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- 0 such that o = (oA - B)
- 0, e\IDI - ILl - IVI) > o.
2
55
M-MATRICES
Obviously, for a diagonally positive Z-matrix, these criteria are equivalent to, respectively, Me .> 0, Me > 0, MTe >- 0, and MTe > o. Corresponding generalized criteria are obtained when e is replaced by an arbitrary vector z
>- O,
2.5.2
Regular splittings
A splitting M = A - B will be called regular if A is inverse-positive and A -1 B ~ O. Obviously, then A and B satisfy Condition (R) of Section 2.4, so that the results of that section apply. In particular, M = A - B is a regular splitting if A is inverse-positive and B ~ O. The more general condition A - 1B ~ 0 holds if and only if for all u, v E [R" Au = Bo, v
~ 0
=>
u
~
(2.13)
o.
Theorem 2.19. For a regular splitting M = A - B of a given matrix M, the following three properties are equivalent: (i) (ii) (iii)
The iterative procedure corresponding to the splitting converges. M is inverse-positive. z ~ 0, M z >- 0 for some Z E [R".
Proof
According to Theorem 2.14 and Corollary 2.14a the inequality 0
peA - 1 B) < 1 is equivalent to each of the properties (ii) and (iii).
Corollary 2.19a. For strictly diagonally positive ME Z, the Jacobi procedure converges if and only if M is inverse-positive. The same statement holds for the Gauji-Seidel procedure. Proof
Both procedures correspond to regular splittings.
0
The above theorem shows that each criterion for the inverse-positivity of M is also a criterion for the convergence ofeach iterative procedure corresponding to a regular splitting M = A - B. A series of criteria for inverse-positivity has been given in previous sections. In particular, for diagonally positive M E Z, the row-sum criterion and the column-sum criterion are sufficient for the convergence of each procedure corresponding to a regular splitting. The same statement holds for the generalized row-sum criterion and column-sum criterion. If, in addition, M is irreducible, the corresponding weak criteria are also sufficient. Furthermore, we recall that each positive definite symmetric ME Z is inverse-positive. While the convergence of an iterative procedure corresponding to a regular splitting is independent of the special choice of the splitting, the ratio of convergence -log p(A - 1 B), in general, does depend on the splitting M = A-B. The same is true for norms of the iteration matrix. Such norms are used for sufficient convergence criteria and error estimations. We collect some related results.
56
II.
INVERSE-POSITIVE LINEAR OPERATORS
Theorem 2.20. Suppose that both M = Al - B 1 and M = A z - B z are regular splittings, and let T; = A i- 1 B;, Pi = peT;) (i = 1, 2). (i)
Ifip A, - B 1)z ~ ofor some z
(p - 1)(B 1
-
>- 0, then IIT111, :( p and, moreover,
B z ) :( 0
IITzll,:(
~
p.
(ii) If B 1 - B z > 0 and if both matrices o.A, - B, (i = 1,2) have the property (2.11) (in place of oA - B), then
either
P: < PI < 1,
Proof
(i)
or
or
PI = pz = 1,
The inequality pA lz -
BIZ ~
0
1 < PI < Pz'
implies T1z :( pz, so that
IIT11I,:( p, since TI is positive. For (p - 1)(B 1 - B z ) :( 0 we have also pAz z - B z z ~ 0 and hence II Tzll, :( p. (ii) follows from Corollary 2.17b. 0
Example 2.12. Let M = D - L - U E Z with DE M and consider the GauB-Seidel procedure, that means, choose A = D - Land B = U. In order to obtain the norm of the corresponding iteration matrix T = A-I B, one may proceed as follows. Calculate U e and solve the triangular system (D - L)v = U e for v. Then p = max, Vi is the smallest number with (pI - T)e ~ o. Therefore, II Tile = p. For example, if M is the matrix in Example 2.2, then Vi = 1 - r i (i = 1, 2, ... , n - 1) and Vn = tV n _" so that p = 1 - r n + 1 < 1. 0 Example 2.13. Let M be as in Example 2.12 and consider again the GauB-Seidel procedure. Now we want to calculate the norm liT 111' Solve the triangular system (D - 10) TZ = e for z and compute W = U T z. Then, p = max, w, is the smallest number with (pI - UT(D - L T)-l)e = (p(D - L T) - UT)z ~ o. Therefore, IITlll = II(D - L)-lUI11 = fIUT(D - L T )- l lloo = p. Observe that the inequality (p(D - L T) - UT)z ~ 0 also yields
II(D - L T ) - l U T II,
:( p. 0
Corollary 2.20a. Suppose that M is a diagonally strictly positive Z-matrix. Let 1J and TG denote the iteration matrices of the Jacobi procedure and the Gauli-Seidel procedure, respectively, and let PJ = p(1J), PG = p(TG)· (i)
(ii)
If 111JII, :( 1 for some z >- 0, then I TGiI, :( 111J I aIf M is irreducible, then either PG < PJ < 1, or PG = PJ = 1, or
1 < PJ < PG'
Proof The statements follow from Theorem 2.20 applied to Al = D, B 1 = L + U, A z = D - L, B z = U, where M = D - L - U. 0
2
57
M-MATRICES
2.5.3 Majorization The theory of the preceding section can also be used to obtain results on iterative procedures corresponding to nonregular splittings. One derives sufficient convergence criteria which are useful in some cases. Theorem 2.21. For a given splitting M matrices such that
(i) (ii) (iii)
= A - B, let A o, B o denote (2.14)
IA-IB/~A(jIBo,
M o = A o - B o is inverse-positive, M 0 = A o - B o is a regular splitting of Mo.
Then the iterative procedure corresponding to the splitting M verges. Moreover, p(A -1 B) ~ p(A(j 1 B o) and for each
z
=
>- o.
A - B con-
(2.15)
The proof is left to the reader. (See Problems 2.22-2.24.) In order to obtain a bound for the norm of A-I B, one may apply the results of Section 2.5.2 to M o = A o - B o and then use the estimation (2.15). Obviously, the relation (2.14) holds if IA - I I ~ A(j I and IB I ~ B o . For example, the inequality IA-11 ~ A(j 1 is satisfied for A o = Do - Co
if this A(jl
with
Do = ID(A)I,
CO = IL(A)
+
U(A)I,
matrix A o is inverse-positive. This follows from
=k~o(D(jlCo)kD(j1
~ !k~}D-l(A)(L(A)
+
U(A»]kD-l(A)! = lA-II.
Thus, when we use A o = ID I, B o = IL + U I for the Jacobi procedure, and A o = ID I - I L I, B o = I U I for the Gaub-Seidel procedure (where M = D - L - U), we obtain Corollary 2.21a. The Jacobi procedure and the Gauii-Seidel procedure for the equation Mu = s converge if M 0 = ID I - IL I - I U I is inverse-positive. In particular, this is true if the (generalized) row-sum criterion or the (generalized) column-sum criterion is satisfied. If, in addition, M is irreducible, the corresponding weak critera are also sufficient for convergence.
2.5.4
Monotonic approximations
We turn now to iterative procedures for two sequences of vectors which not only approximate a solution of the given equation Mu = s, but yield also bounds for this solution. The starting vectors of the iterative procedure have to satisfy conditions which are closely related to the inverse-positivity of a certain matrix M.
58
II.
Theorem 2.22. Suppose that M there exist vectors x o, Yo E IRn with
INVERSE-POSITIVE LINEAR OPERATORS
= A - B is a regular splitting and that Xo ~
Yo·
Then the sequences {xv} and {Yv} defined by AX v + l
= Bx; + s,
AYv+l
= By; + s
(v = 0, 1,2 ... )
converge toward limits x* and y*, respectively, and
(2.16) Proof The iterative procedures can be written as x v + 1 = Tx.; Yv+ 1 = Ty; with Tu = A - 1 Bu + A - 1 s. Since A - 1 B ~ 0, the results stated follow from Theorem 1.3.4 (see also Remark 3). D
The above theorem can be generalized to certain nonregular splittings. Theorem 2.23.
Let A, B, A b A 2, B b B 2 E IRn.n such that
M=A-B and define
Ai, A, jj E 1R 2 n . 2n by with
Assume that the following two conditions are satisfied.
(i) (ii)
Ai = A -
jj is a regular splitting. There exist vectors xo, Yo E IRn with
A1Xo - A 2yo - Blx O
+ B 2yo Xo
~ S ~
Alyo - A2Xo - Blyo
~ Yo.
+ B 2x O ,
(2.19)
Then the sequences {x.}, {Yv} defined by
Alx v + 1 -A 2 x v + l
= Blx v - B 2yv + S, + A1Yv+l = -B 2xv + Blyv + s
-
A 2yv+ 1
= 0, 1, 2, ...) converge toward limits x*, y* and the inclusion statement (2.16) holds.
(v
Proof
This result follows when Theorem 2.22 is applied to the procedures
with
_ Yv
=
(-Xv) . Yv
D
2
59
M-MATRICES
Introducing Uv
=
t(x v + Yv),
+ y*),
u* = t(x*
we can reformulate Theorem 2.23 in the following way. Corollary 2.23a. Suppose that assumption (i) of Theorem 2.23 is satisfied, and that there exist vectors uo, Zo E IRn with (2.20)
Zo ~ 0,
where
M = A - B,
A=
A1
+ A 2'
(2.21)
Then, the sequences {u.}, {z.} defined by Au v + 1 = Bu;
+ s,
(v
= 0, 1, ...)
converge. Moreover, the sequence {z.} is antitonic and
(v
= 0, 1,2, ...)
with u* = lim u,;
Obviously, Theorem 2.22 is' contained in Theorem 2.23 as a special case where A = A 1 and B = B r- Consequently, if M = A - B is a regular splitting, then Corollary 2.23a holds with A = A, B = B, and M = M. Now we shall describe assumption (i) of Theorem 2.23 in a different way, and also formulate some special cases. In the following propositions, the occurring matrices shall be related as above (see (2.17), (2.18), and (2.21». Proposition 2.24. (i) Ai = following two properties hold: (a) (b)
A - B is a regular
splitting
if and only if the
u, VE IRn, IAul ~ Av = lui ~ v. x, y, U, vE IRn, Au = Bx, Av = By, [x] ~ y
(ii) If Ai = splitting.
A- B
is a regular splitting,
= lui ~ v. then M = A -
B is a regular
Proof (i) By applying A to the vector C~~~), it is easily seen that property (a) is equivalent to the inverse-positivity of A. In a similar way, one can show that property (b) is equivalent to (2.13) applied to A, B instead of A, B. The proof of (ii) is left to the reader. 0
Proposition 2.25.
Let A, BE IRn, n be given matrices with M = A-B.
(i) If A is inverse-positive, then the choice A 1 = A, A 2 = 0, B 1 = B + , B 2 = B-, yields a regular splitting Ai = A - B.
60
II.
(ii)
INVERSE-POSITIVE LINEAR OPERATORS
The choice
= D(A) - (L(A) + U(A»+, A 2 = - (L(A) + U(A»-,
Al
yields areqular spliuinq M = is inverse-positive.
B I = B+, B 2 = B-,
A -B ifandonlyifA =
D(A) -IL(A)
+
U(A) I
The proof is left to the reader. In applying the preceding results, the problem arises of how to obtain suitable vectors x o, Yo or uo, zoo Such vectors shall be called admissible starting vectors for the corresponding iterative procedures provided the inequalities (2.19) or (2.20) hold. In this section, we shall discuss only the theoretical conditions under which admissible starting vectors exist. Since all conditions on Xo = Uo - zo, Yo = Uo + Zo can easily be obtained from corresponding (equivalent) conditions on uo, zo, the following proposition is formulated only for the latter vectors. Proposition 2.26. If assumption (i) of Theorem 2.23 is satisfied, the following statements hold. (i) There exist admissible starting vectors uo, Zo with Mz o >- 0 ifand only if M is inverse-positive. (ii) Let M be strictly pre-inverse-positive. Then, there exist admissable starting vectors uo, Zo with Mz o > 0 if and only if M is inverse-positive.
Proof (i) According to Proposition 2.24(ii), M = A - B is a regular splitting, so that the matrix M is weakly pre-inverse-positive. Thus, if a vector Zo ~ 0 with Mz o >- 0 exists, the matrix M is inverse-positive. On the other hand, if M is inverse-positive, there exists a vector z ~ 0 with Mz >- 0, so that Zo = mz satisfies (2.20) for given Uo and large enough m. (ii) is proved in a similar way. 0 Unless the vector U o satisfies at least one ofthe given n algebraic equations (described by Mu = s) exactly, each vector Zo with (2.20) has the property MZo >- o. Thus, exceptional cases excluded, there exist admissible starting vectors if and only if M is inverse-positive. According to Theorem 2.14, this property is equivalent to the condition p(A - I B) < 1, which implies the convergence z, ..... o. Problems
2.19 The point successive relaxation method is defined by (2.12) with A = l)D + U, w 1= O. Prove that the statement in Corollary 2.19a holds also for this procedure, if 0 < w ~ 1. What is the situation for w > I? w-ID - L, B = (w- I
3
61
DIFFERENTIAL OPERATORS OF THE SECOND ORDER
2.20 Use Theorem 2.20(ii) for comparing point successive relaxation methods with different parameters W E (0, 1]. 2.21 (2.13)means that Au E C = U ~ 0, whereC = {w E [R": w = Bv, v ~ a}. Try to apply Theorem 1.2 to this type of inverse-positivity. 2.22 Prove p(T) = min{ II Tllz: z ~ o} for positive T E [R"'" by first verifying this relation for positive irreducible T. 2.23 For TI , Tz E [R"'" derive p(T I ) ~ p(Tz ) from I TIl ~ Tz . 2.24 Prove Theorem 2.21, observing the preceding two problems. 2.25 Prove Proposition 2.25. 3 DIFFERENTIAL OPERATORS OF THE SECOND ORDER, POINTWISE INEQUALITIES
The abstract theory of Section 1 will here be applied to second order linear differential operators related to boundary value problems. In particular, inverse-positivity statements of the following type will be derived: L[u](x) ~
0 (0 < x < I)}·
Bo[u] ~ 0 BI[u] ~ 0
=
u(x)
~
0 (0
~
x
~
1).
(3.1)
.
Here L denotes a (formal) differential operator in the usual sense and Bs, B I are boundary operators. These three terms will be combined in one operator M = (L, B o , B I ) such that (3.1) is equivalent to the inverse-positivity of M, i.e.,Mu ~ 0 = u ~ o.NotethatthenMu = r e [R[O, 1] constitutes a boundary value problem. In many of the results below no regularity conditions on the coefficients of L are required. In cases, however, where due to "smooth" coefficients a suitable existence theory for boundary value problems Mu = r is available, a richer theory on inverse-positivity can be developed. We shall, in particular, often consider regular operators M, as defined in Section 3.1. Section 3.2 mainly provides sufficient conditions and necessary conditions for (3.1) in terms of differential inequalities for a majorizing function z. In Section 3.3 sufficient conditions on the coefficients of the operators L, B o , B I are formulated such that a majorizing function exists. The actual construction of such afunction can often be avoided by considering connected sets of operators, as in Section 3.4. This method leads to several characterizations of the set of all regular inverse-positive operators M. Section 3.5 is concerned with property (3.1) under constraints such as Bo[u] = BI[u] = O. Often, the inequality Mu ~ 0 implies stronger positivity properties of u than u ~ o. Corresponding results are contained in Sections 3.6 and 3.7. For regular operators, Theorem 3.17 is the strongest and most general statement.
62
II.
INVERSE-POSITIVE LINEAR OPERATORS
An operator M is inverse-positive if and only if M has a positive inverse. Under suitable regularity conditions, the inverse is an integral operator with a Green function r:§ as its kernel. Section 3.8 is concerned with the relation between inverse-positivity of M and the positivity (and similar properties) of ~. The results on inverse-positivity can be used for obtaining other interesting statements. For example, boundary maximum principles are formulated in Sections 3.2, 3.6, and 3.7. For regular operators, we show in Sections 3.9 and 3.10 in which way statements on inverse-positivity, in particular Theorem 3.17, can serve as a tool for deriving oscillation theorems and results on eigenvalue problems. It is only for simplicity of notation that we mainly consider the special interval [0, IJ in (3.1). Analogous results for an arbitrary compact interval [IX, fJJ are easily derived from the theory below. In many cases one may simply replace [0, 1J by [IX, fJl We shall sometimes make use of this fact. Differential operators M on noncompact intervals, such as [0, (0) and (- 00, (0), are studied in Section 3.11. In many cases, one can either develop a theory similar to that for operators on compact intervals, or obtain results by using the theorems for compact intervals. Throughout this Section 3 we consider only functions u in (3.1) such that all derivatives u'(x), u"(x) which occur are defined in the usual sense. Functions with weaker smoothness properties will be considered in the theory for nonlinear operators in Chapter III. 3.1
Assumptions and notation
Our main interest is in operators M: R L [ UJ (X) Mu(x) =
{
Bo[uJ B 1[uJ
--+ S
of the form
for for for
x=O
O<x
0
~
u
u(x) ~ 0
(0
x ~ 1); u(x) = 0 (0 ~ x ~ 1); u ~ 0, u(~) > 0 for some ~
1].
~E[O,
The corresponding strict order on R is given by u
>- 0
~
u(x) > 0
(0
~
x
1).
~
Whenever other order relations or order relations on intervals different from [0, 1] occur, we shall point this out explicitly. According to our notation an operator M = (L, B o, B I ) is inverse-positive if for all u E R, Mu ~ 0 => u ~ 0, which means, explicitly, that (3.1) holds for all u E R. Sometimes, when the choice of R is essential, we shall use the term inverse-positive on R. An operator M = (L, B o, B I ) is called regular, if a, b,
CE
Co[O, 1]
a(x) >
and
°
(0
~
x ~ 1).
(3.5)
A regular operator is said to be formally symmetric or formally self-adjoint if L[u] = -(au')' + cu with a E ClEO, 1]. To each regular M = (L, B o, B I ) we adjoin a formally symmetric regular operator M, = (L s , Bs, B I ) , where Ls[u] = -(pu')'
+ qu
= dL[u]
(3.6)
= cd,
(3.7)
with p = exp[ - fob(t)a-l(t) dtJ
°
q
and some Xo E [0, 1]. Since d(x) > (0 ~ x ~ 1), it would theoretically suffice to consider only regular operators which are formally self-adjoint. For example, a regular operator M is inverse-positive if and only if M; is inversepositive.
II.
64
INVERSE-POSITIVE LINEAR OPERATORS
Our concept includes also differential operators of the first order
M ( ) ux
=
{u' + c(x)u u(O)
for for
0 < x :( 1 x = 0
(3.8)
defined on R = CoCO, 1] n CI(O, 1], where c E !R(O, 1]. This operator has the form (3.2) with IXI = 1, YI = c(l). 3.2
Basic results on inverse-positivity
For a fixed operator M = (L, B o , B I ) : R ~ S, as described above, we formulate conditions such that M is inverse-positive or has similar properties. These conditions are formulated in terms of differential inequalities for a function Z E R. Proposition 3.1. The given operator M is pre-inverse-positive with respect to the order relation ~ defined in S by
V ). that means, u Proof
~ 0
0
-=-
and Mu ).
