Multivalued Differential Equations 9783110874228, 9783110132120

Table of contents :
Chapter 1: Multis
§ 1 Upper Semicontinuity
1.1 Some Basic Notation
1.2 Upper Semicontinuity
1.3 Properties of Use Multis
1.4 Other Tests for Usc
1.5 Remarks
Problems
§ 2 Lower Semicontinuity
2.1 Lower Semicontinuity
2.2 Continuous Selections
2.3 Continuity
2.4 Locally Lipschitz Approximations of Usc Multis
2.5 Remarks
Problems
§ 3 Measurability
3.1 Measurable Multis
3.2 Measurable Selections
3.3 Approximation by Step-Multis
3.4 Some Consequences
3.5 Multis of Two Variables
3.6 Remarks
Problems
§ 4 Mishmash
4.1 Tangency Conditions
4.2 Bochner Integrals
4.3 Monotone Multis
4.4 Accretive Multis
4.5 Some Basic Facts about Banach Spaces
4.6 Remarks
Problems
Chapter 2: Existence Theory in Finite Dimensions
§ 5 Upper Semicontinuous Right-Hand Sides
5.1 The Usc Case
5.2 Counter-Examples
5.3 The Carathéodory Case
5.4 Some Consequences
5.5 Remarks
Problems
§ 6 Lower Semicontinuous Right-Hand Sides
6.1 The Lsc Case
6.2 The Carathéodory Case
6.3 Some Consequences
6.4 Remarks
Problems
Chapter 3: Solution Sets
§ 7 Topological Properties of Solution Sets
7.1 Elementary Properties
7.2 Invariance
7.3 Connectedness in the Usc Case
7.4 Connectedness in the Lsc Case
7.5 Funnels
7.6 Remarks
Problems
§ 8 Comparison of Solutions
8.1 Preliminaries
8.2 Extremal Solutions I
8.3 Extremal Solutions II
8.4 Related Problems
8.5 Gronwall’s Lemma
8.6 Convexification
8.7 Remarks
Problems
Chapter 4: Existence Theory in Infinite Dimensions
§ 9 Compactness Conditions
9.1 Two Examples
9.2 Measures of Noncompactness
9.3 The Usc Case
9.4 The Lsc Case
9.5 Remarks
Problems
§10 Noncompactness Conditions
10.1 Baire Category
10.2 Extreme Points
10.3 Proof of Theorem 10.1
10.4 Lipschitz Conditions
10.5 Monotonicity
10.6 Hyperaccretivity
10.7 Remarks
Problems
Chapter 5: Fixed Points and Qualitative Theory
§ 11 Fixed Points
11.1 Some (Geo-)metric Results
11.2 Weakly Inward Maps
11.3 Set-Contractions
11.4 Degree Theory
11.5 An Example
11.6 Remarks
Problems
§ 12 Boundary Value Problems
12.1 A Comparison Result
12.2 Sturm/Liouville Problems
12.3 Solutions in Closed Sets
12.4 Remarks
Problems
§ 13 Periodic Solutions
13.1 Reduction to the Regular Case
13.2 Another Fixed Point Problem
13.3 Examples
13.4 Remarks
Problems
§ 14 Stability and Asymptotic Behavior
14.1 Stability
14.2 Stability Tests
14.3 Asymptotic Behavior
14.4 Perturbations
14.5 Remarks
Problems
Appendix: Related Topics
A 1. Discontinuous Differential Equations
A 2. Implicit Differential Equations
A 3. Functional Differential Equations
A 4. Perturbations of Dissipative Right-Hand Sides
References
Index

Citation preview

de Gruyter Series in Nonlinear Analysis and Applications 1

Editors A. Bensoussan (Paris) R. Conti (Florence) A. Friedman (Minneapolis) K.-H. Hoffmann (Munich) M. A. Krasnoselskii (Moscow) L. Nirenberg (New York) Managing Editors J. Appell (Würzburg) V. Lakshmikantham (Melbourne, USA)

Klaus Deimling

Multivalued Differential Equations

w DE

G Walter de Gruyter · Berlin · New York 1992

Author Klaus Deimling Fachbereich 17 der Universität Warburger Straße 100 D-4790 Paderborn, Germany 1991 Mathematics Subject Classification: Primary: 34-02; 34A60 (34A40, 34B15, 34C25, 34D20), 34G20, 47H10, 49E10, 93C15 Secondary: 26E25, 28B20, 35R15, 34K05, 47H06, 49A10, 49A36, 49A50, 49E15, 54C60, 70K20, 90A16, 93B05, 9 3 D 0 5 Keywords: Multivalued maps, multivalued differential equations, discontinuous differential equations, fixed points of multivalued maps, control theory, treshold and dry friction problems ® Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability. Library of Congress Cataloging-in-Publication

Data

Deimling, Klaus, 1943 — Multivalued differential equations / Klaus Deimling, p. cm. — (De Gruyter series in nonlinear analysis and applications ; 1) Includes bibliographical references and index. ISBN 3-11-013212-5 (acid-free paper): 1. Differential inclusions. 2. Differential equations, Nonlinear. 3. Fixed point theory. 4. Control theory. I. Title. II. Series. QA371,D35 1992 515'.352-dc20 92-18953 CIP

Die Deutsche Bibliothek — Cataloging-in-Publication

Data

Deimling, Klaus: Multivalued differential equations / Klaus Deimling. — Berlin ; New York : de Gruyter, 1992 (De Gruyter series in nonlinear analysis and applications ; 1) ISBN 3-11-013212-5 NE: GT

ISSN 0941-813 X © Copyright 1992 by Walter de Gruyter & Co., D-1000 Berlin 30. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Typesetting: Asco Trade Typesetting Ltd., Hong Kong. Printing: Gerike GmbH, Berlin. Binding: Dieter Mikolai, Berlin. Cover design: Thomas Bonnie, Hamburg.

Preface

The title of the book reveals that our subject (also named differential inclusions) has something to do with differential equations, say x' = f(t, x) on some interval J 2,

maybe the abstract version of a partial differential equation in case X is an infinitedimensional Banach space. So we assume that the reader has seen some simple examples (dim X > 2 is sufficient) from applied science where the unknown x( ) also has to satisfy certain initial/boundary conditions, or is required to be periodic in case / ( · , x) is periodic, etc. In such elementary cases / is at least continuous, if not Lipschitz or even continuously differentiable. This kind of regularity of / is no longer available in certain situations where one tries to control, say, the motion of a particle, determined by the "law" x' = f(t, x), through manipulation of parameters contained in / or by application of additional external forces at certain times, to achieve an optimal performance with respect to some given criterion, say reaching a target not at any but in minimum time. In this case we have a family of laws, say / ( · , ·, u) for all u from some U ) cz X are the values of F ; allowing F(GJ) = 0 is (very seldomly) convenient for purely formalistic reasons only. Talk of "the single-valued case" means F(w) = {/(ω)} on Ω, in which case we omit { }. An / : Ω X with /(ω) e F(oj) on Ω will be called a selection of F. We let F(Q) = (J F((o) and

graph(F) = {(ω, χ) e Ω χ X\ ω e Ω, χ e F(o)},

ωeΩ

the range and the graph of F. The only thing one always has to be aware of is F ~

(1)

1

( A ) =

{ ω ε Ω : Ρ ( ω ) Γ > Α φ 0 }

for A

α

X;

clearly, we write F _1 (x) if A — {x}, hence F _1 (x) = {ω e Ω: χ e F(co)}. (ii) In most cases X will be a real Banach space, i.e. a vector space over "the reals" [R with a norm |·|: = { r e l R : r > 0 } such that every Cauchy sequence is convergent, i.e. s u p \ x n + p — x„| -* 0 implies |x„ — x| -»• 0 (as η -*• oo) for some x e X. p> ι Then X* is the dual space, i.e. the Banach space of continuous linear χ*: X ->· IR with norm |x*| = sup{|x*(x)|: |x| < 1}. This is the special case Y = IR of L(X, y), the Banach space of continuous linear Τ from X into the Banach space Y, with | T\ = sup {I Tx|: |x| < 1}; we write L{X) for L{X, X). —

ο

For A 1} υ {0}, / : Ω —• Κ2 defined by /(Ο) = (0, 0) and /(1/n) = (η, 1/n), and A = {/(ί/η): η > 1}; here we have F ( l / n ) = {/(1/n)} and F(0) = R + χ {0}, hence F(l/n) η Α Φ 0 but F(0) η A = 0. However, F is use if / is locally compact, i.e. to ω 0 e Ω there exists r = r(a> ) > 0 such that f ( B , ( o j ) η Ω) is compact. Indeed, if A is closed, ω ω 0 and y e F(w„) η A then, given δ e (0, r), we have y„ in the compact set cönv f ( B ( c o ) η Ω) for all η > n , hence w.l.o.g. ( = without loss of generality) y„->y0, and therefore obviously y 0 e F( 0 such that F( 0, hence F(a>0) η Α Φ 0, a contradiction. The 0

0

δ

η

0

0

§ 1 Upper Semicontinuity

6

converse is not true; consider, for example, F(ct>) = {ω} χ M+ on Ω = [0, 1] 0}. Therefore, we introduce Definition 1.2. Let Ω φ 0 be a subset of a Banach space and X be a Banach space. Then F: Ω -»• 2 * \ 0 will be called ε-δ-usc if to every ω 0 e Ω and ε > 0 there exists δ = δ(ω0, ε) > 0 such that F{oo) 1

and define F : y ^ 2 x \ 0 b y J F(tu) (4)

on Ω

F(co) Σ Ψ Μ Ρ Μ

on Υ \ Ώ

If F has bounded values then they are uniformly bounded, since Ω is compact; hence it is easy to see that F is ε-d-usc on 7 \ Ω . In fact, F is ε-δ-usc on Y, since if ω 0 e 5Ω, (y*) c and yk ω0 then il/„(yk) Φ 0 means |j fc - ω π | < 2 p ( y k , Ω) < 2\yk - ω 0 |

0

as k -»· oo,

hence ω„ is close to ω 0 if k is large, hence F(con) cz F(co 0 ) + ß £ (0) for large k, and therefore F(yk) a F(cu 0 ) + 5 ε (0) since the latter is convex. Notice that the values of F

1.3 Properties of Use Μ ultis

7

on Y\Q may not be closed, since already the sum of two closed bounded convex sets need not be closed if dim X — oo. In case Ω is only closed, we get the same result by a similar formula,

(5)

F(co)

on Ω

Σ ΨΛω)Ρ(ωλ) λεΛ

on Υ\Ω

F(a>) = \

where Λ may be an uncountable index set. More precisely, consider the open covering Κ\Ω = U By, where By is an open ball around y with r\n diam By = sup{|x — x|: χ, χ e By} < inf{p(x, Ω): χ e By}, and let (UX)XSA be a locally finite refinement, i.e. Ux is open, Ux (1) and every y e 7 \ Ω has a neighborhood which meets only finitely many Uk\ then (x{y) < 1, £ ε 0 and p(x 0 , F(w„)) < \x„ — x 0 | -> 0, a contradiction, since p(x0, F(·)) is lsc; see also Problem 4. (ii) Upper semicontinuity can also be checked by properties of graph(F), to some extent. For example, if F is e-) compact for all ω e Ω, we have F use iff graph(F) is compact. Let us also summarize these parts. Proposition 1.2. Let Ω # 0 be a subset of a Banach space, X be a Banach space and F: Ω -»• 2 * \ 0 have closed values. Then (a) If F is ε-δ-usc then p(x, F(·)) is lsc for every χ e X. The converse is true if F ^ ) is compact. (b) If F is ε-δ-usc and Ω is closed then F has closed graph. If graph(F) is closed and F(Ω) is compact then F is use; in particular, this is the case if graph(F) is compact. 1.5 Remarks. 1. Proposition 1.1(d) can be used to get a so-called "topological degree" for use multis. This will be explained in § 11. Another useful approximation will be given in Lemma 2.2.

