value problem (1.1), (1.2) and
x(b) = 0, is uniquely solvable in the space can be represented in the form
D
(2.3)
j=k,k+l, . .. ,m. -+
Rt, absolutely
(2.4)
for every function
J E L = L[O,bj, then its solution
b
x(t) =
JG(t, s)J(s) ds, o
where the Green's function G( t, s) of this problem is
G( t ,s )=C( t,s )_ C(b,s)C(t,O) C(b, O)
(2.5)
°
and C(t,s) = for t < s. If the periodic problem (1.1), (1.2) and
x(O) - x(b)
=
°
(2.6)
is uniquely solvable in the space D for every f E L, then its solution can be represented in the form b
x(t) =
JP(t, s)J(s) ds, o
where
P(t s)=C(t s)- C(b,s)C(t,O) , , l-C(b,O) '
(2.7)
The boundary value problem
x'(t) = J(t), x(tj)
= (3jx(tj -
t E (0, b],
0),
j
= 1,2, .. . , m,
(2.8)
x(b) = 0, has the unique solution for every summable f. Denote by Go(t, s) the Green's function of problem (2.8). Note that Go(t,s) = for ~ s ~ t ~ band Go(t,s) < 0 for o ~ t < s ~ b.
° °
111 FIRST ORDER IMPULSE EQUATIONS
Define the operator K : D
f-+
!
D by the equality
b
=-
(Kx)(t)
o
t-
m+l
'£[1' ,=1 t._
Go(t,s)
(2.9)
x(r)dTr;(s,r)]ds. 1
Theorem 2.1. Let r;(t,·),i = 1, .. . ,m + 1, be nondecreasing. Then the following assertions are. equivalent: (I) the Cauchy function C(t, s) of (1.1), (1.2) is positive for 0 ~ s ~ t ~ b, (2) a nontrivial solution of the homogeneous equation LX = 0, (1 .2) has no zeros on
[0, b], (3) the spectral radius of the operator K is less than one, (..f) problem (1.1), (1.2), (2 ...1) is uniquely solvable for every f ELand its Green's function G(t,s) is negative for 0 ~ t < s ~ band nonpositive for 0 ~ s ~ t ~ b, (5) (only in the case (31 < 1, . . . , (3k < I) the periodic problem (1.1), (1.2), (2.6) is uniquely solvable and its Green's function P( t, s) is positive for t, s E [0, b], (6) there. exists a nonnegative function v E D such that v(b) -l(Lv)(s)ds > 0,
(LV)(t) ~ 0,
t E [O,b] .
Proof. We prove Theorem 2.1 according to the following scheme:
(6)
=?
(3)
(4)
=?
=?
(6),
(3)
=?
(1)
=?
(5)
=?
(2)
=?
(6).
(6) =? (3). The function v satisfies the integral equation v(t) - (Kv)(t) = t/J(t),
where
t E [0,
b],
b
t/J(t)
!
= v(b) -
Go(t, S)(LV)(S) ds.
t
Since 1/J(t) > 0, the spectral radius p(I
D[v,Jjj, where [v, J.Lj ~ [0, b], by the
equality
where
G~Jj(t,
s) is Green's function of the problem x'(t) = f(t),
t E [v,J.Lj,
k is an integer such that tk-l < s < tk, t8 = J.L, where D[v,Jj] is the space of functions x E D[O,b](t1, ... ,t m ) with support [v,J.Lj . The proof is based on the following assertion. Claim. If p(K) < 1, then p(I{VI') < 1 for [v, J.Lj ~ [0, bj. To verify the claim, by virtue of (3)=?( 4) problem (1.1), (l.2),(2.4) is uniquely solvable and its Green's function G(t,s) is nonpositive for 0 :::; s :::; t :::; b and negative for 0:::; t < s :::; b. The function b
v(t) = -
JG(t,s)ds °
is a positive solution of the boundary value problem
(.cx)(t)=-l,
t E [0, b], j = 1,2, ...
x(b) =
,m,
o.
It is clear that
v'(t)
+ J~' x(s)dsrk(t, s) + L:f=k+1 JiLl x(s)d.r;(t, s)
:::; -1 ,
t E [v, J.L]'
113 FIRST ORDER IMPULSE EQUATIONS
where k is an integer such that tk-l < s < tk, to = J.L . Now, following the proof of the assertion (6)=>(3), we obtain that the spectral radius p(Kv ,,) is less than one. This establishes the claim. We continue the proof of the assertion (3) => (1). Let us assume the contrary. Then there exist v and J.L (v < J.L) such that C(v,J.L) = o. In this case u(t) = C(t,v) is a characteristic function of the operator K v" and Lemma 2.1 implies that p(/{v,,) < 1. (1) => (5). The periodic problem (1.1), (1.2), (2.6), is uniquely solvable if and only if C(b,O) #- 1. Since C(t,s) > 0 for 0::; s::; t::; b, then obviously C(b,O) is also positive. Since 131 < 1, . .. , 13k < I, C(·, 0) is nonincreasing and C(b,O) < C(O,O) = 1. Now (2.7) implies positivity of P(t,s). (5) => (2) . Setting t < s in (2 .7), we obtain that the function C(t,O) can not have a zero on [0, b] . In order to prove (2) => (6) we set v(t) = C(t,O). Note that the assertion (1) => (2) is obvious. Theorem 2.1 is proved. Denote R.;(t,s) = Vare=ori(t,e),
+( ri
_(
)_ R.;(t , s)+ri(t,s) t, s 2 '
ri
)_R.;(t,s)-ri(t,s) t, s 2 .
Let C+( t, s) be the Cauchy function of the equation
x'(t) +
El~,
X(ti)
x(s)d.rt(t, s) = f(t),
= f3iX(ti -
0),
t E [0, b],
i = l, .. . ,m.
Theorem 2.2. IfC+(t,s) > 0 forO::; s::; t::; b, then C(t,s)
~
o ::; s ::; t ::; b. Proof. Using the substitution t
x(t) =
JC+(t, s)z(s) ds, o
we obtain the following equation in the space L,
z(t) - (Hz)(t) = f(t), where the operator H : L
f-+
(Hz)(t) =
t E [0, b],
L is defined by the equality
%l~,[1'
(3.10)
C+(S,T) Z(T) dT]d.r;-(t, s) .
C+(t,s) > 0 for
114
BVPs FOR FDEs The spectral radius of the Volterra operator H: L Neumann series, we obtain
z
= (I -
Htl f
= (I +
t-+
L is equal to zero. Using the
H + H2 + ... )f
and t
x(t) =
JC+(t, s)[(I + H + H2 + . .. )f](s) ds . o
The positivity of C+(t , 3) and of the operator H implies that C(t, s) ~ C+(t, s) > 0. Remark 2.1. Obviously, letting b tend to infinity, we can obtain, with the help of Theorem 2.1 and Theorem 2.2, the properties of C(t, s) and nonoscillation of the solutions on the infinite interval [0, +00) . Consider the following impulse equation of neutral type
(.cx)(t) == x'(t) + p(t)x(t - r(t)) - q(t)x'(g(t)) = f(t) , x(tj) = (3jx(tj - 0) ,
(3j > 0,
x(o = x'(O = 0,
~
t E [0, +00), (2.11)
j = 1,2, . .. ,
< 0,
where g,p,J : [0,+00) --+ Rl are locally summable, r : [0,+00) --+ [0, 00) is measurable, g(t) ~ t . Let q and 9 be such that the spectral radius p(s) of the operator
S : L--+L
(Sy)(t) = {q(t)y(g(t)) , g(t) ~ 0, 0, g(t) < 0,
is less than one unifromly for every b E (0, +00) . For the necessary and sufficient conditions for the inequality p(S) < 1 to be valid see [8] . In the case p(S) < 1 the neutral equation can be rewritten as follows
(Cx)(t) == x'(t) + ((I - StITx)(t) = (I - S)-I f(t) ,
t E [0, 00),
where
(Tx)(t) = { p(t)x(t - r(t)), t - r(t) ~ 0, 0, t-r(t) 0 for 0 S; s S; t < + 00, we set
I
exp ( -e p+(s) dS),
I
/31 ex p ( -e p+(s) dS), v(t) =
(2.13)
Theorem 2.2 implies that C(t,s) ~ C+(t,s) > 0 for 0 S; s S; t < +00 . Obviously, that nontrivial solution x(t) = C(t,O) is positive on [0,+00). The corollary has been proved, since all nontrivial solutions to (2.11) are proportional. Remark 2.2. From the proof of Corollary 2.1 it is clear that inequality (2.12) implies existence of a nonosciliating solution to equation (1.5), (1.6), (1.2), considered by K. Gopalsamy and B. G. Zhang [3J. On the other hand their result (see inequalities (1. 7)) obtains new applications. For example, inequalities (1.7) guarantee that the Cauchy function C(t,s) is positive. Remark 2.3. In the case /3j > 1, for all j, impulses can improve nonoscillation properties of solutions. Let us formulate a result for the equation (1.3). Denote ~ = sup(tj+l - tj), /3 = inf . h = In/3. j /3J, j
116
BVPs FOR FDEs Corollary 2.2. Let there exist a natural k such that t 7"
> k6.,
J
t-T(t)
1 + kh p +() s ds~--, e
t
E [0, +00).
Then the homogeneous equation (1.3), (1 .4), (1 .2) has a nonoscillating solution.
Consider equation (1.3), (1.4) and j = 1, 2, ...
(2.14)
Theorem 2.3. Let m = 1, Q; 2: 0 for j = 1,2, . .. and inequality (2. 12) be fulfilled. Then the inequalities f(t) 2: 0 for t E [0, +00) and x(O) > 0 imply that x(t) > 0 for t E [0,+00). Proof. The proof of this theorem is based on the following idea: Let a function x be a solution of the equation (1.3) , (1.4), (2 .14), and let bI be the first zero of x, i.e. x(t) > 0, t E [0, bI)' x(bd = O. Then there exist 1;, j = 1,2, ... , such that x is also a solution of the the equation (1.3), (1.4) and j = 1,2, ...
(2.15)
for the interval [0, bI)' The inequalities Q; 2: 0, j = 1,2, . .. , imply that 1; 2: (3; > 0 for j = 1,2, .. .. Obviously, for the equation (1.3) , (1.4) , (2.15) the conditions of Corollary 2.1 are fulfilled for the interval [0, bd . Corollary 2.1, Theorem 2.1 and Theorem 2.2 imply that the Cauchy function C-y{t,s) of the equation (1.3), (1.4), (2. 15) is positive for o ~ s ~ t < +00 . The solution x of the equation (1.3), (1.4), (2.14) can be represented in the form t
x(t) =
J
C-y(t, s)f(s) ds
+ C-y(t,O).
(2.16)
o
Now the inequalities C-y (t , s) > 0, f(t) 2: 0, x(O) > 0 imply that x(t) > 0 for t E [0, btl. This contradicts the assumption x(b I ) = O. Remark 2.4. It is clear from the proof of Theorem 2.3 that nontrivial solutions of the homogeneous equation (1.8) , (1.4), (2.14) have no zero on [0, +00) . References
1. V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. 2. S. Hu and V. Lakshmikantham, Periodic boundary value problems for second order impulsive differential systems, Nonlinear Analysis, Theory, Methods and Applications 13 (1989), 75-89.
117 FIRST ORDER IMPULSE EQUATIONS
3. K.Gopalsamy and B.G.Zhang, On delay differential equations with impulses, J.Math . Anal. and Applications 139 (1989), 110-122. 4. A.Domoshnitsky and M.Drakhlin, Nonoscillation of first order impulse differential equations with delay, submitted.
5. P.Zabreiko, et al., Integral Equations, Moscow, Nauka, 1968.
6. V. Lakshmikantham, G. S. Ladde and B. G. Zhang, Oscillation Theory of Differential Equation!; with Deviating Arguments, Dekker, New York, 1989. 7. I. Gyori and G. Ladas, Oscillation Theory of Delay Differential Equations, Clarendon Press, Oxford Mathematical Monographs, 1991.
8. M.Drakhlin, The inner superposition operator in the spaces of summable functions, Izvestiya VUZov, Matematika 30 (5) (1986), 18-24 (in Russian).
119
LINEARIZED PROBLEMS AND CONTINUOUS DEPENDENCE
JEFFREY A. EHME Department of Mathematics, Spelman College 350 Spelman Lane S. w., Atlanta, GA 30314 USA
1. Introduction
In this paper, a technique for establishing the continuous dependence of solutions on boundary data will be considered. This technique can be used for many types of different problems. The types of problems considered in this work include functional boundary value problems, boundary value problems for ordinary differential equations, along with boundary value problems for quasi-differential equations and difference equations. The techniques used in this paper possess two properties that are very useful. The first property is no Green's function is required. This means the problems do not need to be converted into integral equations before this technique can be applied. Since the existence of a Green's function is equivalent to uniqueness for the homogeneous problem, it is not necessary to make this assumption in order to obtain continuous dependence. However, it will be necessary to make a uniqueness assumption on a linearized problem. There also will be no need to assume solutions are bounded a priori. If one makes use of a priori bounds, then similar continuous dependence theorems can be proved using topological transversality methods. For related ideas, see [4,7]. Beginning in section 2, a series of preliminary definitions will be considered. In order for these definitions to apply to the four diverse areas to be considered, these definitions will necessarily have to be abstract. Following these abstract definitions, concrete examples will be considered. The third section contains the proof of the general continuous dependence theorem. In section 4, the general theorem is applied to the specific cases previously considered.
120
BVPs FOR FDEs 2. Definitions and Examples In this section, the spaces and functions used in the proof of the main theorems are defined. After the abstract definitions are completed, several specific examples are considered. Let P denote a finite or infinite dimensional Banach space. This Banach space will represent the space of parameters. Let X, Z, and W also denote Banach spaces. In the most of the applications, these spaces will be function spaces. The space X is the solution space, while the spaces Z and Ware test spaces used to verify that potential solutions satisfy the necessary properties. The norms on these spaces are denoted I . Ip, I · Ix, I · Iz , and I . Iw respectively. The space P x X is a Banach space under the product norm I (p,x) Ipxx= max{1 pip, I x Ix} . Assume
F,L: P
X
X -t Z
are n-times continuously differentiable functions. Also assume for fixed pEP, L(p,·) is a linear transformation. Further assume
B:PxX-tW is n-times continuously differentiable. Define
G:PxX-tZxW by
G: (p,x) -t (L(p,x) - F(p,x),B(p,x)) . It is important to note that zeros of the function G are equivalent to solutions of the abstract system of equations
L(p,x) B(p,x)
F(p,x) O.
(1) (2)
Also note there is no loss of generality in taking all the parameters from the same Banach space. Before we continue the development, we point out that there are several special cases which have been of interest to various researchers.
121 CONTINUOUS DEPENDENCE
2A. Boundary Value Problems for Ordinary Differential Equations The first special case to be considered involves boundary value problems for the nth-order differential equation
yIn)
= f(t,y,y', ... ,y(n-I),A),
(3)
where f(t, y, y', .. . , y(n-I), A) is n-times continuously differentiable on I x Rn+l, and I = (a - E, b+ E) is an interval of the reals containing [a, b]. Let i l , . .. , im be positive integers with 2 :::; 2:i:,1 i l = n and assume tl < t2 < ... < tm are contained in the interval (a, b), then a boundary condition of the form (4) where Ij = O, ... ,ij -l,j = 1, ... ,m is called an (il, ... ,i m ) conjugate boundary condition. If instead we are given points tl :::; t2 :::; ... :::; tn, then boundary conditions of the form
(5) are called right focal boundary conditions. Various properties of these boundary value problems have been studied extensively. See [11,15,16,17] . For this special case, X = C(n)[a, b], Z = C[a, b] and W = Rn. The norm on C(n)[a,b] is the usual norm I y 1= maXo$'$n{maXtE[a,bj{1 y{i)(t) I}}. The space of parameters is given by P = R x Rn x Rn The parameters in this example represent boundary points, boundary values, and A. The maps Land F, which map
L, F : R x R n x R n x C(n)[a, b]--> C[a, b] are defined by
L(A, t l , ... , tn, YI, ... , Yn, y(t)) = yIn) and
F(A, t l , ... , tn, yl, ... , Yn, y(t)) = f(t, y, y', .. . , y(n-I), A). For conjugate boundary conditions, the mapping B can be defined by
B(A, t l , ... , tn, yl, ... , Yn, y(t)) ( y (t) 1
-
. C1,O, ···, Y ('I-I)(t 1 )-C1,ll-t,· ·"
Y
=
'm-I(t Tn )-C· ) m,lrn-1'
For a right focal boundary condition, the mapping B is defined by
B(A, tl, ... , tn, YI,···, Yn, y(t)) = (y(tl) - YI,···, y(tn)(n-I) - Yn). Nonlinear boundary value problems for nth order differential equations have also been considered in [2,18]. These involve problems with boundary conditions of the form
9.(A, y(t.), y'(t.), ... , y(n-I)(t;))
= Y"
(6)
122
BVPs FOR FDEs where 9i(XO' Xt, ... , xn) : Rn+l --+ R is n-times continuously differentiable. boundary value problems of this form, the mapping B is defined by
B(A, tt, . .. , tn, yt, . .. , Yn, y(t)) =
For
(91 (A, y(td, y'(t 1 ), . .• , y(n-l)(t 1 ))
- Yl,· ·· , 9n(A, y(t n}, y'{tn), .. . , y(n-l)(t n)) - Yn).
2B. Functional Differential Equations For this case, we consider a problem of the form
(7)
y'(t) = f(t,y(t),y(tP(t))), My(r(t))
+ My(u(t)) =
f(t),
(8)
where f(t,yt,Y2) : [a,b] x R 2n --+ Rn is a continuously differentiable function, tP E C([a, b], R), r, 17 E C([Q, a], R), f E C([Q, a], Rn), and M, N are real constant n x n matrices. These boundary conditions generalize the usual initial delay conditions. The boundary conditions also generalize the functional boundary conditions given by Hale in [8]. Problems of this sort were considered by Ehme and Henderson in [3]. For this case, X = C 1 ([Q, b], Rn) . The space of parameters is given by P = C([a,b],R) x C([Q,a],R)2 X C([Q,a],Rn). Notice that tP,r,u, and f are being considered parameters. The test spaces are given by Z = C([a, b], Rn) and W = C([Q, a], Rn). The functions Land F, which map
are defined by
L(tP,r,u,f,y(t)) = y'(t) and
F(tP,r,u,f,y(t))
= f(t,y(t),y(tP(t))) .
The functional boundary condition becomes the map
B(tP,r,u,f,y(t)) = My(r(t))
+ Ny(u(t)) - f(t).
It is easily verified that the functions L, F and B are well defined and map the correct domain spaces into the correct range spaces.
2C. Quasi Differential Equations Quasi differential equations of the nth order are equations of the form
123 CONTINUOUS DEPENDENCE
where pi(t) > 0 and Pi E en-i([a, b], R) for 0 :::; i :::; n - 1. These problems have been studied by [5,12, 14J. Notice that if Pi(t) = 1, for 0 :::; i :::; n - 1, then these problems reduce to ordinary boundary value problems. The boundary conditions considered in this work are conjugate conditions. Let il, ... ,im be positive integers with 2 :::; E~l i, = n and assume tl < t2 < ... < tm are contained in the interval (a, b), the boundary conditions at tj become
(10) For these problems the parameter space becomes a rather complicated looking Banach space. Set
The spaces X,Z, and Ware the same as for the example with ordinary differential equations. The maps L, F, and B are defined by
L(Po, ... ,Pn-l, tl, . . . , t m, Yl, ... ,Ym,im-l, y( t)) = (Pn-l (. .. PI (Poy)')' ... )'
F(Po,··· ,pn-l, tl, ... ,tm,Yl, .. . ,Ym,im-l,y(t))
=
- f(t,PoY,Pl(PoY)',· . . , Pn-2( .. . PI (poy)')' ... )')
B(Po, ' " ,Pn-l,tl, . . . , tm,yl,·· · ,Ym,im-l,y(t)) =
(POy(tl) - Yl,O,··· ,Pim-l('" PI (POy(tm))')' ... ) - Ym,im-I)'
Notice that although L is not linear, it is linear in y(t) for fixed Po,·· · ,pn-I.
2D. Vector Difference Equations For this example, we consider an n-dimensional vector difference equation of the form
u(m + 1)
= f(m, u(m))
(11)
where u satisfies the multipoint boundary conditions k
EM,u(m,) = r,
(12)
'=1 where ml < m2 < ... < mk belong to the integers Z, r = (rl, ... , rn) E Rn, and Ml, ... ,Mk are constant real n x n matrices We will use the notation Mr to denote
124
BVPs FOR FDEs the 8th row and tth column entry of the matrix Mi . We assume f(m, x) : ZxRn -+ Rn is continuously differentiable for each fixed value of m. Problems of this type have been considered in [13,17] We are only interested in solutions on finite length discrete intervals of the form [mo, mk+l] where mo < ml < . . . < mk < mk+l ' In this case, our solution space X = Rnm o x··· x Rnm k+l where each Ri is a copy of R. The notation u(i) will be used to denote a vector in Rf, where mo ~ i ~ mk+l' The solutions are being viewed as a sequence of finite dimensional vectors. The parameter space P is defined by P = Matnxn(R)k x Rn. Here the parameters are the components of the matrices M; and the components of r. The first test space is given by Z = Mat nX (mk+l-mo)(R). That is, each column of the test space is an n vector and there are mk+l - mo columns, each representing the solution at a different time. Also, W = Rn. The functions L,F are defined by n L, F : Matnxn(R? x R x R::'" x . .. x R::'k+l
-+
Mat nX (mk+1-mo)(R)
with
L(M1 1, ... ,Mrn ,rl " u(mo), u(mo
'r n ,
+ 1), ... , u(mk+t})
(u(mo + 1), .. . , u(mk+l))
and
F(Mf\ ... , Mr n , rl
' " rn ,
u(mo) ,
u(mo + 1), ... , u(mk+l)) =
(f(mo, u(mo)), ... ,f(mk+l - 1, u(mk+l - 1))
The boundary function B is defined by n x··· x R n B : Matnxn(R)k x R n x Rmo -+ R n mk+l with
k
u(mo), u(mo
+ 1), .. . , u(mk+l))
LMu(mt}-r 1=1
3. Proof of the General Theorem In this section we will prove the main theorem . The theorem is an application of the Implicit Function Theorem, see [1] . We assume that the spaces considered have the properties defined in section 2. Suppose G:PxX-+ZxW
125 CONTINUOUS DEPENDENCE
is defined by
G(p, x) = (L(p,x) - F(p,x),B(p,x)). Lemma 3.1 The function G is continuously differentiable. Proof: Each of the component functions of G is assumed to be continuously differentiable, thus G is continuously differentiable. QED
Lemma 3.2 If the linear problem
dLx(Po, xo)( x) - dFx(Po , xo)( x) = z dBx(Po, xo)( x) = w. has a unique solution for each choice of z E Z, w E W, where Xo is a fixed solution to (2.1),(2.2) with parameter Po, then dGx(Po, xo)(') is a linear homeomorphism. Proof: From Lemma 3.1, G is differentiable at the point xo, hence the partials exist. We compute the partial with respect to the x slot as follows. Let hEX, then using the definition of the partial derivative coupled with Taylor's Theorem, we obtain
G(po, Xo + h) - G(po, xo) (L(po, Xo + h) - F(po, Xo + h), B(po, Xo + h))(L(po, xo) - F(po, xo), B(po, xo)) (dLx(Po, xo)(h) + el (h) - dFx(Po, xo)(h) + e2(h), dBx(Po, xo)(h) + e3(h)) (dLx(Po,xo)(h) - dFx(Po,xo)(h),dBx(Po,xo)(h))+ (el(h) + e2(h),e3(h)). The first expression is the partial of G at Xo with respect to the x slot. By definition it is a continuous linear function. The second expression is easily verified to be an error term of the correct order. Thus the partial of G is given by
dG x(Po , xo)(h)
= (dLx(Po, xo)(h) -
dFx(Po, xo)(h), dBx(Po, xo)(h)).
Upon inspection, this expression is seen to be closely related to the linearized problem. From our assumptions on the linearized problem, this partial is seen to be one to one and onto. The Open Mapping Theorem yields dGx(Po, xo) is a linear homeomorphism. QED The two lemmas above are used to prove the main theorem.
126
BVPs FOR FDEs Theorem 3.1 Assume Xo is a solution to (2.1),(2.2) corresponding to the parameter Assume also the hypothesis of Lemma 3.2 concerning the linear problem are satisfied. Then solutions of (2.1),(2.2) depend continuously on Po. That is, given f > 0, there exists Ii > 0 such that 1po - PI Ip< Ii implies there exist a solution XI satisfying L(PI,xI) = F(PI,XI) and B(Pt,XI) = 0, and moreover, 1Xo - Xl Ix< f .
Po.