0
Vex) > 0 (0:( x :( 1), imply u
>- o.
It suffices to show that for each u
~ 0,
u(~)
=0
=>
~ E
[0, 1]
Mu(~):(
0,
(3.9)
since then the assumption of Proposition 1.3 is satisfied for IV = V(~). First, let ~ E (0, 1). Since u assumes its minimum at ~, the derivatives satisfy the relations u(~) = u'(~) = 0 and u"(~) ~ 0 (provided u"(~) exists). Therefore, Mu(~) = -a(~)u"(~) :( O.For~ = Owehaveu(O) = Oandu'(O) ~ O(provided u'(O) exists), so that Mu(O) = -lXo u'(O) :( o. The case ~ = 1 is treated analogously. 0 From this proposition and the monotonicity theorem the following result is immediately derived. Theorem 3.2.
If there exists a function Z ~
0,
Mz(x) > 0
ZE
R such that (0 :( x :( 1),
(3.10)
then M is inverse-positive. A function z E R with this property (3.10) will be called a majorizingfunction for M. (This definition is consistent with that of a majorizing element in Section 1.2, since (3.10) implies z >- 0.) In general, the existence of a majorizing function is also necessary. If for any rex) > 0 (0 :( x :( 1) the problem Mu = r has a solution z E R, then this solution satisfies (3.10), provided M is inverse-positive. For a regular inversepositive M, the equation Mu = r with rex) == 1 has a solution z E C 2[0, 1]. This yields
3
65
DIFFERENTIAL OPERATORS OF THE SECOND ORDER
Theorem 3.3. A regular operator M = (L, s.; B d is inverse-positive ifand only if there exists a majorizing function for M. Moreover, a regular operator M is inverse-positive on R M if and only if Mis inverse-positive on C 2[0, 1]. Before we discuss these results, we shall consider some modifications, which will be used later. Theorem 3.2 can also be derived from Theorem 1.7, as the proof of Proposition 3.1 shows. One chooses 2 to be the set {t x: ~ x ~ 1} of all point functionals {xu = u(x) on R, 2 1 = 2, and 2 2 = 0. Then the implication (3.9) is equivalent to (1.3) with { = {sand IV = V«(). Moreover, with these notations, Corollary 1.7a yields the following result.
°
Corollary 3.2a. Let the assumption of Theorem 3.2 be satisfied. If Mu ~ 0 and Mu«() > Ofor a given u E R and some (E [0, 1J, then u ~ 0 and u«() > 0. This statement remains true iffor all (with (Mu)«() > only the inequality M z( () ~ is required instead of M z( () > 0, and if z >- o.
°
°
In particular, we obtain Theorem 3.4.
z
>- 0,
If there exists a function z E R such that
L[zJ(x) ~
°
(~
0,
B 1 [zJ > 0, (3.11)
then for all u E R L[uJ(x) > Bo[uJ ~
B 1 [uJ ~
° (0 < x < 1)} ° °
=>
u(x) >
° (0 < x < 1).
(3.12)
According to these results, the inverse-positivity of M and similar properties can be established by solving certain (differential) inequalities such as (3.10) and (3.11). Example 3.1. (a) If c(x) > 0(0 < x < 1), Yo > 0, Y1 > 0, then z(x) == 1 satisfies the inequalities (3.10) and, therefore, M is inverse-positive. (b) If c(x) ~
°
(0 < x < 1),
Yo > 0,
Y1 > 0,
(3.13)
then z(x) == 1 satisfies the inequalities (3.11) and, therefore, statement (3.12) is true. D
°
The condition c(x) > in part (a) of Example 3.1 is rather restrictive. In particular, this inequality is not satisfied for the simple operator Mu = (_un, u(O), u(l». The conditions in part (b) hold for many more operators which are of practical interest. However, property (3.12) does not imply inverse-positivity. Neither do the inequalities (3.13) alone imply inversepositivity.
66
II.
Example 3.2.
INVERSE-POSITIVE LINEAR OPERATORS
The inequalities (3.13) hold for M = (L, B o , B l ) with
L[u](x) = -(1 - 2x)V' - 4(1 - 2x)u',
Bo[u] = u(O),
Bl[u] = u(I).
However, this operator is not inverse-positive, since u = 1 - 8x(1 - x) satisfies Mu ~ 0, but u(t) < o. D In Section 3.3 we shall derive results on inverse-positivity by actually constructing suitable functions z. In particular, we shall show that M is inverse-positive if (3.13) holds and the coefficients a(x) and b(x) satisfy certain boundedness conditions or sign conditions. These conditions on a(x) and b(x) may be relaxed when the following result is used, where
Bo[u](x)
=
-aou'(x) + You(x),
Theorem 3.5. satisfied. (i)
Suppose that the following assumptions (i) and (ii) are
There exists afunction zo(x) > 0
Zo E
Cl[O, 1] ('\ DiO, 1) such that
(0 ::0; x ::0; 1), Bo[zo] > 0,
(0 < x < 1),
L[zo] (x) ~ 0 Bl[zo] > o.
(ii) For each subinterval [«, P] C: (0, 1) with o: < P, there exists afunction zE Cl[oc, P] ('\ Dioc, /1) such that
z(x)
~
0
::0; x ::0; /1), Bo[z](oc) > 0,
(OC
L[z](x) > 0 Bl[z](P) >
o.
(oc < x < /1),
Then M = (L, Eo, B l) is inverse-positive. Proof Let u E R satisfy Mu(x) ~ 0 (0 ::0; x ::0; 1). For reasons of continuity, there exist numbers OCm' Pm (m = 1,2, ...) such that oc m + 1 < OCm < Pm < Pm+ 1; OCm -> 0, Pm -> 1, for m -> co ; and Bo[u + m-lzo] (oc m) ~ 0, Bl[u + m-lzO](Pm) ~ O. Moreover, we have L[u + m-lzo](x) ~ 0 (ocm < x < Pm)' If we now apply Theorem 3.2 to u + m - 1Zo instead of u and to the interval [ocm, Pm] instead of [0, 1], we conclude that u(x) + m-lzo(x) ~ 0 (ocm ::0; x ::0; Pm)' Since this is true for all m, the desired inequality u(x) ~ 0 (0 ::0; x ::0; 1) follows. D Example 3.2 has shown that conditions (3.13) are not sufficient for M to be inverse-positive. They are not necessary either, as seen in the following example. Example 3.3. Let Mu = (-u" + c(x)u, u(O), u(I». If c(x) > _n 2 + J (0 < x < l)for some J > 0, then M is inverse-positive, because for e > 0 small enough z = cos(n - e)(x - t) is a majorizing function. For c(x) == _n 2 ,
3
67
DIFFERENTIAL OPERATORS OF THE SECOND ORDER
however, the operator M is not inverse-positive, since Mcp = a sin 1tX with arbitrary a E
IR.
0
0
for cp =
Example 3.4. The operator (3.8) is inverse-positive if c(x) > - p(x) (0 < x ~ 1) for some p E Co(O, 1] such that So p(t) dt (0 < x ~ 1) exists. Under this condition, the function z(x) = exp{Jo p(t) dt} satisfies (3.10). The condition on c cannot be dropped completely. For if c(x) = -2x- 1 (0 < 2 X ~ 1), then u = _x satisfies Mu ;;:: 0, but u < o. 0 Example 3.5. The operator Mu = (-u" + u, -u'(O) + yu(O), u'(I) + yu(I)) is inverse-positive ify > -~. This is shown by using z = ex - x(l - x) with 1 + exy > 0, a > t. 0 In several of the preceding examples we proved inverse-positivity by constructing a suitable function z, and we shall continue to do this in a more systematical way in the subsequent section. The question arises, exactly under what conditions on c(x), Yo, Yl (and the other coefficients) do there exist suitable functions z. This question will be answered in Section 3.4. The property of inverse-positivity is sometimes referred to as a boundary maximum principle, the reason being that, in a special case, inverse-positivity indeed is equivalent to a boundary maximum principle in the usual sense. This special case is described by
Mu(x) = L[u](x)
0 < x < 1,
for
Mu(O) = u(O),
For this operator, we may use the corresponding space R R = Co[O, 1] n C 2 (0, 1).
Theorem 3.6.
Let there exist afunction (0
~
x ~ 1),
Zo E
=
Mu(l) = u(l). (3.15) R M , or simply
R such that
L[zo](x) ;;:: 0
(0 < x < 1). (3.16)
Then the following two properties are equivalent: (i) The operator M in (3.15) is inverse-positive. (ii) For each uER, the inequality L[u](x) ~ 0 (0 < x < 1) implies that u(x) ~ f.1zo(x) (0 ~ x ~ 1) with
f.1 = max{O, f.10},
f.10 = max{u(O)/zo(O), u(I)/zo(l)}·
When L[zo] (x) = 0(0 < x < 1),thenumberll in (ii) can be replacedby f.1 Proof. (i) =;> (ii) follows when we apply (3.1) to v = - u To derive (3.1) from (ii), apply (ii) to - u. 0
= f.10.
+ f.1zo instead of u.
For the case zo(x) == 1, property (ii) is called the weak boundary maximum principle. For the general case, (ii) is the generalized weak boundary maximum principle. For example, zo(x) == 1 can be used if (3.13) holds.
II.
68
INVERSE-POSITIVE LINEAR OPERATORS
Notice that, for the operator (3.15), condition (3.16) is equivalent to assumption (i) in Theorem 3.5. Thus the corresponding boundary maximum principle holds if assumption (ii) of Theorem 3.5 is satisfied. Example 3.6. We have L[u] := -(1 - sin nx)u" + nZu ~ 0(0 < x < 1) for u = sin ttx - 1. Since Mu = (L[u], u(O), u(l» is inverse-positive and zo(x) == 1 satisfies (3.16), Theorem 3.6 asserts that u(x) ~ max{O, u(O), u(1)} = (0 ~ x ~ 1). The considered function u attains this value only at x = 1, while u(O) < and u( 1) < 0. This shows that, in general, the constant Jl cannot be replaced by Jlo' 0
°
°
°
Problems
3.1 Suppose that M = (L, B o , B 1 ) is regular and inverse-positive. Prove that then M = (L, Eo, E 1 ) with L[u](x) = L[u] (x) + d(x)u(x), Eo[u] = Bo[u] + Pou(O), E 1[u] = B 1 [u] + Plu(1), d(x) > 0(0 < x < 1), Po> 0, Pi > is also inverse-positive. 3.2 Suppose that M = (L, B o' B 1 ) is regular and d, Po, Pi are chosen as in Problem 3.1. Show that M is not inverse-positive, if the eigenvalue problem L[ 1, ~ 1
for case (iv);
p=o
and
~
< -1
for case (v).
for case (iii),
Then, for large enough Y > 0, z(x) has all required properties.
0
The following example will explain condition (iii), and it will show again that the inequalities (3.17) alone are not sufficient for inverse-positivity. Example 3.7.
We consider two operators M = (L, B o , B 1 ) and
(L, B o , B 1 ) given by
L[u] = -u"
+ b(x)u', = u(O),
Bo[u]
M=
+ b(x)u', = u(l),
L[u] = -u" B1[u]
with b(x) = -b(x) = bo(l - 2X)-1 for x:/;!, b(!) = b(t) = 0, and b o ~ 6. The first operator M is inverse-positive, since b(x) sgn(x -
t) =
-boll - 2xl- 1 ~
°
for x :/; t, and hence condition (iii) is satisfied for Xo = t, b = O. The second operator M, however, is not inverse-positive, since u = 2(2x - 1 1 satisfies Mu ~ 0, but u(t) < 0. Consequently, condition (iii) does not hold. Here, the quotient b(x): a(x) = bo(l - 2X)-1 has the "wrong sign." For
t -
70
II.
INVERSE-POSITIVE LINEAR OPERATORS
bo = 6 the operator M is not even one-to-one, since M v with u mentioned above. 0
= 0
for v
=
u- 1
The assumptions on the coefficients a, b, c in conditions (ii) and (iii) of Theorem 3.7 can be relaxed further. Corollary 3.7a.
(iii')
a(x)
Theorem 3.7 remains true, if(iii) is replaced by
+ c(x)
> 0 (0 < x < 1), Yo > 0, Yl > 0,
and if for each subinterval [IX, f3] c (0, 1) with IX < f3 there exist a real and a real b such that b(x) sgn(x - xo) ~ ba(x) (a < x < f3).
Xo
Proof Now the inequalities in condition (i) of Theorem 3.5 hold for zo(x) == 1. Moreover, for each subinterval [a, f3] c (0, 1), there exists a
function z which satisfies the inequalities in condition (ii) of that theorem. Choose z in (3.18) with p = b and ~ = Xo. 0 As we have already seen in Example 3.3, the condition c(x) ~ 0(0 < x < 1) is not necessary for inverse-positivity. A more careful examination of Mz(x) with z as in (3.18) shows that this function z, in general, is a majorizing function also if c(x) assumes certain values < O. This fact is used in the following example. Example 3.8. Now let a, b, c E C 2[0, (0) and a(x) > 0 (0 ~ x < (0), so that L[u] (x) is defined for u E C 2(0, (0) and all x > O. Moreover, suppose that Yo> O'Yl > oand define Bl[u] (x) as in (3.14). Then there exists an I > Osuch thatforall u E ClEO, l] n C 2(0,
f)
L[u] (x) ~ 0 Bo[u] ~ 0
(0 < x
0.
(3.21)
Proof of the theorem. Let the order relation ~ in S be defined as in Proposition 3.1. The above statement follows from Theorem 1.8, if we show that assumptions (i)-(iv) of that theorem are satisfied. Property (i) follows from Proposition 3.1. If (IX) holds, then each MEA is one-to-one and its range contains the function r(x) = 1 (0 ~ x ~ 1), so that (ii) and (iii) are satisfied. For given M 0, M 1 E A, let {M(t) : ~ t ~ I} c A be a connecting family as described above. Because of the continuity of the coefficients of these operators M(t), the solutions z(t) of M(t)z(t) = rare o-continuous on [0, 1]. Thus, (iv) is also true. D
°
Example 3.9 (Continuation of Example 3.8). Let the assumptions of Example 3.8 be satisfied. For small enough I > the operator M 1 in (3.20) is inverse-positive and hence invertible. Consequently, there exists a greatest 10 with < 10 ~ 00 such that M 1 is invertible for all I < 10 , The set .41 = {M 1 : < I < lo} satisfies all assumptions of Theorem 3.9. Therefore, all M 1 E A are inverse-positive. We reformulate this result:
°°
°
3
73
DIFFERENTIAL OPERATORS OF THE SECOND ORDER
The statement (3.19) holdsfor alii E (0, 10 ) , where 10 > 0 is the smallest ofall values 1 > 0 such that the problem L[u] (x)
=0
(0
Bo[u] = 0,
< x < I),
has a nontrivial solution, or 10 =
00,
if no such 1exists.
An immediate consequence of Theorem 3.9 is
Corollary 3.9a. CE Co[0, 1], Yo
E
Let M = (a, b, c, Clo , Yo, ClI, YI) be IR, YI E IR sati~f'y
c(x) ~ c(x) > 0
(0 < x < 1),
Suppose that for 0
~
t
~
Yo ~ Yo > 0,
~
YI > O.
and
let
(3.22)
1 each boundary value problem
+ bu' + [tc + (1 - t)c]u = 0 Cl o u'(O) + [tyo + (1 - t)yo]u(O) = 0, ClIU'(1) + [tYI + (1 - t)YI]u(1) = 0 -au"
-
YI
regular
(0 < x < 1),
has only the trivial solution u = o. Then the operator M is inierse-positioe. While the above theorem yields sufficient conditions for operators M to be inverse-positive, the following theorem characterizes the set of all (regular) inverse-positive differential operators M = (L, B o , B 1 ) . Theorem 3.10. Let % denote the largest piecewise linearly connected set of invertible regular operators M which contains all regular operators M with c(x) > 0 (0 < x < 1), Yo > 0, Y1 > O. Then a regular operator M is inverse-positive if and only if ME %.
Proof To justify the above definition of %, we observe that the set of all operators M with property (3.21) is convex. According to Theorem 3.9, each M E % is inverse-positive. Thus, we have only to show that each regular inverse-positive operator M = (a, b, c, Cl o , Yo, Clio YI) belongs to %. For given M consider a regular operator Nt = (a, b, C, Cl o , Yo, ClI, }II) such that (3.22) is satisfied and define
vii = {M(t) = (1 - t)M
+ tNt: 0
~ t ~ 1}.
(3.23)
For each M(t) the solution z E R of Mz(x) = 1 (0 ~ x ~ 1) satisfies z ~ 0 and M(t)z = Mz + t(Nt - M)z ~ Mz 0, so that all these operators M(t) are inverse-positive. Since Nt belongs to X and M is linearly connected with Nt by the family vii, the operator M belongs also to %. Otherwise, X would not be the largest set as described in the theorem. 0
>
II.
74
INVERSE-POSITIVE LINEAR OPERATORS
Often one may not want to consider all regular operators M = (L, B o , B I), but only a certain subset. The following corollary is proved in the same way as the above theorem. Corollary 3.10a. Suppose that .,I{0 is a set of regular operators M = (L, B o, B I) with the following two properties.
The set of all M E.,I{0 with (3.21) is piecewise linearly connected. For any M = (a, b, c, 1X 0, Yo, lXI' YI) E.,I{0' there exists an Nt = (a, b, C, Yo, lXI' YI) E.,I{0 with (3.22) such that the set (3.23) belongs to .,I{o-
1X 0,
Under these conditions Theorem 3.10 remains true if the term "regular operator Mil is replaced anywhere by "operator ME.,I{ 0'" Example 3.10. Consider the set .,I{0 of all operators Mu = ( _u" + cu, u(O), u'(l) + yu(I» with c E IR and Y E IR. Each M E.,I{0 is described by a point in the (c, y)-plane, M = M(c,y). The operator M(c, y) is one-to-one if and only if (c, y) does not belong to one ofthe infinitely many eigencurves r l' r 2, . . . of the two-parameter eigenvalue problem M(e,y)u = o. The eigencurve r 1 , which passes through (c, y) = (0, -1) and (c, y) = (-n 2 /4, 0) and approaches c = _n 2 for y -+ + 00, is the boundary of an unbounded domairr O in the (c, y)-plane such that M(c.y) is invertible for all (c, y) E n. In particular (1, 1) E n. Thus, according to Corollary 3.10a, the set of all inverse-positive M E.,I{0 is equivalent to the set of all M(c, y) with (c, y) E n. 0
The next theorem characterizes inverse-positive regular operators M = (L, B o , B I) by positive definiteness. This characterization is closely related to
the theory of eigenvalue problems. (See Problem 3.11 for a different proof of the main part of the theorem below.) Similar results will be derived by a different approach in Section 111.3. To each regular M = (L, B o , B 1 ) and each 9 E CoCO, 1] with g(x) > (0 ~ x ~ 1), we adjoin an operator A: D -+ CoCO, 1] which is defined by
°
Au(x) = g-l(x)L[u](x),
D = {u E C 2[0, 1] : Bo[u] = BI[u] = O}. (3.24)
Obviously, Au(x) = k-1(x)Ls[u](x) with k = qpa:? and L; and p defined in (3.6), (3.7). This operator A is symmetric with respect to the inner product Cu, v) =
f
k(x)u(x)v(x) dx,
(3.25)
that is, 0,
for for
CX I
= 0
CX I
CX o
(3.26)
> O.