Problems

9

2. As mentioned before, in most applications where use multis play a role it is essential that F has convex values. For such F, certain properties can also be characterized in terms of all "support functions" s(x*, F ( · ) ) , defined by s(x*, F(ci>)) = sup{x*(x): χ s F(ω)}

for x* e X*,

which is no surprise in the metric case, since for convex C the distance p(x, C) can be represented in terms of functionals x*, but may also be helpful if one considers X with its weak topology σ(Χ, X*); see Problem 12. 3. The two use concepts introduced here are not lonesome in the vast literature. An old one of Kuratowski [1], sometimes called "property K " , is F(a>) = Π Ρ ( Β δ ( ω ) η Ω )

on Ω,

0

while those influenced by Cesari [2], [ 3 ] often prefer his "property β " , i.e. F(co) =

P ) c ö n v F(B0(OJ) η Ω )

δ>0

on Ω;

we will not go into this here, but simple relations are easy to check in Problem 13. N o w , we recommend to solve the following problems, since some routine is expected in subsequent sections.

Problems. 1. Find an ε-ό-usc F with closed bounded values such that F is not use. Hint: Consider e.g. Ω = {l/n: η > 1} υ { 0 } 1} be the i standard base of X, i.e. eni = 1 for i = η and eni = 0 for i # n, define F{l/n) — E_ + - e n + 1 > and find the smallest possible F(0). " J 2. Find an e- ) φ 0 on Ω. Then Fj η F 2 is use. Hint: Consider Μ = { ω e Ω: F^co) η F2(cü) c V} with open V, notice that F ^ c o ) η ( F 2 ( c o ) \ F ) = 0 for ω e Μ and consider disjoint open neighborhoods of F ^ w ) and F 2 ( ( t f ) \ F , respectively. 3.

4. Find a compact Ω c: R and an F : Ω ->• 2 R \ 0 such that p(x, F(·)) is lsc for every χ e IR and F has compact values, but F is not e-)): χ e F(cu0)} = oo for ω Φ ω0, F is not ε-ό-lsc, but F is obviously lsc. Example 2.1. As a prototype of an lsc F we may choose F(t) = [0,1] for t e IR\{0} and F(0) = {1/2}. It is also obvious that the general lsc F: IR 2 R \ 0 with compact convex values is given by F(i) = [i

disjoint with (J A( = Ω and the C ; are nonempty closed subsets of X. i> 1 Definition 3.2. Let (Ω, si, v) be a measure space, X be a Banach space and F: Ω -»· 2X\ 0 have closed values. Then F will be called strongly measurable if there exists a sequence of step-multis Fn such that dH(F„(co), F(co)) -> 0 v-a.e. on Ω as η oo, where v-a.e. means v-almost everywhere, i.e. on Ω \ Ν with v(N) = 0. An / : Ω X is said to be strongly measurable if there are step-functions /„ such that |/„(ω) — / ( ω ) | 0 v-a.e. on Ω. We always assume that (Ω, si, v) is complete. It is then clear that a strongly measurable / has /(Ω) contained in a separable subspace of X, possibly after a change on the exceptional null set. Obviously such an / also satisfies f~l{B) e si for open B, hence strongly measurable implies (si, ^ - m e a s u r a b l e . e si for open Β is also clear for strongly measurable multis, hence they are ^/-measurable, but, as one may expect, the converse is not true, even if F is bounded and X is separable. Consider /

y/2

Example 3.1. Let X = I2, the sequences χ = (x„) c 1R with norm |x| = I £ xf 1


J 2k with cok e {0, 1}, let

24

§3 Measurability F(co) = {χ e X: \x\ < 1, xk = 0 if wk = 0}.

Since χ = £ xkek with ek as in Problem 1.1, we have Jt>l p(x, F(a>)) =

Xjej,F(a))\,

ρ

Xjej, F(o)) I = cn i(x)

on

i- 1 i 2" '2'

for i = 1 , . . . , 2", hence F is .«/-measurable by Proposition 3.1. Suppose F is strongly measurable. Then we find a μ-null set Ν 1. Then F(co) has an em which is not in F{(on), or vice versa, hence dH(F(co), Ρ(ω")) = 1 for all n, a contradiction. This is Example 3.4 in Hiai/Umegaki [1]. Now, let us concentrate on (J, i f , μ) with J = [a, b] c [R, since this is the main case we need in later chapters, and since other complete (Ω, si, v) with "regular" ν can be handled similarly. Let us say that F is Lusin if, given ε > 0, there exists a closed Je a J with /i(J\J £ ) < ε such that F\Jc is continuous. Evidently, a step-multi F = £ χΑ. C{ i> 1 has this property, since the At c J can be approximated from inside by compact sets, as close as we want w.r. to μ. Therefore it is also clear that a strongly measurable F is Lusin. Conversely, if F is Lusin with closed values then F is strongly measurable; notice that we have J = (J J„ with disjoint J„, μ(7 0 ) = 0, Jn compact and F\Jn continuΠ>0

ous, hence uniformly continuous, for η > 1. So, being "Lusinian" is nothing else than strong measurability, but lets us hope for retreat to the continuous case. Now we have Proposition 3.3. Let J = [a, b] c [R, X be a separable Banach space and F: J -> 2 X \ 0 have compact values. Then F is strongly (S^-)measurable iff F is measurable iff F_1(ß) e ¥ for closed B. Proof. If F is strongly measurable then F is Lusin, hence F _ 1 ( ß ) η Jc, with Jt as above, is closed for closed B, and therefore F~l(B) e Z£ for closed B, which implies measurability; see Problem 1. So it is sufficient to show that the latter implies strong measurability. By Problem 2.11, the space (%d H ) of nonempty compact subsets of X is complete and separable. Let g(t) = {^(i)} on J. Then g~l(B) e i f for open Β 0, there exists a closed Jecz J with / i ( J \ J e ) < ε such that F\JexD is use (continuous). F will be called almost Isc if, given ε > 0 and 0 φ C c= D compact, there exists a closed Jta J with ß(J\Je) < ε such that F | Jt;XC is lsc and span F(Je χ C) is separable. We always write w( ) e F(·, υ(·)) instead of "w is a strongly measurable selection of F(',v()T. Let us now indicate how we find such selections under appropriate conditions. As usual, CX(J) denotes the Banach space of continuous v: J->X with norm |υ| 0 = max \v(t)\, and in case X = IR the index IR is omitted. j Proposition 3.5. Let X be a Banach space, J = [0, a] 2X\0 have compact values. For ν e CX(J) we then have (a) Let F(t, •) be use and F(-,x) have a strongly measurable selection. Then there exists w(·) e F(·, D(·))· (b) If F is almost lsc and z: J -*• X is strongly measurable then there exists w( ) e F(·, i>(·)) such that \z(t) - w(f)| = p(z(t), F(t, v(t))) a.e. on J.

3.6 Remarks

27

Proof, (a) Consider step-functions vn-+v on J. Then there exists w „ ( - ) e F ( · , u„(·))· Possibly after a change on a null set we have X0 = span (J wn(J) separable, n> 1

G„(t) = {wft(i): k > n} compact,

since F(t, u(f)) is compact. It is also obvious that p(x, G„(·)) is measurable for every χ g X0, hence G„ is strongly measurable by Proposition 3.3. Let G(i) = Q G„(£). n> 1 Since a decreasing sequence of compact sets has nonempty intersection, we have G(i) φ 0 on J. So, we either use a simple extension of Proposition 3.4(a) to the present case, or simply observe p(x, G( ·)) = sup p(x, G„(·)) η

for all χ e X0

to get measurability of G. Hence Proposition 3.2 yields a measurable selection w of G, which is strongly measurable since w(J) c= X0 and satisfies w( ·) € F( •, ·)), since F{t, ·) is use and v„(t) v(t) as η GO. (b) Evidently, this part follows from Proposition 3.4(b) if G = F{·, r(·)) is strongly measurable. Applying the definition of almost lsc for the compact C = v(J) and using continuity of ν once more, it is clear that F(·, u('))L range in a separable XE cz X, hence measurable by definition of lsc, and therefore strongly measurable by Proposition 3.3. Hence G is Lusin. • Let us finally remark that Proposition 3.4(b) and Proposition 3.5(b) can also be considered as simplest versions of "implicit function theorems", since if we let f(t, χ) = Iz{t) - x|,

W(t) = F(t, i>(i)),

q>(t) = p(z(t), W(t))

then φ is measurable and 0 e f(t, W{t)) — φ(ί), and the results tell that we find a measurable w with w(t) e W(t) such that 0 = f(t, w(i)) — (p{t) a.e. Clearly, using the methods of this and the previous sections one can and has produced more general results of this sort. We shall not go into this; see Remark 3 below.