Proof: It has already been established that G(po, xo) = 0, G is continuously differentiable, and dGx(Po, xo) is a linear homeomorphism. These are the hypotheses necessary for the Implicit Function Theorem. The Implicit Function Theorem gives the existence of a continuous function X defined on an open set U C P, containing Po, such that X : U -+ X and G(p, X(p)) = 0 for all P E U. This yields X(p) is a solution to the problem
L(p,X(p)) = F(p,X(p)) B(p, X(p)) = o. Since X is a continuous function of p, we obtain the continuous dependence result required. QED
4. Applications The abstract continuous dependence theorem proved in section 3 will now be applied in succession to the four concrete examples considered in section 2. The benefit of the abstract approach taken in section 2 is that the proofs of Theorems 4.1-4.4 are all applications of Theorem 3.1. Because of this fact, we will only give ,the proof for the first theorem. Theorems 4.1 and 4.2 have appeared in print [2,3] although they were not presented as a special case of a more general construction. Theorems similar to Theorem 4.4, but involving a different method of proof, have appeared in print [13]. The application to quasi-differential equations, Theorem 4.3, is a completely original application of the techniques used in this paper. The first case considered is that of boundary value problems for ordinary differential equations. Here we consider only the case of nonlinear boundary conditions, as the other two types considered are special cases of these conditions. Theorem 4.1 Assume f,g and the ti are as in section 2A. Let y(t) be a solution to the BVP (2.3),(2.6). Assume the linearized problem Z (n) --
"n L.Ji=i !!L(t 8 Yi
~,
-y, """y ,
... ,
-y(n-I) , A') z (i-I)
Ei'=l ~(A, y(tk)' y'(tk)' . .. , y(n-l)(tk)) z(i-l)(tk)
=
0,
1 ::;
k::; n,
has only the trivial solution. Then given f > 0, there exists Ii > 0, such that if 1ti - Xi 1< Ii, 1Yi - Zi 1< Ii, and 1A - N 1< Ii, then there exists a solution y(t) to the
127 CONTINUOUS DEPENDENCE
BVP
yIn) = f(t, y,y', ... , y(n-I), N) 9i(N, y(Xi), y'(Xi), ... , y(n-I)(Xi)) = Zi,
Moreover, 1y 0, there exists 6 > 0, such that if 1,p - ~ 1< 6,1 7" - 7' 1< 6,1 u 1< 6,1 r - f 1< 6, then there exists a solution y to (2.7),(2.8) corresponding to the data ~, 7', u, and f. Moreover, MZ(7"(t))
(1' -
1y(t) -
y(t)
1< f
for all t E [a, b).
To simplify the statement of the next theorem, we define a series of operators by = PoZ,QI(Z) = PI (PozY, Qi(Z) = P2(PI(POZYY, and so on, where the Pi are as in section 2C.
Qo(z)
Theorem 4.3 Assume f, the Pi, and the ti are as in section 2C. Let y(t) be a solution to the quasi-boundary value problem (2.9),(2.10). Assume the linearized problem (Pn-I ( ... , PI (poz)'Y . . . , )' =
Poz(t l )
=
E;=I ~(t, Poy, PI (poYY, ... , (Pn-2( . .. PI (Po (y)')' ... Y)Qi-1 (z(i-I»)
O, ... ,Pim-I(···Pl(POZ(tm))')'···)'=O
has only the trivial solution. Then given f > 0, there exists 6 > 0, such that if 1ti - Xi 1< 6,1 Yl,O - Zl,O 1< 6, ... ,1 Ym,im-I - Zm,im-l 1< 6, 1Po - PI 1< 6, ... , and
128
BVPs FOR FDEs
1Pn-l - Pn-l problem
1< 6,
then there exists a solution y(t) to the quasi-boundary value
(Pn-l ( ... PI (Poz )')' ... )' = I(t, Poy, Pl(POZ)', . . . , (Pn-2( .. . PI (poz )')' . . . )')
satisfying the quasi-boundary conditions
Moreover, 1Qiy(t) - Qiy(t)
1< €, for
°
~ i ~ n - 1, and tEla, h).
For the following theorem, we will use the notation Ix to denote the matrix of partials of I with respect to the components of x. Theorem 4.4 Assume I, the matrices M i , the integers mi, and the vector r are as in section 2D. Let u(m) be a solution to (2.11),(2.12). Assume the linearized problem
z(m
+ 1) =
Ix(m, u(m))z(m)
k
LM1z(ml) = 0, 1=1
has only the trivial solution. Then given € > 0, there exists 6 > 0, such that if 1MIl - M!1 1< 6, ... ,1 Mrn _1Vf';n 1< 6,1 rl - 1'1 1< 6,···,1 rn - Tn 1< 6, then there exists a solution u( m) satisfying
u(m
+ 1) =
I(m, u(m))
k
L M1u(ml) =
T.
1=1
Moreover, 1u(m) - u(m)
1< € for all integers in 1m}, mk+l).
References
1. M. Berger, Nonlinearity and Functional Analysis, Academic Press, 1977.
2. J. Ehme, Differentiation of solutions of boundary value problems with respect to nonlinear boundary conditions, J. Diff. Eqns . 101 (1993),139-147. 3. J. Ehme and J. Henderson, Functional boundary value problems and smoothness of solutions, Nonlinear Anal. Theory, Methods & Appl., to appear.
129 CONTINUOUS DEPENDENCE
4. P.W. Eloe, and J . Henderson, Nonlinear boundary value problems and a priori bounds on solutions, SIAM J. Math Anal. 15 (1984), 642-647. 5. P. W . Eloe and J. Henderson, Singular boundary value problems for quasidifferential equations, Int. J. Math. & Math. Sci. 18 (1995), 571-578 . 6. P.W. Eloe, and J. Henderson, Multipoint boundary value problems for ordinary differential systems, J. Diff. Eqns. 114 (1994), 232-242. 7. A. Granas, R. B. Guenther and J . W . Lee, Nonlinear boundary value problems for some classes of ordinary differential equations, Rocky Mountain J. Math 10 (1980), 35-58. 8. J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York-Berlin, 1977. 9. J. K. Hale and L. A. C. Ladeira, Differentiability with respect to delays, J. Diff. Eqns. 92 (1991), 14-26. 10. D. Hankerson, An existence and uniqueness theorem for difference equations, SIAM J. Math . Anal. 20 (1989),1208-1217. 11. J. Henderson, Disconjugacy, disfocality, and differentiation with respect to boundary conditions, J. Math. Anal. Appl. 121 (1987), 1-9. 12. J. Henderson and E. Kaufman, Integral conditions for disfocality of .. linear differential equation, Dyn. Sys. & Appl. 3 (1994), 405-412. 13. J. Henderson and B. Lawrence, Smooth dependence on boundary matrices, J. Difference Eqns. Appl., to appear. 14. Z. Nehari, Disconjugate linear differential operators, Trans . Amer. Math . Soc. 129 (1967), 500-516. 15. A. Peterson, Existence-uniqueness for ordinary differential equations, J. Math. Anal. Appl. 64 (1978),166-172. 16. A. Peterson, Existence-uniqueness for focal-point boundary value problems, SIAM J. Math. Anal. 12 (1981), 173-185. 17. A. Peterson, Existence and uniqueness theorems for nonlinear difference equations, J. Math. Anal. 125 (1987), 185-191. 18. G. Vidossich, On the continuous dependence of solutions of boundary value problems for ordinary differential equations, J. Diff. Eqns. 82 (1989), 1-14.
131
POSITIVE SOLUTIONS AND CONJUGATE POINTS FOR A CLASS OF LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS
PAULW. ELOE Department of Mathematics , University of Dayton Dayton, Ohio 45469-2316 USA JOHNNY HENDERSON Department of Discrete and Statistical Sciences Auburn University, Alabama 36849-5307 USA
Abstract The first conjugate point for a class of second order linear functional differential equations is characterized by the existence of a nontrivial solution that lies in a cone. To apply the cone theoretic arguments, linear, monotone, compact maps are constructed. To construct such maps, a standard representation theorem for Green's functions is obtained, known sign properties of Green's functions are employed and some comparison properties of Green's functions are obtained. The functional differential equation is restrictive and examples are presented to illustrate the restrictions.
1. Introduction
Let m be a positive integer and let Pi E C[O, 00), i = 1, . .. , m. Let hi E C[O,oo),i = 1, .. . ,m, be such that 0 S hi(t) S t,i = 1, .. . , m. We define the linear functional differential operator, L, by m
Lx(t)
= x"(t) + q(t)x'(O) + LPi(t)x(hi(t)), Os t, i=1
1991 Mathematics Subject Classification. Primary 34K10; Secondary 34B27.
132 BVPs FOR FDEs
where q E C[O, 00). For each b > 0, we consider the homogeneous two-point boundary conditions of the form (l.1b) x(O) = 0, x(b) = o. We shall, at times, denote the boundary conditions, (l.lb), by TbX = o. Let r E C[O, 00). For each b > 0, we consider boundary value problems (BVPs) of the form, (l.2b) Lx(t) = r(t)x(t) , O S t S b,
nx=O.
(1.3b)
We shall say that bo is the first conjugate point of (1.2b) corresponding to (1.3b) if bo = inf{b > 0 :(1.2b), (1.3b) has a nontrivial solution} . Azbelev [1] has shown that for linear functional differential equations , uniqueness of solutions of initial value problems can be violated; in particular, bo = 0 is possible. We shall make assumptions such that if bo < 00 , then 0 < bo o In this paper, we shall assume a sign condition on q and a sign condition on r, and characterize the existence of bo by the existence of a nontrivial solution of the BVP, (1.2b), (1.3b), for b = bo, that lies in a cone. This paper is motivated by the work in [13], [8] or [9] where sign properties and comparison properties of Green's functions have been employed in cone theoretic arguments. Also, the results in this paper are closely related to some of those found in [1] . Sign properties of Green's functions for BVPs for linear functional differential equations have received considerable attention ([1-4],[6]). In this paper, we shall obtain an appropriate characterization and representation of Green's functions in order that the the integral operator is a compact map from a Banach space of continuous functions into itself. We will then obtain comparison properties for Green's functions and apply cone theoretic arguments in a now standard way to characterize the existence of bo. In what follows, we shall provide the cone theoretic preliminaries in §2 in order that this paper be self-contained. In §3, we shall obtain the representation theorem, provide the appropriate sign properties and obtain the appropriate comparison properties of Green's functions . Finally, in §4, we shall define appropriate Banach spaces and cones and apply the cone theoretic arguments to characterize bo. 2. Cone Theoretic Preliminaries In this section we provide the definitions and results from cone theory which we shall apply in §4. For a thorough development, we refer the reader to Krasnosel'skii [10] . Lemma 2.3 is proved in [13] . Let B be a Banach space over the reals. A closed , nonempty set, PCB, is said to be a cone provided (i) ax + {Jy E P, for all x , yEP and all a , {J ~ 0, and (ii) x, -x E P implies x = O. P is a reproducing cone if for each x E B there exist u, v E P such that x = u - v. P is a solid cone if intP# 0 where intP denotes the interior of P. Every solid cone is reproducing [10] .
133 LINEAR
FDEs
Let PCB. Define the partial ordering on B induced by P by u :S v if, and only if, v - u E P. If M, N : B -+ B are bounded linear operators, we say that M :S N with respect to P provided Mu :S Nu for all u E P . A bounded, linear operator N : B -+ B is positive with respect to P if N : P -+ P. We say that a bounded linear operator N : B -+ B is /-Lo-positive with respect to P if there exists /1-0 E P, /1-0 "# 0, such that for each u E P , there exist positive real numbers, kl(U), k2(U), such that kl /1-o :S Nu :S k2/1-0 . Finally, if N : B -+ B is a bounded, linear operator, let s(N) denote the spectral radius of N .
Lemma 2.1. Let B be a Banach space over the reals, and let PcB be a solid cone. If N : B -+ B is a linear operator such that N : P \ {O} -+intP, then N is /1-o-positive with respect to P. Lemma 2.2. Let B be a Banach space over the reals, and let PcB be a reproducing cone. Let N : B -+ B be a compact, linear operator which is /-Lo -positive with respect to P. Then N has an essentially unique eigenvector in P, and the corresponding eigenvalue is s(N) and s(N) is simple. Lemma 2.3. Let Nb, a :S b :S /3, be a family of compact, linear operators on a Banach space such that the mapping b -+ Nb is continuous in the uniform operator topology. Then the mapping b -+ S(Nb) is continuous. Lemma 2.4. Let N : B -+ B be compact, linear and positive with respect to P . Assume s(N) > O. Then s(N) is a simple eigenvalue of N and there is a corresponding eigenvector in P Lemma 2.5. Let N 1 , N2 : B -+ B be compact, linear and positive with respect to P. If NI :S N2 with respect to P, then s(Nt} :S s(N2). Lemma 2.6. Let N : B -+ B be compact, linear and positive with respect to P. Suppose there exists /1- > 0, u E B, -u f!. P such that Nu 2 /1-U . Then N has an eigenvector in P which corresponds to an eigenvalue, A 2 /1-. 3. Properties of Green's Functions In this section we shall obtain a representation theorem and provide the sign properties and comparison properties of Green's functions which will be employed in the cone theoretic applications of §4. Azbelev and Rakhmatullina [3] have shown the existence of a Green's function for BVPs where the linear functional differential operator is far more general than our operator, L . By a Green's function, G(bj t, s), we mean that if Lx = J, TbX = 0 is uniquely solvable, then the unique solution is given by x(t) = G(bjt,s)J(s)ds. The first theorem that we shall employ, Theorem 3.1, is a special case of a result due to Bainov and Domoshnitsky [4] who require a concept of a semi-interval of nonoscillation, a concept analogous to that of disfocal intervals for ordinary differential equations [11] . An interval, I , is a semi-interval of nonoscillation for
J:
134
BVPs FOR FDEs
the operator, L, if x is a solution of Lx to, tl E I implies x = o.
= 0 on I and x(ta) = x'(tt} = 0 for some
Theorem 3.1. Assume [0,00) is an interval of semi-nonoscillation for the functional differential operator, L. Assume q(t) ~ O. Assume there exists a solution of the differential inequality, Lx(t) :::; 0,0:::; t, such that x(t) ~ 0,0:::; t and x(O) > o. Then G(b;t,s) < 0 on (0, b) X (O,b) for each 0 < b. Remark. The condition, q ~ 0 is a necessary condition as seen by the example = 0,0:::; t :::; b, x(O) = x(b) = o. Then
[4], x"(t) + qx'(O)
(t(2-qt)(b-s)/b(qb-2)
G(b·ts)=~
"
+ (t -
s),O:::; s < t:::; b,
It(2-qt)(b-s)/b(qb-2),0:::;t<s:::;b.
If q < 0, then G < 0 for t < s and G > 0 for s = 0, t > o. Note that G changes sign regardless of the length of the interval, [0, b] . Recall that in the case of disconjugate ordinary differential equations, there exists fj > 0 such that G maintains constant sign on (0, b) x (0, b) if 0 < b < fj [5] . The next result is a characterization theorem for G(b;t,s). Azbelev [1] first observed the representation (3.2). Domoshnitsky [6] points out that the construction of Green's functions for functional differential equations is difficult . It is here that we employ the restrictive delay conditions, h;(t) :::; t, i = 1, ... , m. It is also here that we require L to be so restricted in x'. It is for the purpose of defining Banach spaces and cones in §4 that we employ the other inequalities, 0 :::; h;(t). Theorem 3.2. Assume [0,00) is a semi-interval of nonoscillation for the functional differential operator, L. Assume 0 :::; h;(t) :::; t, i = 1, . . . ,m. Then for each 0 < b, there exists a uniquel),' determined, G(b; t, s), defined on [0, b] x [0, b] satisfying the following properties:
(i) G is continuous on [0, b] x [0, b], G t and G tt are continuous on triangles, o :::; t :::; s :::; band 0 :::; s :::; t :::; bj (ii) Gt(s+,s) - Gt(s-,s) = l,s E (0, b); (iii) as a function of t, LG = 0 for 0 :::; t < s, and s < t :::; bj (iv) as a function of t, nG = 0,0 < s < b. Moreover, if f is continuous on [0, b], then the unique solution of Lx = f,O :::; t :::; b, TbX = 0 is given by x(t)
= 1b G(b;t,s)f(s)ds .
(3.1)
Outline of Proof. Let u( t, s) be the Cauchy function satisfying (as a function of t), Lu(t,s) = 0 for s:::; t,u(t,s) = 0 for t < s,u(s,s) = O,Ut(s,s) = 1. Let {Xl,X2} denote a fundamental system of solutions of Lx = 0 on [0,00). Azbelev [1] has obtained the existence of u and {Xl, X2}. Define
135
LINEAR
FOEs
(3.2) on [0, b] x [0, b], where Cl and C2 are chosen to satisfy (iv). Note that Cl and C2 are uniquely determined since [0,00) is an interval of semi-nonoscillation. Clearly, G satisfies (ii)-(iv) and direct substitution (keeping in mind that u = for t < s) shows that (3.1) satisfies Lx = j, 0:::; t :::; b, TbX = 0. To see that G satisfies (i), we first argue that G is continuous in s. Note that each of Cl and C2 are continuous in s if u(b, s) is continuous in s. Thus, the continuity of Gin s follows from the continuity of u in s. This requires the continuous dependence on parameters of solutions of initial value problems. To see that u is continuous in s, note that
°
u(t, s)
+
= (t -
I
s) +
/.t
(t - a)q( a )Ut(O, s )da
(3 .3)
mo
t
(t - a)(LPi(a)u(hi(a),s)da. i=l
8
Since Ut(O, s) = 0, the Gronwall inequality can be applied in a standard way to lIu(t,sl) - u(t,s2)11 to obtain the continuity of u in s, where the norm denotes the supremum over t E [0,1] . The Gronwall inequality also implies that Ilu( t, s) II is uniformly bounded for s E [O,b] . It now follows by differentiating (3 .3) with respect to t that Ilut(t,s)11 is uniformly bounded for s E [O,b]; since u is continuous in sand Ilut(t,s)1I is uniformly bounded in s, it follows that u is continuous in (t,s). Finally, since Cl (b, s )Xl (t) + C2 (b, s )X2 (t) is separable in s and t, it follows that G is continuous in (t, s). Similarly, Ut and Utt are continuous on triangles s < t and t < s. Remark. It is interesting to note that in the case where m
Lx(t) = X"(t)
+ q(t)x'(g(t)) + LPi(t)x(hi(t)),O :::; t, i=l
a Green's function satisfying (ii)-(iv) exists. However, the Gronwall inequality cannot be employed to obtain (i) and in general, the Green's function does not satisfy (i). Condition (i) is sufficient, although not necessary, to imply by the Arzela-Ascoli Theorem that the integral maps defined in §4 are compact operators. Condition (i) is a usual condition that is satisfied in the case of ordinary differential equations. The following two examples show that Condition (i) is not to be expected in the case of functional differential equations. Example 1. Let
get)
={
t,0:::;t:::;1/2, 1/2,1/2:::; t :::; 1.
136 BVPs FOR FDEs
Consider X"(t)
+ x(g(t»
= 0,
°:S t :S
1. Then
0, t < s, sin(t - s),s < t < 1/2,
u t s -
(,)-
{
v(t,s),s 0 such that X2 - Xl 2 au, where this inequality is with respect to the cone, P(bI) . Since XI(t) = u(t) = 0 for t > bI and X2 E P, it follows that X2 - Xl 2 au where the inequality is now with respect to the cone P Thus, X2 2 Xl + au = (S(Nb 1) + a)u . It follows from Lemma 2.6 that S(Nb,) 2 S(Nb 1) + a > S(Nb1 ).
a
We now state and prove the main result of the paper. Theorem 4.4. The following are equivalent: (i) bo is the first conjugate point of (1.2b ),(1.3b); (ii) there exists a nontrivial solution, x, of the BVP, (1.2b),(1.3b), for b = bo, such that X E P(bo); (iii) S(Nb o) = 1. Proof. (iii) implies (ii) follows from Lemma 2.4. To see that (ii) implies (i), let u E P(bo ) \ {O} satisfy (1.2b) ,(1.3b) , for b = bo o Extend u by u(t) = O,bo < t. Then s(Nbo ) 2 1. If S(Nb o) = 1 the proof is complete by Theorem 4.3. See the details provided in the proof of Theorem 4.2 [8] . So, assume S(Nb o) > 1. Let v E P \ {O} be such that NboV = s(Nbo)v. Note the restriction of v to [0, bo] is in the interior of p( bo ). Thus , there exists > 0 such
a
139 LINEAR
FDEs
that u ;:::: liv where the partial order is with respect to F (bo). Since, u = v = 0 for > bo, u ;:::: liv where the partial order is now with respect to P Assume now that li is maximal. Then
t
= lis(Nbo)V
U = NboU ;:::: Nbo(liv)
which contradicts the maximality of li. Hence, S(Nb o) = 1. (i) implies (iii) follows from Lemma 2.3, Theorem 4.3 and the fact that IINbll-+ 0 as b -+ 0+ . Details are provided in [7] or [8] . Of course, to apply Lemma 2.3, we require that the map b -+ Nb is continuous. Let Ilull = 1. Assume b2 - bl > O. Then (Nb. - NbJU =
l
b
o
'
(G(b 2 it,S) - G(blit,s))r(s)u(s)ds
+
lb.
G(b 2 it , s)r(s)u(s)ds
=
b,
l
b '
o
v(t,s)r(s)u(s)ds +
lb.
G(b2 ;t,s)r(s)u(s)ds,
b,
where vet,s) is the solution of Lx = 0,0 ~ t ~ bl,x(O) = O,x(bd = G(b 2 ;bl ,s). Note that Ilv(t,s)1I = -v(bl,s) = -G(b2 ib l ,s), since [0,00) is a semi-interval of nonoscillation. G is uniformly continuous on compact domains and so v( t, s) is uniformly small in s for b2 - bl sufficiently small. Thus, I J:' (G(b 2 i t, s ) G(blit,s))r(s)u(s)dsl is uniformly small if b2 - bl sufficiently small. The second term is uniformly small since G is uniformly bounded on compact domains . Remark. Azbelev [1] points out that Wronskians offundamental systems of linear functional differential equations can vanish and so conjugate points of (1.2b) , (1.3b) can generate multiple zeros. He proves (Lemma 2 [1]) that the first conjugate point is simple. Theorem 4.4 offers another proof of this observation. The nontrivial eigenfunction, x, that lies is a cone satisfies x E intF(b o). In particular, x'(bo) > 0 and bo is a simple zero. Example. Consider, again,
x"(t) + qx'(O) = 0,0
~
t
~
b,x(O) = x(b) = O.
Then
. _{t(2-qt)(b-S)lb(qb-2)+(t-S),0~S 0. Then for any c: > and x E i(B) we consider five cases: Proof: (i)
°
Case 1: s, t E [-r,O]. Since x(t) = X1(t) for t E [-r,O] and since X1(t) is uniformly continuous on [-r ,O), there exists 61 = 61 (c:) such that Ix(t) - x(s)1 = IX1(t) - x1(s)1 < c:/2 whenever It - sl < 61 . Case 2: s, t E [T, T + h]. Since x(t) = X2(t) for t E [T, T as in Case (i) gives 6~ = 62 (c:) such that
It - sl < 62 Case 3: S;
II,t Ix'(u)ldul
Ix(t) - x(s)1
x2(s)1 < c:/2.
°
II:
C:2·
If s E [-r,O] and t E (O,T), then xes) = X1(S) and
Ix(t)-x(s),
= Ix(t)-x1(0)+x1(0)-x1(s)1 S; Ix(t) - Xl (0)1
if It -
= IX2(t) -
s, t E [O,T] . In this case we have Ix(t) - x(s)1 = x'(u)dul S; Mit - sl so that there is a 63 = 63 (c:) > such that It - sl
..f(t, x t , x'(t)),
t E [0, T], A E [0,1]
ax(t) - (3x'(t) = aCt), ,x(t) + 6x'(t)
tE[-r,O]
= bet),
tE[T,T+h]
we have IIxllr,h $ M, then the BVP (I) has at least one solution. Proof: Let Ml be defined as in (2.10) and set
V:= {x E K1[O,T]: IIxlll $ Ml We define a map H: [0,1] x
V
+ I} C K1M[O,T].
(2.16)
~ K1[O, T] by
(2.17) Then for every A E [0,1], H(A,.) is a compact map by Lemma 2.2. Moreover, the fixed points are precisely the solution to (I>.). Further, H( A, .) has no fixed points on BV by the choice of V and the map H(O, ·) = Cl +C2t where Cl, C2 are given by (2.8) is essential. Therefore, the topological transversality theorem shows that H(l,·) has at least one fixed point in V which gives a solution to (I) in K1[-r, T + h]. This completes the proof. 0 3.
Applications of the Existence Principle
The applicability of Theorem 2.3 depends on finding suitable a priori bounds for the solutions of the BVP (I>.), independent of A E [0,1]. We next state two lemmas whose proofs are essentially the same as those in [7] . Lemma 3.1.
Suppose that the function
° °
f
satisfies the following conditions:
(HI) There exists a constant M > such that for every (t,u,v) E [0, T] xCxRn with lu(O)1 > M and u(O)· v = implies u(O) · f(t, u, v) > where te." denotes the dot product.