For a regular operator M the following three properties
M is inverse-positive, M is positive definite, lEu] > Ofor all u E Cz[O, 1] which do not vanish identically and satisfy u(O) = 0 ifcxo = 0, u(l) = 0 ,= O.
(i) (ii) (iii)
u«,
Proof It is sufficient to consider only formally symmetric operators M = (L, B o , B I ) , where a = p, b = -a'. We shall only carry out the proof for the case CX o > 0, CX I = O. All other cases are treated similarly. For the considered case we shall even assume CX o = 1, CX I = 0,1'1 = 1, without loss of generality. Obviously, (iii) => (ii). Now suppose that (ii) holds, so that lEu] ~ K 0 and all UE D. If (iii) is not true, then lEv] < K(V, v) for some v E Cz[O, 1] with v(l) = O. One can modify this function v in a sufficiently small neighborhood of x = 0 such that the modified function U belongs to D and still lEu] < K (i), we apply Theorem 3.9. Let.#0 denote the set of all positivedefiniteregularM = (L,Bo,BI)withcx o = I,CXI = 0,1'1 = l.Thisset is convex, since lEu] depends linearly on p = a, q = c, and Yo. Each positive definite operator is invertible. Moreover, M E .#0 is inverse-positive for Yo > 0, c(x) > 0 (0 ~ x ~ 1). Thus, all ME.#0 are inverse-positive, by Theorem 3.9. To prove (i) => (ii), let M = (a, b, c, 1, 1'0,0, 1) be inverse-positive. Moreover, consider an operator M = (a, b, C, 1, Yo, 0,1) with c(x) ~ c(x) > 0 (0 ~ x ~ 1) and Yo ~ Yo > 0, so that ME.# o- Then all operators M(t) which belong to .# in (3.23) are inverse-positive, as shown in the proof of Theorem 3.10. Let ll[u] be the functional (3.26) which corresponds to M(t). Then J1(t):= min{Jl[u]/ 0, we have either jl(t) > 0(0 ::::; x ::::; 1), or jl(t) = for some t E [0, 1). The second case cannot occur, because jl(t) = implies that M(t) is not invertible. Thus, jl(O) > 0, so that M is positive definite. 0
°
3.5
Inverse-positivity under constraints
We take up the question under what conditions the inverse-positivity statement (3.1) holds ifthe functions u are subjected to certain constraints. Here we shall only consider the case that all functions u satisfy the boundary conditions Bo[u] = Bl[u] = 0. Let M be a given operator (3.2), but assume now that
a, b, C E
1]
~[O,
and
a(x) ~ 0
(0::::; x ::::; 1).
(We still require lXo ~ 0, 1X 1 ~ 0; now these conditions do not mean any loss of generality.) Define an operator M: R -. S = ~[O, 1] by Mu(x)
=
(0 ::::; x ::::; 1),
L[u](x)
where R denotes the set ofall functions u E D 2[0, 1] which satisfy the boundary conditions Bo[u] = Bl[u] = O. This operator M is inverse-positive if and only if for all uED 2 [ 0, 1] L[u] ~
Bo[u]
0,
=
Bl[u]
=
°=
(3.27)
u ~ o.
We shall see in Section 3.8 that a regular operator M is inverse-positive if and only if the corresponding operator M is inverse-positive. To prove this fact, the following results will be needed. Lemma 3.12. The operator M: R -. S is pre-inverse-positive with respect to the strong order in S defined by U '> 0 U(x) > 0(0 ::::; x ::::; 1).
Proof
In R, the relation u >u(x) > u'(O) >
u'(1)
u(1)
>
°
°
if if
lXo
1X 1
> 0, > 0.
(3.28)
°
for some u E R, one of these inequalities is wrong.
°
In each case one obtains u(O = u'(O = for some ~ E [0,1]. For example, if lXo > and u(O) = 0, we have u'(O) = -lXolBo[u] = 0. If u(~) = u'(~) = and u ~ 0, then also u"(O so that Mu ~ o. 0
~
0. These relations together imply L[u] (~)
This lemma and the monotonicity theorem yield the following result.
::::; 0,
3
77
DIFFERENTIAL OPERATORS OF THE SECOND ORDER
Theorem 3.13.
If there exists afunction z ;:::
L[zJ(x) > 0
0
in D2[0, IJ such that
(0 ,,; x ,,; 1),
(3.29)
then M is inverse-positive, i.e., (3.27) is true. In general, it will be more difficult to construct a function z as in (3.29), than to construct one which satisfies Mz(x) > 0 (0"; x s; 1). However, the conditions (3.29) yield some new possibilities. Example 3.11.
Suppose that the eigenvalue problem
L[(pJ(x) = A 0 (0 ,,; x ,,; 1) and z(x) = g(x) for all x E [0, IJ, except in sufficiently small neighborhoods ofthe "jump points" of g'(x). This function z satisfies (3.29) if x is large enough. D Problems
3.7 Let R denote the set of all u E Co[O, IJ r, D2(0, IJ such that Bt[uJ = 0 and u'(O) exists when (Xo > O. Define an operator M: R ~ !R[O, IJ by Mu(x) = L[uJ(x) for 0 < x ,,; 1, Mu(O) = Bo[uJ; where now a, b, c E !R[O, IJ and a(x) ;::: 0 (0 < x ,,; 1). Prove the following statement. The operator M is inverse-positive if there exists a function z E R such that z ;::: 0 and M z(x) > 0 (0 ,,; x ,,; 1). 3.8 The monotonicity theorem can also be applied to treat periodic boundary conditions. Let R be the set of all u E D2[0, IJ such that u(O) = u(1). Define an operator M: R ~ S = !R[O, IJ by Mu(x) = L[uJ(x) for 0,,; x < 1, Mu(1) = -u'(O) + u'(1), where now a, b, c E !R[O, IJ and a(x) ;::: 0 (0"; x ,,; 1). Prove
78
II.
INVERSE-POSITIVE LINEAR OPERATORS
the following statement. The operator M is inverse-positive if there exists a function zER such that z ~ 0, L[z] (x) > 0 (0 ~ x < 1) and -z'(O) + z'(L) ~ O. 3.6
Stronger results on positivity in the interior
Most operators M = (L, B o, B 1) which are inverse-positive have even stronger monotonicity properties. Such properties will be formulated in the following theorem, from which, for example, strong boundary maximum principles can be derived. These investigations are continued in the following section. Theorem 3.14. (A) TheoperatorM = (L,B o, B 1 ) is inverse-positive if the following two conditions are satisfied: I.
For each
u(x) L[u] (x)
~
~
~ E
(0, 1) and all u E R
0 (0 < x < 1), 0 (0 < x < 1)
=
u(~)
O}
=>
u(x) = 0
(0 < x < 1). (3.30)
II.
z
~
There exists a function 0,
L[z](x)
~
Z E
R such that
(0 < x < 1),
0
Bo[z] > 0,
(B) If M is inverse-positive and condition I holds, then for all u E R the inequality Mu ~ 0 implies that u = 0 or u(x) > 0 (0 < x < 1). Proof This result may either be derived immediately from the monotonicity theorem, or from Theorem 1.7, which itself follows from the monotonicity theorem. When the latter theorem is used for the proof, one defines thestrongorderinSby U > 0-= U(x) ~ 0(0 < x < 1), U(O) > 0, U(l) > O. To derive the result from Theorem 1.7, let
2 = {tx:O 2
1
~
x
= {to,td,
~
I}
with
txu = u(x),
2 z = {tx:O < x < I}.
Then condition Ia of Theorem 1.7 is satisfied, since (3.9) holds for ~ = 0 and ~ = 1. Moreover, it is immediate that the assumptions of Theorem 3.14 are sufficient for assumptions Ib and II of Theorem 1.7 and that the statement of the above theorem follows. 0 Corollary 3.14a. conditions hold: (i)
Condition I of Theorem 3.14 is satisfied
a(x) > 0 (0 < x < 1),
if the following
3
79
DIFFERENTIAL OPERATORS OF THE SECOND ORDER
(ii) for each interval [IX, /3] c (0, 1), there exist constants b, c such that Ib(x) I ::::: ba(x),!c(x)l::::: ca(x)(ex::::: x::::: fJ).
Proof u
~ 0,
Suppose that condition I is not satisfied, so that u(~)
= 0,
L[u] (x) ~
°
(0 < x < 1),
u(1]) >
° (3.31)
for someu E R and some ~,1] E (0, 1). First, let ~ < 1]. Without loss of generality, we may also assume that 1] = ~ + s with some arbitrarily small e > O. We choose s so small that a function ZEC2[~,1]] with z(x) > 0, L[z](x) ~ 0 (~ ::::: x ::::: 1]) exists. Such a function can be constructed as in the proof of Theorem 3.8. After having fixed s, we find a function tp E C2[~' 1]] such that cp(~) = 0, cp'(~) > 0, cp(x) > 0 (~ < x ::::: 1]), L[cp](x) < 0 (~ < x < 1]). Because of the preceding assumptions, cp(x) = exp K(X - ~) - 1 satisfies these relations for large enough K. Finally, for given cp a sufficiently small fl > 0 is chosen such that v = u - J.1cp satisfies v(~) = 0, v(1]) > 0, L[v] (x) > 0 (~ < x < 1]). These relations yield v(x) > 0 (~ < x < 1]), according to Theorem 3.4, applied to the interval [~, 1]] in place of [0, 1]. The properties of v together yield v'(~) ~ 0, so that u'(~)
= v'(~)
+ J.1cp'(~)
~ J.1cp'(~)
> 0.
(3.32)
However, the inequality u'(~) > 0 cannot hold, since u attains its minimum at ~. Therefore, (3.31) cannot be true for 1] > ~. For 1] < ~, one proceeds similarly, using ljJ(x) = exp K(~ - x) - 1 instead of cp. 0 Remark. Obviously, the assumptions ofthis corollary can be generalized. Instead of requiring the preceding conditions on the coefficients a, b, c, one may assume that functions z, cp, ljJ as used in the proof exist.
Under the stronger assumptions that R = Cl[O, 1] n C 2(0, 1);
a(x) >
°
a,b,cEC o[O,l];
(0 < x < 1)
(3.33)
we shall provide a second proof. The ideas involved will later be used in other connections. Condition I of Theorem 3.14 holds if and only if for given u, ~, and 1] there exists a functional y'I/ = U(O with (E (0, 1) such that the relations u ~ 0, u(~) = 0, u(1]) > 0 together imply I Mu < O. We could not use this fact in the above proof since, in general, ( is unknown. In the following proof, an integral functional I ~ 0 will actually be constructed such that the above implication holds and IMu < contradicts L[u] (x) ~ 0(0 < x < 1).
°
80
II.
INVERSE-POSITIVE LINEAR OPERATORS
Second proof (under the assumption (3.33)). Suppose again (3.31) holds. We shall present two possibilities for choosing a suitable functional j/, First, let
for V
E
IV
= fljl(X)E(X)V(X) dx
when
~
< 11,
IV
= frp(X)E(X)V(X) dx
when
~
> 11,
M R, where
E = a-1p,
and K > IMu
for ~ - )1 :::; x < ~, or u(x) > for ~ < x :::; ~ + )1, where )1 > is sufficiently small. Then let (a, P) = (~ - s, ~ + s) with arbitrarily small s E (0, )1] and define for V E MR
°
°
IV
fp
= ~ g(x)E(x)V(x) dx
°
.
wlthg(x) =
°
{
ItMu < 0,
and I = It depends only on /: (Compare this property with
(1.3), (1.4), and (3.9).)
0
3
81
DIFFERENTIAL OPERATORS OF THE SECOND ORDER
The stronger monotonicity statements of Theorem 3.14 yield also stronger maximum principles, as stated in the following theorem. This theorem is proved in essentially the same way as Theorem 3.6. Property (i) of the following theorem is equivalent to condition I in Theorem 3.14; Property (ii) is called the (generalized) strong maximum principle. Theorem 3.15. Let R = CoCO, IJ r-. Cz(O, 1) and suppose there exists a junction z E R such that z(x) > 0 (0 ~ x ~ I), L[zJ(x) ~ 0 (0 < x < 1).
Then the followinq two properties are equivalent: (i) For all u E R, the inequalities u ~ 0 and L[uJ (x) ~ 0 (0 < x < 1) imply that either u = 0 or u(x) > 0 (0 < x < 1). (ii) For all u E R, the inequality L[uJ (x) ~ 0 (0 < x < 1) implies that either u = IlZ or u(x) < Ilz(x) (0 < x < 1), where 11 = max{O, Ilo}, 110 = max{u(O)/z(O), u(1)/z(l)}.
When L[zJ(x) = 0 (0 < x < 1), the number 11 in (ii) can be replaced by 11
=
110'
Strictly inverse-positive operators
3.7
As we have seen in the preceding section, the inequality Mu ~ 0 often implies the positivity statement u(x) > 0 (0 < x < 1). Now we shall derive even stronger positivity results which also describe the behavior of u at the endpoints and 1.These results include statements on strict inverse-positivity as defined in Section 1.1. We shall have to require a stronger type of preinverse-positivity, while at the same time the assumptions on the occurring function z can be weakened. This is important for several applications in the following sections. Throughout this section, R will denote the set Dz[O, 1Jof twice differentiable functions on [0, 1J, if not otherwise specified.
°
Theorem 3.16. (A) The operator M = (L, B o , Bd is inverse-positive the following two conditions are satisfied.
I.
For each
~ E
[0, IJ and all u E R
O}
u ~ 0, u(~) = u'(~) = L[uJ(x) ~ (0 < x < 1)
°
II.
=
u = o.
if
(3.35)
There exists a function z E R such that
z
~ 0,
Mz
z(O) > 0
Mz(() > 0
~ 0,
(f
c(o
= 0,
(B) IfM is inverse-positive and property (P) holds:
for some
z(l) > 0
if
if M satisfies condition
(E [0, IJ; C(!
= O.
(3.36)
I, then the following
82
II.
(P) For all u E R, Mu inequalities are satisfied: u(x) > u(o) > u(1) >
° °
~ 0
INVERSE-POSITIVE LINEAR OPERATORS
(0 < x < 1),
if if
°
Cto Ct1
=
implies that either u
> 0, > 0,
u'(O) >
u'(1)
0, Ct1 > 0. We choose 2 1 = {t~} and 2 z = 2. For t = t~ the implication (1.3) holds with IV = V«), since (3.9) is true for ~ = (. In order to prove (1.4) for each t ~ E 2, let u E R satisfy u > 0, Mu ~ 0, and u(~) = 0. Then u'(~) = holds also. For example, we have 0::::; u'(O) = - Ct 1 Bo[U] ::::; for ~ = 0. Thus condition I implies that u > 0 is false. This verifies condition Ib of Theorem 1.7. (b) Let cc, = Ct1 = 0. Then ro1Bo[z] = z(O) > and rl1Bl[Z] = z(1) > 0, so that all assumptions of Theorem 3.14 are satisfied. That means, we can proceed as in the proof of that theorem; condition I of Theorem 1.7 holds for 2 1 = {t o J d ,2 z = {tx:O < x < 1}. Consequently, the inequalities Mu ~ o and u "# 0 imply u(x) > (0 < x < 1). Moreover, if u "# 0 and u(O) = (u(1) = 0), then u'(O) > (u'(1) < 0), by condition I of this theorem. (c) For the remaining cases we proceed similarly and choose
°
°
o
°
°°
2 2
1 1
= {to}, = {Cd,
2 z = {tx:o < x::::; 1} 2 z = {t x : 0::::; x < I}
Corollary 3.16a. a(x) > 0,
°
Cto
= 0,
Ct1
> 0,
if
Cto
> 0,
Ct1
= 0. 0
Condition I of Theorem 3.16 is satisfied
Ib(x)1 ::::; ba(x),
for some constants
if
b~
0,
c~
Ic(x) I ::::;
ca(x)
if
(0 < x < 1)
(3.38)
0.
Proof Let (3.31) hold for some u E R and some ~, IJ E [0, 1]. First, it is shown as in the proof of Corollary 3.14a that ~ cannot belong to (0, 1). For ~ = it suffices to establish u'(O > 0. This inequality is derived in the same way as (3.32) in the proof of Corollary 3.14a. For ~ = lone proceeds similarly.
°
o
If M is regular and R = Cz[0, 1], the result can also be derived as in the "second proof" of Corollary 3.14a. Now the occurring functionals are also defined for ( = and ( = 1, respectively.
°
3
83
DIFFERENTIAL OPERATORS OF THE SECOND ORDER
(L,
Example 3.13. Property (P) of Theorem 3.16 holdsfor each operator M e; B 1 ) which satisfies (3.38) and
c(x) ~ 0 (0 < x < 1), Yo ~ 0, c«() + Yo + Yl > 0 for some
'E
Notice that z(x)
=
Yl ~ 0, (0, 1).
== 1 satisfies condition II of Theorem 3.16. 0
Under suitable assumptions, condition (3.36) in Theorem 3.16 may be dropped. Also, the statement of the theorem can be generalized by including the possibility that M z = o. For simplicity, the following result is only formulated for regular operators. (See, however, the subsequent corollary.) Theorem 3.17. Let M = (L, B o , B 1 ) be a regular operator such that z > Mz ~ 0 for some z E R = Dz[O, 1]. Under these assumptions, the following statements hold:
0,
(i) If Mz > 0, then M has property (P) in Theorem 3.16. (ii) If Mz = 0, then each solution u E R of Mu = 0 is a multiple of z, and z satisfies the inequalities (3.37) in property (P). Proof We shall derive the above result from Theorem 1.9, where again, we have to distinguish several cases. Observe that condition I ofTheorem 3.16 is satisfied, according to Corollary 3.16a.