3.6 Remarks. 1. For separable X the situation is better than indicated in the first subsection. If F~l(B) est for all Be & then, in particular, for closed B. By Problem 1, the latter implies F _ 1 ( ß ) e si for open Β which in turn implies graph F e si ® From this property one can return to if (Ω, «sa/, v) is complete for some measure ν or if si is replaced by its completion si\ notice that F~i(B) = proj n [(Q χ Β) η graph F ] with proj fi (/4) = {ω e Ω: (ω, χ) e A for some x}, hence one needs the following remarkable "projection theorem". Proposition 3.6. Let (Ω, si, v) be α measure space, (Ω, si, v) be its completion and (X, d) be a complete separable metric space. Then every A e si ® $ has proj n (^l) e si. In particular,

§3 Measurability

28

A e JS? ®

has proj.,(,4) e

in case (Ω, si, v) = (J, l

* {ω: p(xh F((ü)) < r - 1/m},

where X = {x;: i > 1}. (c) If X is separable and F is measurable with closed values then graph(F) 6 si ® Hint: χ € F(cu) iff p(x, F(co)) = 0.

Problems

2. Let 0 < c e Ll(J)

29

and A 1} on J then F is if-measurable, and F is strongly measurable if it has compact values. 6. Let Ω cz R" be Lebesgue measurable, 0 φ Β 0, there may not be a closed Jc a At — {ω: (ω, χ) e A for some χ e R} with μ(Αι\^) < ε such that φ restricted to A η (J£ χ R) is continuous (hint: consider A = {(s, s): s e [0, 1]} and a non-measurable function on [0, 1]). However, φ is Scorza-Dragoni if it is measurable in ω, continuous in x, and A2 has a countable dense Μ c A2 such that Μ η Α(ω) is dense in Α(ω) for almost all ω s A1; see Lemma 2 in Deimling [1], 8. Let A0 c J be not in S£, A — {(s, s): s e A0} and F = [0, Υ]χΑ. Then F: J x J -> 2 R \ 0 is use in each variable, but not almost use. 9. Find an F: J x J but not almost lsc.

2 K \ 0 with compact convex values which is lsc in each variable

10. Let F: J χ X -* 2 * \ 0 be almost use with compact values, and let ν e CX(J). Then F ( · , t»(·)) has a strongly measurable selection. 11*. Let X be separable and F, G: J 2 * \ 0 be measurable with closed values and such that F(t) η G(t) φ 0 on J. By Problem 3.1(c) and Proposition 3.6 it is clear that F η G is (si, ^ - m e a s u r a b l e , since graph (F η G) = graph (F) η graph (G). Is it possible to prove ^ - m e a s u r a b i l i t y of F η G without using the projection theorem? 12. Let J = [0, 1] c R and F(t) = {{x, y) e R 2 : χ > 0, 0 < y < ix}. Then F is lsc but not Lusin. This is from Cesari [1],

30

§ 3 Measurability

13. Let X — IR", F: Ω 2 * \ 0 have closed convex values, and Ρ{ω, ·) be the metric projection onto F(co). Then F is measurable iff Ρ(·, χ) is measurable for every χ e X iff there exist measurable functions uk: Ω -» X and l

on Ω.

This is contained in Theorem 3 of Rockafellar [1], Hint·. Let X = {xk:k>l],

uk = xk - P(\xk),

ctk = (uk(·), Ρ(·,

xk)),

and remember that a closed convex set C is the intersection of all closed half-spaces containing C. 14. Let X = IR", J = [0, a] c IR,/: J χ X -> X be such that / ( · , x) is measurable and f{t, ·) is continuous, [/: J - > 2 * \ 0 be measurable with closed values, and )>(-)e /(·, U(·)). Then y(t) = f(t, u(i)) on J for some u( ) e U(-); remember Definition 3.3. Hint: Consider {x eU(t): f(t, x) = >>(f)}. 15. Let A' be a real separable Banach space, J = [0, a] a R and F: J 2X\0 have compact convex values. Then F is immeasurable iff s(x*, F(·)) is measurable for every χ* ε X*; remember Remark 2 in §1.5. This is contained in Theorem III. 15 of Castaing/Valadier [1]. Hint: Corollary 3.1 for "only if"; for "if" notice that p(x, C) = sup {x*(x) — sup x*(y): |x*| < 1} c

if C is closed bounded convex, and {x*| c : |x*| < 1} is separable if C is compact. 16. Let X = R" and {ey,..., e„} be its standard base, i.e. eki — 1 for i — k and = 0 for π ι / k, hence χ = £ xiei for χ e IR"; we mention this only since the following concept i=l depends on the choice of base. Define the lexicographical order χ < y by x x < or x,· = y, for i — 1 , . . . , k and χ λ + 1 < yk+1 (if k < n) for some k e { 1 , . . . , n}. Notice that a compact Α Φ 0 has a unique minimal element w.r. to < , denoted by lex min A. Now, let J = [α, fr] t= f j and F: J -»• 2 * \ 0 be =5f-measurable with compact values. Then f(t) = lex min F(t) is a measurable selection of F. This was used in Filippov [7], remember Remark 3, for F(t) = {ueU(t):f(t,

u) = y(t)}

and can also be used to get more sophisticated things; see e.g. the references in Problem 9.17.

§4 Mishmash

This section is a collection of various things, some of which are essential for subsequent chapters, while others are mentioned more or less "for completeness", or to make certain facts about infinite dimensions easier to understand. X will always be a real Banach space, "real" only to avoid writing Re x* instead of x* when it comes to inequalities in terms of x* e X*, the dual space of X.

4.1 Tangency Conditions. Let D e 2 * \ 0 be closed and f . D ^ X be continuous. Suppose u: J = [0, 0 and y e D, hence λ -> oo yields x*(x) = supx*(z), and also x*(;y) > 0 since 0 £ TD(x). Thus, D (c) is true. Finally, if xn -> x 0 and x„ e D then lim p{z, TD(x„)) < lim (\z - A(y - x 0 )| + λ\χΗ - x0|) = \z - k{y - x 0 )|

for ζ e Χ, λ > 0 and y e D, hence TD( •) is obviously lsc.

•

The following pictures reveal more about TD( •); notice that we draw χ + T D (x) instead of TD(x).

iΓ VVW'AWW'sV'AiiJiWA *0

/ ^X

Figure 4.1

D

J

, jLt-H" W

Figure 4.2

Fig. 4.1 shows that 7^(x 0 ) is not convex and TD( ) is neither lsc nor use at x 0 . In Fig. 4.2 one can see that sup{p(z, 7^(χ)): ζ e 7^(χ 0 )} = oo, hence TD( ) is not e-ι

Ι-χ,-Ιμ^) < oo,

in which case we define

J

v(t) dt = Σ

;>ι

x^(Ai).

Now, if ν is strongly measurable and |υ(·)| e L1 (J) and (v n ) is a sequence of stepfunctions with vn(t) -* v(t) a.e. as η -*• oo then

v(t)=lVn{t) " [0

w n ^ otherwise

+ W)!

defines another sequence of step-functions with the same properties, but also J |0„(t) — u(i)| dt -> 0 as η

j

oo, by the dominated convergence theorem. We say that

v: J -> X is (Bochner-) integrable

if ν is the a.e.-limit of step-functions v„: J

|υ„(·)Ι e L ^ J ) such that J WO - t>„(OI dt -*• 0 as η

j

J v(t) dt = lim J v„(t) dt,

notice that

J

X with

oo, and define

dt = J χ χ ( 0 » ( 0 dt

for Α ε

υ„(ί) dt J is Cauchy and that, as usual, the limit is independent of the

choice of (υ„) in the class of step-functions specified above. Evidently, ν is integrable over J iff υ is strongly measurable and W ' ) l e Ll(J). Now we can do the usual things, say, identify two such functions which are equal a.e., introduce the Banach spaces LPX{J) with norm \v\p = ^J\v{t)\p dt^

for 1 < ρ < oo, and remember that conclusions

drawn from integrals of ν hold only for almost all t, since we actually should consider elements of L £ ( J ) as equivalence classes w.r. to the relation "equal a.e.". Later we shall use several times the connection of these abstract spaces to the "real world", namely, if |i?„ — -> 0 then vkn -*• ν a.e. for some subsequence of ( u j . We also let (J) be the Banach space of essentially bounded strongly measurable ν with

IwU = inf| sup \v(t)\: Ν e & and μ(Ν) = ol. UW

J

(ii) If ν s Lx(J) then u(t) = f φ ) ds(=

ο

^

Jx [CM] (s)d(s) ds j defines a function u e

j

'

CX(J)

which is a.e. differentiable with u'(t) = u(0; in particular, u is absolutely continuous

35

4.3 Monotone Multis

(a.c.) in the usual elementary sense, i.e. to ε > 0 there exists 0 such that m £ I w(i; + j) - u(ij)| < ε

whenever

1= 1

m £ l*(Ji) < 1=1

for disjoint Jt = (i„ t i + 1 ] c J and msN. The converse is true, for example, if X is reflexive; that one has to worry about this in infinite dimensions is made obvious by Example 4.2. Let X = (c)0 be as in Problem 1.1 and u{t) = £ l/n sin(nt)^«· If " is π> 1 differentiate at t e J then u'„(t) = cos(nt) for all η > 1, but cos(nr) 0 as η -*• oo is wrong for every t e R, since cos(2/c + l)r = cos(2kt) cos t — sin(2/cf) sin t. Hence, u is nowhere differentiable, but |«(r) — u(s)| < |t — s| for all t, s e 1R. (iii) Let F: J 2 * \ 0 have closed values and an integrable selection. Then one may consider the "integral" of F over J as the set

|F(s)ii5 =

w( ) e F ( ) a n d w e L i ( 7 ) | .

t If we let G(t) — j-F(s) ds = | χ [ 0 (](s)F(s) ds for t e J then G(t) is nothing else than the ο J set of points attainable at time t by solutions of u' e F(t) a.e. on J, u(0) = 0, given that one defines the solution concept for such initial value problems this way, as we do in the sequel. Of course, we shall try to prove theorems for cases where F depends nontrivially on u too, and this also requires different tools. Therefore, this special case will only be considered in problems at the appropriate place; see e.g. Problem 6 for additional training to § 3.