°
150 BVPs FOR FDEs
Then every solution X of(l>.),'\ E [0,1] satisfies
Bd
max
-r9$T+h
Ix(t)l:::; Mo
= max{M,AI,
where AI=SUp{XI(t): -r:::;t:::;O,
BI
= SUp{X2(t): T:::; t
:::; T
and
Ix(O)I:::;M}
+ h,
Ix(T)I:::; M}
and XI(t) , X2(t) are given in (2.2) and (2.3). Lemma 3.2. Assume that there exists a constant Mo >
°
such that
max
-r" and {y~}n>" are { uniformly bounded and equicontinuous on [0, k].
(6)
156
BVPs FOR FDEs This is clear for {y.. }..~k. We need to show that {y~} ..~k is equicontinuous on [0, k]. Let t, t" e [0, k]. Then for n ~ k, we have for some w" e F(t, y", y~) and for almost all t e [0, k] that
Iy~(t) - y~(t*)1 =
It w,,(s)dsl ~ J: hk(s)ds,
where hI. e Ll[O, k] is determined from the bounded sets in the theorem and the fact that F is locally LP- Carathoodory. The equicontinuity of {y~}">k is now a direct consequence of the absolute continuity of the Lebesgue integral. The existence of the function w" used in the preceding argument is a consequence of Kuratowski- RyllNardzewski theorem; see [1]. From (6) and the Arzela-Ascoli theorem it follows that there exists N l" C N+, the set of all positive integers, and Zl e Cl[O, 1] such that
Yn(;) -+
z(j) 1 , J" --
° ,
1 as n -+
00
through N"1·
H Nl = Nt - {I}, then (6) and Arzela-Ascoli imply that there exists N; C Nl , the set of all positive integers, and Z2 e C l [0,2] such that Y"(;) -+ z(j) 2 , Evidently,
Z2
= zion
J" =
° ,
1 as n -+
00
through N."2·
[0,1]. Proceed inductively to obtain
Nl 2 N2 2···2Nk 2···, z"
e
Nd;;{k+1,k+2, ... }, ZkeCl[O,k] with Cl[O, k] with y~) -+ zf) uniformly on [0, k] as n -+ 00 through N k •
Clearly, Z/o+l = Zk on [0, k] for k = 1,2, .... Define y : [0,00) -+ R" by yet) = Zk(t) on [0, k]. For any k, y e ClIO, k], ay(O) + .By'(O) = ,"(, { yet) e 11;., y'(t) e B;' for t e [0, k]. Finally, we show that y satisfies the differential inclusion y" e F(t, y, y') almost everywhere on [0,00). To this end, fix k. For ~ t ~ k and n e Nk we have
°
y~(t) -y~(O) e [F(s,y,,(s),y~(s))ds = (NFy,,)(t) where NF : Cl[O, k]-+ 200 [0, 101. Consequently, y~ - y~(O) e NFY".
FUrthennore, NF is point-compact and u.s.c. by Lemma 3. Since y~(t)
- y~(O) -+ z~(t) - zHO) and y,,(t)
-+
Zk(t)
157 THE FILIPPOV ApPROACH
as n -+
00
Since z"
through N Jr, it follows from Lemma 5 that
= y on [0, k], we conclude that y'(t) -:- y'(O) e
Thus, y e W'l,p[O, k]
~d
l
F(s, y(s), y'(s))ds for t e [0, k].
y"(t) e F(t, y(t), y'(t)) for a.e. t e [0, k] and for each
k= 1,2, ....
Remark 1: As the proof of Theorem 6 reveals, the Sturm-Liouville boundary condition ay,,(t) + py~(t) = '"Y can be replace by any boundary condition of the form tp(y" (0) , y~(O)) = 0 with tp : R2N -+ ~ continuous. Furthermore, if each y,,(t) satisfies several such boundary conditions, the global solution y(t) in the conclusion of the theorem also satisfies the same conditions. In particular, if y,,(O) = A and y~(O) = B for all n, then y(t) will satisfy the same initial conditions.
Remark 2: Combining the results of Theorems 4 and 6 gives an existence principle for Filippov solutions to certain initial and or boundary value problems.
Remark 3: The proof of Theorem 6 follows lines of reasoning first used in [9], where existence theorems are established for classical solutions to boundary value problems on infinite intervals, especially for semi-conductor devices. The results in [9] are restricted to differential equations. Further work [5] and [11] establishes similar results for Caratheodory solutions for certain classes of differential equations, including the Thomas-Fermi equation. 3. Applications In this section, the foregoing existence results are applied to a pendulum that experience both viscous damping and dry friction and is driven by an external force. Periodic solutions for such pendulums were established in [13]. Here we establish global solutions in time for the initial value problem. After existence is established, we present and discuss some interesting numerical results. Consider the pendulum equation
y" + by' + ksgn(y') + csin(y) = e(t),
y(O) = Yo,
y'(O) = y~,
(7)
BVPs FOR FDEs where b, k, and c are positive constants, e(t) E L~,[o, oo) , and e(t) is further restricted according to the cases treated below. Of course, yo and y; are given initial data. For the moment restrict t to the interval [O, TI.The following reasoning establishes the p e n ' bounds need to guarantee the existence of a solution to (7) on [0, T ] and also to establish the estimates needed to apply Theorem 6 and hence obtain global solutions to the initial value problem for (7). Let X E (O,1] and y = y(t) be a solution
to
and define yn E(t) = 2
+ Xc(l - cosy),
Y?
Eo = 2 + c(1 - cos yo).
Note that E(0) 5 &. Multiply the diffktential equation in ( 7 ) ~by y' to obtain E1(t) = -Xbyn - Xk lyll+ Xyle(t).
(8) Now we consider three dXerent cases corresponding to t h e e interesting physical situations. Case 1. le(t)l 5 k; dry Mction predominates. Then the right member of (8) is nonpositive, E(t) is decreasing, and hence, E(t) 5 E(0) 5 Eo. It follows that 1y1(t)l 5 c = (yf
+ 241 - aa
for 0 5 t
T
Notice that the constant C does not depend on X E (O,1] or on T. Case 2. e(t) E LZIO,oo); the driving force has finite energy. Observe that, for any E > 0,
-
Lt
byndr
+
y'edr 5 -
/i 0
by'%
+ 2 ityadr + 51 1 e2dr. t
NOWtake E = b, use the foregoing estimate in (8), and integrate to find that
It follow that there is a constant C independent of A E (0,1] and of T sueh that lyl(t)( 5 Cfor 0 St ST.
(10)
159 THE Fn.IPPOV ApPROACH
Case 9. le(t)1 :5 k for t ~ T 1 ; dry friction ultimately dominates. Since e(t) is locally in L2, a bound for Y'(t) as in (10) holds on [O,T1] and, if T ~ TI , the bound for Y'(t) as in case 1 holds on [TI' TJ. Thus, once again, IY'(t)1 :5 C on [0, TJ for any T, where C is a constant independent of T. Under the assumptions in cases 1, 2, or 3, there is a constant C independent of A E (0,1] and of T such that
ly'(t)1 :5 C for 0 :5 t :5 T.
(11)
It follows that
Iy(t) I :5 Iyol + 1o'IY'(r)dr l :5 Iyol
+ CT for 0:5 t :5 T,
ly(t)l:5 IYol + Ck for 0 :5 t :5 k for any k :5 T.
(12) (13)
Apply (11) and (12) with T = n and the existence results in [5] or [11] to obtain the existence of a solution y,,(t) to (7) on [0, n] . Furthermore, it follows from (11) and (13) that the hypotheses in Theorem 6 hold with BIc = [-lyol- Ck, Iyol + Ck] and B: = [-C, C). Consequently, (7) has a globally defined solution y(t) such that
ly(t)l:5 IYol + Ck for 0:5 t:5 k for any k ,
Iy'(t) I :5 C for 0:5 t.
Further important qualitative information about the solution y(t) is provided by the following estimates. Equation (8) with>' = 1 holds for the solution y(t) . Now consider case 1 in which le(t)1 :5 k and, hence, E'(t) :5 _byt2. Integrate to obtain
E(t) :5 It follows that, for any t
~
-l
t2 by dr + Eo.
0,
yt2 ft t2 2 :5 - 10 by dr + Eo, yt2 + 2l byt2 dr :5 2Eo,
~ (e 2bt lot yt2dr)
:5 2Eoe 2bt ,
10' yt2dr:5 ~. Consequently, in case 1, y' E L2[O, 00).
160
BVPs FOR FDEs Finally, we consider the case of a bounded forcing term that may not be dominated by frictional terms. Case -/. le(t)1 ~ Kj bounded forcing. As in cases 1, 2, and 3 assume first that 0 ~ t ~ T. From (8),
E'(t) ~ K Iy'l , and, hence,
E(t) 1
2Y'(t)2 ~
~ Eo + K lIY'(r)1 dr,
Eo + Ktma.x {ly'(r)1 : 0 ~ r ~ t}.
It follows that there is a constant CT independent of ). such that
ly'(t)1 ~ CT for 0 ~ t ~ T,
(14)
and that for any 0 ~ k ~ T there is a constant Ck (dependent on k but not on T ) such that (15) ly'(t)1 ~ Ck for 0 ~ t ~ k. Then
ly(t)1 ~ Iyol + CTT for 0 ~ t ~ T,
(16)
ly(t)1 ~ Iyol + Ckk for 0 ~ t ~ k for any k ~ T.
(17)
Just as in cases 1-3, use (14)-(17) to obtain the existence of solutions y,,(t) to (7) on [0, n] and which satisfy the hypotheses in Theorem 6 with Bk = [-IYol-Ckk, Iyol+ Ckk] and BZ = [-Ck, Ck]. Consequently, just as in the other cases, (7) has a global solution y(t) defined on 0 ~ t < 00.
Remark 4: Figures 1 and 2 show particular pendulum motions. In Figure 1 the forcing term dominates the frictional effects and the pendulum rotates counterclockwise with ever incrasing angular coordinate. The plot of the angular velocity indicates sticking pericds where no motion occurs due to the dry friction. In Figure 2 the pendulum starts near its unstable equilibrium and is driven by a force with variable sign. The pendulum oscilates about its unstable equilibrium and then falls doum and oscillates about its stable equilibrium. As in Figure 1 we can observe periods of
sticking.
161 THE Fn.IPPOV ApPROACH
8 7 6
angls 5
3 angular velocity
2.
a -1
a
0.5
1. 5
2.
2.5
3
3.5
4
4. 5
5
timet
Figure 1: A solution of the equation y" + 2y' + 3siny + 2signy' = 7Isin2?rt/51
3.5,---------.--------.----------,
2.5
angle
1.5
timet
Figure 2: A solution of the equa.tion y" + 2y' + 3siny + 2signy' = 5sin2?rt
162
BVPs FOR FDEs References 1. J.P. Aubin, A. Cellina, Differential inclusions, A series of comprehensive stud-
ies in mathematics, No. 264, Springer-Verlag, Berlin - Heidelberg - New York, (1984). 2. C. Castaing, M. Valadier, Convex analysis and measumble multifunctions, Lect. notes in math., N 0.580, Springer-Verlag, Berlin - Heidelberg - New York, (1977). 3. J. Dugundji, A. Gra.na.s, Fi1:ed Point Theory, Vol 1, Monographie Matematyczne, PNW, Warsza.a.wa, 1982. 4. A.F.Filippov, Differential equations with discontinuous right-hand side, Translat. Amer. Math. Soc. 42 (1964), 199-230. 5. M. Frigon, Application de la thforie de la tmnsversaliU topologique Ii des problemes non lineares pour des equations differntielles ordinaires, Dissertationes Math. 296, (1990). 6. M. Frigon, A. Granas, Theoremes d'existence pour des inclusions diff&eD.tielles sans convexite, C.R. Acad. Sci. Paris, Serie 1, (1990)., 819-822. 7. M. Frigon, A. Granas, and Z.E.A. Guennoun, Sur l'intervalle ma.ximal d'existence de solutions pour des inclusions diff ere ntielles, C.R. Acad. Sci. Paris 306, Serie I, (1988), 747-750. 8. A. Granas, R.B. Guenther and J .W. Lee, Some existence results for the differential inclusions y(k) E F(x, y, •.. , y(k-l)), y E 8, C.R . Acad. Sci. Paris 307, Serle I (1988), 391-396. 9. A. Granas, R.B. Guenther, J.W. Lee, D. O 'Regan, Boundary value problems on infinite intervals and semiconductor devices, Journal for Mathematical Analysis and Applications 116(2) (1986), 335-349. 10. J. Kurzweil, Ordinary Differential Equations (Translation from the Czech edition by Michael Basch), Elsevier, Amsterdam - Oxford - New York - Tokyo, (1986). 11. J.W. Lee, D. O'Regan, Existence principles for differential equations and systems of equations, Proceedings, Topological Methods in Differential Equations and Inclusions, NATO ASI Series C, Kluwer Academic Publishers, Dordrecht, NL, (in press). 12. M. Senkytik, A topological approach to dry friction and nonlinear beams, Ph.D. Thesis, Oregon State University, Corva.llis, Oregon, (1995).
163 THE Fn.IPPOV ApPROACH
13. M. Senkytik and R.B. Guenther, Boundary value problems with discontinuities in the spa.tial varia.ble, Journal of Mathematico1 Analysis and Applications (in press).
165
AN EXISTENCE RESULT FOR DELAY EQUATIONS UNDER SEMILINEAR BOUNDARY CONDITIONS
G. HETZER
Department of Mathematics, 304 Parker Hall Auburn University, AL 36849-5310, USA
1. Introduction
Let us begin with describing an example from applications that leads to a "nondegenerate" boundary value problem in the sense of Waltman and Wong (d. [9]). Example 1.1. Diaz has studied in [4] the controllability of the climate in the framework of a Budyko-type energy balance climate model. The resulting equation is a reaction-diffusion equation with a discontinuous reaction term. If one additionally accounts for the long response times within the climate system, memory terms arise (cf. [8]). In particular, the continental ice-sheets have memory spans of tens of thousands of years. For the purpose of illustration, let us restrict ourselves here to the "global case" and denote by u = u(t), say, the ten-year mean of the globally averaged surface temperature in Kelvin. We account then for expansion or retreat of these ice-sheets by employing a long-term mean f~Tj3(S)U(t +s,x)ds with T:=::;j 104 ys., the memory span of the system, and j3 E COO 0- T, 0]) satisfying j3( s) > 0 Vs E [- T, 0], j3(j)(-T) = 0 Vj E Z+ and f~T j3(s)ds = 1. The albedo a will be a function of u(t) as well as of J~T j3( s )u( t+s, x )ds. Thus, the absorbed radiation flux at time t > 0 is given by Q(t)[l- a( u(t) , f-T j3(s )u(t+ s, x )ds )] , where Q = Q(t) is the ten-year mean of the globally averaged solar radiation flux. Finally, the outgoing terrestrial radiation flux is denoted by g(>., u(t)), e.g. g(>., u) = e(>., u)u 4 according to the Stefan-Boltzmann law with e an infrared opacity function and>' ERa control parameter which models the human impact on the emitted radiation flux (keywords: greenhouse effect, CO 2 , trace gases). The energy balance equation yields
u(t) = Q(t)[l- a(u(t), lTj3(s)u(t+ s,x)ds)]- g(>.,u(t))
(1.1)
for t > O. Selecting a climate history {) E C([-T, 0], R+), a time b > 0 and a temperature p E (0,00) one asks e.g. for a >. E R and a solution u of Eq. (1.1) satisfying Uh-T,O) = {) and u(b) = p in order to study, whether human impacts could drive the
166
BVPs FOR FDEs climate system to the temperature p at time b. Of course, the outlined setting is just that of a "toy-model", and e.g. the framework considered in [8] would immediately lead to a corresponding boundary value problem (in time) for a functional reactiondiffusion equation on the sphere. In this note, we intend to describe some basic existence results for first order semilinear functional differential equations subject to semilinear boundary conditions of the form
x(t) + l(t,xt) = f(t,xt) 00xo + 0bXb = g( XO , Xb),
t E [O,b] ,
(1.2)
where I: [O,b] x C([-I,O],RD) --> Rn is continuous, l(t ,· ) is linear and 0 0 , 0b are bounded linear mapping from C([-I, 0], Rn) into a Banach space V. Throughout, we also require f : [0, b] xC([-I, 0], RD) --> R n to be continuous and 9 : C([-I, 0], RD)x C([-I, 0], Rn) --> V to be completely continuous and write Ut for s t--+ u(t + s) E C ([-1, 0], R n) . The modifications necessary to deal with the par ameter-dependent problem Eq. (1.2) are mostly technical, and hence we are not going to dwell upon this aspect, all the more so as realistic modeling anyway requires to consider functional reaction-diffusion equations, a topic beyond the scope of this paper. In the special case, where g is a constant, Eq. (1.2) has been studied by Waltman and Wong [9] using Granas's version of the Schauder fixed point theorem and a shooting method and by Hale [6], who employed a reduction process that rewrites Eq. (1.2) as an operator equation via the variation of constant formula. Let us briefly indicate the latter. Consider the homogeneous linear equation
Ii + l(·,y.) = 0.
(1.3)
We denote by 3 = 3(t,s)if> (s E [O,b], t E [s,b], if> E C([-I,O],Rn)) the solution operator of Eq. (1.3), i.e. 3(t, s)if> = Yt with y the solution of Eq. (l.3) satisfying y. = if>, and write Y = Y(t,s) for the fundamental solution of Eq. (1.3), i.e. Y(t,s) is the zero matrix for -00 < t < sand s E [0, b], and given z E Rn and s E [0, b], t t--+ Y(t, s)z is the solution of Eq. (1.3) on [s, b], which satisfies y(s) = z. Assuming that the initial value problem
x(t)+I(t,Xt) =f(t,xt) { Xo = if> has a unique solution x = x(t,if» for all if> E C([-I , O],RD) (we write x. (if» for the mapping T t--+ X(T + Sj if» on [-1,0]), one obtains for the special case 9 E V, b 21 that Eq. (l.2) is equivalent to (1.4) which can be studied by methods from functional analysis. It was shown by Waltman and Wong [9] for b 2 1 and in [6] for the general case that Eq. (1.4) is solvable
167 SEMILINEAR BOUNDARY CONDITIONS
°
provided that 9 0 + 9b 0 3(b,0) is onto and liIIltltPlioo-oo IIfl~~t~l,", = uniformly for t E [0, b]. For Eq. (1.2) it appears to be more convenient to write the boundary value problem as an operator equation Lx = Nx from a subspace of C([-l,b],RD) into C([-l,O],RD) X C([O,b],RD) and to employ solvability results for semilinear operator equations in place of fixed point theorems. Thereby, one also avoids to impose "uniqueness conditions" on f and g. 2. Boundary Conditions and Preliminaries Throughout the following notations are used: For n E Nand b E (0,00) set C := C([-I,O],RD), Y := C([O,b],RD) and Z := C([-I,b],RD) and equip each of these spaces with the respective maximum norm 11 ·1100' We begin with some remarks about the linear initial value problem u(t) { Uo
+ l(t, Ut) = y(t) =q
t E (0, b]
(2.1)
assuming y E Y, q E C and I : [0, b] x C --+ R D continuous with l( t, .) linear for t E [O,b]. It is well-known (d. [6,7] e.g.) that Eq. (2.1) has a unique global solution, which will be called z = z(q,y), and that the mapping z : C X Y --+ Z is continuous. Operators S : Y --+ C and T : C --+ C are defined by Sy = z(O, y h and Tq = z(q,Oh, respectively. Clearly, Sand T are bounded linear operators, S is completely continuous and T is completely continuous provided that b ~ 1. Actually, Tit)+! is completely continuous ([p) := sup{m EN: m ::; p} for pER). Before formulating standing basic hypotheses for Eq. (1.2) let us take a look at another example. Example 2.1. (Perturbed periodicity) Consider u(t) = f(t,ud 0::; t ::; b subject to Ub = uo+g( uo), where f : [0, b) xC --+ R Dis continuous, 9 : C --+ C is completely continuous and limll",lIoo_oo IIg(cp) 1100 / IIcplioo = 0. The homogeneous linear problem is UlIo,b) == 0, Ub = uo,which has the constants as solutions. The nonhomogeneous problem UlIo,b) = Y E Y, Ub = Uo + p with p E C is solvable, iff p( b) = y( s )ds, thus the classical Fredholm alternative holds. What we are going to suggest is to rewrite the original semilinear problem as an operator equation Lu = Nu, where L : Z :::> dom( L) --+ Y x C defined by Lu := (u IIo,b], Uo - Ub) is a Fredholm operator of index and Nu := (J(·,u.),g(uo)) is continuous. An extended version of this example arises from a simple economic model.
J;
°
Example 2.2. (Controllability of inflation rate) Kehong Wen proposed the following model to describe monetary growth cycles (d. [10) e.g.).
x(t) { u(t)
= -ax(t - T) + u(t = (J - 6x(t).
0") + (x(t - T))
(2.2)
168
BVPs FOR FDEs Here, notations have been adapted for our purposes, Q , f3 and fI are positive reals, and T belong to (0,1] . The functions x and u stand for, respectively, the deviation of inflation rate and money supply rate from their growth trend and cI>(1]) := 1]e-~2. Economists sometimes consider downsizing the inflation trend by controlling the money supply rate. A first step in dealing with controllability in the framework of Eq. (2.2) is to study this system subject to boundary conditions u,
uo)
Xb
=
Ub
= Uo -1'2(XO , UO, Xb,Ub)
Xo -1'1(XO ,
(2 .3)
(b> 1). Here,1'1 would be a target reduction function for the period [b-l , b] , which should of course depend on the histories Xo and Uo, whereas 1'2 would stand for the change in the money supply rate. Clearly, this leads to the same type of operator equation as in example 2.1. These and other examples fall into the following scope: (HI) I: [0, b] x C (H2)
f : [0, b]
x C
~
~
R D continuous, I( t, .) linear for t E [0, b] ; RD continuous and bounded on bounded subsets;
(H3) V Banach space, 0 p : C (H4) g : C
X
C
----t
----t
V bounded linear operator for p E {O, b} ,
V completely continuous .
We are going to employ the following results about quasi bounded perturbations of Fredholm mapping of index 0, which can be found in [5]. Recall that a mapping r : E ----t F between Banach E and F is said to be quasibounded, iff its quasinorm IIlflll := lim sUPIi'Ii_co ~.i, < 00 . Another tool which we will use later is the so-called topological degree, which goes back to Brouwer. This concept associates with triples (F,O,z) an integer deg(F, O,z), which is a measure for the number of solutions of F(x) =Z in O. Here, 0 is an open bounded subset of, say, RD, z E RD and F : cl(O) ~ RD ;s continuous with z ¢ F(aO). We have
S'ttili
deg(F,O, z ) =
L:
sgn(detJF(x))
x EF-'( . )
in the special case where F E C 2 (cl(O) , RD) and the Jacobian hex) is invertible for all x E F- 1(z). Of course, the set F-1( Z) is finite under these assumptions. The extension to the general case occurs by means of approximation and Sard's theorem. We refer to [3] for details. Moreover, recall that a continuous mapping F from a subset M of a normed space B into another normed space E is called completely continuous, iff F maps bounded subsets of M into relatively compact subsets of E . Now, let L : B :J dome L) ~ E be a Fredholm-operator and P : B ~ Band Q : E ~ E be projections with ran( P) = ker( L) and ker( Q) = ran( L) , respectively. We say that F is L-completely continuous, iff QoF is continuous and (Llker(p)ndom(L))-1 0 (Id- Q)o F
169 SEMILINEAR BOUNDARY CONDITIONS
is completely continuous. The following degree continuation theorem for semilinear operator equations with quasi bounded nonlinear parts is due to J . Mawhin.
Proposition 2.3. (d. [5; Theorem VIII. I. , p. 135)) Let E, F be Banach spaces, L : E :J dom(L) --+ F be a Fredholm operator of index 0 and N : E --+ F be L-completely continuous. Moreover, denote by P : E --+ E and Q : F --+ F continuous projectors with ran(P) = ker(L) and ker(Q) = ran(L), respectively, and by J: ran(Q) --+ ker(L) a linear isomorphism. Assume that (a) (Llker(p)ndo771(L»)-l
0
(Id - Q) is quasibounded;
(b) There exist a, r E (0,00) such that each solution of Q a lI(Id - P)xll + r; (c) (1
+ a) 111(Llker(p)ndo771(L»)-l 0
(d) deg(J 0 Q 0 Nlker(L)' B(r) radius r and center
(Id - Q)
n ker(L), 0)
0
0
N x = 0 fulfils IIPxll
( s) O} . Here, Mk,n denotes the vector space of real k x n matrices .
=
=
+
172
BVPs FOR FDEs (ii) Thereisap>O withf;d1/(s)f(s,u.) + f~ld((s)g(UO,Ub)(S) f; u E dom(L) which satisfies lu(s)l:::: p for all s E [-1, b].
o for
each
(iii) d:= min{lu(s)1 : u E ker(L), lIulioo = I} > O. (iv) There is an expansive linear isomorphism w from Rk onto ker(L) such that deg(G, int(Bpd -')' 0) f; 0, where B pd -' is the closed ball in Rk of radius pd- 1 and center 0 and G : Rk --+ Rk is given by G(x):=
l
d1/(s)f(s,w(x).)