(a) Let (;(0 > 0, (;(1 > O. We have shown in the proof of Theorem 3.16 (part a) that (1.4) holds for all I E 2 = {Ix: 0 ~ x ~ I}. However, this property is equivalent to (1.7) in Theorem 1.9, so that the above statements follow from that theorem, since u >- 0 ¢> u(x) > 0 (0 ~ x ~ 1). (b) Let (;(0 = (;(1 = 0 and z(O) > 0, z(1) > O. Then the above statement (i) follows from Theorem 3.16. Statement (ii) is also true, since Mz =I- o. (c) Let (;(0 = (;(1 = 0 and z(O) = z(1) = O. Here we shall apply Theorem 1.9 to the corresponding operator M: R ~ S in Section 3.5, where R = {u E Dz[O, 1] : u(O) = u(1) = O}. The positive cone K = {u E R: u ~ o} in R is completely described by 2 = {lx:O < x < I} u {t(O),t(1)} with t(O)u = u'(O), t(1)u = -u'(1). Now u >- 0 is defined by (3.28). Since condition I of Theorem 3.16 is satisfied, the implication (1.4) holds for each IE 2, with M in place of M and R in place of R. The element M z vanishes if and only if Mz vanishes. Thus, statement (ii) follows immediately from Theorem 1.9(ii'). Moreover, if M z > 0, then M z > 0, so that Mu ~ 0 => u ~ o. In this case, let ~ 0 be the' solution in R of L[ 0 (x) = 1 (0 ~ x ~ 1). Then the function' = + e satisfies M'(x) > 0 (0 ~ x ~ 1) if e > 0 is small enough. Consequently, M is inverse-positive and statement (i) follows from Theorem 3.16B.
e
e
84
I I.
INVERSE-POSITIVE LINEAR OPERATORS
(d) All remaining cases are treated by properly combining the methods which have been used above. For example, if lXo > 0, 1X 1 = 0, z(l) = 0, one considers an operator M as in Problem 3.7. D The above proof shows that the conditions on M can be relaxed. Corollary 3.17a. The statements of Theorem 3.17 remain true if the assumption that M be regular is replaced by the following two conditions: (i)
There exist constants a(x)
~
a>
a, b, c with
Ib(x)1 ~
0,
b,
Ic(x) I ~ c
(0 < x < 1).
(ii) If M is invertible, there exists afunction , E D 2[0, 1] with L[,] (x) ~ 1 (0 ~ x ~ 1) and '(0) = 0 e(l)=O
if if
lXo = 0, 1X1=0,
Bo[e] ~ 1
if
B1U]~1
if
> 0, 1X1>0.
lXo
When lXo > 0 and 1X 1 > 0, property (P) means that the operator M = (L, B o , B 1) is strictly inverse-positive. If one of the numbers lXo , 1X 1 equals zero, then a regular operator M: R -> S cannot be strictly inverse-positive. For example, if M is inverse-positive and.cq, = 0, then the solution of L[u] (x) = 1 (0 < x < 1), u(O) = 0, B 1[u] = 1 is not a strictly positive element of R. However, property (P) includes the statement that the "restricted" operator M: R -> S defined in Section 3.5 is strictly inverse-positive, whether or not the numbers lXo, IX1 vanish. Property I in Theorem 3.16 can be considered as a certain type of preinverse-positivity. Again, this pre-inverse-positivity is equivalent to a boundary maximum principle as formulated in the following theorem. This theorem is proved similarly as Theorem 3.6. Theorem 3.18. If there exists afunction z E R with z(x) > 0, L[z] (x) ~ 0 (0 < x < 1), then the following properties are equivalent: (i) Condition I of Theorem 3.16 holds. (ii) For all u E R the inequality L[u](x) ~ 0 (0 < x < 1) implies that either u = uz or u(x) < flz(x) (0 < x < 1) and u'(O) < flz'(O)
if
u(O) = flz(O),
u'(L) > flz'(l)
if
u(l)
= flz(I),
where fl = max{O, flo}, flo = max{u(O)jz(O), u(l)jz(I)}. WhenL[z](x)
= 0(0 < x < l)thenumberflin(ii)canbereplacedbYfl = flo.
Problems
3.9 Modify the assumptions of Corollary 3.17a such that the implication in condition I of Theorem 3.16 holds for ( E [0, 1).
3
85
DIFFERENTIAL OPERATORS OF THE SECOND ORDER
3.10 What statement can be proved if the implication in condition I of Theorem 3.16 holds for all ~ which belong to certain intervals contained in [0, 1] and if a suitable condition II is satisfied? What is a suitable condition II?
3.8
Equivalence statements
Theorem 3.17 of the preceding section can be used to prove a series of different results for regular operators. Several of these results will be derived in the following four sections. In Section 3.5 we considered monotonicity statements for functions u which satisfy boundary conditions. The question arises whether such statements hold under less strong conditions than those which are required for inversepositivity of M. An answer to this question is given in the following theorem.
Theorem 3.19. For a regular operator M = (L, Bo, B 1 ) thefollowingfive statements are equivalent (where conditions (i)-(iv) are interpreted to hold for all u E R = ez[O, 1]). (i) (ii) (iii) (iv) (v)
°°
Mu ~ o=>u ~ 0, L[u](x) ~ 0(0 < x < 1"), Bo[u] = 0, B 1[u] = L[u] (x) = 0(0 < x < 1), Bo[u] ~ 0, B 1[u] = L[u] (x) = 0(0 < x < 1), Bo[u] = 0, B 1[u] ~ 0 M has Property (P) in Theorem 3.16.
=> =>
=>
u u u
~
0,
~
0,
~
o.
Proof Obviously, it suffices to show that (v) follows from each of the other four properties. Suppose that one of the implications (i)-(iv) holds (for all u E R) and, for the moment, call this implication (n). Then define z E R by Mz = r where rex) = 1 (0 < x < 1) if L[u] (x) ~ 0 (0 < x < 1) is assumed in the premise of (n), rex) = 0 (0 < x < 1) otherwise; reO) = 1 if Bo[u] ~ 0 is assumed in the premise of (z), reO) = otherwise; and r(l) is defined analogously. This function satisfies z ~ 0, M z > o. Thus, according to Theorem 3.17, the operator M has property (P). 0
°
Now suppose that the regular operator M = (L, B o , B 1 ) is invertible. Then since R = C z[O, 1], the range of M consists ofall V E !R[O, 1] such that Vex) = Vex) (0 < x < 1) with some VEe0[0, I]. Moreover, for such V E S, M- 1 V assumes the form M- 1 Vex) = V(O)IjJ(x) + U(I)cp(x) + .1'6 ~(x, OD(~) di; where cp, IjJ E R satisfy
L[cp](x) = 0
(0 < x < 1),
Bo[cp]
=
0,
B 1[lp]
L[IjJ] (x) =
(0 < x < 1),
B o [ljJ ]
=
1,
B 1 [1jJ] =
°
= 1;
°
86
II.
INVERSE-POSITIVE LINEAR OPERATORS
and '!I denotes the Green function
° °
~ x ~ 1; ~ 1
for for
'!I x _ _ 1 {
°
( - 00 < x < 00).
(3.39)
The results are derived from Theorem 3.17 applied to the operator Mu (L[u] (x), u(O), u(l)): C 2[0, 1] -+ !R[O, 1]. Proposition 3.20.
v> 0; Then w(1;)
0.
°
= for some
v(l)
= 0;
17 EO (0, 1).
Proof Define M as above and suppose that w ~ o. Then also Mw > 0, so that M is inverse-positive, according to Theorem 3.17. Consequently, the inequality Mv ~ 0 implies v ~ 0, which contradicts the assumption v > o.
o
3
DIFFERENTIAL OPERATORS OF THE SECOND ORDER
87
Example 3.14. Let L[u] = - u" - lOu. Then, v = sin 1!X and w = 1 8x z satisfy the required inequalities. Obviously, the above statements hold with '1
= 1 : 2)2. 0
°
Theorem 3.21 (Sturm ian oscillation theorem). Suppose that v, WE Cz(I) are linearly independent solutions of the differential equation L[u](x) = (x E 1), where I denotes any given interval and (3.39) is satisfied. Then the zeros of v and w separate each other, that is, between two consecutive zeros of v in I there exists exactly one zero of w, and vice versa.
Proof Notice first that the zeros of v or w do not have a finite point of accumulation. Let a, [3 E I denote two subsequent zeros of v and suppose that v(x) > (a < x < [3). (Otherwise, consider - v.) If w(a) = 0, there would exist constants c., Cz with let I + Iczl > such that y:= Ct v + Cz w satisfies y'(a) = 0, y(a) = 0. This would imply y(x) == on I, which is not possible, since v and w are linearly independent. Therefore, we may suppose w(a) > 0. (Otherwise, consider -w.) Then w(~) = for a ~ E (a, [3), by Proposition 3.20 applied to [a, [3] in place of [0, 1]. 0
°
°
°
°
Example 3.15. (a) The functions v = sin x and w = cos x are solutions of - u" - u = on ( - 00, 00). These functions show the" typical" oscillatory behavior explained in the above theorem. (b) The functions v = sinh x and w = cosh x are solutions of - u" + u = on ( - 00, 00). None ofthese functions has two zeros. Here, the statement of Theorem 3.21 is empty, but nevertheless true. (c) For L[u] = <xu" + u' the assumptions of Theorem 3.21 are not satisfied on [ -1, 1]. Moreover, the statement of this theorem is false, since v = 1 - X Z vanishes at x = -1 and x = 1, but w(x) == 1 has no zero. 0
°
°
The above theorem describes the relation between the oscillatory behavior of two solutions on any interval I. The question whether such solutions "oscillate" at all on a fixed interval is important for inverse-positivity. An arbitrary interval I is called an interval of nonoscillation for L, if each solution u E Cz(I) of L[u] (x) = either vanishes identically or has at most one zero in I. It is easily seen that Mu = (L[u] (x), u(O), u(1)) is invertible if [0, 1] is an interval of nonoscillation for L.
°
Theorem 3.22. The regular operator Mu = (L[u] (x), u(O), u(l)) is inverseof nonoscillation for L. positive if and only if [0, 1] is an int~rval Proof If [0, 1] is an interval ofnonoscillation, then the solution z E C Z [0, 1] of L[z] (x) = 0(0 < x < 1),z(0) = 0,z(1) = 1 satisfiesz(x) > 0 (0 < x:::;; 1). Thus, M is inverse-positive, according to Theorem 3.17.
88
II.
INVERSE-POSITIVE LINEAR OPERATORS
Suppose now that M is inverse-positive and that there exists a nontrivial solution v E Cz[O, IJ of L[vJ(x) = (0 < x < 1) which has two zeros in [0, 1]. If v(O) = 0, then v(l) -# 0, because M is invertible. Without loss of generality, we may assume v(l) > 0, so that Mv > and vex) > 0(0 < x ~ 1), by Theorem 3.17. If v(O) -# 0, we choose wE Cz[O, IJ such that L[wJ(x) = (0 < x < 1), w(O) = - v(O), w'(O) -# - v'(O). The functions v and ware linearly independent and y = v + w satisfies L[yJ(x) = (0 < x < 1), yeO) = 0. Thus, we have y(x) -# 0(0 < x ~ 1), by the arguments applied to v in the last paragraph. On the other hand, the zeros of v and w separate each other (cf. Theorem 3.21), so that ychanges its sign on (0, 1). Consequently, v cannot have two zeros. D
°
°
°
°
°
Example 3.16. Each nontrivial solution of L[uJ(x) = (0 ~ x ~ 1). We shall prove the existence of a lowest eigenvalue with a positive eigenfunction and derive bounds for this eigenvalue. Moreover, the set of all inverse-positive regular operators will be characterized in a way different from that in Section 3.4. Let R = Cz[O, IJ throughout this section. For each real A, we define an operator M A : R --+ S by
°
g(X)U(X) MA
= M - AD
with
Du(x) = ~ {
for for for
°< x < 1, x = 0, x = 1.
(3.41)
Obviously, the above eigenvalue problem can then be written as M Aq> = o.
Proposition 3.23. The eigenvalue problem (3.40) has a smallest eigenvalue Ao > - CXJ. The operator M A is inverse-positive if and only of A < Ao .
Proof Consider the set A of all A E IR such that M A is inverse-positive. We shall show that (a) A -# 0, (b) A is bounded above, (c) A = ( - 00, Ao) for
3
89
DIFFERENTIAL OPERATORS OF THE SECOND ORDER
some eigenvalue . 1. 0 . Then . 1.0 is the smallest eigenvalue, since M). is invertible for ..1.< . 1. 0 . (a) There exists a function z E C 2[0, 1] such that z(x) > (0 ~ x ~ 1), Bo[z] > 0, BI[z] > (for example, such a function z can be constructed similarly as in Example 3.12). Consequently, M "z(x) > (0 ~ x ~ 1) for A < with sufficiently large 1AI. These A belong to A (cf. Theorem 3.17). (b) Choose any c e Aand anyr e Cjj'O, 1] such that r(x) > 0(0 < x < 0, reO) = if (;(0 = 0, r(l) = if (;(1 = 0. Then define v E Cl[O, 1] to be the solution of L[v] - gv = rex) (0 < x < 1), Bo[v] = BI[v] = 0. This function satisfies the inequalities (3.37), since M I' has property (P). Therefore, we have L[ -v] - Ag(-v) = -rex) + (..1."'- ,u)g(x)v(x) > (0 < x < 1) for large enough A, so that M;.( -v) ~ o. For these A, M" cannot be inverse-positive, since - v ~ o. Consequently, A is bounded above. (c) Let AE A and suppose that z E C 2[0, 1] satisfies z ~ 0, M "z > o. Then for each ,u < A we have also M I'z > 0, so that ,u E A. Consequently, all A < . 1.0 : = sup A belong to A. Suppose now that M "0 is invertible and let ~ = {M,,: . 1.0 - /; ~ A ~ Ao} with some s > 0. This set satisfies the assumptions of Theorem 3.9, so that M"o is inverse-positive. Therefore, the solution (E C 2[0, 1] of M"o(x) = 1 (0 ~ x ~ 1) satisfies ( ~ o. Moreover, we have M,,((x) > (0 ~ x ~ 1) for all A in a sufficiently small neighborhood of . 1. 0 . Consequently, this neighborhood belongs to A, which contradicts the definition of ..1.0 . D
°
°
°
°
°
°
°
°
The given eigenvalue problem (3.40) can be written as A
0 for 0 < x < 00, and z(x) --+ 0 for x --+ 00. Therefore, we are interested in relaxing this condition. Theorem 3.26. The statements of Theorem 3.14 and Corollary 3.14a hold also for the operator M in (3.45) (when the endpoint x = 1 is replaced by x = I). Proof This result is proved in very much the same way as the theorem and corollary mentioned, except that (3.47) is needed to verify condition I of Theorem 1.2. 0
Example 3.18. Suppose that a, b, C ECoCO, I), Yo > (0 < x < I). Thenfor each u E Cl[O, I) n CiO, I)
° °
L[u](x) ~ Bo[u] ~ 0 limx~l
u(x) ~
(0 < x
or
u(x)
and a(x) > 0, c(x) ~ 0
° > ° =
For proving this statement choose p(x) == 1 and z(x) == 1. Example 3.19. (-u" limx~ro
(0 < x < I).
0
Let k > O. Thenfor each u E CoCO, 00) n Cz(O, 00),
+ Pu)(x) ~
u(O) ~ 0
(0 ~ x < I)
°
u(x) exp( -kx)
(0 < x
~
0
or
u(x) = 0 (0 ~ x < 00) u(x) > 0 (0 < x < 00).
This statement follows from Theorem 3.26, when z(x) = p(x) = exp(kx) is used. D The above theory can be generalized in several ways. For example, boundary operators B 1 other than (3.44) can be used. The only essential condition on B, is that (3.47) holds. Moreover, with essentially the same ideas one can also treat problems on open intervals with generalized boundary operators at both endpoints. The generalization is straightforward.
3
93
DIFFERENTIAL OPERATORS OF THE SECOND ORDER
3.11.2
Approximation by compact intervals
This method, too, will be explained by treating a special case. Now let (0) be the given interval. Suppose that {am}' {13m} are given sequences with am+ 1 < am < 13m < 13m + 1 (m = 1,2, ...), and am -> -00, 13m -> 00 (m -> (0). ( - 00,
Theorem 3.27. (i)
Suppose that the following conditions hold:
For each interval [am, 13m] (m = 1,2, ...) and all u E CZ[am, 13m]
(ii) There existfunctions Zm E CO[a m, 13m] n CZ(am, 13m) (m = 1,2, ...) and a function Z E IR( - 00, (0) such that
o
0, we reach the desired conclusion that u(x) :;;, 0 ( - 00 < x -cco), D This theorem, too, can be generalized in many ways. Suppose for example, that a stronger condition than (i) is assumed where u(x) :;;, 0 (am ~ x ~ 13m) is replaced by the property that either u(x) == 0 or u(x) > 0 Cam < X < 13m)' Then the inequality u(x) :;;, 0 ( - 00 < x < (0) in the statement of the theorem may be replaced by the statement that either u(x) == 0 or u(x) > 0 ( - «: < X < (0). To verify condition (i) or the stronger condition just mentioned, the results of the preceding sections may be used. For example, the following corollary can be derived with Theorem 3.14 and Corollary 3.14a. Corollary 3.27a. If a, b, c E C o( - 00, co) and a(x) > 0 (-Xl < X < (0), then condition (i) of Theorem 3.27 may be dropped. Moreover the inequalities (3.48) imply that either u(x) = 0(-00 < x < Xl)oru(x) > 0(-00 < x < (0).
94
II.
Example 3.20. -u"
+ k 2u ~ 0
Let k > O. Then for each (-00
< x
00
U
E
C 2( -
00,
(0)
either
u(x) = 0 (-oo<x 0 ( _
00
< x
0 for some z ~ 0 in R, then M is inverse-positive andfor uER, Mu '> 0 = U >- o.
96
II.
INVERSE-POSITIVE LINEAR OPERATORS
Proof Since Mu ~ 0 => PMu ~ 0 and Mu follows from Theorem 1.2 applied to PM. 0
> 0 => PMu > 0, the result
The theorem essentially describes the reduction method: Find an operator P: S ---> Y and a strong order relation in Y such that the assumptions of the theorem hold, i.e., PM is pre-inverse-positive (with respect to the strong order in Y) and a suitable z exists. In general, the main purpose of the reduction method is to show that M is pre-inverse-positive. Since both strong orders in Sand Yare involved here, reduction means not only the choice of an operator P, but also that of suitable strong orders. Example 4.1. Let the sets R, S, and Y all equal the vector space [R4 with denoting the natural order relation. However, we shall distinguish between Sand Y by defining ~
>0
U
¢>
U >
0,
for
U
E
S;
Y
>0
¢>
Y .>
0,
for
y E Y.
We have
for
3 -1 o -3 3 -1 M= 3 [ 1 -3 o 1 -3 The matrix PM satisfies assumption I of Theorem 4.1, since it is a Z-matrix. To verify assumption II, we can, for example, show that Mz > 0 for ZT = (1,3,6,10) or show that PMz >- 0 for ZT = (1,2,3,4). Theorem 4.1 then yields that M is inverse-positive and, moreover, Mu > 0 => u >- o. Consequently, M is strictly inverse-positive. Observe that the strict pre-inverse-positivity of M here follows from the weak pre-inverse-positivity of PM, since U > 0 => PU >- o. 0 If P satisfies (4.1) and if PM=M1-N
with a linear operator M 1: R then M is pre-inverse-positive, this property, too.