4.3 Monotone Multis. Let X be a real Hilbert space with inner product ( ·, ·). Besides the multis characterized by semicontinuity and compactness properties there is another interesting class, characterized by a certain behavior w.r. to (·, ). An F: D c X 2 * \ 0 is said to be monotone if (4)

(Fx - Fy,x - y)> 0

on D χ D,

which is understood as (u — ν, χ — y) > 0 for all u e Fx and all ν e Fy. In case — F is monotone we say that F is dissipative. If AT = IR and F is single-valued, this kind of monotonicity is evidently the usual one, i.e. χ < y implies / ( x ) < f(y); in case dim X > 1, it has little to do with monotonicity relative to some partial ordering, but is a generalization of positive semi-

§4 Mishmash

36

definite linear operators, since (4) is the same as ( T x , χ) > 0 on X if Fx = {Tx} and Τ is linear on D = X. The main results about such multis are already in several books. Therefore, we concentrate on a few basic οthings, and mention a little bit more at the , appropriate place later on. For the case D Φ 0 some remarkable properties are contained in Proposition 4.2. Let X be a real Hilbert space, D c X with D φ 0, and F: D -*• 2 * \ 0 be monotone. Then ο (a) F is locally bounded on D. (b) There exists a Gs-set D0 a D, dense in D, such that F\Dq is single-valued. Remember that Ga-set is the name for intersections of countably many open sets. Both parts are easy consequences of Baire's category theorem, see §4.5, Theorem 23.2 and Theorem 23.6 in Deimling [3], for example. On dD such an F may be unbounded; see Problem 7. However, in many situations one is also led to monotone multis defined on a set D which has empty interior and is not closed. For example, it may help to consider boundary value problem (5)

u"=f{t,u)

on J = [0,1],

K(0) = H(1) = 0

in X = L2(J), i.e. to write this as Fx = 0 with Fx = — x" + / ( · , x(·)) on D = {x e C 1 ^ ) : x" e X, x(0) = x(l) = 0 , / ( · , x(·)) e X}; notice that F: D -*• X is monotone if / is increasing in the second variable, since partial integration yields ι (Fx - Fy, χ - y) = J [x'(s) - / ( s ) ] 2 ds ο 1 + J U(s, x(s)) - f(s, y(s))](x(s) - y(s)) ds > 0. 0 In such cases one can only hope that F: D-+ 2 X \ 0 is maximal monotone, i.e. such that there is no proper monotone extension F of F, which means that "F monotone and graph(F) ψ graph(F)" is impossible. Evidently, F has this property iff (6)

(Fx — y 0 , χ — x 0 ) > 0 on D

implies

x 0 e D and y 0 e Fx0;

notice that if x 0 φ D then F = F on D and F x 0 = {y 0 } defines a proper monotone extension of F, and if x 0 e D but y0 φ Fx 0 , then Fx = Fx for χ φ x 0 and Fx 0 = F x 0 u {y 0 } is such an extension. Consequently, every monotone operator has a maximal monotone extension; consider all monotone F with graph(F) cz graph(F), partially ordered by graph inclusion, and apply Zorn to see this. If F is single-valued, its

4.3 Monotone Multis

37

maximal extension may be a multi; consider e.g. / = on IR which has the maximal extension F = [0, 1]*{0} + X(o,oo)· Now, the usefulness of maximal monotonicity is due to the following characterization. Proposition 4.3. Let X be a real Hilbert space and, F: D cz X Then F is maximal monotone i f f

2 * \ 0 be

(F + XI)(D) = X for some λ > 0 i f f (F + XI)(D) = X for all

monotone.

λ>0.

See e.g. Theorem 12.5 in Deimling [3], specialized to Hilbert space. Let us only remark that "if" is easy since, given (Fx — y 0 , χ — x 0 ) > 0 on D, we find an x t e D such that λχ0 + _y0 e F x ! + λχγ, hence we get A|x 0 — Xj | 2 < 0 by letting χ = X! in the inequality. "Only if" is established by showing (7)

(Fx + x 0 , χ — x 0 ) > 0

on D for some x 0 e X,

which is possible by means of a tricky reduction to Brouwer's fixed point theorem (see § 11); by (6) we then have x0e D and — x 0 6 F x 0 , hence 0 e (F + I)(D), and therefore (F + I)(D) = X since, given y e X, F — y is also maximal monotone. Finally, the last equivalence in the proposition follows easily from Banach's fixed point theorem: notice that (Fx + λχ — Fy — Xy, χ — y) > A|x — y\2 hence (F + A/) - 1 : X

on D χ D,

D is Lipschitz of constant \/λ and

y e Fx + μχ

iff

χ = (F + kl)~y{y + (λ -

μ)χ),

where the right-hand side is a strict contraction for μ sufficiently close to λ\ hence the result follows for arbitrary μ by repetition of this argument. Let us illustrate by means of Example4.3. (i) Let / = and λ > 0. Then ( / + XI){R) = R \ [ 0 , 1), h e n c e / + λΐ is not onto, but F + XI with F = [0, 1]χ { 0 } + χ ( 0 αο) is. (ii) Concerning the abstract setting of (5), i.e. F x = —x" + / ( · , x(·)) on D α X = L2(J), we see that (F + λΙ)χ = 0 with λ > 0 does not appear to be easier to solve than F x = 0. So, it is good to notice that F is the sum of two monotone F1 and F 2 , namely F x x = — x"

on

F2x = / ( · , * ( · ) )

D 1 = { j c e C 1 ( J ) : x " e X , x ( 0 ) = x ( l ) = 0}1 on

D2 =

{xeX:f(-,x(-))eX}.

Now, Fl is maximal: either we write down the general solution of χ" — λχ. — —y and adjust to the boundary condition x(0) = x(l) = 0 or, since we don't want to work that

§4 Mishmash

38

hard, observe that {sin(/ra·): η > 1} is an orthogonal base for X, hence x(t) = Σ απ δϊη(ηπί) n>1

and

j(r) = £ ß„ sin(wri) n>1

yields a„ = β„/(η2π2 + λ), in particular Σ η2 1 ing F2, let / ( · , p) be measurable, f(t, ·) be continuous increasing and f(-,0)eX. Consider y e X and g(t, p) = f(t, ρ) + λρ with λ > 0. Since g(t, p) > f(t, 0) + λρ for ρ > 0

and

g(t, p) < f(t, 0) + λρ for ρ < 0,

we get g{t, R) = R for all t, hence a unique measurable χ such that g(t, x(i)) = y(t), and χ s X, since |χ(ί)| < \/λ\γ{ί) — f(t, 0)|. Thus, F2 is also maximal; notice that D2 may be much smaller than X, say D2 = L6(J) for f(t, p) = p 3 . So, F is the sum of two maximal monotone Ft and the question is: under which conditions is Fx + F2 maximal monotone o n D = D 1 n D 2 ? Some reasonable criteria exist; see e.g. Theorem 11.4 and the exercises to § 11 in Deimling [3]. The simplest one is Di η D2 φ 0 which, in our case, is satisfied if \f(t, ρ)| < c(l + |p|), since then D2 = X. To see that F t + F2 is not always maximal if the Ft are, change F2 to F2 χ = —χ" on D2 = {xe Cl{J)\ x'(0) = x'(l) = 0, x" e X}, and notice that χ" — λχ = t has no solution satisfying x(0) = x(l) = 0 and x'(0) = x'(l) = 0; see also Problem 9. By what we have mentioned so far, use of maximal monotonicity rests on an observation much older than this, namely on the fact that the "resolvent" of an operator may be much better than the operator itself. Therefore, let us mention how good it really is. For this purpose it is convenient to write (8)

Rxx = (/ + AF)"1 x,

Fxx = λ~ι (I - Rx)x e FRxx

for λ > 0 and

xeX.

Proposition 4.4. Let X be a real Hilbert space and F: D α X 2 * \ 0 be maximal monotone. Then (a) F has closed convex values and closed graph. D and F(D) are convex. (b) Rx.X->D is Lipschitz of constant 1 and Fx is Lipschitz of constant 1/Λ. (c) Rxx χ on conv D and |F^x| oo if χ φ D and λ -* 0 + . (d) |FAX| < |/(x)| and Fxx^f{x) on D as Λ - • ( ) + , where /(x) e F x has |/(x)| = p(0, Fx); remember Problem 2.10. This follows from the later Proposition 4.7 by specialization and simple modifications.

4.4 Accretive Multis. While Hilbert space is good enough for some examples, other models suggest to develop similar concepts for more general real Banach spaces X, and this is possible to some extent. Notice first that we have identified a Hilbert space

4.4 Accretive Multis

39

with its dual space, hence it was sufficient to use the identity I. In the more general situation the substitute for I is the so-called duality map X 2**\0, defined by (9)

= {x* e X*: |x*| = |x| and x * ( x ) = |x|2},

where &χ Φ 0 is a consequence of Hahn/Banach; see the next subsection. It is not difficult to prove that has the following properties. Proposition 4.5. Let X be a real Banach space and !F\X

2x'\0

be defined by (9).

Then (a) is convex and σ(Χ*, X)-closed. JF(Ax) = X!Fx for all A e R . (b) BF is single-valued iff X* is strictly convex. (c) X* is uniformly convex iff X X* is uniformly continuous on bounded sets. For a proof see e.g. Proposition 12.3 in Deimling [3]. Some particular examples are contained in Example 4.4. (i) Let Ω cz IR™ be measurable and X = L P ( Q ) with 1 < ρ < oo. Then X* = Lq(Ω), with 1 jp + l/q = 1, is uniformly convex, hence strictly convex. Consequently, it is easy to verify that the single-valued & is given by (Ρχ)(ζ)

= \χ\1~ρ\χ(ξ)\'-2χ(ξ)

and

supp ν is the smallest closed A c Ω such that Jy(£)dv(£) = 0

for all ye X with

supp y = {ξ e Ω: j>(£) Φ 0} «= Ω\Α,

Ω

and ν sgn χ > 0 means J yfö) sgn χ(£) άν(ξ) > 0 for all y e X with yfä) > 0 on Ω; see e.g. Example 12.2 in Deimling [3]. By means of SF it is possible to define substitutes for the inner product, namely the so-called semi-inner products ( · , •)+: Χ χ X -*• IR, given by

40

§ 4 Mishmash

(10)

(χ, y)+ = max{y*(x): y* e !Fy)

and

(x, y)_ = min{y*(x): y* e iFy)

or equivalently /(11) m

t(x,y)\ = I|y|i i -lim ^ + +

ίχ

f->0+

Ι ~ Μ,

,(χ, y)_ χ = ι|y|ι >· lim b l - b - ^ l

t

o+

t

These limits exist since + x| is convex in χ for fixed y. By definition of ^y it is also clear that (x, y)_ < y*(x) < (x, y)+ for all y* e !Fy since for example, Μ \ y + txI - \y\2 > y*(y + tx) - y*(y) =

ty*(x).

For a complete proof of the equivalence see again Proposition 12.3 in Deimling [3]. The following result shows that (·, ·)+ have at least some properties of an inner product. Proposition 4.6. Let X be a real Banach space.