+
II
d((s) g(w(x)o, w(x)b)(s)
for x E Rk. Then there exists at least one solution of Eq. (1.2) . Proof. We want to apply Propositions 2.4. to Land N as defined in Section 3 and observe that (HI) (H5) and Proposition 3.2. ensure that N is L-completely continuous. Thus, we only have to realize (1) (4) of Proposition 2.4. The growth assumptions made for f and 9 immediately imply (3) . Moreover, we obtain from (iii) for u E ker(L)\{O} that d- 1 Iu(s)1 = d- 1 lIulioo ,',:\i~ lIull oo, hence (1) is fulfilled with A = d- 1 • Now, let 1 .:5 j .:5 k and 1/j and (j denote the j-th row of "I and (, respectively, then rj(y,tP) = f;d1/j(s)y(s) + f~ld(;(s)tP(s) for (y,tP) E Y X C defines a continuous linear functional rj : Y X C --+ R. Choosing (Yj , tPj) E Y x C with f;(Yj, tPj) = 6i ,i for 1 .:5 i, j .:5 k we can define a bounded linear projector Q in Y x C by Q(y , tP) = Ej=lri(Y,tP)(Yi , tPj) . It follows from (i) that ker(Q) = ran(L) and from (ii) that Q 0 Nu f; 0 for u E dom(L) with lu(s)1 :::: p Vs E [-1, b]. Thus (2) is derived. Setting B := {u E ker(L) : Ilulloo.:5 pd- 1 } we get w- 1 (B) C B pd -' thanks to w being expansive. Furthermore, B contains all the zeroes of Q 0 Nlker(L)' since u rt B yields lIulioo > pd-t, i.e. lu(s)1 :::: d lIulioo > p for all s E [-1, b] by definition of d, thus f;d1/(s)f(s,u.) + f~ld((s)g(UO,Ub)(S) f; o in view of(ii), which leads toQNu f; o. Now, choose linear isomorphisms J : ran(Q) --+ ker(L) and A : Rk --+ ker(L) with J 0 Q 0 Nlker(L) = A 0 G 0 w- 1 , then all zeroes of G are contained in Bpd-l . Therefore additivity and product formula for the Brouwer degree yield
::
Ideg(J 0 Q 0 Nlker(L)' int(B), 0)1 = Ideg( G, int(Bpd- 1 ), 0)1 f; 0, which implies (4) . Remarks 4.2. (i) There exist always matrix-valued functions "I and ( satisfying hypotheses (i) of Theorem 4.1., since ran(L) has finite codimension in Y x C. One can determine them by observing that Eq. (3.2) implies (y, tP) E ran(L) , iff 0 b 0 Sy - tP E
173 SEMILINEAR BOUNDARY CONDITIONS
ran(0 0 + 0b 0 T) . Hence following the arguments of the proof of [7; Theorem 4.1., p. 181] one obtains (y,1jJ) E ran(L), iff
l
v(t)y(t) dt
+
/1
dcp (s)1jJ(s) =
°
for all solutions v, cp of an adjoint problem that we are not stating here. (ii) If 9 is a nonzero constant in C, i.e. nonhomogeneous boundary conditions are of concern, then it is necessary for the solvability of Eq. (1.2) that 9 E ran(0 0 + 0b 0 T) + 0b 0 S. Suppose now that 9 E ran(0 0 + 0 b 0 T), then J~l d((s)g(s) = 0, and hypotheses (ii) reduces to a condition for f alone of the type one would get in the homogeneous case when using the approach described in the introduction. (iii) Hypotheses (iii) can be equivalently formulated as a "unique continuation property" for ker(L), i.e. u E ker(L) and u(s) = for some s E [-1,0] imply u == 0.
°
A direct consequence of Theorem 4.1 . is the following extension of the Waltman-Wong result: Corollary 4.3. Let (HI) (H4) be satisfied and 0 0 + 0 b 0 T be bijective. Further suppose liIllJlqlioo-+ oo If(t, q)1 / IIqlloo = 0, uniformly fort E [0, b], and IIg(q,p)lIoo /(lIqlloo + IIplloo) - - as IIqlloo + IIplioo - - 00. Then Eq. (1 .2) has at least one solution.
°
Proof. L is an injective Fredholm operator in view of Eq. (3.1), hence all assumptions of Theorem 4.1. are fulfilled . Now we are going to illustrate the applicability of Theorem 4.1. to problems with "perturbed periodic boundary conditions" . Corollary 4.4. Let k : [0, 1] x R - - [-1, 1] be a continuous function with k( t, x) ::; t for all t E [0,1] and x E R, F: [0,1] x R - - R be continuous, and 9 : C x C - - C be completely continuous and bounded. Further suppose:
(i) limlxl-+oo IF(t, x)1 / Ixl
=
°uniformly for t E [0,1],-
(ii) (3v E {-1, I} )(3r ~ O)(Yt E [0, 1])(Yx E R : Ixl ~
r) : vF(t, x)sgn x
~ 0,-
(iii) There exists a continuous function h : [0,1] -- R+ with J~ h(t)dt > > sup{g(q,p)(O) : q, p E C} and liminflxl-+oo IF(t,x)1 ~ h(t) uniformly for
t Then
E
[0,1]. u(t) = F(t, u(k(t, u(t))) Uo
=
U1
+ g(uo, ud
(4.1)
174
BVPs FOR FDEs
has at least one solution. Proof. We set 0 0 = Ide, 0 1 = -Ide and f(t , q):= F(t,q(k(t,q(O» - t) \1O:::; t:::; 1, \lq E e and apply Theorem 4.1. We have (0 0 - 0 1 0 T)q(s) = q(s) - q(O) \1-1 :::; s :::; 0, hence ker(0 0 - 0 1 0 T) consists of the constant functions on [-1,0]' thus ker(L) = {u E Z : u constant} in view of Eq. (3.1). Eq. (3 .2) yields that (y,p) E YxC belongs to ran(L), iff there exists a q E e with q(s)-q(O)- J~-' y(T)dT = p(s), which is equivalent to - J~ y(T)dT = p(O), i.e. ran(L) has codimension 1. Consequently, L is a Fredholm operator of index 0. Obviously, f satisfies (H2) and is quasi bounded with IIlflll = 0. We set TJ = -t for t E [0,1], (It-l,O) == -1 and ((0) = 0, and obtain (i) of Theorem 4.1. Now choose t: = ~[J~ h(t)dt - sup{lg(q,p)(O)1 : q, pEe}] and find 6 according to (iii) such that IF(t , x)1 ~ h(t) - dor t E [0,1] and x E R \ [-6, 6]. Select r > according to (ii) and set p := max{h,r} , then (ii) implies for all U E dom(L) with lu(s)1 ~ p that
°
fl dTJ(t) f(t, Ut) +
Jo
1 d((s) g(uo, ud(s) 0
-1
= -
fl F(t, u(t , k(t, u(t))) dt + g( Uo , ud(O)
Jo
is either smaller than -;< or greater than ~ . Thus (ii) of Theorem 4.1. is fulfilled and clearly d = 1 in (iii) of that theorem. Finally, let w be the inverse of the identification map between constants on [-1,1] and their value, then G(u) := J~ F(t , w(u) dt + g(w(u), w(u»(O) for u E R, and (ii) and our previous considerations show that vuG(u) > for lui = p. Therefore deg(G, (-p, p), 0) f 0, and (iv) of Theorem 4.1. is also established. Consequently, Eq. (4.1) has a solution.
°
Remarks 4.5. (i) Corollary 4.4. does not apply to Example 2.2., but it should be clear from the proof how to proceed in that case. The linear operator L would be given by
L(x,u)
= ((x+Qx( ' -T)-u(. -u) , idhx),(XO-Xb,UO-Ub»)
for (x, u) E Z, R(x, u) continuously differentiable, which is a Lo-completely continuous perturbation of Lo : (x,u) ...... ((x , u),(xo - Xb,UO - Ub»), hence a. Fredholm operator of index O. The new difficulty consists in determining the kernel of L, which depends on the choice of the parameters involved. (ii) There are many other existence results for semilinear Fredholm operator equations, and most of them provide further useful existence theorems for Eq. (1.2) via the approach outlined in Section 3. We refer to [5] for a collection of such abstract solvability assertions. (iii) Our approach applies also to various classes of neutral functional differential equations. It may be necessary, though, to replace C by another space of "initial conditions" and to use related existence results for L-condensing nonlinearities.
175 SEMILINEAR BOUNDARY CONDITIONS
(iv) We mention [1) and [2) for other results on problems with nonlinear boundary conditions. Most applications in these papers focus on ~point boundary" conditions, though.
References 1. A. R. Abdullaev, Solvability of boundary value problems for functional-differential equations with Lipschitz nonlinearities, Differential Equations 25 (1989), 1283 1287.
2. A. R. Abdullaev and A. B. Burmistrowa, Solvability of boundary value problems in the case of resonance, Differential Equations 25 (1989), 1437 1442. 3. K. Deimling, Nonlinear Functional Analysis, Springer-Verlag 1985. 4. I. J . Diaz, to appear in The Mathematics of Models for Climatology and Environment, Proc. NATO ASI, Tenerife, 1995. 5. R. E. Gaines and L. L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Mathematics, vol. 568 Springer-Verlag, 1977. 6. J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, 1977. 7. J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993. 8. G. Hetzer, S-shapedness for Energy Balance Climate Models of Sellers-Type, to appear in The Mathematics of Models for Climatology and Environment, Proc. NATO ASI, Tenerife, 1995. 9. P. Waltman and J. S. W. Wong, Two point boundary value problems for nonlinear functional differential equations, Trans. Amer. Math. Soc. 164 (1972), 39 - 54. 10. Kehong Wen, Pin Chen, and J. S. Turner, Bifurcation in a Lienard equation with delays, in Proc. Dynamic Systems Appl. 1 (1994) , eds. G. S. Ladde and M. Sambandham,. 377 - 384.
177
PERIODIC SOLUTIONS OF FUNCTIONAL DIFFERENTIAL EQUATIONS OF RETARDED AND NEUTRAL TYPES IN BANACH SPACES
LE HOANHOA Dai Hock SuPhan-School of Education 280 An Duong Vuong HoChiMinh City, Vietnam KLAUS SCHMITT Department of Mathematics University of Utah Salt Lake City, UT 84112 USA
Abstract
The paper presents some applications of fixed point theorems for operators of the form U + C on '" bounded closed convex subset of a locally convex space to the existence of periodic solutions of functional differential equations of retarded and neutral types in a Banach space.
1. Introduction Let E be a Banach space and let r > 0 be given. We denote by C = C ([ -r, 0], E) the Banach space of all continuous functions from [-r,O] to E equipped with the usual norm. For each continuous function x : IR -+ E and for t ~ 0 we let Xt E C be defined as in [5]. In this paper we shall consider the following functional differential equations
X'(t) = A(t)xt + g(t,Xt),
(1)
[X(t) - A(t)x(t - r)l' = g(t, Xt),
(2)
and
where {A(t)} is a continuous time periodic (with period w) family of bounded linear operators from C to E or E to E and g : [0,00) xC -+ E is completely continuous and
178
BVPs FOR FDEs time periodic with period w . We shall employ fixed point results from [9J to deduce the existence of periodic solutions of (1) and (2). We shall adopt the notations and conventions of [9J , the latter paper hence is requisite for the reading of the present one. In case E is a finite dimensional space these problems have been extensively investigated, see e.g. [2J, [3], [4J, [6], [7], [l1J and also many references in [5J . In this case [2J employs homotopy arguments for semilinear Fredholm maps to, in fact, reduce (2) to (1). This reduction is not possible in infinite dimensions. This paper is organized as follows . We first state the relevant fixed point theorem and provide, without proofs, some conditions for the relative compactness of subsets of some function spaces used in our discussion. We then use these criteria together with some existence and continuous dependence theorems for initial value problems and the fixed point results of [9J to obtain existence theorems for periodic solutions of (1), and (2). The proofs given here, for lack of space, will only be sketched. Complete proofs are contained in the authors' longer (unpublished) article [10J . 2. A Fixed Point Theorem In this section we shall state a fixed point theorem from [9J for operators of the form U + C on a bounded closed convex subset of a locally convex space, where C is a completely continuous operator and U satisfies condition (A) (defined below) . This fixed point theorem will be the principal tool to establish periodicity theorems for the functional differential equations stated . Condition (A) is defined precisely as follows: Condition (A): Let X be a locally convex topological vector space and let P be a separating family of seminorms on X. Let D be a subset of X and let U : D ..... X . For any a E X, define Ua : D -+ X by Ua(x) = U(x) + a . The operator U : D --t f{ is said to satisfy condition (A) on a subset 0 of X if: (A.l) For any a E 0, Ua(D) C D. (A.2) For any a E 0 and pEP there exists ka E Z+ with the property: for any f > 0, there exist r E Nand 5 > 0 such that for x, y E D with Q~(x , y) < f + 5 implies Q~(U;(X), U;(y))
0 (depending on p) such that p(U(x) - U(y))
~
179 RETARDED AND NEUTRAL EQUATIONS
(iii) There exists Xo E X with the property: for any pEP there exist r E Nand A E [0,1) (r and A depending on p) such that p(U;o(x) - U;o(Y))::; Ap(X - y). (iv) C is completely continuous p( C (A)) < (v)
lim p(C(x))/p(x)
p(:r)_oo
= 0 for
00
whenever p( A)
(O)+J;f(t,xt)+J;g(t,xd,
=
4> E C.
(4)
Since [-r , 00) = Uf[-r, n], one may think of X as a Fnkhet space with metric defined by a sequence of seminorms as considered above. For x E X one defines x E X by _( ) = { x(s)
x s Further define U, G : X
+ 4>(0) -
4>(s), -+
x(O), if if
~ 0 s E [-r,O].
s
X as follows:
10' f(t,xt), Gx(s) = 10' g(t,Xt).
Ux(s) =
One then sees that a fixed point x of the operator U + G will yield x as a solution of the initial value problem. In order to see that U + G has a fixed point, one employs the fixed point theorem (Theorem 1) of [9]. The verification is technical and lengthy. Concerning continuous dependence of solutions on initial conditions and the equations we have the following result.
181 RETARDED AND NEUTRAL EQUATIONS
Theorem 3 Let {A(t)} be a family of bounded linear operators from C to E depending continuously upon t. For each n E N let gn be a mapping satisfying the conditions imposed on 9 by the previous theorem and assume gn converges uniformly to go. Further let {4>n} c C with 4>n -+ 4>0 in C. For each n E N let Xn E X be a solution of X'(t) Xo
and assume that for n = Then Xn -+ Xo in X .
= A(t)xt + gn(t, Xt), = 4>n E C,
t ~
°
(5)
°the problem is uniquely solvable.
5. Periodic Solutions
Let us now consider the problem of the existence of periodic solutions of the equation X'(t) = A(t)xt + g(t, Xt),
(6)
where {A(t)} is a family of bounded linear operators from C to E depending continuously upon t which is also w- periodic in t. Further the mapping 9 satisfies the conditions imposed on 9 earlier and is also w- periodic in t. We further shall assume that initial value problems associated with this equation are uniquely solvable (existence follows from our earlier theorem, we hence assume uniqueness here) . If we denote by x( 4» E X the unique solution with initial condition 4>, then the continuous dependence theorem guarantees that the mapping
is a continuous mapping. One next establishes the existence of two-parameter families of bounded linear operators S(t,s):C-+C, V(t,s):E-+E, t~-r, s~t such that a solution x of (6) is given by Xt(4))( · ) = S(t, 0)4>
where S(t+w,s)
and V(t
+
l Yt(.,
s)g(s, x s (4)))ds,
= S(t+w,t)S(t,s)
+ w,s) = V(t + w, t)V(t, s),
One then defines the operator
T : C -+ C, T
= S + F,
t ~ s.
182
BVPs FOR FDEs by F(4))
= 10'" Vw(-,s)g(s, x s (4)))ds
and
S
= S(w,O),
and one obtains periodic solutions of the equation as fixed points of the operator T. One obtains the following theorem. Theorem 4 Under the above assumptions the equation has an w-periodic solutions, whenever the family {A(t)} is uniformly asymptotically stable. The proof proceeds via a sequence of steps: • One shows that the operator F is completely continuous. • One shows that F is quasibounded with quasinorm 0, i.e.
IFI =
lim sup lIull-oo
IIF(u)1I = O. lIuli
• The uniform asymptotic stability of {A(t)} implies that there exist positive constants a, b such that
IIS(t,O)4>1I :::; bll4>lIe- at and hence One hence may apply the fixed point theorem (Theorem 1) to deduce the existence of a fixed point. 6. The Neutral Equation Let us now consider the neutral equation [x(t) - A(t)x(t - r)]' = g(t, Xt),
(7)
under assumptions similar to the above, i.e. 9 : [0, 00) x C --+ E
is completely continuous and lim Ig(t,x)1 - 0 11"11_00 IIxll - , uniformly with respect to t on bounded intervals with {A(t)} a continuous family of bounded operators from E to E. Under these assumptions one has the following existence theorem.
183 RETARDED AND NEUTRAL EQUATIONS
Theorem 5 The initial value problem [x(t) - A(t)x(t - r)l' Xo
(8)
has a solution.
One first shows that the initial value problem (8) is equivalent to the integral equation x(t) Xo
= =
A(t)x(t-r)-A(0)x(-r)+1/>(0)+J~g(s,x8)ds, t~O I/> E C.
(9)
One then defines the operators Zx(t)
=
A(t)x(t - r) - A(O)x( -r), t ~ 0 0, -r ~ t ~ 0,
(10)
and Gx(t)
= =
1/>(0) + J~ g(s, xs)ds, t ~ 0 I/>(t), -r ~ t ~
o.
(11)
Then Z is a bounded linear operator on X and G will be compact. Furthermore one may verify that the conditions of the fixed point theorem (Theorem 1)) hold and obtains a fixed point of the operator F + G and hence a solution of the initial value problem. One next obtains a continuous dependence theorem analogous to the one above, namely.
Theorem 6 Let {A( t)} be a family of bounded linear operators from E to E depending continuously upon t. For each n E N let gn be a mapping satisfying the conditions imposed on 9 by the previous theorem and assume gn converges uniformly to go. Further let {I/>n} c C with I/>n -+ 1/>0 in C. For each n E N let Xn E X be a solution of [x(t) - A(t)x(t - r)l' = gn(t,Xt), t ~ 0 Xo = I/>n E C,
(12)
and assume that for n = 0 the problem is uniquely solvable. Then Xn -+ Xo in X.
These results, in turn, may be used in a very similar manner to our earlier procedure to deduce an existence theorem for periodic solutions of the neutral equation. One obtains the following result assuming the w- periodicity of the operators involved.
184 BVPs FOR FDEs Theorem 7 Under the above assumptions the equation has an w-periodic solutions, whenever the family {D(t, 4» = 4>(0) - A(t)4>} is stable. (See [5] for the concept of stability of such a family.) A result similar to the one just stated also holds for the more general case of operators of the form
D(t,4» = 4>(0) where t 2": 0, 4> E C,
ri
L
Ai(t)4>( -ri),
> O. Note that this family is stable, whenever
L
IIAi(t)11 < 1,
(see e.g. [6], [11]). References 1. J. Dieudonne, Foundations of Modern Analysis, Academic Press, New York, 1969.
2. L. Erbe, W. Krawcewicz, and. J. Wu, Leray-Schauder degree for semilinear Fredholm maps and periodic boundary value problems of neutral type, Nonl. Anal., 15 (1990), 747-764. 3. G . Gustafson and K. Schmitt, A note on periodic solutions for delay-differential equations, Pmc. Amer. Math. Soc. 42 (1974), 161-166. 4. J . Hale and M. Cruz, Asymptotic behavior of neutral functional differential equations, Arch. Rat. Mech. Anal. 5 (1969), 331-353. 5. J . Hale and S. Lunel, Introduction to Functional Differential Equations, Springer Verlag, New York, 1993 6. J. Hale and J. Mawhin, Coincidence degree and periodic solutions of functional differential equations, J. Diff. Eq. 15 (1974), 295-307.
7. G. Hetzer, Boundary value problems for retarded functional differential equations, Comment. Math. Univ. Carol. 16 (1975),121-137. 8. L. Hoa, On a fixed point theorem of Krasnosel'skii and its applications, Acta Math. Vietnamica 14 (1989), 3-17. 9. L. Hoa and K. Schmitt, Fixed point theorems of Krasnosel'skii type in locally convex spaces and applications to integral equations, Results in Math. 28 (1994),
290-314.
185 RETARDED AND NEUTRAL EQUATIONS
10. L. Hoa and K. Schmitt, Periodic solutions of functional differential equations of retarded and neutral types in Banach spaces, preprint. 11. O. Lopes, Forced oscillations in nonlinear neutral differential equations, SIAM J. Appl. Math. 29 (1975), 195-207.
187
EXISTENCE OF TRAVELING WAVES FOR REACTION DIFFUSION EQUATIONS OF FISHER TYPE IN PERIODIC MEDIA
W . HUDSON AND B. ZINNER Department of Discrete and Statistical Sciences Auburn University, Alabama 36849-5307 USA
Abstract
The existence of periodically varying wavefronts is established for a class of reaction-diffusion equations of Fisher type where the reaction term is periodic in the space variable.
1. Introduction Fisher's equation (Fisher 1937) Ut
= u"'''' +u(1- u),x E (-oo,oo),t ~ 0
was proposed as a model for the propagation of an advantageous gene in a homogeneous unbounded space. Since it is one of the simplest nonlinear reaction-diffusion equation that admits traveling waves, it has been the focus of a large number of studies. Fisher's equation and extensions thereof have been analyzed, among others, by Kolomogorov, Petrovskii, and Piskunov (1937), Aronson and Weinberger (1975), Hadeler and Rothe (1975), Murray (1977), Fife (1979). Most of the work is concerned with traveling waves propagating in a homogeneous media. The problem of propagation speed in a heterogeneous media was investigated by Gartner and Freidlin (1979), Freidlin (1986), Shigesada, Kawasaki, and Teramoto (1986) . The existence of periodically varying traveling waves for the bistable reactiondiffusion equation with periodic coefficients was established recently by Xin (1991, 1992, 1993, 1994). In this article we consider the equation
(1.1) where
Ut
f
= U""" + f(u,x),u = u(t,x),x E (-oo,oo),t ~ 0,
satisfies the conditions
BVPs FOR FDEs ( C l ) f is jointly continuous and Lipschitz continuous in u . (C2) f is periodic in x with period 1, and (C3) f is of Fisher type, i.e., there exists a 1-periodic continuous function G ( x ) > 0 such that f ( u ,x ) > 0 for u E (O,ii(x))and f ( u ,x ) 5 0 for u > ~ ( x ) . We will show the existence of a critical speed c, such that for all c 2 c, there is a periodically varying wavefront of (1.1) of speed c. More precisely, let c, = inf sup y"(x)
"'
z~[O,1]
+P(X)Y(X) Y(X)
where p ( x ) = s u p { y : u E ( 0 ,~ ( x )and ) the infinum is taken over all r > 0 and y E C 2 ( R )for which y ( x ) > 0 for all x and y ( ~ ) e is- ~periodic ~ with period 1.
Theorem 1.1 For all c 2 c, there exists a function u ( t , x ) which is a solution of (1.1) in the distributional sense. This solution u may be written in terms of a wave profile U , namely u ( t ,x ) = U ( x + ct, x ) , ( t ,x ) E R 2 where U ( x ct, .) is periodic with period 1. Furthermore, u(., x ) is nondecreasing for each x , u ( t , .) is uniformly Lipschitz u ( t ,x ) exists and is zero, lim,+m inf u ( t ,x ) 2 min C ( x ) , and continuous, liw,-, lim,,+, sup u ( t ,x ) 5 m a x i i ( x ) for all t E R .
+
The lower speed c, agrees with the speed of propagation obtained by Gartner and Freidlin (1979) for compactly supported initial data.
2. Existence of Approximating Wavefronts In this section we will show the existence of approximating wavefronts V m . The wave profile U of our Theorem will be obtained by transforming a subsequential limit V of these approximating wavefronts. The wavefronts V m satisfy the equations
.m
( k t ,k2-m)
(2.1) V,c,
= 22mAmVm(c,,,t, k2-m)
+ f ( V m ( c m t ,k2Tm),k2-m),
k E Z , t E (-oo,m), m = 1,2,3,. . . where A m V ( t , x ) abbreviates V ( t ,x - 2-m) -
+
+
2 V ( t ,X ) V ( t ,x 2-") and c ~ ( c tx ,) denotes $ ~ ( c tx,) . In the remainder of this article V m represents a solution of (2.1) with the following properties: Vm(O,0 ) = min i i ( x ) .m
V ( t , k2-"') 2 0 , (2.2)
+
+
V m ( t ,k2-m 1 ) = V m ( t 1, k2-m), limt,-, V m( t , k2-m) = 0 , and V m ( t ,k2-m) 5 m a x i i ( x ) , min i i ( x ) 5 limt,,
189
EQUATIONS OF FISHER TYPE IN PERIODIC MEDIA
for all k E Il, t E IR. Such a solution exists for all positive
(2.3)
Cm :::::
c.(m), where
c.(m):= infsuplj(a,r) , r ,a
i
the supremum is over j E {I, 2, . .. , 2m },
°
ao := a2me-r, a2m+1 := al er , and the infimum in (2.3) is taken over all r > and all a = (at, a2, . . . ,a2m) such that all the coordinates of a are positive. (See Corollary 4.1 in Hudson and Zinner 1994). Lemma 2.1 Let ~:= inf{JL(x) : x E IR}. Then ~> 0.