--->
Y and a positive linear operator N: R
(4.2) --->
Y
if M 1 is pre-inverse-positive, since then PM has
4
97
ABSTRACT THEORY (CONTINUATION)
A reduction may also be achieved through a (multiplicative and additive) splitting M = LM 1
-
Q.
(4.3)
Corollary 4.ta. Let (4.3) hold with a positive linear operator Q: R --+ S, a pre-inverse-positive linear operator M 1: R --+ Y and an inverse-positive linear operator L such that LY = S and for all y E Y, the inequalities Ly '> 0 and y ~ 0 imply y '> O. Then M is pre-inverse-positive. Proof Applying P = L -1 ~ 0 to (4.3) yields (4.2) with N = PQ ~ O. Moreover, if U = Ly '> 0, then Ly ~ 0 and hence y ~ 0, so that PU = y '> 0 by the preceding assumptions. Thus, the second implication in (4.1) holds also, so that M is pre-inverse-positive according to Theorem 4.1. D
Example4.1
(continued).
We have M = LM 1 with
All assumptions of Corollary 4.1a are satisfied. Observe that L is strictly preinverse-positive, so that the inequalities Ly > 0, Y ~ 0 imply y >- 0 (see Example 2.2). 0 A simple but sometimes quite useful application of the reduction method is described in the next two corollaries (see also Problem 4.1). Corollary 4.tb. Suppose that M = D + A - B - G with positive linear operators D, A, B, G such that D and D - B are inverse-positive, DR = S, (D - B)R = S, and A ,;( BD- 1G. Then M is weakly pre-inverse-positive. Proof Here, (4.3) holds with L = D - B, M 1 = I - D- 1G, and Q = BD- 1G - A ~ O. Observe that M 1 and L are weakly pre-inverse-positive,
due to Proposition 1.4 and Theorem 1.6(i).
0
In particular, this statement may be applied to n x n matrices, as in the next result, where ,;( denotes the natural order relation in [R". Corollary 4.tc. Let M denote a diagonally strictly positive n x n matrix and D = (mii (ji)' Suppose there exist matrices B, G with the following properties: (M - D)- = B + G, B ~ 0, G ~ 0, D - B is inverse-positive, and to each mij > 0 with i #- j an index k existsfor which mijmkk < bikg kj. Then M is weakly pre-in verse-positive.
98
II.
Example 4.2.
For 1 -1 2 -1 1 -16 1 30 -16 1 -16 30 -16 1 -16 1 30 -16 1 -1 2 -1 1
M=
0 0 0
B=
0 8
INVERSE-POSITIVE LINEAR OPERATORS
0 8
16 0 8 16 0
c= 8 0 0 0
0 1 0 1 0 8 0 0 8 0 0
the assumptions of Corollary 4.1c are satisfied. Choose k = i = 1, for example. D
8 0 8 1 0
1 0
+ 1 for elements
mi,i+2
Problems
4.1 Generalize Corollary 4.1b in the following way. Suppose that :::;1 is a strong order in S, ~ 2 a strong order in R, and that U '> 1 0 = (D - B) - 1 U 1G >20; u >20 = (D+ I)u >- o. Prove that then M is pre-inverse-positive with respect to ~ l ' 4.2 If M is inverse-positive and (4.2) holds with positive N or (4.3) holds with positive Q and inverse-positive L, then for all u E R, Mu ~ 0 implies that u ~ 0 and M lU ~ o. This property can be considered as inverse-positivity of M with respect to an order relation in R given by the two inequalities on the right-hand side of the above implication. Apply the monotonicity theorem to this type of inverse-positivity. 4.1.2
Local reduction
To prove that M is pre-inverse-positive, certain conditions have to be verified for each v E oK (i.e., v E R with v ~ 0, v >I- 0). Sometimes, a reduction for proving the pre-inverse-positivity is carried out more easily by using a family of operators P(v): S ~ Y(v) with v E oK as parameter. We assume that in each ofthe linear spaces Y = Y(v) which will occur, a linear order ~ and a further stronger linear order :::; are given, which, of course, may depend on
4
99
ABSTRACT THEORY (CONTINUATION)
the parameter v. Again, whenever a symbol ~ or < occurs in the following, it will be clear from the context to which space this relation belongs. Theorem 4.2.
r.
Suppose that the operator M has the following two properties:
To each v E R with v ~
0
and v
*
0,
a linear operator P = P(v): S
--+
Y
= Y(v) exists such that (4.1) holds and P(v)Mv :} o.
II. There exists a z ~ 0 in R such that Mz operators P(v) occurring in I. Then M is inverse-positive and Mu
> 0, or P(v)Mz > 0 for
all
> 0 = u >- o.for all u E R.
Proof Let Y be the product space Y = DVEOK Y(v) with elements {y(v)} and define y ~ 0 and y 0 componentwise, so that, for example, y 0 ¢ > y(v) 0 for all v E oK. Moreover, define P: S --+ Y by Pu = {P(v)U}. Then the preceding statements follow from Theorem 4.1 with
y=
>
>
>
D
p=P.
In particular, all spaces Y(v) may equal ~, in which case we obtain from Theorem 4.2 the following result, which essentially was already obtained in Section 1 (cf. Proposition 1.3 and Theorem 1.7). Corollary 4.2a. Suppose that for each v E R with v ~ 0 and v;} 0 a and U 0 implies positive linear functional I on S exists such that 1Mv ~ IU > 0, for all U E S. Moreover, let a function z E R exist which satisfies z ~ 0, Mz o. Then M is inverse-positive and for u E R, Mu 0 = u >- o,
°
>
4.1.3
>
>
A differential operator of the fourth order
We shall illustrate the reduction method by applying it to a particular differential operator of the fourth order. The results obtained are formulated in Propositions 4.3 and 4.4. We shall first discuss the problem in an elementary way and then describe several possibilities to carry out a reduction. The emphasis is not only on the results themselves but as well on the methods to obtain them. The ideas involved can also be used to derive results on more general fourth order operators (see Section 6.1). The boundary value problem L[u] (x)
= r(x)
(0
~ x ~
with
1), L[u](x)
u(o)
=
= ul/(O) = u(1) = ul/(l) =
uiV(x) + cu(x)
°
(4.4)
describes the bending of a beam which is attached to an elastic support; u denotes the deviation ofthe beam under the (continuous) load function r, and c is an elasticity constant. (Later we shall also consider functions c E Cora, 1].) Our question is: For which c does a load r ~ 0 always result in a deviation
II.
100
INVERSE-POSITIVE LINEAR OPERATORS
u ?: o? In other words, for which c is the following operator M inversepositive: M: R
----)0
R = {u E C4[0, 1]: u(O) = ul/(O) = u(1) = ul/(l) = O},
S, S
= CoCO, 1],
(0 ~ x ~ 1).
Mu(x) = L[u](x)
(4.5)
Before we apply our theory (to the more general case where c = c(x», we shall first discuss the preceding simple problem in a more direct way. This will help us in understanding the meaning of the assumptions in Propositions 4.3 and 4.4. If M is invertible, then the inverse is an integral operator with the Green function corresponding to M as its kernel. M is inverse-positive if and only if the Greenfunction exists and is positive (?: 0). For the above operator M the Green function ~(x, 0 can be calculated explicitly. In particular, one sees by formal calculations that ~(x, ~) = ~(x, ~, c) (exists and) satisfies ~(x,
~,
c) ?: 0
(0
~
x,
where Co = 4K4 ~ 951 with ing K > O. For example, ''''( 1_ ";<J x,
X,4K
4) _ ~ sin -
K
K
~ ~ K
1)
if and only if
_n: 4
the smallest solution of tan
cosh K - 'cos K sinh 2 h cos 2 K - cos K
K
X
2
K
< c ~ co, (4.6)
= tanh K satisfy-
+ 0 (3) X
for l'
x
----)0
O.
Since n:4 is the smallest eigenvalue of the problem u iv = AU associated with the preceding boundary conditions, the operator M is not invertible for 4 4 C = _n: , but it is invertible for all C > _n: • This explains the lower bound 4 - n for c in (4.6); the upper bound co, however, has to be explained differently. We point out that the necessary and sufficient condition in (4.6) for the inverse-positivity of the fourth order operator M differs from corresponding conditions for second order operators in an essential way (cf. Proposition 3.23). We shall describe the behavior of ~(x, ~, c) in the neighborhood of c = Co in more detail. If _n: 4 < c < co' one calculates G(x, ~, c) > 0 (0 < x, ~ < 1) and cp(x) > 0 (0 < x < 1), cp(l) = 0, cp'(1) < 0 for cp(x) = ~~(x, 0). If c varies continuously from values co, the function ~(x,~, c) changes from a function > 0 to a function which assumes also negative values. For c = Co all the preceding inequalities remain true except that now cp'(1) = O. For c in a right-hand neighborhood of co, one has cp'(1) > 0, so that ~(x, ~, c) assumes negative values near (x, ~) = (1,0) (and, analogously, near (x, ~) = (0, 1». Let us explain these facts somewhat differently without using the Green function. For each c > Co there exists a load r ~ 0 with deviation u } o. It turns out that for c only "slightly larger" than Co the load r has to be
4
101
ABSTRACT THEORY (CONTINUATION)
"concentrated near one end" of the beam in order to achieve values u(x) < 0, which then occur "near the other end." To study the "limit behavior" for c -> Co - 0, we consider the functions , 'P which for c > - n 4 are defined by L[](x)
=0
L['P](x)
=0
(0 : 0,
(0 < x < 1). (4.12)
t/J(x) > 0
For u E R one obtains by partial integration, using the boundary conditions, PMu(x) = Mlu(x) - Nu(x)
(4.13)
with M lU(X)
= -
p(x)u"(x)
+ p'(x)u'(x) + q(x)u(x) + ljJ"(O)u'(O» + cp"(I)u'(I»,
- cp(x)(t/J(O)u"'(O) + ljJ(x)(cp(1)u"'(I) p = cpt/J' - cp'ljJ,
f
Nu(x) = -
For each u
~ 0
f
cp(x)L[ljJ] (t)u(t) dt -
in R we have Nu(x)
L[cp] (x)
q = cp",t/J - cpt/J""
~
~
0 (0
~
x
ljJ(x)L[cp] (t)u(t) dt.
L[ljJ](x) ~ 0
0,
1),provided the relations
~
(4.14)
(0 ~ x ~ 1)
hold, which will be assumed in the following. In order to apply the results of Section 4.1.1, we have to determine a suitable space Y and strong order relations in Sand Y. Let Y = {y U
'>
0
¢>
U > 0,
E
ClEO, 1]: y(O) = y(l) = OJ,
for
U E S;
y
'> 0
¢>
Y >- 0,
for
y E Y, (4.15)
i.e., y '> 0 ¢> y(x) > 0 (0 < x < 1), y'(0) > 0, y'(I) < o. Then M .R c Y, PS c Y, and (4.1) hold if cp(1)
= 0,
ljJ(O) = 0
(4.16)
ljJ'(O) > 0,
(4.17)
and cp'(l) < 0,
besides the conditions already imposed on cp, ljJ.
4
ABSTRACT THEORY (CONTINUATION)
103
In order to prove that M: R ~ S is (strictly) pre-inverse-positive, it suffices to show that M 1: R ~ Y is (weakly) pre-inverse-positive. M 1 has this property if p = (M lU)'{O) > O. The case ¢ = 1 is treated similarly. 0 Summarizing the requirements on
_n 4 (0 ~ x ~ 1), the function z = sin zx satisfies the required assumptions. Thus, it remains to choose suitable 0 => (j) >- 0 can be derived for (j) in (4.26), if (M, N) belongs to the class Z(C). For such operators M, N the results of Section 1.5 on noninvertible operators can be applied to M + AN. Proposition 4.11. Suppose that (M, N) in R. Then for each A E IR and all u E R
u>
>
0,
(M
E
+ AN)u ;" 0
Z(C) and Nu > =>
U
0
for all u >
0
>- o.
Proof. The inequalities u > 0 and (M + AN)u ;" 0 imply (M 0, for all A' > A. Thus, u >- 0 follows, since (M, N) E Z( C). D
+ A' N)u
Theorem 4.12. Suppose that (M,N)EZ(C), Nu > o for all u > 0 in R, K is finite, and (M + KN)(j) = 0 for some tp > o. Then K is a simple eigenvalue of problem (4.23) and (j) >- o. Moreover, K = min {A E IR : (M + AN)z ;" 0 for some z >- o} K = max{A E IR: (M + AN)z ::::; 0 for some z >- o}. Proof. We apply Theorem 1.9 to M + KN (in place of M). According to Proposition 4.11, assumption I is satisfied. Moreover, assumption II holds with z = tp, Thus, K is simple and (j) >- o. The preceding formulas for K then follow from Corollary 4.8a. D Finally, we shall describe certain sufficient conditions for (M, N) being a Z(W)-pair. These conditions are particularly useful to prove that (M, N) E Z. We assume that !f is a given set of linear functionals t > 0 on R which describes the cone K completely (see Section 1.1.3) and that :IF is
4
109
ABSTRACT THEORY (CONTINUATION)
a nonempty set of linear functionals U'>o
Proposition 4.13a. for all u E R
{
u~o}
tu = 0 ~
~ 0
o »»
IU> 0
If to each
then (4.22) holds for all it E
I
t E fil =>
on S such that for all
I
there exists an
E
:F.
(4.27)
I = It E:F such
/ MU ~ 0 { INu = 0,
that
(4.28)
and u E R.
The proof is obvious. Formula (4.27) may be used to define the strong order ~ by choosing = It (t E fil) which occur in (4.28). Let us consider a special case, where it is convenient to redefine some terms. Suppose that (4.21) holds and denote by fil a set of linear functionals t > 0 on S, but use the letter fil, for simplicity, also for the set of the restrictions of these functionals to R. Then, as before, assume that fil describes the cone in R completely. For 1= t property (4.28) is here equivalent to :F to be the set of alII
u
~ 0,
tu
=0
=>
tMu
~
O.
(4.29)
A linear operator M such that this implication holds for all u E R and all fil will be called fil-quasi-antitone. (Compare the analogous definition for nonlinear operators in Section IV.2.1.3.) For such operators we obtain the following statement under the above assumptions.
t
E
Proposition 4.13b. An fil-quasi-antitone operator M is a Z(W)-operator, where the cone W corresponds to the relation U >. 0 defined by U ~ 0 and /U > 0 for alll E fil. If; in particular, the set fil also describes the cone in S completely, then M is a Z-operator. 4.2.2
Quasi-antitone vector-valued differential operators
Many results of Section 3 can be carried over to certain vector-valued differential operators. (See, for example, Section IV.2.3.) Here we shall apply the theory of the preceding section to such operators under the condition that they have sufficiently smooth coefficients. Let R = C~[O, 1] and S be the space of all U E ~n[o, 1] such that the restriction of U to (0, 1) can be extended to a function in q[O, 1]. In both spaces, ~ shall denote the natural order relation. Then U E S is strictly positive if and only if inf{Uj(x): 0 ~ x ~ 1; i = 1, .... n} > O.
II.
110
We consider operators
ME
Mu(x) =
{
INVERSE-POSITIVE LINEAR OPERATORS
L[R, SJ, N
L [ UJ(X) Bo[uJ Bl[uJ
g(X)U(X) Nu(x) = gOu(O) { glu(1)
E
L[S, SJ given by
for 0<x 0 ¢ > V >- o. Proposition 4.14.
Each pair (M, N) of the above kind is a Z-pair.
Proof. Using the methods in the proof of Proposition 3.1, one verifies that (4.28) holds for tu = ui(~)' IV = Vj(~), ~ E [0, 1J,j = 1,2, ... , n. D
4
111
ABSTRACT THEORY (CONTINUATION)
If the operator M + AN is one-to-one for some A, then the inverse (M + AN)-l is given by a vector-valued integral operator with the Green matrix as its kernel. One shows that this inverse is defined on all of Sand that (M + AN) - 1N E L(S, S) is a compact operator with spectral radius > 0 (the order topology being given by [u] = sup{lulx)I:O;:( x;:( 1; i = 1,2, ... , n}). Hence, by Proposition 4.7, K = K(M, N) is the largest (real) eigenvalue of the problem
L[ep](x) Bo[ep]
+ Ag(X)ep =
+ Agoep =
0,
0
Bl[ep]
(0 < x < 1), (4.32)
+ Aglep = o.
Moreover, for each A which is not an eigenvalue, the range S of M + AN contains a strictly positive element. Thus, we can apply many results of the preceding section. For example, the five properties in Theorem 4.8 are equivalent for the differential operators M, N and V 0 ¢> V >- o. Moreover, the estimates in Corollary 4.8a can be applied, and (4.25) holds. In particular, we obtain
>
The differential operator M is inverse-positive if and only eigenvalue A of the, eigenvalue problem (4.32) satisfies A < O.
Theorem 4.15.
if each (real)
The following simple example will explain the theory.
Example 4.3. Let n = 2 and L[u](x) = -u"(x),
Y= (_ g(x) = 12 ,
~
u
-(1)
Bo[u] = yu(O), with
Bl[u] = yu(l), (4.33)
0 > 0, with s
~
O.
(a) If 0 < 1, we may choose I: = 0 (observe (4.31)). For this case, the boundary conditions of the eigenvalue problem (4.32) do not depend on A, and the eigenvalues are Av = - v2n 2 < 0 (v = 1,2, ...). Thus, the corresponding operator M is inverse-positive. (b) For arbitrary 0 > 0, we may choose G = 1. Then the numbers Av are again eigenvalues of the corresponding problem (4.32). However, A = - 1 - 0 and A = - 1 + 0 are also eigenvalues, so that the largest eigenvalue is K = -1 + o. Thus, the operator M given by (4.30), (4.33) is inverse-positive if and only if 0 < 1. D Using the results of Sections 3.6 and 3.7, one can also derive sufficient conditions such that (M, N) E Z(W) for W different from C s ' For example,
112
II.
INVERSE-POSITIVE LINEAR OPERATORS
W may be given by
U
'> 0
{U;(~:
~
U;(l) > 0
if if
a? = 0 !Xl = 0
(i (i
= 1,2, = 1,2,
,n)
(4.34)
, n).
In the following Tesult we restrict ourselves to the special case in which the cone W so defined equals C.
Proposition 4.16.
Under the assumption
a? > 0,
al > 0
the operator pair (M, N) belongs to Z(C), is irreducible.