Then

(a) (x, z)± + {y, z)_ < (x + y, z)± < (χ, z)± + ( y , z)+

and |(x, y) ± | < |x| |y|, (x + ay, y)± = (x, y ) ± + a | y | for a e R, (ax, ßy)+ = 0

on D χ D,

and dissipative if — F is accretive, i.e. {Fx — Fy, χ — means again (u — ν, χ —

< 0 on D χ D; of course, this

< 0 for all ue Fx and all ν e Fy. A n accretive F is called

maximal accretive if there is no proper accretive extension, i.e. (Fx — y, χ — x 0 ) + > 0 on D

implies

( x 0 , y ) e graph(F),

and hyperaccretive (or /n-accretive in some papers) if (F 4- λΙ)(Ω)

= X for some λ > 0.

This time " h y p e r " is stronger than "maximal", i.e. a hyperaccretive F is maximal accretive but the converse is not true even if X and X*

are uniformly convex; see

Calvert [ 1 ] for a counter-example. Therefore, we concentrate on hyperaccretive F, in which case " f o r some λ > 0 " is again equivalent to " f o r all λ > 0", Rx and Fx are again defined by (8) and have at least some of the properties mentioned in Proposition 4.4; of course, we cannot get all of them since ! F is nonlinear if X is not Hilbert, i.e. ( · ,

)±

are nonlinear in the second argument, but we still have Proposition 4.7. Let X be a real Banach space, F: D α X and RK,Fxbe

hyperaccretive

defined by ( 8 ) . Then

(a) graph(i 7 ) is closed; Fx is closed convex if X* is uniformly

is strictly convex; D is convex if X

convex.

(b) Rx is Lipschitz (c) Rxx-^x

2X\0 be

of constant 1; Fk is accretive and Lipschitz

of constant 2/Λ.

on D as Λ . - • ( ) + , hence on conv D if X is uniformly

convex;


/ ( x ) [ e F x as in Proposition

4.4(d)] if X is uniformly

convex

too. Proof.

1. Since ( · , ·)+ is use and F is maximal accretive, it is easy to check that

graph(F) is closed and F x 0 = { y e X: (Fx — y, χ — x 0 ) + > 0 on D}, hence F x 0 is closed convex if X*

is strictly convex, since then & is single-valued. Next, it is no

problem to verify (b). Then 0 < ( F x — Fxx, Fxx)+ p(0, Fx), and therefore Rxx

is obvious by (8), hence |FAx|
|x| imply x„ x. So, we still have to prove that Rxx χ on cönv D in such an X. 3. Let x,yeD and x, = tx + (1 — t)y for t e (0, 1). Then, as λ -»• 0 + and t is fixed, we may assume Rxxt-^z, have | R x x t — x| < |x ( — x| + λρ{0, Fx) and the same for y, hence |z — x| < (1 — t)\x — and \z — y\< t\x — but also \z - y\ > \x - y\ - \z - x| > i|x - y\ = |x t - y| and similarly for |ζ — x|. Thus, \z — x| = |xf — x| and |ζ — y\ — |x t — y|, and therefore ζ = χ, if ζ e {xs: s e R}. In case ζ φ {xs: s e R} strict convexity implies l / M I < (1 - 01* - y\and

\f(y)\ < t\x - y\

hence |x — >>| = | / M — / ( y ) | < |x —

for

/ ( x ) = t{z - χ) + (1 - t)(xt - x)

a contradiction. So, Rxx,-^xt

and

|x ( — x| < lim IR x x t — x| < lim | R x x t — x| < |xf — x|, Λ-Ό+ Λ-0 + hence Rxx -+ χ on conv D, and therefore on cönv D since Rx is Lipschitz.

•

This result certainly made clear that the situation does not become much worse if we replace a Hilbert space by a uniformly convex X such that X* has the same property, but the more we leave this case the less we are able to get in general. As mentioned before, we did not intend to give a survey of monotone/accretive multis, but wanted to introduce some things that are inevitable if one runs into mixed cases, say Fj + F2 with Fi maximal dissipative and F2 of the sort considered in previous sections. Let us finally make sure, by a kind of appendix to this basic chapter, that also a reader with little experience in infinite dimensions will understand what we mean by Hahn/Banach, Baire category, etc.

4.5 Some Basic Facts about Banach Spaces, ο(i) A complete metric space (M, d) is of second category, i.e. Μ = [ j M„ implies M„ φ 0 for some η eN. This is usually n> 1

called Baire category theorem (though there are many of them) and can be found in every reasonable introduction to functional analysis, since it is basic for several important things in this area. Sometimes one can use its equivalent version in terms of open sets, i.e. On = Μ if On is open and On = Μ for every η e N, in other words: π> 1

4.6 Remarks

43

a countable intersection of open dense subsets of Μ is dense; notice that M„ = 0 iff M\M„

= M.

(ii) Let X be a real Banach space, XQ be a subspace, x£: X0 IR be continuous linear and φ: X - * Κ be sublinear, i.e. φ(λχ) = λφ(χ) for λ > 0 and ψ(χ + y) < φ(χ) + φ{γ). The Hahn/Banach theorem says that if |xjj(x)| < φ(χ) on X0 then there exists an x * s X* such that χ*|χ0 = x * and |x*(x)| < φ(χ) on X. (iii) Let I b e a real Banach space, x * e X* and r e R. Call Η = { χ e X: x * ( x ) = r } a hyperplane and { x e X\ x * ( x ) < r}, { x e X: x * ( x ) > r} the sides of H. Say that Η separates subsets A and Β of X if A is in one side of Η and Β is in the other one, and call Η a supporting hyperplane for Α c X if A is in one side of Η and Η η Α φ 0. As an easy consequence of Hahn/Banach one then obtains Mazur's separation theorem ο for convex sets, namely: Let C be convex with C φ 0. Then we have ο , (a) If (x 0 + X 0 ) η C = 0 for some x0e X and some subspace X0 then there exists an Η => x0 + X0 such that Η η C = 0. In particular, to x 0 e C\C there is a supporting Η containing x 0 . (b) If Cj φ 0 is also convex with C1 η C = 0 then there is an Η which separates C1 and C. (iv) Let Χ , Y be real Banach spaces. As immediate consequences of Baire one gets the open mapping theorem, saying that Τ e L(X, Y) with T(X) = Y maps open sets onto open sets, and the closed graph theorem which says that a linear Τ: X -*• Y with closed graph is already in L(X, Γ). (v) Let us also recall that a Banach space X is uniformly convex if to every ε e (0, 2 ] there exists 0 such that |x| = = 1 and |x — }>| > e imply |(x + y)/21 < 1 — 0. A uniformly convex X is also reflexive, i.e. if χ e X is considered as J ( x ) e X** by means of ε 0 (for some subsequence) would imply 1 = |x| < lim |(xm + x)/2| < 1 — 5(ε 0 ), a contradiction. m->oo Proofs for these and other facts can be found e.g. in Day [1], Dunford/Schwartz [1], Hille/Phillips [1], Köthe [1], Yosida [1], while Diestel [ 1 ] and Lindenstrauss/ Tzafriri [ 1 ] contain more about special Banach spaces. So, whenever no specific reference is given, we always have one or all of these in mind.

4.6 Remarks. 1. TD(x) from (2) is called "Bouligand contingent cone" by Aubin [ 1 ] and followers, while TD(x) has the name "intermediate cone" in Aubin/Frankowska [1], though these sets may only have λΚ α Κ for λ > 0 in common with what is

44

§4 Mishmash

generally understood by a cone, see § 8. The existence result mentioned in front of (2) was established by M. Nagumo in 1942 and "rediscovered" several times; see e.g. the remarks in §4 of Deimling [2], 2. A real inflation of such "cones" started shortly after F. Clarke's observation that it makes sense to develop a "generalized gradient" calculus for functions φ : X -»• (R which are only locally Lipschitz. He introduced his "tangent cone" (13)

f D ( x 0 ) = ] y e X: lim sup λ'1 [p(x + ky, D) - p(x, Z>)] = 0 f

for x 0 e D;

see Chap 2 in Clarke [1], Evidently, T D (x 0 ) is closed with 0 e f D ( x 0 ) , and it is easy to see that T D (x 0 ) is convex, even if D is not; in case D is convex it coincides with TD(x0) from (1); see also Problem 4. To make the difference between T D (x 0 ) and TD(x0) more transparent, notice that ye TD(x0) iff x 0 + kn yn e D for some l n - + 0 + and some yn y, but y e T D (x 0 ) iff to every (x„) y such that x„ 4- kny„ e D for all η > 1. For the role played by these two cones in qualitative study of monotonicity properties of solutions to x' = f(x) see Aubin [5]. More than 40 "cones" can be found in Ioffe [2], Except for some remarks, we shall only use TD(x) for obvious reasons; see Problem 5.1. 3. Though integrals for functions with values in a Banach space can also be found in some of the references given in §4.5, we shall mainly refer to Diestel/Uhl [1]. Early papers about integration of multis are Aumann [1], Debreu [ 1 ] and some references given there. Hiai/Umegaki [ 1 ] contains a collection of results for separable X, and for X = IR" one may also consult part IDII in Hildenbrand [1]; more will be mentioned in § 9. Some possibilities to define differentiation of multis will be considered much later in the problems to § 14. 4. The material contained in subsections 4.3 and 4.4 is standard today and can be found in several books. Besides the references already given, Brezis [ 1 ] is still a basic text for Hilbert space (and French-reading people), also Chap III of Pascali/Sburlan [1], or § 7 - § 10 of Browder [ 1 ] though some of the results given there are out of date; Barbu [1], Martin [ 2 ] or Pavel [ 2 ] are "more accretive". Problems. 1. (a) If D = Br(x0) c X and χ e dD then y e TD(x) iff (>*, χ — x 0 ) + < 0. (b) Let D be closed convex and kD c D for all λ > 0. Let x 0 e dD. Then y e TD(x0) iff x*(y) > 0 whenever x* e X * , x*(x 0 ) = 0 and x*(x) > 0 on D. 2. Let D c l b e closed and x 0 6 3D. Call ν e X an outer normal to D at x 0 if ν Φ 0 and ß|v|(xo + ν) η Ζ) = 0, and let Ν{χ0) be the set of all such v. If y e TD(x0) then (y, v)_ < 0 for all ν e N(x0), given that N(x0) φ 0. For X = IR" this is in Redheffer [1]; for general X see Proposition 5.1 in Deimling [2]. 3. If y 6 TD{x) then (_y, v)+ < 0 may not be true in the previous problem. Hint: Consider X = IR2 with |x| = \xt \ + |x2|,D = {x: x 2 < 0},x o = Oandy = (1, 0). This is Example 5.1 in Deimling [2].