Proof: Since f( u, .) has period 1, the definition of JL( x) implies that JL(-) has period one. Hence there exists a sequence (xn) in [0,1] such that
Let Xo be a limit point of x n. We may and do assume that Xn converges to Xo. Now by the definition of JL there exists a point Uo E (0, u(xo)) such that
°< ~JL(xo):S f(uo,xo). Uo 4
Since f( uo, ·) and u are continuous, there exists 0 > 0 such that for all x E (xoo,xo + 0), ~JL(xo) :S J(':'"x) and Uo < u(x). So for n so large that IXn - xol < 0,
~JL(xo) :S
! O.
Lemma 2.2 Let c ::::: c. . Then there exists a sequence of positive numbers {c m } such that C m > c.(m) , m_oo lim em = c. , and system (2.1) has a solution Vni with the above properties.
Proof: It suffices to show that c. > 0 and that 2~ sup c.(m) :S c•. Let I!:. be as in Lemma 2.1. Then ~> O. Choose sequences {Yn} and {rn}, rn Yn E C 2(1R), Yn(x) > 0, Yn(x)e- rnX periodic with period 1, such that sup
xe[O,I]
y~(x)
> 0,
+ JL(x)Yn(x) < +1 ) c.-. Yn(x
-
n
Since Yn(O) < Yn(I), t~ere exists b E (0,1) such that y~(b) > O. Therefore we may assume without loss of generality that y~(O) > OJ otherwise consider Yn( b + x) . Then
190
BVPs FOR FDEs
and therefore
y~(1) - y~(O)
t
+ 10
1
fl
p(x)Yn(x)dx ~ (c. +;;) 10 Yn(x)dx .
Since and
fl p(x)Yn(x)dx ~ P fl Yn(x)dx
10
-10
e.>
it follows that Co + ~ ~ i!: Hence Co ~ o. Let a'J = Yn(j2- m) for j = 1, ... , 2m. Then for an := (ai', a~, ... , a~~), x'J := j2- m,
for j = 1, ... ,2 m • Note that there exists mo such that for all m
1lj(an, rn) _
Y~(Xj)
+ p(Xj )Yn(Xj) Yn(xj)rn
1
x-oo
x~-oo
min ii( x). We will assume in the following without loss of generality that 1 is bounded.
Lemma 3.1 For every t E JR., mEN, and k E Z
jVm(t, k2- m) - Vm(t, (k - 1)2-m) where M :=
lIuli oo + 11/1100.
I~
M2- m
191 EQUATIONS OF FISHER TYPE IN PERIODIC MEDIA
Proof: Let t and m be chosen arbitrarily and then fixed. For brevity write Wk(t) := V m(t,k2- m), k E Z. Since for each k and m, V m(-,k2- m) is nondecreasing, 0::; Wk and consequently 0::; 22m(Wk+1 - 2Wk + wk-d + f(Wk,k2- m ). Let j ::; r be any two integers. Then r-l
0::; L:{Wk+1 - 2Wk + Wk-l + 2- 2m llflloo} k=j
and hence
(3.1) Sum over r = j to r = j
+ 2m
-
1 to see that
2m(wj - wi-d ::; Wj+2~-1 -
Wj-l
+
(2m _ 1)2m 2
.T
2m
llflloo
::; 11111100 + IIflloo Hence (3.2)
Similarly, use (3.1) to see that
r
::;
L:
=
2m( Wr
{wr -
-
WT-I
Wr-l
+ (r - j)T2mllfIl00}
) + (2m -2 1)2m2-2mllfll
::; 2m(wr - Wr-l)
00
+ IIflloo.
So
(3.3) The Lemma follows from (3.2) and (3.3). Definition 3.2 For real numbers x such that k2- m ::; x < (k
o:m(x)
:=
2m((k
+ I)Tm
Then for such an x and s E JR, let
+ 1)2-m, define
- x) and am(x) :== 1- o:m(x).
192
BVPs FOR FDEs Lemma 3.3 For every t E IR, x, Y E IR,
Proof: The proof will be divided into two cases. In the first case we suppose that for some integer k, k2- m x, y (k + 1)2- m • In this case
:s
:s
wm(t, x) - vm(t, y)1
I(am(x) - am(y))(vm(t, kTm) - vm(t, (k
:s :s
2m{(k + 1)2-
Mix -
m
-
x - (k
+ I)Tm + y}MTm
yl
:s
:s
In the second case suppose that (k - 1)2- m x k2- m (k + r + 1)2-m . Then by the triangle inequality and case 1, Wm(t,y)-Vm(t,x)l
:s
+ I)Tm))l
:s
(k
+ r)2-m
:s y :s
Wm(t,y)-Vm(t,(k+r)Tm)1 T
+ L wm(t, (k
+ j)Tm) -
vm(t, (k
+j
- I)Tm)1
j=l
:s
T
M{y-(k+r)T m +LT m +(k2- m -x)} j=l
Mlx-yl· Lemma 3.4 Let x E IR be fixed . Then any subsequence Vm'( .,x) of Vm(·,x) con-
tains a further subsequence Vm"(·,x) which converges to u unique right continuous nondecreasing function V(·,x) at each point of continuity ofV(·,x).
Proof: Each V m (., c) is a continuous nondecreasing function which satisfies the conditions lim Vm(s,x) = 0 and lim Vm(s,x) exists. The Lemma now follows by the "_-00
5-+00
Relly Compactness Theorem (e.g. see R. G. Tucker [11], Theorem 2, p. 83). The next task is to show that there is a subset D of IR and a subsequence Vmo(k) of vm such that (a) Vmo(k)(.,x) converges to say V on DC for each binary rational x, and (b) D is countable. Let B denote the set of all binary rationals in IR and write the elements of B as a sequence (x n ). By Lemma 3.4, there is a right continuous non decreasing function V(·, xt} and there is a subsequence Vm,(k) of vm such that V(t, Xl) := limk~oo Vm,(k)(t, xtl exists for each point t of continuity of V(·, xt}. Let DI denote the countable set of discontinuities of V(·,XI) . Then for each t ft Db Vm,(k)(t,XI) converges to V(t,xt}. Now use Lemma 3.4 again to choose a subsequence V m2 (k) of Vm,(k) which converges
193 EQUATIONS OF FISHER TYPE IN PERIODIC MEDIA
to V(·,X2) say, at each point of continuity of V(·,X2). Continue this process. At the ph step, choose a subsequence Vmj(k) of V ffli - 1 (k) such that Vmi(k)(t,Xj) converges to a limit V(t,Xj) at each point t of continuity of V(·,Xj) . Let D j denote the set of discontinuities of V( ·,Xj). Then for t i U{D i , vmi(k)(t,Xl) converges to V(t,Xl), l = 1,2, ... ,j. Now use the diagonal argument to see that there is a subsequence Vmo(k) such that for each tiD := Ur'D i , lim Vmo(k)(t,x)
k-+oo
= V(t,x),
x E B.
Since each Di is countable, D is countable. Lemma 3.5 For each tiD and for each x E IR., the sequence vmo(k)(t, x) converges.
Proof: It is enough to show that Vmo(k) (t, x) is a Cauchy sequence when tiD and x E JR. Let I" > O. Choose a binary rational q such that Ix - ql < f/(3M). Choose k so large that if rl 2 K and r2 2 K, then Ivmo(r,)(t,q) - vm o(r 2 )(t,q)1 < 1"/3. Then for such rl and r2,
IVmoh)(t,x) _ vmo(r')(t,x)l::; JVmo(r,)(t,x) _ vmo(r,)(t,q)1 +Ivmo(r')(t,q) _ vmo(r')(t,q)1 +IVmo(r»(t,q) _ v mo (r 2 )(t,x)1 I"
::; Mix - ql + 3" + Mix - ql I"
f
0 so V(-oo,O) Xl as long as u(x) < minu(x) because f(s,t) ~ 0 for all s E [O,minu(x)) . Since u(xd is the minimum of 1.1 and since 1.1 is periodic it follows that u(x) = U(XI) for all x. It follows that u(x) = 0 for all x . Now u( -00, x) and u( 00, x) are solutions of (3.4) . Furthermore 0 :::; u( -00 , 0) :::; ~minu(x) :::; 1.1(00, 0). Therefore by Lemma 3.18, u(-oo , x) = 0 and u(oo , x) ~ min u( x) for all X E JR. References 1. D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion and nerve propagation, Lecture Notes in Mathematics 446, SpringerVerlag (1975) , pp. 5-49.
2. P. Fife, MathematIcal Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics 28, Springer-Verlag (1979) . 3. R. A. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics 7 (1937) , 355-369. 4. J. Ga.rtner and M. I. Freidlin, On the propagation of concentration waves in periodic and random media, Soviet Math . Dokl. 20 (1979) , 1282-1286. 5. M. I. Freidlin, Limit theorems for large deviations and reaction-diffusion equations, Ann. of Probability 13 (1985), 639-675. 6. K. P. Hadeler and F. Rothe, Traveling fronts in nonlinear diffusion equations, J. Theor. BioI. 52 (1975), 251-263. 7. W. Hudson and B. Zinner, Existence of traveling waves for a generalized discrete Fisher's equation, Com. Appl. Nonlin. Analysis 1, No. 3, (1994), 23-46.
199 EQUATIONS OF FISHER TYPE IN PERIODIC MEDIA
8. A. Kolmogorov, 1. Petrovskii, and N. Piskunov, Etude de l'equation de la quantite de la matiere et son application a un probleme biologique, Moscow University Bull. Math. 1 (1937), 1-25. 9. J. D. Murray, Lectures on Nonlinear Differential-equation Models in Biology, Clarendon Press, Oxford, 1977. 10. N. Shigesada, K. Kawasaki, and E. Teramoto, Traveling periodic waves in heterogeneous environments, Theor. Populo Biology 30 (1986), 143-160.
11. H. Tucker, A Graduate Course in Probability, Academic Press, 1967. 12. X. Xin, Existence and stability of travelling waves in periodic media governed by a bistable nonlinearity, Journal of Dynamics and Differential Equations, 3, No. 4, (1991), 541-573. 13. G. Papanicolau and X. Xin, Reaction-diffusion fronts in periodically layered media, J. Stat. Physics 63, Nos. 5/6, (1991), 915-931. 14. J. X. Xin, Existence of planar flame fronts in convective-diffusive periodic media, Arch. Rat. Mech. and Anal. 121 (1992), 205-233. 15. J. Xin, Existence and nonexistence of traveling waves and reaction-diffusion front propagation in periodic media, Journal of Stat. Physics 73 (1993), 893-925. 16. J. X. Xin, Existence of multidimensional traveling waves in transport of reactive solutes through periodic porous media, Archive for Rational Mech. and Analysis 128 (1994), 75-103.
201
PERMANENCE OF PERIODIC SOLUTIONS OF RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS
LUIZ A. da C. LADEIRA AND SUELI M. TANAKA Instituto de Ciencias Matematicas de Sao Carlos Universidade de Sao Paulo Campus de Sao Carlos 13560-970 - Sao Carlos - SP - Brazil
Abstract For
a
retarded
functional
differential
equation
given
by
x(t) = F(x(t),x(t - (7),x(t - T)) with an w-periodic solution p(t), it is proved that if we take delay sufficiently close to (7 and T we still obtain periodic solutions near to p(t) of period close to w. An illustration of this result is given.
1. Preliminaries Theorem 1.1 Let E be an open subset of a Banach space Y , F be a closed subset of a Banach space X, where interior of F is a non-empty set. Assume that T : F x E -+ F satisfies the following hypotheses: (i)T(x,.) : E -+ F is continuous. (ii) T( . , y) : F -+ F is continuous and, for each y E E , has a unique fixed point x(y) that depends continuously on y. (iii) If x(E) Fl C F, then T(x, y) is continuously differentiable in y for (x,y) E Fl X E. (iv) There is an open set F2 C X such that FeF2 and the derivative DxT(x,y) of T(x,y) with respect to x is continuous and satisfies the condition: I DxT(x,y) I::; 0 -1, we take s
y(s)
= x(t), therefore y(s) must satisfy:
. y(s) = (1
+ (3)F(y(s),y(s -
t = --f3
1+
U+E r+E 1 + (3)'y(s - 1 + (3)) .
and
(2.4)
Note that if we find an w-periodic solution of (2.4) then we get a we take another change of variable
+ (3) w-periodic solution of (2.3)j so that y(s) = p(s) + z(s), and thus z(s) satisfies:
(1
z(s) = (1
+ (3)F(p(s) + z(s),p(s -
U+E 1 + (3)
+ z(s -
U+E r+E 1 + (3)'p(s - 1 + (3)
r+E +z(s - --(3)) - F(p(s),p(s - u) , p(s - r)) .
(2.5)
1+
1 Let us take f3 such that 1f31 ::s 2' r functional differential equation in C.
n = {u : IR -+ IRnj 1.1 is = sUP tE IR IIu(t)IIIR we define
For
111.1110
n
,
= 2( r + Eo)
.
> 0 and we consIder (2.5) as a
continuous and w-periodic } with the norm
204 BVPs FOR FDEs
G .• lR x 0 x (-'f. 0 ,f.) 0
X
(-~2'2~) --> lR
n
G(s,U,f.,(3) = (1
+ (3)F(p(s) + u(s),p(s -
a+f. 1 + (3)
a+f. T+f. 1 + (3) ' p(s - 1 + (3)
+ u(s -
+ u(s -
r+f.
1 + (3))
of
-F(p(s),p(s - a),p(s - T)) - ox (p(s) , p(s - a),p(s - T)) u(s)
of
,
of
- oy (p(s),p(s - a"p(s - r))u(s - a) - oz (p(s),p(s - a),p(s - r))u(s - r) It is shown that the equation (2 .5) is equivalent to
i (s) = L(s) z.
+ G(s, Z,f.,(3)
(2.6)
and thus we interpret it as a perturbation of linear equation (2 .2) . Using (H .2) , we can decompose C relative to the characteristic multiplier 1 as C=EffjK, we also take P the projection induced by this decomposition such that P takes C onto K . Note that the equation (2.2) has a unique w-periodic I.i ., then the formal adjoint equation associated has only an w-periodic solution I.i. denoted by q(t) and assume that f;;' q( a) qT (a) da = 1 with qT the transpose of q. We know that if hE 0, f;;' q(a) h(a)da = 0, the equation
i( s) = L( s) z.
+ h( s)
(2.7)
has a unique solution in 0 whose (I - P)-projection is zero. Let us designate this solution by K hand K is a continuous linear operator taking 0 into O. Consider the continuous linear transformation Q taking 0 into 0 such that Q(h) = U;;' q(a) h(a)da)qT(.), and we can obtain K(I - Q)h the unique w-periodic of i(s) = L(s) z. + (I - Q) h(s) whose (I - P)-projection is zero. Define
1 1
H : 0 x (-f.o,f. o ) x (-2"' 2") -> 0 H(u, f., (3) = K(1 - Q) G( . , U, f., (3). From the Mean Value Theorem, H is a uniform contraction with relation to f., {3 and then we can use the Fixed-Point Theorem to show that there are positive constants Db f.}, {31, u·(f., (3) E OJ Ilu·(f., (3)110 ~ D}, 1f.1 ~ f.}, 1(31 ~ (31, u·(f.,(3) continuous in (f.,{3) , u·(O,O) = 0 such that u·(f.,(3) is solution of
i(s)
= L(s)z. + G(s,u·( f. , (3) , f., (3) -
B(f.,(3)qT(s)
(2.8)
205 PERMANENCE OF PERIODIC SOLUTIONS
with
B(c:,{J)
= low q(a) G(a,u·(c:,{J),c:,{J) da.
(2.9)
The Theorem 1.1 insures that u·(c:,{J) is continuously differentiable in (c:,{J) aI}.d
au·
we can prove that o{J (c:, (J) must satisfy:
v(s)
= L(s)v. + A(s,v,c:,{J) - ~;(C:,{J)qT(s),
(2.10)
where
A(s,v,c:,{J)
=-
of ox (p(s),p(s -17),p(s -
v(s -17) -
aF oz (p(s),p(s -
T))V(S) -
of oy (p(s),p(s -17),p(s -
T))
17),p(s - T)) v(s - T)+
17+C: T+C: (1 + (J)[ a.,p(s) v(s) + b.,p(s) v{s - 1 + (J) + c.,p(s) v(s - 1 + (J) J+ • 17+C:. 17+C: T+C: F(p(s)+u (c:,{J)(s),P(s-l+{J)+u (c:,{J)(s-I+{J),P(s-I+{J) r+c: 17+C: . 17+C: +u·(c:,{J)(s - 1 + (J)) + (1 + (J) b.,p(s) (p(s - 1 + (J) r + c: . +(1 + (J)c.,p(s)(p(s and
a.,p(s)
duo
17+C:
+ Ts(c:,{J)(s - 1 + (J)) r + c: duO r + c: 1 + (J) + Ts(c:,{J)(s - 1 + (J))
= F,,(p(s) + u·(c:,{J)(s),
17+C:. 17+C: T+C:. r+c: P(s-l+{J)+u (c:,{J)(s-l+{J),P(s-l+{J)+u (c:,{J)(s-l+{J))
b.,p(s) = Fy(p(s)
+ u·(C:,{J)(s),
17+C:. 17+C: r+c:. r+c: P(s-l+{J)+u (c:,{J)(s-l+{J),P(s-I+{J)+u (c:,{J)(s-I+{J)) c.,p(s)
= Fz(p(s) + u·(c:,{J)(s),
17+C:. 17+C: r+c:. r+c: p(s - 1 + (J) + u (c:,{J)(s - 1 + {J)'p(s - 1 + (J) + u(C:,{J)(s - 1 + (J))' Thus, the equation (2.10) has an w-periodic solution and so
BVPs FOR FDEs
hence,
In particular, taking e = p = 0, we have that
Let y(t) be a solution of (2.2), i ( t ) = y(t) + tp(t) and then
Suppose that the expression (2.11) is zero, we get y(t) = i ( t ) - tp(t) solution of (2.2), with fj and p w-periodic functions, yo = go - (.)A = cp. And so,
= cp - w up, = cp
+ U(-wp,)
= cp
+ U(Ucp - cp).
Therefore cp E N(I - U)'. We can prove that cp and p, are 1.i. and thus we have a contradiction because the hypothesis (H.2) implies that 1 is a simple characteristic multiplier. And then applying the Implicit Function Theorem for B(E,P) we get strictly positive real numbers ,f3z,~2; Pz 5 PI, 8 2 5 el, a continuous function P(e); I/?(e)I 5 Pz for all s satisfying IeI 5 ez such that P(0) = 0 and B(e, P(E)) = 0. From the equation (2.8) we conclude that u*(e, P(e)) is solution of (2.6), u*(O, 0) = 0. Thus for all e; 0 < lei 5 sz, we have x(e)(t) a solution of (2.3) of period W ( E ) = (1 P(e))w that satisfies x(0) = p, w(0) = w, with x(e) e ~ ( e depend ) continuously on 8.
+
207 PERMANENCE OF PERIODIC SOLUTIONS
Remarks: It can be seen from the proof of Theorem 2.1 that the same conclusion holds if the equation (2.3) is replaced by the equation
i(t)
= F(x(t),x(t -
(0- + £t}),x(t - (r + £2))) '
In this case, the periodic orbit continues to exist for £ = (£1, £2) in a full neighborhood of (0-, r) in the parameter space, and has the same continuity proprierties. The proof also shows that the result is valid for equations with finitely many delays. 3. Application Consider the following scalar functional differential equation:
i(t)
= ~[b2 a
x 2(t -.!.) + x(t)x(t - .!.)] 4 2
(3.1)
with a, b, nonzero real numbers. We can prove that p(t) = a + bcos 211"t is a I-periodic solution of (3.1) and the variational equation associated is given by:
y(t)
= ~[(a -
bcos 211"t) y(t) + (a + bcos 211"t) y(t -
a
~) -
2(a + b sin 211"t) y(t - .!.)] h.2)
In order to obtain a I-periodic solution y(t) of equation (3.2) we use the Theorem 1.2 and so y(t) has the form: y(t)
= Bo(t) + Bl(t) cos211"t + B2(t) sin211"t + B3(t) cos411"t
(3.3)
for B t , i = 0,1,2,3 a scalar 1/4-periodic function. We can show that the function 1, cos 211"t, sin 211"t, cos 411"t are i.i. over the 1/4periodic functions, namely, the unique scalar 1/4-periodic functions Ql(t), Q2(t), Q3(t), Q4(t) such that Ql(t) + Q2(t) cos 211"t + Q3(t) sin 211"t + Q4(t) cos 411"t = 0 are the zero functions. Substituting (3.3) into (3.1) and using that l,cos211"t,sin211"t,cos411"t are i.i. over the 1/4-periodic functions, we conclude that Bo(t), B1(t), B2(t), B3(t) must satisfy:
. Bo(t)
211"b
= --Bl(t) a
(3.4)
(3.5)
208 BVPs FOR FDEs
(3.6)
(3.7) k, k E ffi., Bo Bl B3 0, from (3.3), Therefore, B2 y(t) = k sin 211"t, k E ffi.. It is clear that p(t) generates the l-periodic solutions of (3.2), then, p(t) is a non degenerate solution of (3.1) . It follows that the equation (3.1) satisfies the statements (H.l) and (H.2), hence, the equation
x(t) =
11"
-W - x (t a 2
1
1
(- + c)) + x(t) x(t - (-2 4
+ c)) 1
has a periodic solution x(e) of period w(e) if we take e sufficiently close to zero, x(O) = p, w(O) = I, x(e), w(e) depend continuously on c. 0 References 1. L. A. V. Carvalho, On a method to investigate bifurcation of periodic solutions in retarded differential equations, Preprint.
2. J. R. Clayessen, Effect of delays on functional differential equations, 1. Differential Eqns. 20 (1976), 404-440. 3. J.K. Hale and L.A.C. Ladeira, Differentiability with respect to delays, 1 .Differential Eqns. 92 (1991), 14-26. 4. J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993. 5. L. A. C. Ladeira, Resultados sobre diferenciabilidade com rela!;ao ao retardamento, Tese de Livre-docencia, ICMSC-USP, Sao Carlos, 1992. 6. L. A. C. Ladeira and S. M. Tanaka, A method to calculate periodic solutions of functional differential equations, Preprint. 7. S. M. Tanaka, Urn estudo sobre solu!;oes peri6dicas de Equa!;oes Diferenciais Funcionais (A study on periodic solutions of F.D.E.), Doctoral Thesis, ICMSCUSP, Sao Carlos, 1994.
209 METHOD OF QUASILINEARIZATION FOR BOUNDARY VALUE PROBLEMS OF FUNCTIONAL DIFFERENTIAL EQUATIONS
V. LAKSHMIKANTHAM AND N. SHAHZAD
Applied Mathematics Program, Florida Institute of Technology Melbourne, Florida 92901 USA
1. Introduction
Quasilinearization is a well known technique for obtaining approximate solutions of nonlinear differential equations [1, 2]. It provides a monotone sequence of approximate solutions that converges quadratically to a solution of the boundary value problem (BVP),
-x"(t) = f(t, x(t)) , t E [0,1]' x(o) = a, x(1) = b, if f is uniformly convex. This technique has recently been generalized and extended using less restrictive assumptions so as to be applicable to a large class of functions [8-10] . This generalization is also discussed for initial value problems [5-6]. In this paper, we shall extend the method of quasilinearization to a boundary value problem associated with functional differential equations, that is,
-x"(t) where
f
= f(t , x(h(t))),
t EJ
= [0,1],
(1.1)
E e[J x JR, JR] and h E C[J, JR], subject to the boundary conditions
x(t)
= ¢>(t),
t E Jo = (-00,0], x(1)
= b,
where ¢> E C [Jo,JR]. This we shall do in the framework of the method of upper and lower solutions so that one can obtain a monotone sequence that converges quadratically to the unique solution of the BVP (1.1). We have discussed a simple BVP (1.1) in order to bring out the ideas involved in the method of quasilinearization. One could however, apply our method to the more general BVP 1
-x" = f(t,x(t),x(h(t,x(t))), x(t) = ¢>(t) on Jo and x(t) = tf;(t) on J 1 = [1,00), since necessary existence results
are available. See [3].
210 BVPs FOR FDEs
2. Main Results A function a E B = C[lR,lR]n BC1[Jo,lR]n BC1[{b},lR]n C1[lR,lR] having a continuous second derivative on J will be a lower solution of (1.1), if -a"(t) ::; f(t,a(h(t))), t E J, a(t) ::; tjJ(t) on J o, a(l) ::; b.