(i
= 1,2, ... , n)
if the matrix
yO
(4.35)
+
yl
+ S6 c(x) dx
Proof'. We verify (4.22) for the case U '> 0 U > o. Let u E R and A > 0 satisfy u ~ 0 and (M + AN)u > o. Then we have also (Nt + AN)u > 0 where Nt is obtained from M by replacing all off-diagonal elements of c(x), yO, and yl by O. Since the ith component of (Nt + AN)u depends only on Ui' a linear operator M, on C 2 [0, 1] is defined by MiUi = «Nt + AN)u)i' Obviously, MiUi(X) ~ 0 (0 ~ x ~ 1). When we apply Theorem 3.16 and Corollary 3.16a to this operator M, for sufficiently large A, we see that either Ui(X) = 0(0 ~ x ~ 1)oru;(x) > 0(0 ~ x ~ 1). Because of (Nt + AN)u > 0, not all components Ui vanish identically. If one of them vanishes identically, then there exist indices j and k such that u/x) = 0, Uk(X) > 0 (0 ~ x ~ 1) and, moreover, y& < 0, or ylk < 0, or Cjk(~) < 0 for some ~ E (0, 1), since yO + yl + S6 c(x) dx is irreducible. The above inequalities, however, contradict + AN)u)/x) ~ 0 (0 ~ x ~ 1). Consequently, u >- o. 0
«M
If the assumptions of Proposition 4.16 are satisfied, then the statements of Theorem 4.12 are truefor the pair ofoperators M, N in (4.30). Problems
4.3 Find sufficient conditions such that the operators in (4.30) constitute a pair (M, N) E Z(W) with W given by (4.34), without assuming (4.35). 4.4 Let R = {u E q[O, 1] : u(O) = u(l) = O}, S = CoCO, 1], and Mu(x) = L[u] (x), Nu(x) = g(x)u(x) with Land 9 as in (4.30). Show that (M, N)EZ. (See Section 3.5.)
4.3
Operators AI - T with positive T
Each Z-matrix M can be written in the form AI - Twith a matrix T ~ O. Thus, operators of this form may be considered as generalizations of Zmatrices. Indeed, under appropriate assumptions, they are Z-operators as
4
113
ABSTRACT THEORY (CONTINUATION)
defined in Section 4.2. We shall here investigate this special type of Zoperators in somewhat more detail. In particular, conditions on inversepositivity which involve the spectral radius of T will be discussed. This then leads to results on iterative procedures. Moreover, we shall introduce the concept of W-irreducible positive operators, which generalizes the notion of an irreducible matrix in several ways. Let (R, ~, K) be an Archimedian ordered linear space which contains an element ( >- o. Suppose that R is a Banach space with respect to the norm I lis· Moreover, let 1 denote the identity operator in Rand T: R -> R a positive linear operator. This operator is bounded, according to Theorem I.1.10(ii). Define peT) to be the spectral radius of Tand (J(T) to be the supremum of the (real) spectrum of T. If (J(T) > - 00, this number belongs to the spectrum. Moreover, (J(T) ~ peT) by definition of these quantities. (See Section 1.3.1.) 4.3.1
Inverse-positive operators AI - T
We shall and N = I. with (R, ~), i.e., u ), 0 ~
apply the theory of Section 4.2.1 to the operators M = - T The space (5, ~) occurring in that section will be identified so that, in particular, C = K. Moreover, we choose here W = K., u
>- o.
Proposition 4.17.
For each A E IR, AI - T is a Zi-operator and (J(T) =
K( - T, 1).
This statement, which the reader can easily prove, enables one to apply the theory in Section 4.2.1. We shall here formulate explicitly only a few results which involve the spectral radius peT). Further results, such as estimates of the spectral radius, may then immediately be obtained from the theorems in Section 4.2.1. Theorem 4.18. (i) peT) = (J(T). (ii) For A > peT) the operator AI - T is inverse-positive. (iii) A > peT) if and only if Z ? 0, AZ >- Tz for some z E R.
Proof. For A > (J(T) = K( - T, I) the operator AI - Tis inverse-positive according to Theorem 4.6, and its range equals R, by the definition of (J(T). Consequently, for any ( >- 0 in R the equation AZ - Tz = ( has a solution z ? o. On the other hand, if z ? 0 and AZ >- Tz , then even z >- 0 and hence (J(T) ~ peT) ~ I TIIz < A. Combining these two arguments we see that peT) = (J(T), so that all statements are proved. 0 As a simple consequence of statement (iii) we obtain Corollary 4.18a.
A > peT) if and only if I TIIz < Afor some z
>- 0
in R.
114
II.
INVERSE-POSITIVE LINEAR OPERATORS
The following example will show that, without further assumptions on T, A > peT) is not necessary for the inverse-positivity of AI - T.
Example4.4. Let R = I", and T = (t ik ) be the infinite tridiagonal matrix with (i,i+1 = t i + 1 , i = 1 and t ik = 0 otherwise (i, k = 1,2, ...). (We identify such matrices with operators from I", into 1",,) For A > 2, the inequalities z ~ 0 and AZ >- Tz hold with z = e e, = 1 (i = 1,2, ' , .), Thus peT) ~ 2, and hence AI - T is inversepositive. (b) For A = 2, one verifies that (21 - T)-1 = (J-lik) with J-lik = i for i ~ k, and J-lik = k for i > k. Obviously, this inverse is positive, so that 21 - T is inverse-positive. However, the inverse operator is not bounded (and not defined on all of R). Consequently, A = 2 belongs to the spectrum of T and peT) = 2. According to part (iii) of Theorem 4.18 the range of 21 - T then does not contain any element >-0. Observe, however, that 2z - Tz > 0 for z = e >- o. D (a)
= (e;),
A behavior as in the example can, for instance, be excluded for compact operators T. We derive from Proposition 4.7 and Theorem 4.10: Theorem 4.19.
For compact T, the operator AI - T is inverse-positive peT)·
Example 4.5.
Define an operator T on R = eo[O, 1] by
if and only if A >
(Tu)(x)
=
f
'§(x,
Ou(~) d~
(0
~
x
~
(4.36)
1),
in vector notation. We assume that each element '§ik(X,~) of the n x n matrix '§(x,~) = ('§ik(X, is positive and integrable on [0, 1] x [0, IJ, and that for each E: > 0 a b(G) exists such that
m
whenever
[x -
o~
yl < x, y
b(E), ~
1.
Then Tmaps R into R and is compact, so that Theorems 4.18 and 4.19 can be applied. If, in particular,
f
'§(x,
~)d~ ~
r
(0
~ x ~
1)
(4.37)
for a (constant) n x n matrix T, then AI - T is inverse-positive if the matrix AIn - I" is inverse-positive. To prove this, one verifies AZ>- Tz for z = (z.)
4
115
ABSTRACT THEORY (CONTINUATION)
with zlx) == (; and a vector' E IR n satisfying' ~ 0, A'i > (n); (i = 1, 2, ... , n). Such a vector' exists, according to Theorem 2.6. 0 Example 4.6. Consider the operator (4.36) for n = 1 and ~(x,~) = x) = x(l -~) (0:::;; x :::;; ~ :::;; 1). For A> n- 2 the function z(x) = cos(n - [;)(x - t) satisfies z ~ 0 and AZ>- Tz, if I; is sufficiently small. Consequently, AI - T is inverse-positive for A> n- 2 , and peT) :::;; n- 2 • Since, on the other hand, Tip = ACP for A = n- 2 , cp(x) = sin zrx, we have peT) = n- 2 . Using (4.37) with r = i, the inverse-positivity of AI - Tcan be established for A> l 0 ~(~,
Example 4.7. Consider the operator (4.36) under the additional assumptions that all functions ~;k(X, ~) are bounded above by a constant c and that ~(x, ~) = for x :::;; ~, so that
°
Tu(x)
=
f ~(x,
°
~)u(~)
d~.
°
Now for each A > a constant y > exists such that Z ~ 0, AZ>- Tz for Z = (z.), z;(x) = exp yx (i = 1,2, ... , n). Consequently, peT) = 0, and AI - T is inverse-positive if and only if A > 0. 0 4.3.2
Iterative procedures
Most results of Section 2.5 on iterative procedures in finite-dimensional spaces can be proved to hold also in certain Banach spaces. The preceding section provided the main tools for such generalizations. Since most of these generalizations are straightforward, we shall only state a few basic results. Suppose that (R, :::;;) and (S, :::;;) have the properties required in the first paragraph of Section 4.2.1, let' E C", and assume that (S, :::;;) is an ordered Banach space (with respect to the order topology induced by I lis). The results of the preceding section will be applied to an operator T:· S --> S (instead of T: R --> R). We consider an iterative procedure. Au v + 1
= Bu; + s
(v = 0, 1,2, ...)
(4.38)
for solving an equation Mu = s with given s E S. Here M, A E L(R, S) and BE L(S, S) are operators such that A is inverse-positive, AR = S, A - 1B E L(S, S) is positive and compact, M R = S if M is one-to-one. The procedure (4.38), which can be written as u v + 1 = Tu;
+r
with
T
=
A - 1B : S
will be called convergent if peT) = peA -1 B) < 1.
-->
S,
r
=
A-I S,
116
II.
INVERSE-POSITIVE LINEAR OPERATORS
Theorem 4.20. Under the preceding assumptions, the following four properties are equivalent: (i) The procedure (4.38) is convergent. (ii) II Tllz < 1 for some z >- 0 in S. (iii) M = A - B is inverse-positive. (iv) z ~ 0, Mz >- ofor some ZER. Proof.
We first recall that U
1.2a).
>- 0
=>
A -1 U
>- 0
for U E S (cf. Corollary
(i) ~ (ii) as a consequence of Corollary 4.18a. (i) => (iii): If peT) < 1, then I - T is inverse-positive (cf. Theorem 4.18(ii». Consequently, Mu ~ 0 => (I - T)u ~ 0 => u ~ o. (iii) => (i): If M is inverse-positive, then M z = ( >- 0 for some z ~ 0 and hence (I - T)z = A -1( >- 0, so that peT) < 1 (cf. Theorem 4.18(iii». (iii) => (iv): Choose z satisfying Mz = (. (iv) => (iii): The inequalities u ~ 0 and Mu >- 0 imply u >- A -1 Bu ~ 0, so that M is weakly pre-inverse-positive. Then apply Theorem 1.2. 0 Example 4.8. Let M and N be the differential operators in Section 4.2.2 and consider the splitting M = A ...:.- B with A = M + pN, B = pN with p > 0 so large that A is inverse-positive. Since A - 1 is defined on all of S and the integral operator A - 1 B is compact, the preceding theorem yields that for this splitting the iterative procedure (4.38) converges if and only if M is inverse-positive.
4.3.3
0
Irreducible operators
This section is concerned with operators T: R -> R which have the properties described at the beginning of Section 4.3 and, in addition, are irreducible or W-irreducible in a sense defined below. The notion of an irreducible matrix has been generalized to operators in Banach spaces by several authors in different ways. We do not intend to give a survey of these results here. However, we shall indicate how some of the most important definitions fit into our theory. let a further linear order ,( be given which is In the space (R, ~,K) stronger than ~ and has an order cone W. A (positive linear) operator T: R -> R is said to be W-irreducible if for all u E R and all sufficiently large AE IR
u~ o}
AU - Tu
'> 0
=>
U
>- o.
(4.39)
In particular, a K-irreducible operator is simply called irreducible. For such an operator T the preceding implication holds with > in place of '>.
4
ABSTRACT THEORY (CONTINUATION)
117
(Compare the characterization of an irreducible matrix in Corollary 2.12b.) It is unfortunate that the negating term "irreducible" is generally used for a property with so many "positive" aspects. Neglecting, however, linguistic problems, we have introduced the term W-irreducible which allows one to describe different "degrees of irreducibility." The extreme cases are obtained for W = K and W = Ks, Each (positive linear) operator is K s irreducible. (Concerning this terminology, see also the Notes.) As in Section 4.3.1 we can again apply the results of Section 4.2.1 to the operator pair (M, N) = ( - T, I). Obviously, for each A E IR, T is Wirreducible if and only if AI - T is a Z(W)-operator, and T is irreducible if and only if AI - T is a Z(K)-operator. Therefore, we derive from Proposition 4.17, Theorem 4.18(i), and Theorem 4.12:
Theorem 4.21. If T is irreducible and has an eigenelement q> > 0 corresponding to the eigenvalue A = p(T), then this eigenvalue is simple and q> >- o. An eigenelement q> > 0 corresponding to A = p(T) exists, for example, if T is compact and p(T) > 0 (cf. Theorem 1.3.3). The next two theorems provide a series of necessary and sufficient conditions for W-irreducibility and irreducibility. Obviously, each of the equivalent conditions can also be used as a definition. Let !l' be a set of positive linear functionals on R which describes K completely. Moreover, denote by R A = RiT) for A > p(T) the resolvent operator Rjt T) = (AI - tv:: = If=o A-(j+ l)Tj, the series being convergent with respect to the operator norm.
Theorem 4.22.
The following five properties are equivalent.
(i) Tis W-irreducible. (ii) There exists a A > p(T) such that (4.39) holds for all u E R. (iii) v '> 0 = RiT)v >- o,for each A> p(T) and all v E R. (iv) There exists a A > p(T) such that v '> 0 = RA(T)v >- o,for all v E R. (v) To each v '> 0 in R and each t E fe there exists an integer m ~ 0 such that tTmv > o. Proof. (i) = (ii) and (iii) = (iv) are obvious. (iv) (v) (iii) is derived from tRiT)v = .LJ"=o A-(j+ »rri« (ii) = (iii): For each A > p(T) and each v '> 0, u = RiT)v satisfies u ~ 0 and Au - Tu '> 0, so that u >- o. (iii) = (i): Let A > p(T), u ~ 0 and (AI - T)u = v '> o. Then u = RiT)v >- o. 0
= =
Theorem 4.23. («) (f3)
The following four properties are equivalent. T is irreducible. v > 0 = T RiT)v >- o,for each A > p(T) and all v E R.
118 (y)
II.
INVERSE-POSITIVE LINEAR OPERATORS
There exists a A > p(T) such that v>
0
=
TR;.(T)v
>- 0,
for all
VER.
(b) To each v > such that tTmv > O.
0
in R and each t E!fl, there exists a number mEN
Proof. The equivalence of (a), (y), and (b) is proved similarly as the equivalence of (iii), (iv), and (v) in the preceding theorem. To complete (iii) (6), where now v > 0 ~ the proof, it suffices to show that (13) v > 0 is defined in (iii). (13) (iii) is obvious. To prove (iii) = (6), suppose that the five (equivalent) properties of Theorem 4.22 hold for W = K, and let v> 0, t E!fl, and A > p(T) be given. Then (iii) yields R;. v >- 0 with R;. = R;.(T). Define w = T R;. v. If w > 0, then tTkw > 0 for some integer k :;:: 0 because of (v). This inequality implies 1 A- jtTj+k V > 0, so that tTmv > 0 for some mEN, as stated in (6). Now suppose that w = R;. Tv = 0 and, consequently, Tv = o. If v >- 0, this relation can only hold if T = O. For, given any u E R, there exist a, 13 > 0 with - av ~ u ~ f3v, so that 0 = - «T» ~ Tu ~ f3Tv = o. However, the null operator is not irreducible. If v 0, there exists an t E !fl with tv = O. From this relation and T R A v = 0 we conclude that tRio v = 0, which contradicts R;. v >- o. Consequently, w = 0 cannot hold. 0
= =
=
IJ"=
*'
Example 4.9. T is irreducible, if for each v > exists such that Tmv >- o. Obviously, this property of T implies (b). 0
0
in R a number mEN
We shall explain in the notes following this chapter how several of the above properties are related to known definitions of irreducibility and similar properties. 4.4
Inverse-positivity for the case K'
=
0
The monotonicity theorem for linear operators M: R -> S will here be generalized to the case in which R need not contain a strictly positive element. Let the spaces (R, ~) and (S, ~) again satisfy the requirements (A 1 ) , (A 2 ) in Section 1.1 except the condition K C =I- 0. We assume that p > 0 is a given element in R, and we shall then use the notation >-p and B p introduced in Section 12.3 (see (12.6». Theorem 4.24. I. II.
Let the following two conditions be satisfied.
>0 imply u >- »":
For each u E R, the inequalities u :;:: 0, M u There exists an element Z E R such that Z :;::
0,
Mz
> o.
4
119
ABSTRACT THEORY (CONTINUATION)
Then for each u E Bp , Mu
=
~ 0
u
I
Mu >
~ 0;
0
u >- po.
=
(4.40)
Proof. Suppose that Mu ~ 0, u ~ -yp with some y > 0, but u ~ o. Then there exists a minimal A > 0 such that v = (1 - A)U + AZ ~ o. Because of II, we have M v 0, and hence I implies that v ~ rxp for some rx > O. Therefore, (1 + e)[(1 - f.1)u + f.1z] = (1 - A + e)u + AZ = v + eu ~ (rx - ey)p ~ 0 for (1 + e)f.1 = A and ey = a, which contradicts A being minimal. Consequently, the first statement in (4.40) holds. If Mu 0, then u ~ 0 and hence u >- po by assumption I. D
>
>
We shall, in particular, consider the special case in which (R,
~)
c (S, ~),
A> 0,
M=AI-T,
TEL(R, S),
T~O
(4.41) with I the identity in S. For this case the preceding theorem yields Corollary 4.24a.
(a)
If Z
u E B p, (b) If Z > 0, f.1Z - Tz ~ holdsfor p = z. (c) If Z ~ 0, AZ - Tz > Tz.
(~I
~ 0,
AZ - Tz
- T)u
~ 0
>- po for some Z E R, then
=
u
~
o.
(4.42)
0
for some Z E R and some f.1 E (0, A), then (4.42)
0
for some Z E R, then (4.42) holds for p = AZ -
Example 4.10. Let T = (t i k ) denote the infinite matrix with elements t i k = k- 1 for i < k, t i k = 0 for i ~ k (i; k = 1,2, ...). Choose the spaces R = {u E 12 : Tu E 12 } and S = 12 , furnished with the natural order relation, and consider the operator M = I - T: R ~ S. This operator is not inversepositive, since Mu = 0 for u = (u;) with Ui = rx(l + i)-1, rx E IR. However, statement (4.42) holdsfor p = (p;) with Pi = i" 3 and A = 1. To prove this, one verifies z, - (TZ)i ~ ti- 3 for z, = t:' and applies Corollary 4.24a. D
Theorem 4.24 and its corollary yield sufficient conditions for a restricted inverse-positivity, meaning that (4.40) is only established for all u which belong to a subset of the domain of M. Nevertheless, these results can also be used to prove (unrestricted) inverse-positivity. One has to verify that for each u ERa strong order ~ and an element p > 0 exist such that the assumptions of Theorem 4.24 are satisfied and u E B p • As an example we shall formulate a corresponding result for the case R = S using the following notation. A subset f!J> c R will be called positive, f!J> ~ 0, if p ~ 0 for all p E f!J>. Moreover, f!J> c R is said to be strictly positive, f!J> >- 0,
120
if flIJ
II. ~
0, ()
f/= flIJ,
U {B
and R =
Corollary 4.24a
p :
INVERSE-POSITIVE LINEAR OPERATORS
p E flIJ}. With this notation, we obtain from
Theorem 4.25. Suppose that for the operator M = AI - T: R :!l' c R exists such that
:!l'
and
~ 0
~
R a set (4.43)
M:!l' ;> o.