Problems

45

4. Let D α X be closed, x 0 s dD and TD(x0) be defined by (13). Then (a) TD(x0) is closed convex and TD(x0) cz TD(x0). (b) y e TD{x0) iff to ε > 0 there exist δ = 0 such that (x + (0, A)B,(.y)) η D φ 0

for all λ > 0 and all χ e ^(XQ) η D.

This is Theorem 2.1 in Treiman [1]; here (0, A)A means of course {ta: 0 < t < λ, a e A}. (c) r D (x 0 ) = lim TD(x) if X = R", but for general X we only have lim TD(x) a fD(x0). X-*X0

X-'XO

For the first part see Cornet [1]. The second part is in Treiman [1]. Here, given F: D -*• 2 X \0, lim inf is defined as x-»x0

lim

F(x) =

X~*XQ

Π δ>U Π ( ^ W +

C>0

0

BM).

xeBj(x0)nD

5. Find an example where TD{x0) φ TD{x0) % TD(x0). What is TD(0) in the special case considered in Example 4.1? 6. Let J = [0, a] aa

|j>fc| = r,

inf \xk - )>t| > 0

k~*ao

for some r > 0. Then lim |xt +

k> 1

< 2r. Hint:

If wrong, one may assume r = 1,

k-* oo

1**1 ι. ΙΛΙ

+ ΛΙ

1 and

2·

12. Let X be a real Banach space and 1R = I R u { o o } be lsc convex with Dv = { x e Χ: φ(χ) < GO} Φ 0. The subdifferential δφ(χ0) δφ{χ0)

= { χ * e Χ*: φ(χ) > φ(χ0)

and Dd(/> = { x e Χ: δφ(χ)

is defined by — *o)

+ **(*

on

Φ 0}. Then φ and δφ: Χ -» 2X* have the following properties.

(a) φ is continuous on Dv (even locally Lipschitz on Dv if dim X < oo) ο ο — (b) If X is reflexive then ΌΒφ = ϋφ, ϋΒφ = Dv and δ φ is maximal monotone, i.e. 0 for x * e δφ(χ)

and y* e 1

such that by this kind of continuity problem (0.1) and the convexified problem (0.5)

x ' e f ] conv /(ß a (i, χ ) η D) S> 0

a.e. on J,

x(0) = x 0 e D

have the same solution sets. Hence, the problem is reduced to the use case considered before, convexity of the values F(i, x ) is redundant, but F ( i , x ) c= TD(x) on [0, a) χ D is needed to get (0.4) for this selection /. W e have counter-examples if this stronger condition is violated or F is only measurable in t and lsc in x, but also an easy extension to the situation where F is only almost lsc (remember Definition 3.3). A prominent example for the lsc case is ext cönv F, where cönv F is assumed to be continuous and ext A denotes the set of extreme points of A; see Proposition 6.2 and

Chapter 2 Existence Theory in Finite Dimensions

51

the corresponding consequence Corollary 6.2, the usefulness of which for some control problems will be indicated in the context of the next chapter. One may also try to unify the use and the lsc/continuous case by assuming, say, that the use F be continuous on the set of (i, x) for which F(t, x) is not convex. This is relatively easy if F does not depend on t, see Corollary 6.4, but unknown in the general case, unless one has no constraints, in which case only too complicated proofs exist.

§ 5 Upper Semicontinuous Right-Hand Sides

Given Χ = Rn, J = [0, a] c= R, a closed D^X, we are looking for a.c. solutions of (1)

x' e F(t, x)

a multi F:J

a.e. on J,

χ D

2X\0 and x 0 e D,

x(0) = x 0 .

Evidently the "safest" analogue of (0.2) from the introduction is (2)

||F(i, x)|| = sup{Iy|: y e F(t, x)} < c(i)(l + |x|) on J χ D, with

ceLl(J).

It is also obvious that the weakest form of (0.4) is now given by (3)

F(t, χ) η TD(x)

(4)

TD(x) = \yeX\

on [0, a) χ D

with

lim λ~ιρ{χ + ky, D) = 0 > for xe D.

Under these assumptions we are allowed to replace (2) by (5)

||F(i,x)|| 1 such that |x|0 = max |x(f)| < r — 1 for all j solutions of (1). So, let ψ: IR+ -> [0, 1] be continuous such that φ(ρ) — 1 for ρ < r — 1 and ι f / ( p ) = 0 for ρ > r, and consider (6)

F{t, x) = if/(\x\)F(t, x)

on D = D η 5,(0).

Clearly, (1) with F in place of F has the same solutions as (1), F has the same measurability/usc properties as F, the sets F(t, x) will satisfy the same assumptions as the F(t, x), and we have F(t, χ) η Tfj(x) φ ψ

on [0, a) χ D, since μΤΒ(χ) a TD(x) for all μ > 0.

Therefore we may forget the factor 1 + |x| in (2) and may assume c(t) > 1 on J. Then a the transformation _y(i) = x(c(i)), where c: [0, b]

ι

[0, a] with b = J c(s) ds, is the

0

inverse of J c(s) ds, yields the equivalent problem y' e F(t, y) a.e. on [0, b], y(0) = x 0

ο

with

5.1 The Use Case F(t, y) = [c(c(i))] -1 .F(c(i), y)

satisfying

53

||F(r, y)|| < 1 on [0, 6] χ D;

remember also Proposition 3.4(c). It is, of course, not exciting to start a section by such a "formalistic" remark, but the proofs to follow will show immediately how useful it is. 5.1 The Use Case. The basic existence result for autonomous problems (1) is Lemma 5.1. Let X = FT, D c= X be closed and F\ D ->2X\0 (a) F is use, F(x) is closed convex for all χ e D.

satisfy

(b) ||F(x)|| < c(l + |x|) on D, for some c > 0. (c) F(x) η TD(x) Φ0onD. Then (1) has a solution on ER+, for every x 0 e D.

Proof. Evidently, it is enough to prove existence on J = [0, a] for any a > 0, and we may assume (5) instead of (b). 1. Given e β (0, 1), we find v: J X such that ν is a.c. and r

(7)

»(t) = x 0 + J v'(s) ds, D η Β2ε(υ(ί)) Φ 0 on J, v(a) e D ο

(8)

v'(t) e F(D η B2c{v{t))) + Be(0)

a.e. on J

are satisfied. This can be established as follows. Consider Μ = {(», δ): δ e (0, β], ν: [0, δ]

Χ satisfies (7), (8) on [0, 2*\0 be use with closed convex values and such that (2), (3) are satisfied. Then (1) has a solution on J. Proof. Consider X0 — IR χ X with |(τ, x)| = |τ| + |x|, let D0 = J χ D and F0: D0 2 X °\0 be defined by F0(τ, χ ) = { 1 } χ F ( τ , χ). Then it is easy to check that for (τ, χ ) G D0 with τ < a we have (1, y) e F0(x, χ) η TDQ{x, x )

iff

ye F( τ, χ ) η

TD(x).

By the proof to Lemma 5.1 it is therefore clear that we can apply this result for X0, D0, F0 and initial value (0, x 0 ). Then the first component of this solution is τ ( ί ) = t and the second one is a solution of (1). • Before we proceed to the case where F is only measurable in t, let us consider some clarifying

5.2 Counter-Examples. (i) Classical single-valued examples like χ ' = x 1 + £ , x(0) = 1 with ε > 0 show that (1) may not have a solution on J (choose a > l/ε) if F grows faster

5.2 Counter-Examples

55

than linear. It is also easy to check that (c) in L e m m a 5.1 is necessary; see Problem 1 at the end of this section. (ii) The standard example showing that (1) may not have a solution if only one F(x) is not convex is Example 5.1. Let η = 1, D = IR, x 0 = 0 and '{-1} F(x) = -j { — 1,1} {1}

if χ > 0 if χ = 0; if χ < 0

notice that F is use, solutions cannot leave the initial value a n d χ = 0 is no solution. If we convexify, i.e. let F(0) = [— 1, 1], then there is a unique solution (x = 0) and this solution is C 1 , but if we change to x 0 = 1 then the unique solution is given by x(t) = (1 — ΟΖιο,ΐ]^). which is not C 1 on [0, 2]. (iii) In case η = 1 arcwise connectedness of the sets F(x) is the same as their convexity, hence sufficient for existence. F o r η > 1 this is not the case; consider Example 5.2. Let η = 2, D = IR2, x 0 = 0 and F( ϊ W

=

Η-χ/|χ|} 1^(0)


φ„(ή

on J,

and therefore |z" — (x, i»(t))| > 1, which means (x, r(i)) e G0(t) on J\NX with μ(Νχ) = 0. Therefore (a) is obvious too. If ζ e G 0 (t) then we find (z k ) cr (z n ) with zk -»· z, hence p(z, G(i)) = lim p(zk, G(t)) < lim 7pk{t) < lim \zk - z\ = 0, fc-»oo fc-^oo k->oo

i.e. ζ e G(t).

Evidently we also have F0(t, x) closed. 3. Given ε > 0, we let ε„ — ε/2"+1, find closed Jn cz J with μ ( . / \ J n ) < e„ such that (p„\jn is continuous, and let J0 = Q Jn, which is closed and has ^ ( J \ J 0 ) < ε/2. We n> 1 also find Ν c J0 with μ(Ν) = 0 such that every t e J0\N is a point of density of J 0 , i.e. (20)~^{J0n[t

- δ, t + 1

as 0+

forreJ0\N;

see e.g. p. 274 in Hewitt/Stromberg [1]. So let χ ε D and t e J0\N. Then we find (i,.) cr J0\NX with ti -*• t, and yl s F0{th x) such that y' -> y for some y e X (w.l.o.g.), hence |z" — (x, y')| > φπ(£;) for all i, n > 1, and therefore (x, y) e G 0 (i) since all φ„ are continuous on J0. Thus, we have F0(t, χ) Φ 0 on ( J 0 \ N ) χ D and can choose a closed Λ c Jo\H w i t h μ ( Α Λ ) ^ ε · Then the same argument shows that F0\JexD has closed graph, which is the same as use in the present case. Finally, if the F(t, x) are also convex, we consider conv F0(t, x) instead of F0(t, x). • Now it is easy to prove the following extension of Theorem 5.1.