An upper solution fl(t) of (1.1) is defined similarly by reversing the above inequalities. For further details we refer to [3]. Next we recall a result [3] in a special form which is needed for the main result. Theorem 2.1: Assume that (i) ao, flo are lower and upper solutions, respectively, of (1.1) such that ao ::; flo on lR; (ii) f E C[J x n, lR] and f(t, y) is nondecreasing in y for each t, where n = {x: ao(t) ::; x ::; flo(t), t E lR} . Then there exists a solution x(t) of (1.1) such that ao(t) ::; x(t) ::; flo(t), t E lR. Now we are in a position to prove our main result. Theorem 2.2: Assume that (i) ao, flo are lower and upper solutions, respectively, of (1.1) such that ao ::; flo on lR; (ii) f E C[J x n, lR], fx, fn exists and are continuous on J x n satisfying fn(t,x) 2: 0 on J x n; (iii) f(t,y) is nondecreasing in y for each t; (iv) K < 4, where K is a bound of fx(t,x) on J x n. Then there exists a monotone sequence {wn(t)} which converges uniformly and montonically to the unique solution of (1.1) and the convergence is quadratic. Proof. In view of (ii), we see that f(t,x) 2: f(t,y)
+ fx(t,y)(x -
y),
(2.1)
for x 2: y. Consider the BVP _X"(t)
== Let wo(t)
= f(t,ao(h(t») + fx(t,ao(h(t)))(x(h(t)) g(t, x(h(t»j a(h(t))), x(t)
= ao(t) so that
= tjJ(t), t
ao(h(t)))
E J o , x(l)
= b.
(i), (2.1) and (2.2) yield
-a~(t) ::; f(t, ao(h(t»)
==
g(t, ao(h(t»j ao(h(t))),
(2.2)
211 QUASILINEARIZATION FOR BOUNDARY VALUE PROBLEMS
-f3~(t) ~ ~
f(t,ao(h(t)))
f(t, 130 (h(t)))
+ f,,(t,ao(h(t)))(f3o(h(t)) -
ao(h(t)))
= get, f3o(h(t))j ao(h(t))). Then by Theorem 2.1, there exists a solution WI(t) of (2.2) such that wo(t) ::; WI(t) ::;
f3o(t), t E JR. Since
WI ~
Wo, 130
~ WI,
we get using (2.1) and (2.2),
-w;'(t)
= g(t,wI(h(t))jwo(h(t)))
= f(t,wo(h(t))) + f,,(t,wo(h(t)))(WI(h(t)) ::; f( t, WI (h( t)))
= g( t, WI (h(t))j WI (h( t))),
-f3~(J) ~ ~
f(t,WI(h(t)))
wo(h(t)))
f(t, 130 (h(t)))
+ fx(t, WI (h(t)))(f3o (h(t)) -
WI (h(t)))
= g(t,f3o(h(t))jWI(h(t))). Consequently, we have, as before, a solution W2(t) of the BVP -W~(t)
= g(t,w2(h(t)) jWI(h(t))),
W2(t) = r!J(t), t
E J o,
xCI) = b,
such that This process can be continued successively to obtain the monotone sequence {wn(t)} satisfying where wn(t) is a solution of the BVP -W~
= g(t,wn(h(t))jWn_I(h(t)))
= f(t,wn-I(h(t))) + f,,(t,Wn-I (h(t)))(w n (h(t)) wn(t)
= r!J(t),
t
E J o,
wn-I(h(t))),
wn(l) = b.
Employing standard arguments (as in [4]), it is easy to conclude that the sequence {wn(t)} converges uniformly to the unique solution x(t) of (1.1). Next we shall show that the convergence of wn(t) to x(t) is quadratic. For this purpose, consider Pn+I(t) = x(t)-Wn+I(t) ~ 0 for each n and note that Pn+I(t) = 0, t E Jo and Pn+I(l) = O.
212 BVPs FOR FDEs
Hence we can write
Pn+1(t)
= 10
1 G(t,S)[J(s,x(h(s») - g(S ,Wn(h(S»;W n_1(h(s)))]ds,
where
G(t, X)
= {OG,(t,X) ,
tEJ t rf. J.
Here G(t,x) is the Green's function given by
Gtx ={t(1-X) OS; x S;tS;1 (,) x( 1 - t) 0 S; t S; x S; 1. It is easy to observe that G(t,x) is nonnegative. Now
OS; Pn+1(t) =
10
1 G(t , s) [J(s , x(h(s») - f(s,wn(h(s»)
- fz(s,wn(h(s )))(Wn+1 (h(s» - wn(h(s)))] ds S;
10
1 G(t,s) (Jx(s,x(h(s)))(x(h(s)) -wn(h(s)))
- fz(s,wn(h( s »)(Wn+1 (h(s» - wn(h( s »)] ds =
10
1 G(t , s) [fzx(t,e)(x(h(s» -wn(h(s)?
+fz(s,wn(h(s))(x(h(s)) - wn+1(h(s)))] ds 1
10 G(t,s) [J%X(s,e)p~(h(s)) + fz(s, w (h(S)))Pn+1 (h(s))] ds , where Wn < e< x . Then we get =
n
A
sup IPn+1(t)1 S; - 1 B sup IPn(t)12 ,
tElR
where on J
X
fl, IG(t,s)1 S;
-
t,
tElR
If"z(t,x)1 S; L, Ifz(t,x)1 S; K, A
=t
and B
= f.
213 QUASILINEARIZATION FOR BOUNDARY VALUE PROBLEMS
References
1. R. Bellman, Methods of Nonlinear Analysis, Vol. II, Academic Press, New York 1973. 2. R. Bellman and R. Kalaba, Quasilinearization and Nonlinear Boundary Value Problems, American Elsevier, New York 1965. 3. S.R. Bernfeld and V. Lakshmikantham, An Introduction to Nonlinear Boundary Value Problems, Academic Press, New York 1974. 4. G.S. Ladde, V. Lakshmikantham, and A.S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, Boston 1985. 5. V. Lakshmikantham, An extension of the method of quasilinearization, J. Optim. Theory. Appl. 82 (1994), (in press). 6. V. Lakshmikantham and S. Koksal, Another extension of the method of quasilinearization, Proc. of Dyn. Systems and Appl., Dynamic Publishers, Atlanta (1994), 205-210. 7. V. Lakshmikantham and S. Malek, Generalized quasilinearization, Nonlinear World 1 (1994), 59-65. 8. S. Malek and A.S. Vatsala, Method of generalized quasilinearization for second order boundary value problem, Inequalities and Applications 3, World Scientific, to appear. 9. N. Shahzad and A.S. Vatsala, Improved generalized quasilinearization method for second order boundary value problem, Dyn. Systems and Appl. (1995), to appear. 10. N. Shahzad and A.S . Vatsala, An extension of the method of generalized quasi linearization for second order boundary value problems, Applicable Analysis (1995), to appear.
215
A RESOLVENT COMPUTATION RELATED TO COMPLETENESS FOR COMPACT OPERATORS
SJOERD M. VERDUYN LUNEL* Faculteit der Wiskunde en Informatica, Universiteit van Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam, The Netherlands
Abstract In a recent paper [4) we have studied the problem whether the period map of a periodic delay equation has a complete span of eigenvectors and generalized eigenvectors. There is an abstract theory (see [4) and also [3)) that can be used to verify whether the eigenvectors and generalized eigenvectors corresponding to the nonzero spectrum of a compact operator from a given class of operators are complete. To use the abstract results one needs good estimates for the resolvent operator near infinity and to compute the resolvent explicitly one often has to solve a boundary value problem. In this paper we first give an abstract theorem and then we discuss some explicit examples. 1. A Result About Completeness
Let H be a complex Hilbert space and let T : H -+ H be a compact operator. Let ET denote the span of the eigenvectors and generalized eigenvectors corresponding to the nonzero eigenvalues of T. If ET is dense we call the system of eigenvectors and generalized eigenvectors complete. In general, the nonzero spectrum of a nonselfadjoint compact operator T can be empty and to study T, one first introduces the singular values of T using the compact positive selfadjoint operator T*T . If
denote the eigenvalues of T*T, the singular values of T are defined as follows
sj(T)
:=
}.,j(T*T)1/2,
j ~ 1.
The research of S.M. Verduyn Lunel has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences.
216
BVPs FOR FDEs For example, the integration operator V : prO, 1]
(V.p)(0) = 2i
l
--->
L2[0, 1]
.p(s)ds
is compact but has no nonzero spectrum (such operat~rs are called Volterra operators). The singular values are given by Sj(V) = 4((2j - 1}7r)-I, j ~ 1. (See [2] for details and more examples.)
Theorem 1.1. Let T : H - t H be a compact operator sucb tbat Ker T {OJ . Suppose tbere exists a r > 0 sucb tbat
= Ker T*
=
00
LSj(T),
, then ¢> = 1jJ - izV1jJ. Hence we have to solve 1jJ' - izb1jJ = ¢>'
(2.3) (2.4)
1jJ( -1) = ¢>( -1). If n~(z) denotes the fundamental matrix solution of if (2.3)-(2.4) that
((I - izVtl¢>)(8)
= n~l(iz)¢>(-1) + = ¢>(8)
+ iz
1 8
-1
i:
=
n:(iz)¢>'(s)ds
. J"b( )d
etZ
•
zby, it follows from Eq.
Cf
(2.5) Cfb(s)¢>(s) ds .
219
A
RESOLVENT COMPUTATION
A simple computation shows that ~(iz)
= 1=1=1-
iz - iZT2(1 - iZV)-1izT1 iZn~1(iz)
ize izii ,
where
b=
(2.6)
1° b(s) ds. -1
Thus we have the following representation for (1 - zTt1:
(I - zTt 1(c,tP) = (~(izt111:(iz),0
1-+
iz~(izt1n~1(iz)lI:(iz)
+ «(1 -
izVt1tP)(0)), (2.7)
where
lI:(iz)
= c + iZT2(I -
izVt11jl
i: i: (/
= C + iz l1 b(s)ljl(s) ds + (iz)2 = c + iz l1 b(s)ljl(s) ds + (iz)
b(O) l1
n~(iz)b(s)ljl(s) ds dO
izb(O)n:(iz) dO)b(s)ljl(s) ds
(2.8)
= c + iz l1 n~(iz)b(s)ljl(s) ds. From representation (2.6)-(2.7) we conclude that the nonzero spectrum of T is given by ..!:. E q(T) if and only if e-il'ii - iJ1.
J1.
= o.
If b = 0, the spectrum of T is finite and the eigenvectors and generalized eigenvectors cannot be complete. If b '" 0, we assume that b > O. Let J1.;, j ~ 1 denote the zeros of e- izii - iz ordered according to increasing modulus. It is a simple calculation to show that 00
1
E -.11+6 I < 00
;=1 J1.J Thus we have that
for every Ii
> O.
00
E A;(T)1+6 < 00,
;=1
where A1(T) ~ A2(T) ~ ... denote the nonzero eigenvalues of T. In order to apply Theorem 1.1 we first show that T satisfies (1.1) with r = 2, i.e., the singular values of Tare square-summable. Such operators are called HilbertSchmidt operators. Since T = F + V, where F is a finite rank operator and V an integral operator given by
V(c,tP)
= (0,0 1-+ l1 b(s)ljl(s)ds),
220
BVPs FOR FDEs it follows from the fact that both F and V are Hilbert-Schmidt operators (see [2;Prop IX.1.1]) that the sum T = F + V is a Hilbert-Schmidt operator. Thus T satisfies (1.1) with r = 2 and 0 = 0 1 U O2 , where
0 1 = {reiO 17r/4 ~ 0 ~ 37r/4},
O2 = {reiOI7r/4 ~ 7r - 0 ~ 37r/4}.
Because le-izbl = e!.'lzb and I~zl > ¥Izl in 0 1 U O2 , it follows from Eq. (2.5) that there are positive constants Mb M 2 , M3 and M4 such that Mlexp{
1I(I- izV)-I I1~
min
fO b(O") dO" .J222Izl}
-1~.., it follows that
(Here P>. denotes the spectral projection corresponding to an eigenvalue). of T.) From the Laurent series expansion, it follows that if (c, 4» E E;/; then Z 1--+
(I - izT*tl(C, 4»
is an entire function.
(3.2)
First we compute T*. From
(T*(d, 'I/J), (c, 4»)
= ((d, 1/J), T(c, 4») = d(c + =
1:
tjJ(s) ds)
+
(d + l21/J(S) ds)c +
+
1: 0
1:
1/J(s)(c _1~1 tjJ(u) du + d4>(s) ds
[2 4>(u)du) ds
_1~11/J(S)(J.-l 4>(u) du) ds
1/J(S)(i'1 4>(u)du)ds
= (d+ l21/J(S)ds)c +
1
1:
°
1:
dtjJ(s)ds
+ -1(L 1/J(s)ds)tjJ(u)du,
-1~\1: 1/J(s)ds)tjJ(u)du
222
BVPs FOR FDEs it follows that
T*(d, t/J)
= (d + i: t/J(s) ds, dX[-I,Oj + r
i~ t/J(s) dSX[-2,-lj),
t/J(s) dSX[-I,Oj -
(3.3)
°
where X[a,bj(O) = 1 if 0 E [a, b) and otherwise. Before we can construct a pair (c, 1/» =I (0,0) such that (3.2) holds we have to compute the operator (I - izT*tl. If (d, t/J) = (I - izT*tl(C, '
= t/J' + izt/J with
2
t/J(O) = e- iz (8+ )(_2)
1/>( -2)
j-1 t/J(s)ds.
(3.7)
-2
= t/J( -2), it follows that
+ 1:2 e- iz (8-')I/>'(s)ds
=1/>(0)_izj8 e- iz(8-s)l/>(s)ds
(3.8) -2::;0(0) - iz 1:1 e- iz(8-s)l/>(s)ds
-1::; 0::;
°
and, using boundary condition (3.7) at 0 = -1, we find
1/>(0)
= e- iz(8+l) (izd + iz i: t/J(s) ds -
iz i~1 t/J(s) ds)
+ 1/>(0) (3.9)
8
-izj e- iz (8-')I/>(s)ds At 0 =
-1::;0::;0
-1
°
we must satisfy the boundary condition izd
= t/J(O) -
1/>(0) or,
izd=e-iZ(izd+izjO t/J(s)ds-izj-lt/J(s)ds)-izjO e-zi(-·)(s)ds. -1
-2
-1
(3.10)
223
A
RESOLVENT COMPUTATION
Therefore, by using (3.8),
.
iz(e'Z -1)d - iz
where l(iz; ¢»
1°
'I/J(s)ds
-1
= - i~1 ¢>(s) ds + iz i~1 -eiZ
1°
= izl(iz; ¢»,
l2
e- iz (8-·)¢>(S) ds dfJ
e-zi(-')¢>(s)ds
(3.11)
-1
=-
1 0
-2
-1.( ) dS. e -iz(-1-.) 'l'S
Together with boundary condition (3.4) this yields the following system of equations
.
(etz
-
1)d -
1°
-1
'I/J(S) ds = l(iz; ¢»
(1- iz)d - iz i : 'I/J(s)ds
= c.
This system can be solved for d and J~ 1 'I/J (s) ds if det ~ (z) i- 0, where
= ( el- -'zz1
-1) .
iZ
~(z)
. -zz
The solution is given by d = det
and l1 'I/J(s)ds
/2
+ 1/1(0) -
iz
= (I -
e- iz (9-')I/I(s)ds ,
t/J(O) = e- iz (9+1) (izd + iz
/1
1
0 -1
-2
t/J(s) ds - iz
~ 0 < -1,
1-1 -2
I/I(s) ds
1:
+ (iz)2
1-11 9
-2 ( -2
e- iz (9-')I/I(s)ds
1
iz
(3.13)
izT*)-1(C, 'I/J), then
=ize- iz (9+1)(d+ _.0 t/J(s)ds)_ize - iz (9+1)(
+ l/I(fJ) -
(3.12)
= det~(z)(-(I-iz)l(iz;¢»+c(eiZ -1)).
Thus we found that if (d, 'I/J) t/J(O) = 1/1(0) - iz
~(z) (-izl(iz; ¢» + c)
1-1 -2
I/I(s)ds-iz
e- iz (9-')I/I(s) ds) dO)
1-11 9
-2
(_2 e- iZ (9-' )I/I(s)ds)dO)
e- iz (9-')I/I(s) ds , (3 .14)
224
BVPs FOR FDEs where d and f~1 1jJ( s) ds are given by (3.12) and (3.13), respectively. To construct a pair (c, 4» i- (0,0) such that (I - izT*)-1(c, 1jJ) is an entire function we proceed as follows. If c = 0 and 4> is such that there exists an entire function J with l(iz,4» = det .6.(iz)J(iz), then
i:
d = -izJ(iz) 1jJ(s) ds = - J(iz)
+ izJ(iz) .
So representation (3.14) implies that (I - izT*)-1(0, 4» is entire. Thus it remains to find a function 4> and an entire function J such that l( iz, 4» = det .6.(iz)J(iz). From (3.11), we conclude that O (-iz + e-iz)f(iz) = _ f eizs 4>(s) ds .
(3.15)
L2
If J is given by
then
(-iz+e-iz)J(iz) = -iz
, 1 e,zsa(s)ds+e-''z 1° e,zsa(s)ds 0
.
-1
-1
O
=
Thus if we take a(O) [-1,0] and equal a(·
r-
f eizsa'(s) ds + J-2 1 eizSa(s + 1) ds . a(O) - e-iza( -1) + L1
= a( -1) = 0,
a differentiable on [-1,0] and 4> equal to a' on
+ 1) on [-2, -1]
z ...... (I - izT*t1(0, 4»
then (3.15) holds and hence is a nontrivial entire function.
This completes the proof that the system of eigenvectors and generalized eigenvect~rs of T given by (3.1) is not complete. We conclude with a remark. From the definition of T* it is easy to see that the eigenvectors and generalized eigenvectors of T* are identically zero on the interval [-2, -1] . Therefore, the system of eigenvectors and generalized eigenvectors of T* cannot be complete. Thus in this example we see that both T and T* are not complete. For this type of operators this is a general fact (see [3]). Therefore to show that completeness fails for T, it actually suffices to prove the existence of a pair (c, 4» such that z ...... (I - izTt1( c, 4» is entire. It is not difficult to see that any pair (c, 4» with c = 0, 4>(8) = 0 for -1 ~ 8 ~ 0 and 4> not identically zero, has this property.
225 A
RESOLVENT COMPUTATION
References 1. J.K . Hale and S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.
2. I. Gohberg, S. Goldberg and M.A. Kaashoek, Classes of Linear Operators, Birkhii.user, Basel, 1990. 3. I. Gohberg, M.A. Kaashoek and S.M. Verduyn Lunel, in preparation. 4. S.M. Verduyn Lunel, About completeness for a class of unbounded operators with applications to delay equations, J. Differential Equations (1995) , to appear.
227
THE STUDY OF THE SOLVABILITY OF BOUNDARY VALUE PROBLEMS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS: A CONSTRUCTIVE APPROACH
V. P. MAKSIMOV AND A. N. RUMYANTSEV
Perm State University 614600, Perm, Russia
1. Introduction
A survey is concerned with a constructive approach to the study of the solvability of BVP's. The principal idea of this approach is the use of the computer in establishing solvability on the basis of special theorems and techniques. Broad classes of linear BVP's for FDE's as well as certain classes of nonlinear BVP's are considered.
2. Preliminaries Here we give a brief survey of certain results in the theory of FDE's which form a basis of effective investigation of BVP's (for details see [1] as well as [2]). A FDE is representable in the form
x(t)
= (Fx)(t),
t E [0, T]
(1)
where F is an operator defined on some set of absolutely continuous functions x : [O,T] --+ Rn. If F is the Nemytskii operator (Fx)(t) = f(t,x(t)), then (1) is a K(t, s, x(s)) ds this is an integral-differential differential equation. For (Fx)(t) = equation. Equations with deviating argument
J!
x(t)
= f(t, x[h(t)], x[g(t)]) ,
t E [0, T], if
~
(2)
¢ [O,T]
(here '1',"": Rl\[O,T]--+ Rn and h,g: [O,T]--+ Rl are given functions), can be rewritten in form (1) by putting
(3)
228
BVPs FOR FDEs Here for y: [O,T]-+ RR, r: [O,T]-+ Rl and z: Rl\[O,T]-+ RR :
(srY )(t) -_
{y[r(t)] 0
if r(t) E [0, T]; if r(t) f/. [0, T];
r
z (t) =
{O if r(t) E [0, T]; z[r(t)] if r(t) f/. [0, T].
If h(t) = t - TI, g(t) = t - T2, TI,T2 = const, then (2) is a differential-difference equation. Equation (1) with aftereffect occupies a special place. It is an equation with the Volterra operator F, Le., an operator such that for any T E [O,T] and any XI,X2 from its domain, which are such that Xl(t) = X2(t) on [0, TJ, the equality (Fxt}(t) = (FX2)(t) is fulfilled on [0, T]. Equation (2) is an equation with aftereffect if h(t) :S t, g(t) :S t. The Volterra property of the operator F permits us to study behavior of the solution x to equation (1) on any segment [0, T] C [0, T] ignoring the values of x(t) and (Fx)(t) when t > T, Le., to study the so-called local-solution X'T : [O,T]-+ Rn Let the operator F be a mapping from the space DR of absolutely continuous functions x : [0, T] -+ RR into the space LR of Lebesgue summable functions y [0, T] -+ Rn We define the norms of the spaces DR and LR by T
Ii Xliv n
=
Ix(O)1
+
J1:i:(s)1 ds , o
T
liyliLn =
Jly(s)1 ds, o
where I . I is a norm of Rn. If a continuous operator F : DR -+ Ln is compact and it has the Volterra property, then there are for equation (1) valid analogs of fundamental results of the theory of differential equations: on local solvability of the Cauchy problem :i: = Fx, x(O) = 0; on extendibility of solutions; on a solution's continuous dependence on the initial value 0. Let TJi : DR -+ RI, i = 1, ... , N, be a given system of functionals. An operator
is called a vector-functional. A BVP for equation (1) with an operator F : Dn is the system :i: = Fx, TJX = O.
-+
LR
BVP's are a basis for the modern theory of FDE [1] . Such theory is constructed for the class of "reducible" equations (1) which contains, in particular, all the equations with a correctly solvable Cauchy problem or a correctly solvable BVP. The linear case of equation (1) we shall write in the form
I:-x
=J,
(4)
where I:- : DR -+ LR is a linear bounded operator. Characteristic properties of I:become clear if we rewrite this operator in a form based on an isomorphism between
229 A CONSTRUCTIVE ApPROACH
Dn and Ln x Rn, which is defined by
Jx(s) ds + x(O). t
x(t)
=
o
By applying C to the both sides of the latter, we then obtain the representation
(Cx)(t)
= (Qx)(t) + A(t)x(O).
(5)
The general theory of equation (4) has been constructed under the assumption that the operator Q ("principal part" of C) is Fredholm one, i.e. it is representable in the form of the sum of a completely continuous operator and an invertible one. The equation
x(t) - P(t)x[g(t)]- R(t)x[h(t)] x{e}
= 'P(~),
x(o
= 1fJ(O,
r(t),
=
t
E
[0, T],
for ~ ¢ [0, TJ,
takes form (4) if we put
(Cx)(t)
= x(t) -
j(t)
P(t)(Sgx)(t) - R(t)(ShX)(t),
(6)
= r(t) + P(t)1fJg(t) + R(t)'Ph(t).
Natural assumptions concerning P, R, g, h, r, 'P, 1fJ such that this C is a bounded operator acting from Dn in Ln and j E Ln are given in [1] . In this case the principal part Q of C has the form Q = 1- S - K; where I is the identity operator, K is a completely continuous integral operator, (Sz)(t) = P(t)(Sgz)(t). In what follows the invertibility of Q is assumed. The principal part of C defined by (5) is invertible when h(t) ~ t and there also exists T > 0 so that g(t) ~ t - T, T
E
[O,T]. Under such an assumption the Cauchy problem
Cx
= j,
x(O)
=a
(7)
is uniquely solvable for any j E Ln, a E Rn, and its solution x is representable in the form
x(t)
=
(E-
j(Q-1A)(S)ds)a+ j(Q-lj)(S)dS.
(8)
The matrix X (t) = E - fci (Q-l A)( s) ds is called the fundamental matrix of the homogeneous equation Cx = O. Each column Xi, i = 1, . . . ,n of X is the solution to the Cauchy problem Cx = 0, x(O) = ei, where ei is the i-th column of the identity matrix E.
230
BVPs FOR FDEs
nn -> R 1 \ i = 1, . .. , N, be a linear bounded functional. An operator R N , Ix =col{/lx, ... , INX}, is called a vector-functional.
Let Ii : I:
nn ---t
The system ex
=
j,
Ix
(9) = /1 = /1 is its boundary conditions. If we
is called the linear BVP, and the equation Ix apply the operator I to the both sides of the identity x(t) = J~ x(s) ds + x(O), we then obtain the representation T
Ix =
J~(s)x(s) ds +
I}/x(O),
(10)
o
where each column of the (N x n)-matrix I}/ is the value of I on the corresponding column of the identity (n x n)-matrix E, that is I}/ = I(E). The (N x n)-matrix ~ has measurable and essentially bounded elements and is determined by
I
(j
y(s) dS) =
1~(s)y(s)
ds,
y E Ln.