Then M is inverse-positive. Under certain conditions, the assumption of the theorem can be verified by constructing a single element q> E R, as formulated in the following corollary. Here, the given space (R, ~) is called directed if to each u E R an element v E R, v > 0 exists such that u ~ - v. Corollary 4.25a. Suppose that the ordered linear space (R, ~) is directed, and that the domain TR is a subset ofan ordered linear space (R, ~) c (R, ~) such that (R, ~) contains a strictly positive element q> satisfying
Aq> - Tq> Then M = AI - T: R
~
~
for some v > 0.
vq>
(4.44)
R is inverse-positive.
Proof. For each u E R choose v :> 0 in R such that u ~ - v, and a constant b > such that vq> - o'Tn ~ o. Then Z := q> + bv satisfies z ~ 0, and u ~ - v = - yA bv ~ - Y(AZ - Tz) for Y = A-1b- 1. Consequently, (4.43) holds for the set :!l' of all these elements z (u E R). 0
°
°
Example 4.11. Let R = LiO, 1) and define u ~ 0 to mean that u(x) ~ for a.a. x E [0, 1]. We shall conclude from the corollary that the integral operator AI - T: L 2 (O, 1) ~ L 2 (O, 1) with
Tu(x) =
f
'§(x,
Ou(~) d~
and
'§(x,~)
=
'§(~,
x) = x(l for x
is inverse-positive if A > n-
2
~)
~ ~
.
The space (R, ~) is directed since u ~ - Iu I, and the range T R is a subset of the space R = CoCO, 1] furnished with the natural order relation. Finally, if A = n- 2 + 2p, 0< 13 < n, (n - 13)-2 - n- 2 ~ u, then q>(x) = cos(n - c)(x - 1) satisfies (4.44) for v = u, and this function is strictly positive as an element of R. 0 In Section 1 we were also concerned with strictly inverse-positive operators. Here this term, in general, does not make sense, since K may be empty. One can, however, often prove a stronger type of inverse-positivity which coincides with strict inverse-positivity under suitable assumptions. The C
5
121
NUMERICAL APPLICAnONS
following result is again concerned with the case (R, ::() = (S, ::(), M = AI - T, A > 0, T;::, 0 (the proof is left to the reader). Proposition 4.26. If AI - T: R ~ R is inverse-positive and T has property (b) in Theorem 4.23for the set if of all linear functionals t > 0 on R, then for all u E R
(AI - T)u ;::,
0
=>
either u
=
0
or u is a nonsupport point of K.
(4.45)
Example 4.11 (continued). For this case if is the set of all functionals tu = (WI, u) with W = WI E Lz(O, 1), W > o. Since (w, Tu) > for arbitrary u > 0, W > 0 in Lz{O, 1), the implication (4.45) holds here for A> n- z. D
°
Problems
The following problems are concerned with the special case (4.41) under the general assumptions made above.
°
4.5 Suppose that pet) = L::'=o Yktk is a polynomial with Yk ;::, (k = 0, 1, ... ,m) such that pel) = 1 and the following conditions are satisfied. (i) For all u E R, u > 0 => P(T)u >- po. (ii) There exists a z ;::, 0 in R such that z ). Tz or Z ). P(T)z. Prove that under these assumptions for all u E B p : u - Tu ;::,
0 =>
u ;::, o.
,
4.6 Reformulate (i) in the preceding problem for the case u S, 0 ¢ > U >- po (with some p > 0 in R). 4.7 Generalize Problems 4.5 and 4.6 using a power series pet). 4.8 Show that a set f!J> = {p} c R consisting of a single element p is strictly positive if and only if p >- o. 4.9 Show that Corollary 4.25a remains true, if the assumption T R c R is replaced by TmR c R and, in (4.44), AI - T is replaced by AmI - T". (Here m ;::, 1 denotes a fixed integer.) 5
NUMERICAL APPLICAnONS
For certain problems, for instance, boundary value problems of the second order, the theory of inverse-positivity is a convenient tool to estimate the errors of approximate solutions. In Section 5.1 the convergence of discretization methods is proved by estimating the error by functions of h, where h denotes the discretization parameter. In particular, we treat several difference methods for two-point boundary value problems. The ideas of proof can also be used for partial differential equations. Section 5.2 is devoted to the problem of a posteriori numerical error estimates. It is a trivial statement that approximate numerical solutions are of no value, if one does not have, in addition, some indication of how accurate
122
II.
INVERSE-POSITIVE LINEAR OPERATORS
the approximations may be. Certainly, a rigorous estimate of the error would be the most satisfying answer. Such an error estimate, however, in general is much more difficult to obtain than the approximate solution itself. We shall not try to discuss here for which problems such error estimates could possibly be carried out. We simply report on some work which has been done for initial value problems ofthe first order and boundary value problems of the second order. Sections 5.2.2 and 5.2.3 contain corresponding results for linear problems. The methods can be carried over to nonlinear differential equations and systems of such equations. See Section 111.6 and, furthermore, Examples V.2A and VA.12 for the most difficult problems treated so far by our methods. In Example V.2A we shall make some remarks concerning the computing time needed for the error estimates. One may consider all work on numerical error estimates which one can find in the literature as a beginning, a first step. Certainly much more effort has been put into the development of approximation methods, than into the development of estimation methods. How far one can go in constructing methods for numerical error estimation cannot be answered a priori but only by further investigations. It should also be mentioned that such estimation methods together with comparison techniques as described in Section ilIA may be used to prove the existence of solutions with a certain qualitative behavior in cases where comparison functions cannot be constructed by analytic techniques. 5.1
A convergence theory for difference methods
5.1.1
The difference method
Suppose that an equation (5.1)
Mu=r
with a linear operator M : R ~ S is given, where Rand S are linear subspaces '1] of all bounded real-valued functions on a compact of the space IRb[~' interval I = [~, '1]. We assume that for the given rES a unique solution u* E R exists. Moreover, let e E Rand e E S (where e(x) == I), so that u .> o-=inf{u(x): ~ :( x :( IJ} > 0 for u E Rand u E S. We consider a difference method !!JJ for obtaining approximate solutions for the given problem. This method is described by a family of difference equations (hEH)
(5.2)
for functions qJ = qJh E R h := 1R(l h) defined on a discrete set I h = {Xo, x.} C I. (Of course, the x.and n may depend on h.) Here M h : R h ~ R h and N h : S ~ R h are linear operators, and H is a sequence of numbers Xl"'"
5
123
NUMERICAL APPLICAnONS
hi > 0 (i = 1,2, ...) converging to O. We assume that a constant exists such that for all U E S and all h e H
> 0
K
(5.3) Here and in the following, Uh denotes the restriction of U E IR(I) to I hThus, Uh E lR(Ih) and hence Uh ~ 0 ¢ > u(x) ~ 0 for all x E I h. (By our convention, ~ denotes the natural order relation in the function space under consideration.) For convenience, we shall often write Mhu instead of MhUh, if U E IR(I). We shall use the notation
IlullJ = inf{«
~
0: -ex ~ u(x) ~ ex
for all
x E J}
for any set J c I and any suitable u, so that, in particular,
IIullI = inf'[« ~ 0: -exe ~ U ~ exe} l[cplIl h = inf'[« ~ 0: -exeh ~ cP ~ exeh}
for for
Rand cP E Rho
U E
u E S,
Each operator M h can be described, in a natural way, by an (n + 1) x (n + 1) matrix. This matrix is inverse-positive (pre-inverse-positive) if and only if the operator M; is inverse-positive (pre-inverse-positive). Moreover, the matrix is a Zsmatrix if and only if M h is a Z-operator as defined in Section 4.2.
5.1.2
Sufficient convergence conditions
The difference method ~ is called convergent (more precisely: uniformly convergent on I) if solutions cP = CPh of the difference equations (5.2) exist for all sufficiently small h E Hand for
h ---+ 0 (h E H).
(5.4)
The method has the order of convergence p (is conrerqent of order p) if for
h e H,
where p > 0, and y denotes a constant which may depend on u* and p. The convergence on a subset J c I and the order of convergence on J are defined analogously with I h replaced by I h n J. As a reasonable assumption on the difference method, we require that equations (5.2) approximate the given problem (5.1) "well enough." This requirement will be formulated as a consistency condition. For fixed u E R, the difference method ~ is called consistent at U if for
h
---+
0
(h
E
H),
124
II.
INVERSE-POSITIVE LINEAR OPERATORS
The order of consistency at u is p, if
Ph(U)
:=
II00h(U)ll h
,,;
yhP
for
hEH,
where p > 0, and y > 0 denotes a constant which may depend on u and p. Of course, the numbers p which occur above are, in general, not uniquely determined by the properties mentioned. This could be taken into account by using terms like" the order of convergence is at least p". However, we do not expect any difficulties in using the preceding simpler terminology. In the convergence proofs the inverse-positivity of the difference operators M h will be employed. This property can very often be derived using properties of the given operator M. Proposition 5.1.
Let the following two conditions be satisfied.
(i) M h is weakly pre-inverse-positive for each sufficiently small h E H. (ii) There exists a Z E R such that Z ~ 0, M Z ~ e, and the difference method f?fi is consistent at z. Then M h is inverse-positive for each sufficiently small hE H. Proof.
For sufficiently small hE H Mhz h = NhMz
+ O"h(Z)
~ K-1eh - 1K-1eh
so that M; is inverse-positive by Theorem 2.1.
= 1K-1eh
>- 0,
(5.5)
0
Theorem 5.2 (Convergence theorem). Suppose that assumptions (i), (ii) of Proposition 5.1 are satisfied and the difference method f?fi is consistent at the solution u*. Then Method f?fi is convergent, and for sufficiently small h e H Ilu~
- !Phllh ,,; 2KPh(U*) Ilz111'
(5.6)
Proof. Let h E H be so small that the arguments in the proof of Proposition 5.1 are valid. Then equation (5.2) has a unique solution !Ph, since M h is inverse-positive and hence invertible. Moreover, we estimate Mh(U~
- !Ph) = O"h(U*) ,,; Pheh ,,; Ph 2KMh Zh
with
Ph = Ph(U*),
using (5.5), and obtain from this relation u~ - !Ph ,,; Ph2KZh' Analogously, the inequality - Ph2KZh ,,; u~ - !Ph is derived, so that (5.6) follows when Ilzlllh " ; Ilzlll is observed. 0 Obviously, (5.6) implies that Method f?fi is convergent of order p consistent of order p.
if it
is
Quite often difference methods are used such that not all mesh points Xi lie in the interval on which the boundary value problem is given (see the cases (5) and (6) below, for example). To treat such cases, one could generalize
5
125
NUMERICAL APPLICAnONS
the abstract theory above by not requiring I he I. Instead, we shall extend the given problem to a problem on a larger interval. Then the following result will be used. Corollary 5.2a.
Under the assumptions of Theorem 5.2,
Ilut - CfJhllhnJ
~
2KPh(u*)IlzIIJ
for any J c I and sufficiently small h e H.
The above convergence theory can also be formulated somewhat differently, so that the notion of stability is involved. (The proofs of the results below are left to the reader.) The difference method !!2 is called stable, if a constant y exists such that for all sufficiently small h the operator M h is invertible and M; 1: R h -+ R; has an operator norm !IM; 111 ~ y (with respect to the norm IICfJhllIJ Theorem 5.3. If the difference method !!2 is stable and consistent at u*, then the method is convergent.
Of course, in applying this result one has first to prove the stability, i.e., one has to estimate the norms I M; 111. Our aim here is to replace the assumption on these operator norm~ by simple sufficient conditions, using inversepositivity. Proposition 5.4. The difference method !!2 is stable, of Proposition 5.1 are satisfied.
5.1.3
if the assumptions
Boundary value problems
The convergence theory of the preceding section will here be applied to a two-point boundary value problem
Mu(x) = r(x)
(0 ~ x ~ 1)
(5.7)
with a differential operator M of the form (3.2), (3.4) satisfying the regularity condition (3.5) and a function r
E
!R[O, 1]
such that r(x) = s(x) (0 < x < 1) for some s E CoCO, 1].
(5.8)
Observe that "o = r(O), rl = r(l) are the given inhomogeneous terms in the boundary conditions. Let S denote the space of all functions r of the form (5.8); then M: R = C 2[0, 1] -+ S. (We shall use the letter r not only for the given function in (5.7), but also for an arbitrary element in S.) To explain various ideas in applying the abstract theory, we shall consider More general problems can be treated in an several special cases (l)~(6). analogous way. For all considered cases, the difference method converges, if the differential operator M is inverse-positive.
II.
126
INVERSE-POSITIVE LINEAR OPERATORS
In the following, the notation of the previous sections will be used. In particular, I h = {X o , Xl' ... , x.}, and u* E Cz[O, 1] denotes the solution of the given problem (5.7). (If M is inverse-positive, a (unique) solution exists for each rES.) Moreover, define for h > 0 1
Lh[cp](x) = a(x) hZ [-1
2 -l]hCP(x)
°
1
+ b(x) 2h [-1 where [tx [3 r]hCP(X)
= txCP(x
l]hCP(x)
+ c(x)cp(x),
- h) + [3cp(x) + rCP(x + h).
(1) (Dirichlet problem, ordinary difference method.) Suppose that Bo[u] = u(O), Bl[u] = u(l), and choose for each h e H = {n- l : n = 2,3, ...}: xi=ih=in- l (i=0,1,2, ... ,n), Nhr=rh MhCP(Xi) = Lh[cp](x;) M hcp(xo) = cp(xo),
for
0< i < n,
M hcp(x n) = cp(x n ) ·
For the difference method so described, we obtain the following result. Proposition 5.5. (a) If thedifferential operator M: Cz[O, 1] ~ IRb[O, 1] is inverse-positive, then the difference method is convergent on [0, 1]. (b) The order of convergence is 2 or 1, if u* E C4[0, 1] or u* E C 3[0, 1], respectively.
Proof. For sufficiently small h; the operator M h: R; ~ R; is described by a Z-matrix, and hence M; is weakly pre-inverse-positive, so that condition (i) in Proposition 5.1 holds. Moreover, since M is inverse-positive, there exists a ZER as required in (ii) (choose Z satisfying Mz = e). Finally, a Taylor expansion of ah(u)(x) = Mhu(x) - (MuMx)atxiE[h, 1 - h]shows that the difference method is consistent at each u E Cz[O, 1]. Moreover, the order of consistency is 2 or 1 if u E C4[0, 1] or u E C 3[0, 1], respectively. Thus, the statements of this proposition follow from Theorem 5.2. D (2) (Dirichlet problem, nonconstant mesh width.) Let again Bo[u] = u(O),Bl[u] = u(l),butchoosenowforeachhEH = {1m- l : m = 2,3,4,o ..} for Xi
with h =
tm- l,
i = 0, 1, 2, .
= 1 + (i - 2m)h n = 3m, and define
for
i
0
•
,
2m;
= 2m + 1, ..
°
0
,
n
M hcp(x;) = L h/ Z [cp] (Xi) for < i < 2m, MhCP(Xi) = Lh[cp] (x.) for 2m ~ i < n, MhCP(xo) = cp(xo), MhCP(x n ) = cp(x n ) , Nhr = rho
5
127
NUMERICAL APPLICAnONS
By a proof similar to that in (1), one shows that for this difference method Proposition 5.5 holds also. (3)
(Dirichlet problem, Hermiteformula.)
To a boundary value problem
(5.7) with L[u] (x)
= -ul/(x) + c(x)u(x),
Bo[u]
= u(O),
B 1[u]
= u(I),
we apply a Hermite formula (Mehrstellenverfahren). We choose H, the points Xi> and the values Mh({J(xo), Mh({J(xn ) as in (1), but define now
for for
0 < i < n,
0 < i < n,
for r having the form (5.8). For this difference method/Proposition 5.5 holds also. Moreover, the order of convergence is 4 or 3, ifu* E C6[0, 1] or u* E Cs[O, 1], respectively.
Again, this statement is proved by similar means as the result for (1). Now tTh(u)(x) = h- 2 [ -1 2 -1]hu(x) + /2[1 10 l]hu"(x). (4) (A method of order 1 for Sturmian boundary conditions.) To describe how more general boundary conditions can be treated, we consider the case where
with
0(0
> 0,
B 1[u]
= u(I).
(5.9)
All quantities are chosen as in (1), except that now
For this difference method, part (a) of Proposition 5.5 holds also. Moreover, the order of convergence is 1 ifu* E C 3[0, 1].
The proof is the same as in (1), except that for u* E C 4[0, 1] we can, in general, establish only the order of consistency 1, because MhUh(O)Mu(O)
= O(o(-!hul/(O) + O(h 2 ».
(5) (A method of order 2 for Sturmian boundary conditions.) Consider again the case (5.9). The difference method in (4) was not quite satisfying,
128
II.
INVERSE-POSITIVE LINEAR OPERATORS
since for u* E C 4[0, 1] only convergence of the order 1 could be established. Therefore, we shall now describe another difference method. Choose . - -h + ih
2
(i = 0, 1,2, ... , n)
°
1]h/2 q>(0)
1
+ "2 Ya[l
(0 < i < n), 0< i < n,
for
F or this difference method Proposition 5.5 holds also.