58

§ 5 Upper Semicontinuous Right-Hand Sides

Theorem 5.2. Let X = R", J = [0, a] 2 * \ 0 have closed convex values and be such that (a) F(t, ·) is ε-δ-usc, F(·, (b) F(t, χ) η r(t, x)Bt(0)

x) is measurable. η TD(x)

/ 0 on [0, a) χ D,

c e L1 (J). Then (1) has a solution on J, for every x0 β D.

with r(t, x) = c{t){ 1 + |x|) and

60

§ 5 Upper Semicontinuous Right-Hand Sides

5.5 Remarks. 1. The simplest version of Corollary 5.2, with F bounded and use in (t, x), is already in Zaremba [1], Its extension to the Caratheodory case was considered later by several authors and is simple, since without boundary condition it can be solved as a fixed point problem, like in the single-valued case. Indeed, the usual a priori estimates yield a compact Κ c= CX(J) such that the multi

(11)

(Gu)(t) = j x 0 + J h>(s) ds: w(·) e F(·, M(-))j

t maps Κ into itself; notice that (Gu)(t) = {x 0 + J / ( s , w(s)) ds} in the single-valued ο case. Since F has convex values, Gu is convex. Furthermore, G has closed graph since if u„ -»· u in CX{J) and (vv„) is a corresponding sequence of measurable selections, we may assume (12)

vvn—w,

hence v„ -* w for some v„ e conν{νν*: k > n}

by Mazur's theorem, see e.g. Theorem II.5.2 in Day [1], and therefore vn -> w a.e., without loss of generality. Hence, one can apply the simplest multivalued version of Schauder's fixed point theorem; see § 11. In this light it is a trivial but useful observation made in Davy [1] that existence of a measurable selection for F(-,x) is sufficient. There it was used in the study of the solution set for (1), a problem that will be considered in §7. 2. The simple proof of Lemma 5.1 is the finite-dimensional version of the proof to Theorem 1 in Deimling [5], Theorem 5.2 is from Deimling [7]. The idea of looking for a multi F 0 c F with (a) and (b) in Proposition 5.1 is from Jarnik/Kurzweil [1]. Its construction was simplified in Rzezuchowski [1], Unfortunately, in both papers this problem is considered in such a generality that F0(t, χ) ξ 0 may be the only candidate. 3. A different proof of Theorem 5.2 was given in Tallos [1], under the much stronger assumption that TD(x) be replaced by Clarke's cone (13)

fD(x)=\yeX:

lim

λ'1 p(z + Xy, D) = o i ;

remember Remark 2 in § 4.6. 4. Corollary 5.3 is from Haddad [1], where a complicated typical finite-dimensional proof was given; see also Problems 4 and 5. Its extension to cases where F is not so nice requires additional thinking. Notice first that if, say, ||F(i, x)|| < 1 on G = graph ( K ) a n d u' 6 F(t, u)

a.e. on [τ, α],

η(τ) = χ e Κ(τ)

has a solution for every (τ, χ) e G with τ < a then it is easy to see that (10) and

Problems (14)

Κ (τ) c Κ (τ + h) + hB1 (0)

61

for τ 6 [0, a) and h e [0, a - τ]

are necessary conditions. Evidently, (14) implies that Κ is ε-5-usc from the left on (0, a]. Now, it is possible to prove Theorem 5.3. Let X = R", J = [0, a] c R, K: J 2 * \ 0 be ε-δ-use from the left with closed values, G = graph(K) and F: G -*• 2X\0 be almost use with closed convex values such that (2) holds with G instead of J χ D and, for some Ν c [0, a) with μ{Ν) = 0, (15) 1 '

ί({1} * 1({1} x Χ) η TG(t, x) =t 0

for t e [0, a)\N, χ e K(i) for teN,xeK(t)

is satisfied. Then u' e F(t, u) a.e. on J, «(0) = x 0

e

^(0) has a solution.

This is Theorem 1 in Bothe [2]; see also Problems 6 - 9 . 5. Solutions with values in D are called "viable solutions" by J. P. Aubin and his followers; see e.g. Aubin/Cellina [ 1 ] and Aubin [4]. We do not use such a name since it is trivial that solutions must have their range in the domain of definition of F. 6. Under certain conditions one may find solutions of (1) also if F is not locally bounded by integrable functions. In the situation of Corollary 5.2., assume in addition that F has compact values and F(·, x) is measurable. For ω 0 = (ί 0 , x 0 ) e D and α > 0, consider the "cones" Κ"{ω0) = {ω = (t, χ): 11 - t 0 | < a|x - x 0 | } u {ω 0 } and let Ψ be an at most countable family of locally integrable φ'. IR —* IR+. In V. Filippov [1] it is claimed that local boundedness can be replaced by the following assumption: to every ω 0 e D there exist α > 0 , 0, ι/'βΨ, ζ e X and a locally integrable φ: IR IR+ such that ||F(co)|| < φ(ή

on

K"(co0) π Βδ(οο0) c D

F(co) c { y e X: |y| < φ(ή or > a|y|}

and

on Βδ(ω0).

In V. Filippov [ 2 ] this is "axiomized" and slightly generalized, and some details of proof are given in V. Filippov [3].

Problems. 1. Let X = IR", D 2 * \ 0 be use with compact convex values. Suppose (1) has a solution. Then F ( x 0 ) n TD(x0) Φ 0. Hint: If x'(i) e C a.e. and h

C is closed convex then Jx'(s) ds e hC for h > 0, since C is the intersection of all ο closed half spaces {y e X: y*(y) < a} containing C.

62

§ 5 Upper Semicontinuous Right-Hand Sides

2. Let φ(ή = t c o s ( l f t ) for t Φ 0 and φ(0) = 0. Consider X = U2 and F: X defined by {(0, - 1 ) }

2X\0

if χ 2 > φ ( χ 1 )

F(x) = 1 {(1 - S 2 , 5 ) : S e [ - l , 1]}

if χ2 = φ(χ1)

{(0,1)}

ΐίχ2 IR for i g { 1 , . . . , m}, let P(x)= Π i=\

{yeD-.v^ivM}.

This Ρ has closed graph if D is closed and the üf are continuous, but much more is required to get Ρ lsc too; e.g. the additional assumption "D convex and y, strictly convex for i = 1 , . . . , m" is sufficient. For some applications of preorders in mathematical economics see e.g. Levin [1]. 5. Let X, D and F be as in Lemma 5.1, with (c) replaced by F(x) η TP(X) Φ 0

onD,

where P: D 2 D \ 0 has closed graph, χ e P(x) on D, and y e P(x) implies P{y) 0

η D)

on D.

Then u' e F0(u) a.e., u(0) = x 0 e D has a solution, hence u' ε F(U) a.e. has a slow solution through x 0 if F0(x) η TD{x) = { / ( x ) } on D (why?). The latter is satisfied if F ο

is, moreover, continuous, D φ 0 and F(x) η TD(x) Φ 0 on D; notice that F is lsc in this case. This is from Falcone/Saint-Pierre [1]. 12. Let D c X = IR" be closed and F: D χ X (17)

u"eF{u,u')

a.e. on J = [0, a],

2*\0 be given. T o find solutions of

u(0) = x 0 e D,

u'(0) = x 1 e A r ,

is the same problem as finding solutions of v' e { v 2 } x F(v) a.e., υ(0) = (x 0 , x t ) such that t>(i) e graph(T D ) on J, since u(J) a D implies u'(t) e TD(u(t)) a.e. on J. But G = graph(T D ) is closed for very special D only, hence the approach used in this section for first order systems is very seldomly applicable in the present case. Suppose, for example, that/ e Cjt(X) and f'(x)(X) = X on D = { x : f(x) = 0}. Then an application of the later Problem 11.2 (with Τ = /'(χ)) shows that TD(x) = N(f'{x))

={yeX:

f'(x)y

= 0};

see e.g. § 26.2 in Deimling [3]. Hence G — graph(T D ) is closed in this case, and we find a solution of (17) if F: D χ X -* 2*\0 is, say, use with compact convex values, satisfies a linear growth condition and ( { y } χ F(x, }/)) η TG(x, y) φ 0 on G; notice that TG(x, y) = {(K k) e X2: f'(x)h

= 0,

f"{x)(h,

y) + f'{x)k

= 0}

if / is even C 2 , hence it is not quite hopeless to check the condition in this case. Related results are reported in Haddad [2], Some special single-valued examples can be found in Chap 3 of Pavel [1].

§ 6 Lower Semicontinuous Right-Hand Sides

We look again for a.c. solutions of (1)

x' e F{t, x)

a.e. on J,

x(0) = x 0 e D,

where J = [0, a] c U, D α X = W and F: J χ D -> 2 X \ 0 are given. This time we assume that F is lsc, or F(·, x) is measurable and F(t, ·) lsc. Then we cannot expect to get solutions by means of approximate solutions, since if x„ e Gx„ + yn with yn -* 0, xn -> χ and G is lsc then the best we can get is χ e l i m , , ^ Gx„, but Gx may be smaller than this limit. Now, results like Michael's selection theorem suggest to look for selections / of F which are good enough to get solutions of (1) by solving (2)

x' = f{t, x)

a.e. on J,

x(0) = x 0 e D.

So, if we keep the simplest growth condition (3)

||F(t, x)|| < c(i)(l + |x|)

on J χ D,

with c e L ^ J )

then the cheapest existence theorem is obviously given by Proposition 6.1. Let X = R", J = [0, a] c R, D α X be closed, F:J χ D2*\0 with closed convex values such that (3) and (4)

Fit, x)

TD(x)

be lsc

on [0, a) χ D

are satisfied. Then (1) has a solution on J. Here, we even get a C 1 -solution, since F has a continuous selection / . Compared to (3) in § 5, we now have the stronger condition (4), since otherwise we are not sure that fit, x) e TD(x) on [0, a) χ D; remember Example 5.4, showing that Fit, χ) η TD(x) may be bad as a function of (f, x). However, (4) is not proof-technical; consider Example 6.1. We let D, x 0 and F be as in Example 5.4, except the value of F at x 0 , which we now choose as {JB} — x 0 instead of Δ — x 0 . This new F is lsc with compact convex values and satisfies F ( x ) n TDix) / 0 on D, but (1) has no solution, since (1) means x'2 > 1 — x 2 a.e. and x 2 (0) = 1, hence χ 2 (ί) = 1 and x x (i) = 0, a contradiction to 0 φ {Β} — XQ. This is the counter-example from Deimling [5]. A different one with non-convex values is in Problem 2. The most interesting point in this section is to show that, contrary to the use case, convexity of the values of F is redundant.