Representation (8) implies that the solvability (unique solvability) of BVP (9) is equivalent to the solvability (unique solvability) of the algebraic system
/1 -Ig,
IX · Q =
(11)
where each column of the (N x n)-matrix IX is the value of I on the corresponding column of X, and 9 is the solution to the Cauchy problem ex = j, x(O) = O. In the case N = n (that is, the number of boundary conditions equals the dimension of the system of FDE), BVP (9) is uniquiely solvable if and only if IX is invertible: det IX
i- O.
(12)
A constructive way to verify effectively this criterion without explicit evaluation of the fundamental matrix X is described below (see the next section). 3. The Study of the Solvability of Linear BVPs As we have already noted, the criterion of the uniquie solvability of the linear BVP ex = j,
Ix
= /1, /1 E R n
(13)
with n boundary value conditions is the invertibility of the matrix IX (12). This condition cannot be verified immediately, because the fundamental matrix X(t) cannot be evaluated explicitly. In addition, even if the matrix X were known, then the elements of IX, generally speaking, could not be evaluated explicitly. We offer a method
231 A CONSTRUCTIVE ApPROACH
of the verification of (12), which is based on the theorem about inverse operators (see, e.g. [3]). By virtue of this theorem, the matrix IX is invertible, if one can find an invertible matrix r such that 1
II/X -
rll < IIr-IIi.
(14)
As it has been shown in [4,5) (see also [1)), such a matrix r for the invertible matrix IX always can be found among the matrices r = T X, where T : Dn --+ Rn is a vector-functional near to I, and X is the fundamental matrix of the homogeneous equation lx = 0 with operator l : Dn --+ Ln near to C. Ways to find such a matrix r are constructive if we are able to use Computer Algebra Systems (in particular, the systems of analitical computations JLSIMP-JLMATH, Formac, Reduce, and others) guaranteeing the reliable verification of inequality (14). Use of these systems places certain demands on the operators l and T. Here we describe these demands by the so-called "property C" (computability). Further, we assume that the value T is rational. Let 0 < tl < ... < tm < T, where tI, ... , tm are rational numbers. Denote £1 = [0, t 1 ); £i = [ti-I, t;), i = 2, ... , m; £m+I = [tm' T), £0 = (-00,0). Let Xi be the characteristic function of the set £i. We denote by 'P;:' the set of all functions y of the form m+l
y(t)
=
L
Xi(t)Pi(t),
t E [0, T),
i=1
where the components of vector-functions Pi : [0, T) --+ Rn, i = 1, ... , m + 1, are polynomials with rational coefficients. We say that a linear bounded operator l : Dn --+ Ln satisfies the property C if it maps the set Dn n 'P;:' into 'P;:' . If lx = x + P(· )x, then the operator l is computable, for example, under the condition that the columns of the matrix P are functions of 'P;:'. The operator l: (lx )(t) = x(t) + P(t)(ShX)(t), t E [0, T), satisfies the property C if the columns of the ma.trix P belong to the set 'P;:' and the function h E 'P!. satisfies the property ~q: for any i = I, .. ., m + 1 there exists the unique q, 0 S; q S; j such that h(t) E £q for t E £j. _ We say that a line"r bounded functional I : Dn --+ Rn satisfies the property C if it maps the set Dn n 'P;:' into the set of rational numbers. Our way to prove the solvability of BVPs essentially uses an approximation of operators determined by the original BVP in the class of computable operators. Constructive theorems on the solvability of BVPs are formulated in terms of parameters characterizing a precision of such an approximation. The precision of an approximation l for C is defined by the inequality
l(Cx)(t) - (lx)(t)J S; >'v(t)lIxIiDD Vx E Dn,
(15)
232 BVPs FOR FDEs
= {ajj}, lAJ means the (n x m)-matrix {Iajjl} . Further, let I: Dn -+ Rn, Ix = J 4la(s )x(s )ds + Wax(O), be a vector-functional
where,xv E P;:. . Here for (n x m)-matrix A T
o
T
approximating the vector-functional I : D n -+ Rn, Ix = J41(s)x(s) ds o
+ Wx(O).
We suppose that the columns of the (n x n)-matrix 4la belong to the set P;:', and that the elements of the (n x n)-matrix Wa are rational numbers. The precision of the approximation is defined by inequalities
l41(t) - 4l a(t)J :::; 4lv,
(16)
t E [O,T],
where 4lv and Wv are (n X n)-matrices with rational elements. By means of estimate (15) we can construct (n x n)-matrices Xa(t) and Xv(t) with columns from P;:', and being such that the fundamental matrix X(t) satisfies the inequality t
!lX(S)-Xa(S) J ds:::;Xv(t),
tE[O,T] .
(17)
o
Further, by means of this inequality and inequalities (16) we can construct an (n x n)matrix Mv with rational elements and being such that
(18) Therefore, the elements of the matrix Ma = lXa can be explicitly evaluated because of the property C. This then enables us to verify effectively by means a real computer the conditions of the following theorem. Theorem 1. Let the matrix Ma be invertible and 1
IIMvll < IIM;ll1· Then problem (13) is uniquely solvable Jor any J E Ln and (3 E Rn Below we present some examples of the computer-assisted study of BVPs. These examples show there are many situations, where the known sufficient conditions of the solvability of a BVP (in particular, conditions obtained by the technique described in N. Azbelev's survey, this issue) are inapplicable, and our constructive computerassisted approach gives the only chance to achieve the result. The unique solvability of the following BVP's has been established by a computer-assisted check of the conditions of Theorem 1. Example 1.
0
2t In(1 + O.lt) ~-2
texp(~)
o
-2
x(t) -
[
8t
o "9 -2 - t
1x x(t) = J(t), t E [0,1],
233 A CONSTRUCTIVE ApPROACH
Example 2.
XI(t)] [ . (t) X2 Xl [
X2 Xi
+
[2t - 3 2t (t) X2
4t - 3 + 2t2] 2 sin(!¥) (/+2)
(¥) C'-2 /2) J -- [!I(t)] !2(t) , 1
(e) =0,
e!l'[O,I],
X
t E [0,1], i=I,2;
1
XI(O)
J(2s - l)x2(s) ds + -5-
= (32.
o
Here XI(-) and X2(-) are the characteristic functions on the intervals [0,2-t] and [2-t,l] respectively. 4. Nonlinear BVP's: a priori Inequalities Technique The main point of any research of nonlinear boundary value problems, say of the form
X= Fx,
(19)
= 0,
(20)
'T/X
if! the problem of constructing a priori estimates for solutions. The sentence " ... an a priori estimate IIXilDn ~ d takes place for any solutions to problem (19), (20)" means that there is d E (0,00) such that problem (19), (20) has no solution X such that ilXilDn > d. Thus a statement on a priori estimates of solutions does not include the assumption of the solution existing. We say that for any solution X to equation (19) the a priori inequality
Ix(t)1
~
m(t, Ix(O)I),
t
E
[O,T]
(21)
takes place if there is a function m : [0, T] x R+ ---+ R+ such that m(., T) E LI 'IT and equation (19) has no solution x such that inequality (21) is violated. Here a priori inequalities of form (21) play the role of the source of a priori estimates for solutions to BVP's. The way of using a prio,r i inequalities (21) to obtain a priori estimates is very simple in the case that the vector functional 'T/ satisfies the condition I'T/X - x(O)1 ~ jL(lxl, Ix(0)l) \Ix E nn, (22) where the functional jL : LI x RI ---+ RI does not decrease with respect to the first component. In this case the existence of an a prio,r i estimate of solutions to problem
234
BVPs FOR FDEs (19), (20) is assured by the following condition: the set B = {6 > 0 : 6 :::; Jl[m(., 6), 6)} is bounded. Indeed, let 60 be such that 6 :::; 60 V6 E B. Then for any solution x of problem (19) , (20) the estimate IIXIlDn :::; 60
+
sup IIm(·,6)IILI 6E[0.601
takes place. More detailed information of ways for studing nonlinear BVP's by using a priori inequalities can be found in the book [1] and in the paper [6] . This article is primarily concerned with the constructive aspects of our approach. Let us note that effective use of a priori inequalities needs the construction of m in an explicit form. For broad classes of FDE's, the question of the construction of m is reduced in [7] to the question of the construction of the general solution to an ancillary ordinary differential equation if = w(t,y) . This equation is defined by a majorant of F. Thus our opportunity to CO:lstruct the function m is essentially restricted to within the context of E. Kamke's book [8] . Of course, one can find w such that w(t,y) :::; w(t,y) and a new equation if = w(t,y) is integrable. Most often this way brings either a linear equation or a Bernoulli equation. Note that the roughness of the a priori inequality obtained in such way may be unacceptable. In the case of ordinary differential equations, an ingenious technique for constructing a priori inequalities was proposed by E. Zhukovskii [9] . The difficulties associated with the integration of the majorant equation are essentially growing in the case that we use vector a priori inequalities. We say that for any solution x to equation (19) a vector a priori inequality L:i:(s)J :::; met, Lx(O)J)
t E [o,T]
R+
(23)
R+
takes place if there is a function m : [0, T] X -+ such that m(·, r) E Ln Vr and equation (19) has no solution x such that inequlity (23) is violated. Recall that, for a = col(a l , ... , an), LaJ means col(lall, ... , lanl). In this vector case, the majorant system may in exceptional cases be integrated. The possibilities of both the construction and the use of vector a priori inequalities expand essentially by modern Computer Algebra Systems (Formac, Reduce and so on). To illustrate this we consider the case that the operator F in equation (19) satisfies the condition
L(Fx)(t)J :::; A(t) (LX(O)J
+
j
L:i:(s)J
dS) +,(t) ,
t E [O,T],
Vx E nn .
(24)
Here elements of the (n x n)-matrix A(·) and vector-function ,(.) are polynomials with rational coefficients (belong to Pl). Denote by yet, r) the solution of the Cauchy problem if = A(t)y+,(t), yeO) = r, r E The question of constructing a function m is assumed by
R+.
235 A CONSTRUCTIVE ApPROACH
Theorem 2. Suppose that F satisfies condition (24). Then for any solutions to equation (19), vector a priori inequality (23) holds with m defined by
= A(t)y(t, r) + -.,.(t) .
m(t, r)
Now we formulate the preposition which brings a way of constructing a function ml(t, r) such that ml(t, r) ~ m(t, r) "It E [0, T), r E R~ and ml is a little different from m .
Theorem 3. For every c > 0, there exist a (n x n) -matrix Yi(t), vector ft(t) with elements belonging to PI, and a rational number ~ such that the vector-function Yl(t,r) == Yi(t)r+~ . r+ fl(t), t E [O,T), r E R~ satisfies the inequalities Ly(t,r)Yl(t,r)J :5c · r+col(c, . .. ,c), y(t,r):5Yl(t,r). Such a function Yl and hence ml(t,r) = A(t)[Yi(t) + ~)r + A(t)fl(t) + -.,.(t) may be found practically by using the mentioned above Computer Algebra Systems. For details of such construction see [10). Using a priori inequality L:i:(t)J :5 ml(t, Lx(O)J) established for solutions to (19) we shall study the solvability of BVP (19), (20) . We suppose that T
+
L1/X - x(O)J :5 \If Lx(O)J
J
cfI(s) L:i:(s)J ds
+ a,
"Ix E nn,
o
where a E R~, \If and cfI(·) are (n x n) -matrices with elements belonging to the set of rational numbers and to PI respectively. Denote T
M = E - \If -
J
cfI(s)A(s)[Yi(s)
+ ~. E)
ds.
o
Theorem 4. Let F : nn -+ Ln be a completely continuous operator, and suppose the matrix M is positively invertible. Then BVP (19), (20) has at least one solution. An algorithm for the study of the solvability of BVP's based on Theorems 2-4 is realized as a system of programs. In the case of solvability, the computer outputs the information of component-wise estimates of a solution. Notice that BVP's (19), (20) with an operator satisfying condition (24) arise, in particular, in mathematical economics [11). 5. Supplementary Techniques of the constructive study of linear BVP is developed and extended by authors to impulsive BVP's with boundary inequalities, i.e. problems of the form
.cy = f,
iY :5-.,.
(25)
236 BVPs FOR FDEs with linear bounded operators.c : Dsn(m) - t Ln, I : Dsn(m) - t Rk, and an arbitrary number k of boundary conditions. Here Dsn(m) = DS(m) x . .. x DS(m), DS(m) is the space of piecewise absolutely continuous functions y : [0, T]-t Rl described in N. Azbelev's survey, this issue. One can find details concerning problems (25) and its application in mathematical economics in [12] . For a study of certain classes of FDE, (2) in part, at times the frame work of the space Dn ~ Ln x Rn is too restricted. Thus the question arises for more suitable spaces. One of them may be a space of absolutely continuous functions with derivative belonging to a Orlicz space L M. In this case, the role of the main space of solutions is played by the space DM = LM x Rn. Certain classes of BVP 's with solutions belonging to DM have been studied by L. Kultysheva [13] . For certain BVP's for FDE's with singularity, special spaces of solutions which agreed with the kind of singularity were constructed by A.Shyndyapin [14] . A priori inequalities find applications in the study of BVP 's for ordinary differential equations :i; = f(t, x(t)) in the case that f does not satisfy the Caratheodory conditions. For this case an investigation on techniques based on a priori inequalities has been developed by E. Zhukovskii [9]. A priori inequalities have been used also by A. Bulgakov [15] and M. Karibov [16] to obtain tests of the solvability of BVP's for Functional Differential Inclusions. 6. References 1. N. V. Azbelev, V. P. Maksimov, and L. F . Rakhmatullina, Introduction to the Theory of Functional Differential Equations, Nauka, Moscow, 1991, (in Rus-
sian) . 2. N. V. Azbelev, V . P. Maksimov, and L. F. Rakhmatullina, Functional Differential Equations, World Federation Publishers Company, INC, Tampa, to appear. 3. V. Hutson and J. Pym, Applications of Functional Analysis and Operator Theory, Academic Press, New York, 1980. 4. A. N. Rumyantsev, To the study of the solvability of boundary value problems by using Computer Algebra Systems, Archives of VINITI AN SSSR, N 3373, 1987, (in Russian) . 5. A. N. Rumyantsev, A constructive study of the solvability of boundary value problems for funct ional differential equations, Doctoral thesis, Urals State University, Sverdlovsk, 1988, (in Russian) . 6. V. P. Maksimov, On certain nonlinear boundary value problems, Differential Equations 19, No. 3 (1983), 396-414, (in Russian) .
237 A CONSTRUCTIVE ApPROACH
7. N. V. Azbelev and V. P. Maksimov, A priori estimates of solutions to the Cauchy problem and the solvability of boundary value problems for equations with delayed argument, Differential Equations 15, No. 10 (1979) , 1731-1747, (in Russian). 8. E. Kamke, Differentialgleichungen, Losungsmethoden und Losungen. v. 1, Leipzig, 1959. 9. E. Zhukovskii, Operator inequalities and functional differential equations, Doctoral thesis, Urals State University, Sverdlovsk, 1984, (in Russian). 10. A. N. Rumyantsev, The study of the solvability of boundary value problems by using vectoral a priori inequalities, Archives of VINITI AN SSSR, N 5028, 1987, (in Russian). 11. D. L. Andrianov, A priori estimates in stabilization problems for trajectories of nonlinear economic systems, Doctoral thesis, Leningrad State University, Leningrad, 1985, (in Russian) . 12. V. P. Maksimov and A. N. Rumyantsev, Boundary value problems and problems of pulse control in economic dynamics. Constructive study, Russian Mathematics (Iz. VUZ) 37, No. 5 (1993), 48-62. 13. L. M. Kultysheva, On equations with the internal superposition operator, Doctoral thesis, Alma-Ata, 1981, (in Russian) . 14. A.l Shyndyapin, To the questing on singular functional differential equations, Doctoral thesis, Tbilisi State University, Tbilisi, 1984, (in Russian) . 15. A.1. Bulgakov, V. P. Maksimov, Functional and functional differential inclusions with Volterra operators, Differential Equations 17, No.8 (1981), 1362-1374, (in Russian). 16. M. R. Karibov, Functional differential inclusions, Doctoral thesis, Gorki State University, Gorki, 1981 , (in Russian) .
BOUNDARY VALUE PROBLEMS FOR NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS
S. K. NTOUYAS Department of Mathematics, University of Ioannina 451 10 Ioannina, Greece
1. Introduction Let R" be the n-dimensional euclidean space and let R". For a fixed r > 0, we define
1.1
be any convenient norm of
to be the Banach space of all continuous functions 4 : [-r,O] -t R" endowed with the sup-norm I I ~ I I= ~ U P { I ~:(-r~ )rI 0 5 01. For any continuous function x defined on the interval [-r, 11, and any t E [O, 11 we denote by xt the element of C defined by
In the present paper, we consider neutral functional differential equations of the form d -[zf(t) - g(t, zt)] = f (tl Xt, xf(t)), t E [O, 11
(El
dt
where f : [O,1] x C x R" + R" and g : [O, 11 x C
+ R"
are continuous functions.
This is a particular class of neutral functional differential equations which includes the usual second order functional differential equations of retarded type (g = 0). A good guide to the literature for neutral functional differential equations, is the related chapter in the book of Hale [5] and the references therein. Equations of the above type appear in hereditary control systems. A function x : [-r, l] R" is said to be a solution of (E) with initial function dECatt=O,if
240 BVPs FOR FDEs (ii) XI E C for t E [0,1]' and (iii) the difference x'(t) - g(t, XI) is differentiable and satisfies (~) for t E [0,1] . Boundary value problems for neutral functional differential equations is a very interesting project and its development is much slower as compared with the other topics on neutral functional differential equations. The main difficulty is localized in approaching the relations between equation with boundary value. The main purpose of this paper is to discuss the existence of solutions for the boundary value problem (BVP for short) for neutral functional differential equations
![X'(t) - g(t,Xt)] Xo
where f : [0,1] x C x R:' E C and TJ E R:' .
= --+
=
,x(1) =
f(t,XI,X'(t)),
t E [0,1]
(E) (BC)
TJ
Rn and 9 : [0,1] x C
--+
Rn are continuous functions,
The key tool in our approach is the Leray Schauder alternative [1]. This method reduces the problem of existence of solutions of a BVP to the establishment of suitable a priori bounds for solutions of these problems. We notice that this method is a modification of the well known Topological transversality method of Granas [1] . For applications of this method in boundary value problems for functional differential equations we refer to [6], [9], [12]. For some recent results on initial and boundary value problems for neutral functional differential equations see [7-9]. Also we remark that in [10-11] boundary value problems for neutral functional differential equations have been studied with the aid of the shooting method. We notice that recently in [8], the global existence of solutions for initial value problems for neutral functional integro-diffrential equations of the form
!
[x'(t) - g(t, XI)] Xo
=
= , x'(O) =
L(t, Xt)
+ f(t, Xt),
t
E
[O,T]
TJ
is studied. The paper is organized as follows. In section 2 we present some notations and preliminaries. The basic existence theorems are given in section 3. Finally in section 4 we give two applications of our results. 2. Preliminaries
If m a positive integer we denote by C([O, 1], Rm), Ck([O, 1], Rm) and Lk([O, 1], Rm) the classical spaces of continuous Rm-valued functions, k-times continuously differentiable ~-valued functions and measurable Rm-valued functions whose k-th power of the
241 BOUNDARY VALUE PROBLEMS FOR NEUTRAL EQUATIONS
euclidean norm
1.1 is Lebesque integrable on A. We introduce the following notations: IIxlio =
sup{lx(t)1 :
°: :; t ::; I}
IIx'llo = sup{ lx'(t)1 : 0:::; t :::; IIxll* = max{lIxll, Ilx'lI} IIxlh =
t
I}
Ix(s)lds.
The proofs of our existence theorems in the next section are based on the theory of completely continuous mappings. We recall here that, if X, Yare normed spaces, then an operator A : X --+ Y is completely continuous if it is continuous and the image of any bounded set is included in a compact set . See [1] . Also a function 1 : [0,1] x C --+ Rn is called completely continuous ([5]), if it is continuous and takes dosed bounded sets of [0,1] x C into bounded sets of ~ . Before stating our basic existence theorems, we need the following lemma which is known as "Leray-Schauder alternative". Lemma 2.1 Let S be a convex subset of a normed linear space E and assume Let F: S --+ S be a completely continuous operator, and let
£(F) = {x E S : x = )"Fx
for some
°E S.
0(t) , t E [-r, 0) x(t), t E [0,1) . Now, define the operator F : Co --+ C 1 ([O, 1),~) by
Fx(t) = ['7 - l g(t , xt)dt - l l f( s,X.,x'(s))dsdt)t
+l
g(s,x.)ds
+l
f f(r , XT) x'(r))drds,t
E [0, 1) .
It is obvious that F(Co) c;:::; Co, and x is a solution of the BVP (E)-(BC) if, and only if, Fxl[O, 1) = xl[O, 1) and Xo = 4>, x(l) = '7 .
In order to prove the existence of a solution of the BVP (E) -(BC) we apply Lemma 2.1. First we obtain the a priori bounds for the fun ctions x E Co such that x = )"Fx,).. E (0,1). In other words we shall prove that the set £(F) = {x E Co : x = )"Fx,O < ).. < I} is bounded. Let x E £(F) . Since x(O)
= 0 by the relation x(t) = x(O) + J~ x'(s)ds we have IIxllo:::; IIx'llo.
Also using (HI) and (H3) we get
Ix(t)l:::; 1'71 + 2l[a(t)max{lIxllo, 114>1I} + b(t))dt
+ 2ll[P(s)max{lIxllo, 114>1I} + q(s)lIx'llo + r(s))dsdt :::; 1'71 + 211 alll{11 xllo + 114>11) + 211 bll 1 + 2l1plh(lI xlio + 114>11) + 2l1qllll1x'llo + 211 r lh :::; 1'71 + 2(lIalh + Ilplh)lIxlio + 2l1qlhllx'llo + 2(lIalh + IIplh)II4>1I + 211 bll 1 + 211rlh,
(3.3)
243 BOUNDARY VALUE PROBLEMS FOR NEUTRAL EQUATIONS
and
1'71 + lIalh(lIxlio + 114>11) + IIblll + 2l1pll1(lIxllo + 114>11) + 211qlllllx'ilo + 211rlh + lIallo(lIxlio + 114>11) + IIblio.
Ix'(t)1 ~
Consequently
IIx'llo ~ (lIalh + 211plh + lIalio + 21lqlh)llx'llo + 1'71 + (lIalh + 211plh + lIallo)II4>1I + IIblll + IIbllo+ 211 r lll or
and therefore, by (3.3),
IIxlio ~ IIx'llo ~ c.
(3.5)
From (3.4) and (3.5) we have that
IIxll* ~ c, which implies that the set £(F) is bounded. This proves our first claim. In the second step, we will prove that the operator Fx is completely continuous. Let {hll} be a bounded sequence in Co, i.e for all where K is a positive constant. We obviously have Hence we obtain easily that
v,
IIhlltll
~
K, t E [O,lJ for all v.
1'71 + 2Ulalll + IIplll + IIqlh]K + 2(lI alh + Ilplll)II4>1I + 2(lIb111+ IIrlll), I(Fhll),(t) I ~ 1'71 + Ulalll + lIalio + 2(lIplh + Ilqlh)]K + (lIalh + lIalio + 2I1plh)II4>1I + IIblll + IIbllo+ IIrll!'
IFhll(t)1 ~
which means that {Fh ll } and {(Fhll)'} are uniformly bounded. We rewrite the operator Fx as
Fx(t) = Hx(t) where
Hx(t)
= ['7
-l
+ Tx(t), t E [0, IJ
g(t,xt)dtJt +
l
g(s,x.)ds,
t E [0,1]
and
Tx(t)
= -t
II
f(s, X., x'(s))dsdt +
ll'
f(T, X.,., x'(T))dTds,
t E [0,1] .
244
BVPs FOR FDEs Then we have the estimates:
IThv(t 1 )
-
+ IIqlh)I< + 211plhllt/JID + 211 r lhll t l ~ [(lIplh + IIqlldI< + IIplIlllit/J11) + Ilrlll]lt 1 -
Thv(t2)1 ~ [2(lIplh
I(Thv)'(tl) - (Th v)'(t2)1
t21 t21 ·
From these estimates we deduce that the sequences {Thv} and {(Thv)'} are equicontinuous. Thus, by the Arzela-Ascoli theorem th~ operator Tx is completely continuous. Furthermore, by assumption (H2) the operator Hx is completely continuous and therefore the operator Fx is completely con~nuous . Thus our second claim is proved. Consequently, by lemma 2.1 the operator Fx has a fixed point in Co . Then it is clear that the function z(t) = {t/J(t), t E [-r,O] x(t), t E [0,1] is a solution of the BVP (E)-(BC). If t/J(O)
f
0, by the transformation y
= x - t/J(O)
the BVP (E)-(BC) reduces to the following BVP
!
[y'(t) - g(t, Yt Yo =
for which ~(O) =
+ t/J(O))]
=
f(t , Yt
.p - t/J(O) == ~,y(l) =
+ t/J(O) , y'(t)) ,
t E [0,1]
7J - t/J(O)
o.