Now, I h is not a subset of the interval [0, 1] on which the boundary value problem (5.7) is given. Nevertheless, the meaning of the above statement is clear, the convergence on [0, 1] being defined by (5.4) with I h replaced by t, n [0, 1]. However, for being able to apply Theorem 5.2, we have to incorporate the above difference method into our general scheme. This will be achieved by extending the given problem (5.7) to a problem Mil = r on an interval I = [ - 8, 1], with an operator M: C 2 [ - 8, 1] -> !R o[ - 8, 1] and some 8 > 0. Obviously, I h c I = [ -8, 1] for all sufficiently small h of the above kind. This extension of the given equation is carried out for formal reasons only. For inverse-positive M there exists a z E C 2[0, 1] satisfying Mz = e, z(x) > (0 ::::; x ::::; 1). This function can be extended to a function Z E C 2 [ -8,1] with z(x) > (-8::::; x::::; 1), where 8> denotes a suitably chosen constant. Obviously, ocz(x) ?: 1 (-8 ::::; X ::::; 1) for some a > 0. We define Mu(x) = ocu(x) (-8 ::::; X < 0), Mu(x) = Mu(x) (0::::; x ::::; 1). This operator M maps R = C 2[ - 8, 1] into the set S of all E !Rb[ - 8, 1] such that r(x) = r(x) (0 ::::; x ::::; 1) for some rES. Obviously, Mz(x) ?: 1 for -8::::; X ::::; 1. We also extend N h to an operator Nh on S by defining Nhr = Nhr for r as above. For inverse-positive M the given problem (5.7) has a unique solution u* E C 2[0, 1], which can be extended to a function Il* E C 2 [ -8, 1] such that, for k = 0, 1,2, Il* E C 2 + k [ -8,1] whenever u* E C 2+k[0, 1]. Clearly, u* is a solution of the extended equation Mu = r, where r(x) = ocu*(x) for -8 ::::; x::::; and f(x) coincides on [0, 1] with the given r(x) in (5.7). Now one derives the preceding convergence statement from Theorem 5.2 applied to Mil = r and the difference method Mhq> = Nhr, in essentially the same way as in the other cases. 0
°
°
°
r
°
5
129
NUMERICAL APPLICATrONS
(6) (A modification of (5) where M h is not a Z-operator.) we consider the case (5.9) choosing now Xi= -h+ih
(i=O,l, ... ,n) 1
MhCP(xo) = -ao 2h [-1
0
with l]hCP(O)
Once more,
h=(n-l)-l,
+ roCP(O)
and defining all M hCP(Xi) for i > 0 and N h by the formulas used in (5). For this difference method Proposition 5.5 holds also. The proof is essentially the same as for the case (5), with the following exception. If ro > 0, then M; is not a Z-operator for any h > 0, so that the (weak) pre-inverse-positivity of M h is not immediately clear. However, this property of M; can here be established using the method of reduction explained in Section 4.1. For Ph: R; --+ R h defined by (i
= 1, 2, ... , n)
and sufficiently small h, PhM h is a Z-operator. Consequently, PhM h is weakly pre-inverse-positive and hence M; has this property also (cf. Theorem 4.1).
5.1.4
A generalization and its application
We continue now the abstract convergence theory. To treat more complicated difference methods, Theorem 5.2 will be generalized, replacing condition (ii) (in Proposition 5.1) by a weaker assumption. (Here condition (5.3) will not be needed.) Theorem 5.6. Let M h be weakly pre-inverse-positive for sufficiently small h e H. Moreover, suppose that an element Z E R exists and for all sufficiently small h E H elements 0/ h E R h exist such that (5.10)
and
Iah(u*)I ~ r(h)Mho/h
with some r E CoCO, CD).
(5.11)
Thenfor h e H sufficiently small,
Ilut - CPhlllh In particular, Method (r(h) = ChP ) .
~
~
r(h)llzIII'
is convergent (convergent of order p)
(5.12)
if
r(O)
=
°
This result is proved similarly as Theorem 5.2. While in the proof of that theorem "bounds" o/h of the form o/h = const Zh with z in (ii) were used, the
130
II.
INVERSE-POSITIVE LINEAR OPERATORS
functions t/Jh here may be chosen differently. We shall give an example where this greater flexibility is exploited. (7) (A method convergent of order 4 and consistent of order 2.) consider again the problem in (1), but choose now 1 MhqJ(x;) = a(x;) 12h2 [1 -16 1
+ b(x;) 12h [1
- 8
30 -16
°8
We
l]qJ(x;)
-1]qJ(x;)
for
+ c(xi)qJ(x;) 2::;; i ::;; n - 2,
for
i
= 0, n.
Proposition 5.7. Suppose the differential operator M is inverse-positive. Then the difference method (7) is convergent. Moreover, for k = 2,3,4, the method has the order of convergence k, if u* E C 2 +k[0, 1]. This statement will be derived from Theorem 5.6 in two different ways. First proof (Method of z-splitting). In the following arguments we shall always assume that h is sufficiently small, without mentioning it each time. Obviously, M h is not a Z-operator; but we can prove the weak pre-inversepositivity of M h by the method of reduction. By formal calculation, one verifies that M; ::;; it'hAh for it'h: R; --+ R; and A h: R; --+ R; defined by qJ(X;) (x.)
it' hqJ
I
A hqJ(x;)
= !h-
2a(x;)qJ(xi)
2[ { U2 ah - -1 qJ(X;)
for for
i i
°
= and i = = 1 and i =
8 -1]qJ + /2 bh - I [ -1 0 for 2::;; i ::;; n - 2
= { [_ 1 8 - 1]qJ(x;)
for for
n n- 1 l]qJ) (x;)
i = 0 and i = n 1::;; i ::;; n - 1.
The operator it' h is inverse-positive, because it is a Z-operator and ~ o. Since also A h is a Z-operator, M h is weakly pre-inverse-positive, according to Corollary 4.1a applied to M h, it'h' A h in place of M, L, MI' Using Taylor expansions one shows that the difference method has the order of consistency 2 at u E C 2[0, 1]. Moreover, since M is inverse-positive, there exists a 2 ~ 0 in R such that M2 ~ e. Thus the convergence of the difference method already follows from Theorem 5.2. However, to prove also the stated order of convergence, we shall use the more general Theorem 5.6, which allows us to exploit "local" orders of consistency. For simplicity, we shall consider only the case u* E C 6[0, 1]. it'heh
5
131
NUMERICAL APPLICATIONS
First observe that M hZh ~ ieh for h sufficiently small, due to the consistency of the method. Again using Taylor expansion, we estimate for u* E C6[0, IJ IO"h(U*) I ~ C[h 2(e(l) + e(n-l») + h4 eJ (5.13)
e
= e(2) + e(3) + ... + e(n-2) and eli) = e~i) E R h defined by e(i)(x) = (iij (j = 0, 1, ... ,n). Exploiting this estimate we shall construct suitable t/JhERh as a sum
with
These relations together with (5.13) imply (5.11) for r(h) = Ch 4 . Moreover, if the t/J~j) are positive and uniformly bounded (for h sufficiently small), then (5.10) holds for z = az with a large enough a and hence (5.12) yields the order of convergence 4. Obviously, we may choose t/J~3) = 2z h. Concerning the construction of t/J~l) and t/J~2), we consider first the special case a(x) == 1, b(x) == 1. Here, a formal calculation shows that Mhw(Xi) ~ (h- 2e(l) + ChW)(X;) (1 ~ i ~ n - 1) for WE R h, W(Xo) = 0, w(x;) = 3 - hXj-l otherwise. Consequently, t/J~l) = W + riZ has all desired properties, if a is a sufficiently large constant. The function t/J~2) can be defined in an analogous way. In the general case, one can construct t/J~l) in essentially the same way using now w(Xj) = exp(,ux;)(3 - hXi-1) for i = 1,2, ... , n, where ,u denotes a suitable constant. D Second proof (Perturbation method). The difference procedure (7) can also be written in the form (5.2) with M h, N; replaced by operators M h, N h which are defined as follows: Mhq>(x;) = Mhq>(xj), Nhr(x;) = Nhr(x;) Mhq>(X 1) = h 2M hq>(Xl) + ea(x1)q>(xo), Mhq>(x n- 1) = h 2Mhq>(x n- 1) + ea(xn-l)q>(Xn), Nhr(x 1) = h 2r(xl) + ea(xl)r(xO)' Nhr(xn-l) = h 2r(x n_l) + ea(xn-1)r(x n)·
for
i"# 1, n - 1;
In the first proof we used a particular reduction method to show that M h is weakly pre-inverse-positive (for all sufficiently small h). The same arguments apply to an operator which is obtained from M h when is replaced by [ -(1
+ e)
2 -IJq>(xd
132
II.
and [-1
2 -1Jcp(x n -
1)
INVERSE-POSITIVE LINEAR OPERATORS
is replaced by
[-1
2 -(1
+ /;)JCP(Xn-l),
where s > 0 is a sufficiently small number independent of h. Consequently, for this s > 0 and all sufficiently small h the operator Mh is weakly pre-in versepositive. The perturbation by the number s makes an important difference, when (5.11) is verified. For u* E C 6 [0, 1J we now obtain Io-h(U*) I ~ Ch4(e(l) + e(n-l) + e) instead of (5.13). Therefore, we need only construct a Z ~ 0 in R such that t/Jh = Zh satisfies Mht/Jh ~ eh' This is done in the following way. Let Z >- 0 be the solution of Mz = e. Then Mhz h ~ !eh for all sufficiently small h, because of the consistency. From the latter inequality one derives Mhz h ~ /leh for a /l > 0 satisfying ! ~ u ~ w.p, where 0 < DC ~ a(x), o < fJ ~ z(x) (0 ~ x ~ 1). Observe, for instance, that Mhzh(X1 ) ~ !h 2 + ca(x1)z(xo) ~ fl. Thus, Z = /l-lZ has the desired properties. 0 5.2
A posteriori error estimates
We shall here describe a method to obtain (two-sided) error bounds for solutions oflinear equations Mu = r where M is an inverse-positive operator. After explaining the general approach in an abstract setting (Section 5.2.1) we shall illustrate the method by applying it to linear differential equations (Sections 5.2.2 and 5.2.3). The general ideas involved can also be used for more complicated problems, as indicated at the beginning of Section 5. 5.2.1
General description of the method
Suppose that an equation Mu = r
(5.14)
is given where M denotes a linear operator which maps an ordered linear space (R, ~) into an ordered linear space (S, ~) and rES. We assume that a solution u = u* E R exists and that M is inverse-positive, so that the solution is unique. We are here concerned with methods for providing upper and lower bounds for u* by using the implications Mcp ~ r ~ Mt/J
=
tp ~
(5.15)
u* ~ t/J.
If an approximate solution Uo E R with defect d[uoJ = r - Mu., has been calculated, one will choose bounds of the form cP = Uo + v, t/J = Uo + w and then use the principle of error estimation:
Mv
~
d[uoJ
which is equivalent to (5.15).
~
Mw
=
v
~
u* - Uo
~
w,
(5.16)
5
133
NUMERICAL APPLICAnONS
In order to obtain bounds v, w which are close to the actual error u* - uo, one could try to choose uo, v, and w from appropriate finite dimensional spaces and determine the parameters involved such that the inequalities in the premise of (5.16) are satisfied and w - v becomes "small" in some prescribed sense. This method, however, leads to a problem of (linear) optimization which, in general, will be rather complicated to solve. We are interested in much simpler methods. Our goal is not to obtain bounds which are as close as possible to the actual error, but to obtain sufficiently close bounds with as little effort as possible. We prefer to construct first Uo such that d[uoJ becomes "minimal" in a sense which has to be described and then to construct the bounds v, w. Of course, the details of the procedure, especially the procedure to minimize the defect, will depend on the type of the problem considered. However, the following general remarks concerning the calculation of the bounds will apply to most cases. After having computed uo, one could set up a sophisticated numerical procedure to find elements v, w which yield very accurate estimates of the error u* - Uo' Another possibility would be to choose a very "simple" element z with z ;;:: 0, M z ;;:: and calculate a constant a such that v = - «z, w = az satisfy the inequalities required, where little or no effort is put into the construction of the element z. Of course, for a given uo, the latter method, in general, will lead to less accurate bounds. Nevertheless, in all our examples this method (or a slight modification of it) is proved to be the most effective. This may be explained by the following arguments. If bounds of the special form - v = W = «z with a > 0 are used, the element z has to satisfy the inequalities z > 0, Mz > (except in the trivial case z = 0). Often it is mainly this requirement, which causes the difference w - v = 2az to be much larger than the error u* - uo. In particular, this is true for boundary value problems. Then trying to find art optimal z, in general, is not worthwhile. As an example, let
°
°
Mu(x)
= -u"(x)
(0 < x < 1),
Mu(O)
= u(O),
Mu(1)
= u(1),
and (u* - uo)(x) = s sin mnx for some mEN and c; > O. Then the choice = x(l - x) results in an estimate Iu* - uo/(x)::s; c;tm 2n2x(1 - x). Here, it would not help very much to find better functions z with z > 0, Mz > o. For example, one may want to exploit the estimate 41sin mnxl ::s; 3 - cos 2mnx and choose z = 6x(1 - x) + m- 2n- 2(1 - cos 2mnx). For this function, one calculates Iu* - Uo I(x) ::s; c;{im 2n2x(1 - x) + /6(1 cos 2mnx)}, which presents no significant improvement. For the boundary value problems which will be considered in Section 5.2.3 the error bounds show a similar behavior; they are larger than the error z(x)
II.
134
INVERSE-POSITIVE LINEAR OPERATORS
roughly by a factor «m", where rx is a constant and m the number of free parameters in determining Uo' This factor can be large, in particular, if the accuracy desired can be reached only for a large m. Nevertheless, using such simple bounds v, w can often be recommended, as the numerical results and the following discussion may show. Suppose now that for a given approximation Uo elements v, w satisfying Mv :::;; d[uoJ :::;; Mw have been calculated-with considerable numerical effort -such that v:::;; u* - Uo :::;; w is a very accurate estimate. Then U1 = Uo + t(v + w) in general, is a much better approximation. For this approximation and z = t(w - v) we have - Mz :::;; d[u 1 J :::;; Mz and -z:::;; u* - U1 :::;; z and hence also z ~ 0, Mz ~ o. This estimate for u* - U1 may be very poor. However, the poor estimate - z ~ u* - U1 :::;; z for the error of the better approximation U1 and the accurate estimate v ~ u* - U o :::;; w for the error of the original approximation yield exactly the same bounds for u*, namely, v + Uo = -z + U1 :::;; u* :::;; z + U1 = W + Uo' Because of these relations we suggest that one does not use much additional effort in constructing bounds, but rather calculate a better approximation in the first place. As mentioned before, the way to proceed will depend on the type of the problem considered. Sometimes it is advantageous to use bounds of the form v = - «z, w = f3( with two prescribed "simple" functions z, (. Then one obtains the following method, which we used for all our special algorithms of error estimation for problems of type (5.14). Method of Approximation and Estimation. Choose elements CPi E R (i = 0, 1, 2, ...) for constructing an approximate solution Uo = CPo + L7'=1 'YiCPi' Calculate the constants 'Yi (i = 1,2, ... , m) such that the defect d[uoJ = r - M CPo - L7'= 1 'Yi M CPi becomes" minimal" in a sense which has to be prescribed. Choose appropriate elements z, , in R and calculate constants «, 13 with -rxMz : :; d[uoJ :::;; 13M'. Then the error estimate - az :::;; u* - Uo :::;; 13' holds for the solution u* of the given equation (5.14).
5.2.2
An initial value problem
To obtain exact error bounds for solutions of differential equations by the method described in the previous section, one has to estimate the defect d[uoJ(x) (very accurately) on the whole interval considered. This constitutes a difficult task. Considering a very simple initial value problem, we shall demonstrate how one can proceed. The following method can be generalized to nonlinear equations and systems of nonlinear equations (see Section 111.6.2 and Example V.2.4). Although for those problems additional difficulties will have to be overcome, the choice of Uo and the method to estimate the defect will be essentially the same as explained in this section.
5
135
NUMERICAL APPLICATIONS
Suppose that a given problem u'(x)
+ f(x,
u(x)) = 0
(0 < x
~
u(O) = ro
1),
(5.17)
with f(x, u(x)) = c(x)u(x) - s(x), C E CoCO, 1], s E CoCO, 1] has been "solved numerically," so that approximations iii for the values u*(x;) of the solution at certain points Xi are known, where 0 = Xo < Xl < ... < XN = 1. We want to estimate the errors u*(x;) - ili' The method of error estimation explained in the following can be used for arbitrary approximations ili' However, the estimation method is so constructed that it becomes most efficient for values ili obtained by a fourth order approximation method, such as the standard Runge-Kutta method. To apply the estimation principle of the preceding section, we have to find an approximate solution uo defined for all X E [0, 1]. We now assume, additionally, that N = 2/1 is even, X2k-1 = t(X2k-2 + X2k) for k = 1,2, ... , /1, and that on each interval (X2k-2, X2k) the functions c and shave uniformly continuous derivatives up to the order 5. Then we choose uo to be a piecewise polynomial function with the following properties: on the interval h = [X2k - 2, Xu], the approximate solution uo coincides with a polynomial of the fifth degree such that uo(x;)
=
ili'
=
U~(Xi)
j;
:= f(x;,
il;)
for
i = 2k - 2,2k - 1, 2k.
Obviously, the given problem can be written as Mu = r with the operator M: CI[O, 1] --+ ~[O, 1] defined by Mu(x) = u'(x) + c(x)u(x) for 0 < x ~ 1, Mu(O) = u(O), and r(x) = s(x) for 0 < x ~ 1, r(O) = roo This operator M is inverse-positive (cf. Example 3.4), so that for z E C I [0, 1] Ib[uo](x)1 ~ z'(x) Iro - uo(O) I ~ z(O)
+ c(x)z(x)
(0 < x ~ I)}
=
I(u* - uo)(x) I ~ z(x) (0 ~ x ~ 1),
(5.18) where b[uo] = - u~ - f(x, uo) denotes the defect of Uo with respect to the differential equation. We shall, however, choose here less smooth functions z such that for k = 1,2, ... , /1: z(x)
= Zk(X)
(X2(k-l)
Zk(X2(k-I»)
=
Gk-l
(0 ~ x ~ 1). (Other linear boundary conditions can be treated analogously.) For a given approximate solution Uo E Cz[O, 1] denote by O. For the set At of all M ~ (0 ~ 11 < 1) the assumptions of Theorem 6.3 are satisfied, In particular, one shows by elementary calculations that the above operator M 0 has all properties required of M 0 in that theorem. Consequently, all M" (0 ~ I] < 1) are inverse-positive. The inversepositivity of M = M 1 then follows from the fact that the Green function of M~ depends continuously on 11. 0 derivatives of
(L~,
Vi,~'
~,
Vi,~)
Problem
6.1 Find a set G E 1R 2 such that for each (h, c) E G the operator M determined by L[u] = uiv - bu" + cu and the boundary conditions u(O) = u"(O) = 0, u(l) = uf(l) = 0 is inverse-positive. 6.2
Duality
For many properties which occur in the theory of inverse-positive linear operators one can formulate equivalent dual properties. As examples, we consider here duals for the inverse-positivity itself and for the existence of a majorizing element. In the following, we assume that assumption (AI) in Section 1.1 is satisfied, that (S, ~) contains a strictly positive element and that M: R ---+ S denotes a linear operator. The elements of Rand Swill be denoted by u and V, respectively.
Theorem 6.5. (i) If the range MR contains a strictly positive element in S, then the following properties P and D are equivalent: P:
u E R, Mu
=
~ 0
D: For each t >
0
u
~
o.
in R* there exists an I =
It
>
0
in S* with 1M = t,
(ii)
The following properties IIp and Il , are equivalent:
II p : Il,:
IE S*, I M
There exists a z E R with z ~ 0
= I::j>
~ 0,
Mz
» o.
o.
Proof. (i) If P holds, then for each t > 0 in R* A (\ Bt = 0 where A = {(V, t)ES x IR: V ~ o.t < O},B t = {(V, t)ES x IR: V = Mu.t = /u, u E R}. Since A = {(V, t) E A : V » o} of- 0, there exists (to each such t) a functional (?' - f.1) of- 0 on S x IR with? E S*, f.1 E IR such that