Theorem 3.2 Let f : [0,1] x C x Rn -> Rn and 9 : [0,1] x C functions. Assume that (H2 ) holds and moreover
->
Rn be continuous
(Hd There exist bounded real valued functions u, b defined on [0,1] and real constant k, 0 ~ k ~ 1 such that Ig(t, u)1 ~ a(t)lIuli k
+ b(t), t E [0,1]' u E C.
There exist p, q, r in £1([0,1], R) and real constants f. m 0 < f. < 1 0 < such that ' , - , -
(H3)
m ~ 1
If(t, u, v)1 ~ p(t)lIuli l
+ q(t)lvl + r(t), t E [0,1]' u E C, v E W . m
Then the BVP (E)-(BC) has at least one solution provided
Q(kHllalh
+ lIallo) + 2Q(f.)lIplh + 2Q(m)lIqlh < 1
245 BOUNDARY VALUE PROBLEMS FOR NEUTRAL EQUATIONS
where
Q( j.l ) =
{O,1,
o:s: j.l < 1 j.l = 1.
Proof. As in the proof of the Theorem 3.1 and using (HI), (H3)' instead of (HI)' (H3)' we obtain
IIxllo:S: 1171 + 2!1l alhmax{lIxllo, 11¢>II}k + Ilblh] + 2[lIplllmax{lIxllo, II¢>II V+ Ilqlhllx'lI~ + IIrlll] :s: 1171 + 2l1alh(llxll~ + 1I¢>lI k) + 211bll l + 2l1plh(llxll~ + 1I¢>lI i ) + 2I1qlllllx'lI~ + 211rlh :s: 2l1alhllxll~ + 2l1plhllxll~ + IIqlhllx'lI~ + 1171 + 2l1 alhll¢>lI k+ 211blh + 2l1plldl¢>lI i + 211 r lll :s: 21lalhllxll~ + 2l1plIlllxll~ + Ilqlhllx'lI~ + d, where d = 1171 + 2l1alllll¢>lIk Also, we have
(3.6)
+ 211blh + 2l1plhll¢>lI i + 2l1rlh ·
IIx'lIo:S: 1171 + lIalll(lIxll~ + 1I¢>lI k) + IIblll + 2l1plII(llxll~ + 1I¢>lI i ) + 2I1qlhllx'll~ + 211rlh + lIallo(lIxll~ + 1I¢>ln + IIbllo :s: (lIalh + lIallo)lIx'lI~ + 2I1plIlllx'lI~ + 211qlh Ilx'lI~ + 1171 + lIalhll¢>lI k+ IIblll + 2l1plhll¢>lI i + 211rlll + Ilalloll¢>lIk + IIbllo :s: (lIalh + lIallo)lIx'lI~ + 2I1plIlllx'lI~ + 2l1qlllllx'llm + e, (3.7) i k k where e = 1171 + lIalhll¢>lI + IIblh + 2l1plIIIl¢>lI + 211rlh + Ilallll¢>lI + Ilbllo. Now for the sake of simplicity we consider the case where 0 < k < 1 and f = m = 1. Then, by (3.7) we have
or
211plh - 2I1qlh)lIx'llo :s: (lIalh+ lIallo)llx'll~ + e. (3.8) At this point, we note that if L ~ 0, 0 :s: j.l :s: 2, h > 0 are given constants, then there (1 -
exists a constant N > 0 such that
Lzi' Hence, if we put (3.8) imlpies
(1 -
hZ2
< - + N, - 2
x >_ O.
z = Vllx'lIo,L = lIalll+llallo,j.l = 2k,h = 1-21IplIl-2I1qlll inequality
211plh - 2I1qlh)lIx'llo :s: ~(1 - 211plh - 2l1qlldllx'llo + e+ N
246
BVPs FOR FDEs or ,
IIx 110 :::; Finally, this inequality, (3.6)
2(e+N)
211plh _ 211qlh = Cl· and the obvious relation IIxlio :::; IIx'llo imply that 1_
which completes the proof. The proof for the other cases is similar and is omitted. Corollary 3.3 Let f, 9 are as in Theorem 3.2. Let also (H2 ) holds, and with 0 :::; k < 1,0 :::; f. < 1,0 :::; m < 1.
(Jid, (Ji3 )
Then the BVP (E)-(BG) has at least one solution. 4. Applications
We close this paper by giving two applications of our results. One for a BVP for neutral functional differential equations with a "mixed "boundary condition and another for a "three-point"BVP for functional differential equations. Consider first the following BVP
~[x'(t) Xo =
g(t,xd] = f(t,xt,x'(t», 4>,ax'(O)
t
E [0,1]
+ ,8x(l) =."
where f : [0,1] X G([-r,O],R) x R -+ Rand 9 : [0,1] x G([-r,O],R) -+ Rare continuous functions, 4> E G and a,,8,.,, real constants, with a + ,8 # O. As far as we know this type of BVP is new, even for ordinary differential equations, i.e when 9 = 0 and r = o. Theorem 4.1 Let f : [0,1] x C([-r,O],R) x R -+ Rand g : [0,1] x C([-r,O],R) -+ R be continuous {unctions. Assume that (Ht) , (H2 ) and (H3) hold. Then the BVP (E) - (BCd has at least one solution provided
Proof. As in the proofs of Theorems 3.1 and 3.2 we consider only the case 4>(0) = 0, because by a transformation the BVP with 4>(0) # 0 is reduced to this case. Consider also the subspace Co and the function x as defined in Theorem 3.1. Since x(O) = 0, we obviously have
IIxllo:::; IIx'llo.
247 BOUNDARY VALUE PROBLEMS FOR NEUTRAL EQUATIONS
Therefore it is enough to prove that the IIx'llo is bounded. The BVP (E) - (BCd is equivalent to the following integral equation
x(t)
1 = --13[77 a +
ag(O,.. < 1. Now by a solution to (2.5)>. we mean a function y E C([O, T], Rn) which satisfies the equation in (2 .5}A for t E [0 , T] .
°
Theorem 2.1. Let U be a relatively open subset of C([O,T],Rn) with E U. Suppose (2.4) holds and in addition assume the following condition is satisfied:
N :U
->
C([O, T], Rn) is a continuous and compact operator.
(2.6)
Then either (iJ. (2.1) has a fixed point in U (i.e. N + RN has a fixed point in UJ; or (iiJ . there exists a >.. E (0,1) and ayE au with y satisfying (2.5)>. (i.e. there exists a >.. E (0,1) and ayE au with y = >"(Ny + RNy)). Proof: As above it is easy to see that solving (2.5)>. is equivalent to the fixed point problem y = >..(Ny + RNy) == >..Fy. (2.7)>. Notice F : U -> C([O, T], Rn) is continuous and compact since R is continuous and completely continuous and N satisfies (2.6). Thus theorem 1.2 with Q = E = C([O, T], Rn) guarantees the result. We now use theorem 2.1 to discuss nonlinear integral equations. Consider first the Volterra integral equation
y(t)
= Ly(t) + IT and x E [0, T] we have
°
lim z -x
rz lul~r sup 11« z, s, u) -
Jo
J«x , s, u)1 ds = 0, z E [0, Tj.
(2.9)
In addition assume there is a constant M o, independent of >., with
Iylo =
sup ly(t)1 =1= Mo [O,T)
for any solution y to y(t) = Ly(t)
+
>.l
J«t ,s,y(s))ds for t E [O , Tj
(2.1 0h
for each>' E (0, 1) . Then (2.8) has a solution in C([o,TJ,Rn). Proof: Let N : C([O, T], Rn) ..... C([O, Tj, Rn) be given by
Ny(t) = Also let
l
J«t,s,y(s))ds .
u = {u E C([O,Tj,R n): lulo < Mo}.
The result follows im~ediately from theorem 2.1 (notice alternative (ii) cannot occur) once we show N : U ..... C([O, T], Rn) is continuous and compact . We first show that N is continuous. To see this let Yn ..... Y in C([O,TJ,Rn) . Then there exists r > such that IYnio S; rand IYlo S; r. There exists hr E V [0, Tj such that 1J«x,s,u)1 S; hr(s) for almost all s E [O,Tj and all x E [O,Tj and lui S; r. Also for each x E [0, Tj we have
°
1«x's'Yn(s)) ..... 1«x,s,y(s)) for a.e. s E [O , Tj
255 EXISTENCE PRINCIPLES FOR NONLINEAR OPERATOR EQUATIONS
and this together with the Lebesgue dominated convergence theorem implies that NYn(x) -+ Ny(x) pointwise on [0, T). We next show the convergence is uniform. Fix x E [0, T) and let z E [0, T) with x < z. Then
INYn(x) - NYn(z)1 ::; Ie: IK(x,s,Yn(s)) - K(z,s,Yn(s))1 ds +f: IK(z,s,Yn(s))lds ~f; sUPlul~r IK(x,s,u)-K(z,s,u)lds+f: hr(s)ds.
(2 .11)
Consequently {NYn} is equicontinuous at x for each x E [0, T) and hence uniformly equicontinuous on [0, T). This together with the fact that NYn -+ Ny pointwise on [0, T) implies that the convergence is uniform. Consequently Ny E G([O, T), Rn) and N : G([O, T), Rn) -+ G([O, T), Rn) is continuous. In addition the Arzela-Ascoli theorem guarantees that N is completely continuous. To see this let 0 ~ G([O, TJ, Rn) be bounded i.e. there exists r > 0 with Ivlo::; r for each v E O. Also there exists hr E Ll[O,T) with IK(x,s,v)1 ::; hr(s) for a.e. s E [O,T) and all x E [O,T) and v E O. The equicontinuity of NO on [0, T) follows essentially the same reasoning as that used to derive (2.11). Also NO is bounded since
INv(x)l::; sup xe[O,Tj
I
fX K(x,s,v(s))dsl::; sup fX hr(s)ds
Jo
for each v E O. Consequently N : G([O, TJ, Rn) continuous.
xe[O,Tj -+
Jo
G([O, TJ, Rn) is completely
Essentially the same reasoning as that used in theorem 2.2 establishes the following existence principle for the Urysohn integral equation
y(t) = Ly(t) + loT K(t,s,y(s))ds for t
E
[O,T)
(2.12)
Theorem 2.3. Assume (2.4) holds, K : [0, T)2
X Rn -+ Rn and that K(x, s, u) L1-Garatheodory uniformly in x . Suppose for each r> 0 and x E [0, T) we have
lim fT sup IK(z,s,u)-K(x,s,u)lds=O, zE[O,T) . Jo lul~r
~s
(2.13)
z_x
In addition assume there is a constant Mo, independent of A, with any solution Y to
Iylo =I-
Mo for
y(t)=Ly(t)+AloTK(t,S,y(s))ds for tE[O,T) for each A E (0,1). Then (2.12) has a solution in G([O, T), Rn). Existence theory in the spirit of [1,7,11,12) could be established for the integral equations (2.8) and (2.12). We provide one such result for the Volterra equation (2.8).
256 BVPs FOR FDEs Theorem 2.4. Assume (2.4) holds, K : [0, T]2
X Rn --> Rn and that K(x, s, u) is Ll-Caratheodory uniformly in x. In addition assume (2.9) is true and that
there exists a continuous function 1j;: [0,00) --> (0,00), a continuous function tjJ: [0, T]--> [0,00), a constant 0" ::::: 1, with { I(I + R) (I~ K(t, oS, u(s)) dS) I ~ tjJ(t)1j; (I~ lu(x)ICT dX) , t E [0, T] for any u E C([O, T], Rn) and
fT
Jo
tjJCT(x)dx
(t+81)-x(t+82 )1, ( 14>(t+8t)-4>(t+82 )1, Ix(t + 81) - 4>(t + 82 )1,
(8)1 Xt
2
-
Xt
E
iJ
and for 81, 82 E [-r, 0] ,
°
if t + 81 :2: and t + 82 :2: 0, ift+81 . For the second case of 14>(t + 81) - x(t + 82 )1, t + 81 < ~ t + 82, we can rewrite 14>(t + 8t) - x(t + 82)1 as
°
14>(t + 8J) - x(t + 82)1 ~ 14>(t + 8t) - 4>(0)1 + Ix(O) - x(t + 82)1 . By the equicontinuity of B and uniform continuity of 4>, such that lSI - s21 < hI and SJ, S2 E [0, T] imply that also that, 181 - 821 < h2 and 81, 82 E [-r,O] imply that h = min{hJ,h2}' So, if 81,82 E [-r,O] and if t + 81 < then, It + 81 - 01 < h and It + 82 - 01 < h and so
°
IXt(8t) -
Xt(8 2 )1
~
(81) - 4>(82)1 < c/2. Let ~ t + 82 and 181 - 821 < h,
14>(t + 81) - x(t + 82)1 14>(t + 8J) - 4>(0)1 + Ix(O) - x(t + 82)1 c/2 + c/2 c.
Case 4, Ix(t + 8J) - 4>(t + 82)1, is similar. Thus iJ is uniformly equicontinuous with respect to CT and so there exists a compact DeC such that iJ ~ D C CT' Since B was an arbitrary bounded subset of C, we need to show F(B) is precompact. Choose a sequence {Fxv}:'1 in F(B), (so {xv} C B) . Thus {xv,} c iJ ~ D C CT' Define X = [O,T] X D X [-b,b] x .. · x [-b, b]. X is compact by Tychanoff's Theorem. Let 8 = maxlf(t,uJ,u2, . .. ,un)l . By Jackson [14], there exist constants ",.11 , < X _ i ~ n - 1, such that, for all t E [0, T],
°
fl
ai
Jo laxiG(t,s)lds=,i(T-o)n-l, O~i~n-l.
273 MULTIPOINT BOUNDARY VALUE PROBLEMS
For each 0:5 i :5 n - 1, let Hi v EN,
= SUPO 0 such that IIxlll-r,T] S M and IIx(i)llIo,T] S M , 1 SiS n-l , for all solutions x 0/ (3,\) with the boundary conditions xo(s) = (s), - r S s S 0, y(')(T/j) = 0, 0 SiS mj - 1, y(i)(T) = 0, 0 SiS mk - 1,
I
2 S j S k -1,
(7)
where L:~=2 mt = n - 1, and 0 < T/j < T/j+1 < T , 2 S j S k - 2. The proof of the above theorem follows from Theorems 2.3 and 3.1 in Eloe and Henderson [9) . As a consequence of Theorem 3.1 above, we have an immediate corollary. Corollary 3.2 Assume the hypothesis of the Theorem. Then there exists a solution
0/(1) (7) .
275 MULTIPOINT BOUNDARY VALUE PROBLEMS
References 1. R. P. Agarwal, Boundary value problems for differential equations with deviating arguments, J. Math. Phy Sci. 6 (1992), 425-438.
2. D. C. Angelova, Quickly, moderately, and slowly osciallatory solutions of a second order functional differential equation, Arch. Math. 21 (1985), 135-146. 3. D. C. Angelova and D. D. Bainov, On oscillation of solutions of forced functional differential equations of second order, Math. Nachr. 122 (1985), 289-300. 4. D. C. Angelova and D. D. Bainov, On the oscillation of solutions to second order functional differential equations, Bollettino U. M. I. (6) 1 -B (1982), 797-807. 5. D. C. Angelova and D. D. Bainov, On some oscillatory properties of the solutions of a class of functional differential equations, Tsukuba J. Math. 5 (1981), 39-46. 6. J. Ehme, P. W. Elce, and J . Henderson, Differentiability with respect to boundary conditions and deviating arguments for functional differential systems, Diff. Eqns. Dyn. Sys. 1 (1993), 59-71. 7. J. Ehme and J. Henderson, Functional boundary value problems and smoothness of solutions, Nonlin. Anal., in press. 8. P. W. Eloe and L. J. Grimm, Conjugate type boundary value problems for functional differential equations, Rocky Mtn. J. Math. 12 (1982), 627-633. 9. P. W. Eloe and J. Henderson, Nonlinear boundary value problems and a priori bounds on solutions, SIAM J. Math. Anal. 15 (1984), 642-647. 10. P. W. Eloe, J. Henderson, and D. Taunton, Multipoint boundary value problems for functional differential equations, Pan Am. Math. J., in press. 11. L. J . Grimm and K. Schmitt, Boundary value problems for delay differential equations, Bull. Amer. Math . Soc. 74 (1968), 997-1000. 12. L. J. Grimm and K. Schmitt, Boundary value problems for differential equations with deviating arguments, Arq. Math. 3 (1969), 24-38. 13. L. J. Grimm and K. Schmitt, Boundary value problems for differential equations with deviating arguments, Arq. .Math. 4 (1975) 176-190. 14. L. K. Jackson, Boundary value problems for ordinary differential equations, in "Studies in ordinary Differential Equations" (J. K. Hale, Ed.) pp. 92-127, MAA Studies in Mathematics, Vol. 14, Mathematical Association of America, Washington, DC, 1977.
276
BVPs FOR FDEs 15. G. A. Kamenskii and A. D. Myskis, Variational and boundary value problems for differential equations with deviating argument, Proc. Eqadiff IV, Lecture Notes in Mathematics 703 (1979), 179-188. 16. P. R. Krishnamoorthy and R. P. Agarwal, Higher order boundary value problems for differential equations with deviating arguments, Math . Seminar Notes 7 (1979), 253-260. 17. J. W. Lee and D. O'Regan, Existence results for differential delay equations - I, J. Diff. Eqns. 102 (1993), 342-359. 18. J. W. Lee and D. O'Regan, Existence results for differential delay equations - II, Nonlin. Anal. 17 (1991), 683-702. 19. S. K. Ntouyas, Y. G. Sficas, and P. Ch. Tsamatos, Existence principle for boundary value problems for second order functional differential equations, Nonlin. Anal. 20 (1993), p. 215-222 . 20. D. O'Regan, Existence of solutions to some differential delay equations , Nonlin. Anal. 20 (1993) , 79-95. 21. P. Ch. Tsamatos and S. K. Ntouyas, Existence and uniqueness of solutions for boundary value problems for differential equations with deviating arguments, Nonlin . Anal. 22 (1994), 1131-1146. 22. P. Ch. Tsamatos and S. K. Ntouyas, Existence of solutions of boundary value problems for differential equations with deviating arguments via the topological transversality method, Proc. Royal Soc. Edinburgh USA (1991), 79-89. 23. H. Xia and T . Spanily, Global existence problems for singular functional differential equations, Nonlin. Anal. 20 (1993), 921-934.
277
THIRD ORDER BOUNDARY VALUE PROBLEMS FOR DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS
P. CH. TSAMATOS Department of Mathematics, University of Ioannina 451 10 Ioannina, Greece
1. Introduction
In this paper we deal with the following differential equation with deviating arguments x"'(t)
= f(t, x(t), x(u(t)), x'(t), x'(g(t)), X"(t), X"( T(t))),
a.e t E (0,1),
(E)
where f : [0, 1) X ~ - R is a function satisfying Caratheodory's conditions and u, g, T are continuous real valued functions defined on [0,1]. We suppose that -00
< -r = min {u(t),g(t),T(t)} < tE[O,lj
°
and 1 < max{u(t),g(t),T(t)} = d < tE[O,lj
+00 .
We also consider the following boundary conditions
x(t) x(t)
= (/JI(t), = rP2(t),
t E [-r,O) t E [I,d)
(BCh
t E [-r, 0] t E [I,d)
(BCh
m
L aiX(ei) =
'T/,
i=l
x(t) x(t)
= rP1(t), = rP1(t),
m
L aix'(e;) = 1:=1
'T/
278 BVPs FOR FDEs and
= (/J!(t), = li>2(t), m L aix"((i) = 'T/, x(t) x(t)
tE[-r,O] tE[I,d]
(BCh
;=1
where ~l : [-r,O] -+ R, ~2 : [1, d] -+ R are differentiable functions such that ~~ and ~~ are absolutely continuous functions on [-r, O] and [1 , d] respectively. Also (i E (O,I),ai E R,i = 1, . .. ,m and 'T/ E R. Recently, several papers have appeared which are concerned with the existence and uniqueness of solutions of boundary value problems (BVP for short) for third and higher order nonlinear differential equations (see [1], [3], [7], [12]). Also, during the last five years, boundary value problems for differential equations with deviating arguments were the subject of several articles (see [8-11] and [13-14]) . The purpose of the present paper is to study some multi-point BVP for third order differential equations with deviating arguments . As usual, in BVP for differential equations with deviating arguments we search for solutions which satisfy the differential equation on a compact interval I and identify with an a priori given functions outside of I. Moreover, here we demand the solutions to satisfy a multi-point condition on I. In this way, we enlarge the class of boundary conditions considered in [3] and [7]. So, the results obtained here are new even for the usual case, that is for the case where u(t) = g(t) = T(t) = t, t E [0,1]. We notice that multi-point BVP for second order differential equations have been recently studied by Gupta, Ntouyas and Tsamatos in [4-6]. The existence results here are obtained by the help of the well known LerayShauder Alternative, which follows immediately from the Topological Transversality Theorem of Granas [2].The Topological Transversality Theorem of Granas and its corollaries are widely applied in the theory of boundary value problems. For applications of the Topological Transversality method on boundary value problems for differential equation with deviating arguments we refer the reader to [10], [13] and the references therein.
2. Preliminaries If A is a compact interval ofthe real line, we denote by C(A), Ck(A), Lk(A) and L OO (A) the classical spaces of continuous, k-times continuously differentiable, measurable real valued functions whose k-th power of the absolute value is Lebesgue integrable on A, or measurable functions that are essentially bounded on A. We also use the Sobolev space W 3 ,k ( A), k = 1, 2 defined by
W 3.k(A)
= {x : A -+ R
x, x', x" abs. cont . on A with x E Lk[o,
In
279 THIRD ORDER BOUNDARY VALUE PROBLEMS
1I.lIk' and the norm in Loo(A) by 11.1100.
We denote the norm in Lk(A) by
Definition 2.1. A function x : [-r, d] -+ R is called a solution of the BVP (E) - (BCh (resp. (E) - (BCh,(E) - (BCh) if x, x'I[-r, 0], x"I[O, 1], x'I[I, d] are absolutely continuous, xl[O, 1] satisfies the equation (E) and xl[-r, 0] = rPl, xl[l, d] = rP2, E~l aiX(ei) = "I (resp.E~l aix'(ei) = "I, E~l aix"(ei) = "I). Lemma 2.2.Let ai E R, i
= 1, ... , m
with all of the aj's having the same sign and
ei E (0,1), i = 1, ... , m be given. Let also, x E W ,1[0, 1] be such that x(O) = x(l) = 0 3
and E~l aiX(ei)
= o.
Then
Proof. Since x(O) = 0, we have x(t) = f~ x(s)ds, t E [0,1] and hence
Moreover, since x(O)
= x(l) = 0, there exists 0 E (0,1) such that x'(O) = o.
Therefore
IIx'lloo :::; IIx"lloo. On the other hand the relation E~l ajX(ei) = 0 and the continuity of x imply that there exists e E [minl1(0) 2
E C ([0, 1]) : x(O)
= ¢>2(1) = TJ = O.
= x(l) = O}
is obviously a subspace of C 2 ([0, 1]). Now, for any function
X
E C we define the function x(t) =
x : [-r, dJ ---> R by
¢>l(t), t E [-r,O] x(t), t E [0,1] ¢>2(t) , t E [1 , dJ
1
and, for t E [0,1], we set Tx(t)
=
III
J(z,x(z),x(u(z)),x'(z),x'(g( z )),x"(z),x"(r(z )))drdBds
Next we define the operator F1 : C ---> C 2 ([0, 1]) by
where Al
=
Tx(l) E~l ai~i - E~l aiTx(~;} B E~l ai~i(~i - 1) , 1
=
E?:l aiTx(~i) - Tx(l) E?:l aiel E~l ai~i(~i - 1)
282 BVPs FOR FDEs It is obvious that x is a solution of the BVP (E) - (BCh if and only if F1 x1[0, 1] = xl[O, 1] and xl[-r, 0] = 4>b xl[l,d) = 4>2.
Obviously F1 (C) ~ C and Fl is completely continuous. Now, it will be shown that the set £(Fl) = {x E C : x = AF1 x,0 < A < I} is bounded. Indeed, let x E £(Ft). Then we use the assumptions to get that
IIxllllh = Allf( t, x(t), x( u(t)), x'(t), x'(g( t)), x"(t) , x"( r( t)))II1 ::; IIpl 111 II x 1100 + IIp2Ih max {lIxIl00, 114>11100, 114>21100} + IIp31hll x'iloo + IIp4Ih max {lI x'lIoo, 114>~1I00, 114>~1I00} + IIPslhllx"lloo + IIp6Ih max {lIx"lIoo, 114>~1I00, 114>~1I00} + 1I1lh ::; IIp11hll xll00 + IIp211dllxll00 + 114>11100 + 114>21100} + IIp311dl x'il00 + IIp411dllx'il00 + 114>~1I00 + 114>~1I00} + IIPslhll x"lIoo + IIp6l1dllx"1I00 + 114>~1I00 + 114>~1I00} + 1I1111. Using Lemma 2.2 we obtain 6
II XIII II 1 ::; (I: IIp;Jldll xlll lh + P, i=l
where
+ 114>2I1oo} + IIp4I1dll4>~1I00 + 114>~1I00} + IIp61h {1I4>~1I00 + 114>~1I00} + 1I1111'
p = IIp2 lid 114>1 1100
Then (3.1) implies that
11 X 11111 1