66

§ 6 Lower Semicontinuous Right-Hand Sides

6.1 The Lsc Case. As mentioned earlier, Michael's general selection theorem is sharp, in particular, wrong if the F(t, x) are not convex. But now we are in a special situation, since we have an easy reduction to a compact domain, and need not have continuous selections, since we can also handle some not too badly discontinuous / by means of the usual use convexification and the results in § 5. Before going into details, let us first state what we want to prove. Theorem 6.1. Let X = R", J = [0, a] e R, D c X be closed, F: J χ D 2*\0 be lsc with closed values such that (4) and ||F(f, x)|| < c ( l + \x\) on J χ D are satisfied. Then (1) has a solution on J. N o w , notice first that, by means of the reductions at the beginning of § 5, we may assume w.l.o.g. that D is compact and F satisfies ||F(i, x)|| < 1 on J χ D. Next, let / be a selection of F and consider (5)

x' e F(t, *) = Π ^°nv /((^

x

n

a>o

*))

a e· on

J>

χ(°)

=

χο·

Since f(t, x) e F(t, χ) η T D (x), problem (5) has a solution, by Theorem 5.1, and we are done if we are able to show that this solution is already a solution of (2). Therefore the program is to find conditions on / guaranteeing that (2) and (5) have the same solution set, and to show that F has such a favorable selection. Concerning the first point, notice that a solution of (5) satisfies |x'(f)| < 1 a.e. since I f(t, x)| < 1 on J χ D, hence |x(i) — x(s)| < t — s for t > s, in other words (t, x ( f ) ) e (s, x(s)) + Kx for t > s, where Ka is an "ice-cream" cone in R n + 1 , i.e. (6)

Ka = {(t, x) e R n + 1 : |x| < at}

with a > 1.

N o w , the solution sets coincide, of course, if / is continuous, since then (5) is (2), but it is also easy to see that they do if / is only continuous along Ka with α > 1, i.e. (tn, x j e (t, x ) + Kx and (f n , x„)

(i, x) imply f(tn, x j ^ f(t, x).

For ease of reference, we formulate this as Lemma 6.1. Let X = R", J = [0, a ] 1 and such that \f(t, x)| < 1 on J χ D and f(t, x) e T D (x) on [0, a) χ D. Then (2) has an a.c. solution on J, and the solution sets of (2) and (5) are identical. Let χ be a solution of (5). Then J = \J J„ with μ(7 0 ) = 0 and disjoint closed n>0 __ J„ (for η > 1) such that x'\Jn is continuous and x ' ( i ) e F(t, x ( i ) ) on J„. Let Jn\Nn be the density points of Jn, remember the proof to Proposition 5.1, and let t e J„\N„. Then we find tke J„r\ (t, oo) with tk t, hence x ' ( i k ) -*• x'(t) and Proof.

(s, y) e (t, x ( f ) ) + Ka

if

{s, y) e BSk(tk,

x(tk)),

6.1 The Lsc Case

67

where ök is sufficiently small. Therefore x' = f(t, x) on J\J0\ of (2).

IJ Nn, i.e. χ is a solution •

So, the final step of the proof to Theorem 6.1 is to prove Lemma 6.2. Let X = IR", Μ c= IR χ X be compact, F: Μ -*• 2X\0 be lsc with closed values, and Κ be the cone Kx from (6). Then F has a selection which is continuous along K. The same is true if Μ is only locally compact. Proof. Following the standard procedure of Michael's proof, we first construct a good approximate selection in case Μ is compact, and then we proceed as usual. For ω e Μ we consider neighborhoods W{co) of the form Μ η (ώ + Κδ), with ώ e [R χ X and ο

Κδ = Κ η ([0, relative to M.

χ Χ) for 0, such that ω e W(o)\ where the interior is understood

1. Let V = ΰε(0). Since F is lsc, we have F x{y — V) open in M, and to coe F'Hy - V) we find a W(oo) ) + Vk

for k > 1.

Indeed, the first step with ε = 1/2 yields f t and, given Λ , •··,/*, we may consider f k as the / from the first step with ε = 2~k. Since G(a>) = F((o) η (3/, + V k ) is lsc on the compact M, and need not have closed values for the argument in the first step, we do for Μ i , G and Vk+l what we did in the first step for M, F and V. So we get

ρ

Μ, = ( J Ml

1=1

ρ

with disjoint M\, hence

M, = {J Mj η

l=i

Mh

and we may continue with Mt instead of Μ in the first step to obtain f k + l on M,·, and this for i = 1, . . . , m. Evidently, (7) then implies f k f uniformly on M, for some / which is continuous along Κ and satisfies /(ω) e F(ω) on M. 3. Let Μ be locally compact, i.e. every ω e Μ has an open neighborhood U(ω) with compact closure. Then this open covering of Μ has a locally finite refinement (KA)AeA; remember § 1.3(ii). Since Μ is "normal", (F A ) AeA allows shrinking, which means that there is an open covering (l/A)AeA of Μ such that c= Vx for all λ e A. This is not difficult to prove by well-ordering of A and 2M and use of normality; see e.g. Theorem 6.1 in Chap VII of Dugundji [1]. Remember also that by Zermelo's theorem, an equivalent of Zorn's lemma, every set Ω allows well-ordering, i.e. a partial ordering < such that ω < ώ or ω < ω for all (ω, ώ) e Ω 2 and every 0 φ Ω 0 > e TG(i, x)} < min{ 0, there exists a continuous g: Κ -»L^(J) such that, for every ue K, g(u)eF(-,u(

))

and

\\v(t) - g(u)(t)\ - p(v(t), F(t, u(t)))\ < ε

a.e. on J.

This is Theorem 1 in Chugunov [1]; see also the later remarks to §9.4.

Chapter 3 Solution Sets

We remain in X = FT since some of the questions to be considered now have only found unsatisfactory answers, if any, in infinite dimensional situations so far. In the previous chapter one hopefully got the impression that the solution set S(x0) a CX{J) of (0.1)

x'eF(t,x)

a.e. on J = [0, a],

x(0) = x0

is usually very large, hence uniqueness is a very rare phenomenon, even in very special situations such as those considered in appendix Al. Fortunately, only a small elite of writers tried to make us believe that uniqueness being "generic" is ultimate wisdom in the single-valued case, while many others worried about nonuniqueness long time ago. So, knowing a little bit of the results obtained by the latter, it is certainly no surprise to see that S is use with compact values in the use case, but S{x0) need not be closed if F is only lsc. It is also easy to see that S(x 0 ) is connected in both situations, but need not be arcwise connected, under the assumptions of one or the other basic existence theorems proved earlier, of course. Since a use F can be approximated from above by better ones, little extra effort shows that in this case S(x 0 ) is in fact a compact R3, i.e. the limit of a decreasing sequence of compact contractible sets (even homeomorphic to compact convex ones in the present case). Such assertions about S(x 0 ) have trivial consequences for the sections P t (x 0 ) = {*(*): * e S(x0)}> since χ -» χ(τ) is continuous. Like in the single-valued case one can show, in particular, that if F is use and y e δΡτ(χ0) for some t > 0 then there exists an χ e S(x 0 ) such that x(i) e dP t (x 0 ) on [0, τ] and χ (τ) = y, but one should not believe that such a solution can be extended as a "boundary solution" after time τ; see Problem 7.2 for the latter and Remark 3 in § 7.6 for an indication of the role played by boundary solutions in some basic control problem. Other items of the problem section to § 7 show that continuous F with nonconvex values are good for any surprise concerning properties of S and Ρτ. Usually it will be impossible but also not necessary to know all solutions. To get an approximate picture of the "funnel", one may replace the sets F(t, x) by simpler figures, say inscribed/circumscribed ellipsoids or order intervals with respect to a cone Κ like IR+. If we define χ < y by y — xe Κ and u < ν by u(t) < v(t) on J then we find a minimal solution and a maximal solution u* of (0.1) with respect to this partial ordering provided F is, say, use and (0.2) F(t, x) = [/„.(f, x), f*(t, x)]

on J χ X,

where

[ζ, ζ] = {χ e Χ: ζ < χ < ζ)

and /„,, f* are quasimonotone with respect to Κ (see (3) in §8.1). So, in this case

78

Chapter 3 Solution Sets

(0.3)

Pt(xo) = K W , "*(t)]

on J,

but usay, need not be the minimal solution of x' = χ) a.e. on J with x(0) = x 0 , since this problem may have no solution at all, unless F is continuous in x. Clearly, the order interval in (0.3) is at least an upper estimate for Pt(x0) if F is contained in a multi of type (0.2). In an elementary course one might have seen another (crude) comparison of solutions, sold as Gronwall Lemma, saying (for example) that if / is Lipschitz of constant k with respect to χ and g is continuous with sup|g(t, x) — f{t, x)| < c then every JxX solution of y' = g(t, y)

on J,

y(0) = j;0 e Βδ(χ0)

remains in the tube around the solution x( ) of x ' = f(t, x) with x(0) = x 0 given by (0.4)

χ(ί) + r(i)ßi(0)

on J

with

r(i) = ekt δ +

C

-(l-e-k 0, but u(f) = — t2 is a solution with u(t) φ D for t > 0. We say that D is (forward) invariant for (1) if all solutions starting in D remain in D on J. Evidently, the simplest way of studying this question is to assume that a solution u leaves D and to get a contradiction by showing that p{u{t), D) = 0 on J. In this respect the following facts are useful. Proposition 7.1. Let D α X = [Rn be closed, φ{χ) = p{x, D) for χ e X and PD: X -*• 2 D \ 0 be the metric projection, i.e. PD{x) = {y e D: |_y — x| = · 2 R \0 be defined by F(x) = {0} for χ < 0, F(x) = {1} for χ > 0 and F(0) = [0, 1]. Then S(0) is not convex and the subset of C^-solutions is not connected. 8. Consider polar coordinates χ = (r cos Θ, r sin θ) e IR2, let g: IR -> IR be 2n-periodic and odd with g(6) = 2^/θ(π — θ) on [0, π], and consider x' = f{x) = ( - x 2 ,

x(0) = x 0 = (1, 0).

Χι)9(θ),

Then P t (x 0 ) = dB^O) for all τ > π/2. Hint: Notice that & = g(9), 0(0) = 0 has maximal solution (see § 8) öirt = ί π s i n 2 1 (π

in

for t > π/2

and minimal solution θ(ή = —θ(ί). This is an example of C. Pugh, given on p. 588 of Hartman [1], 9. Let F(x) = {y, z) on IR2 with y = (1, 1) and ζ = ( 1, - 1) e IR2. Then S(0) for u' e F(u) a.e. on J = [0, a] is not closed but Ρτ(0) is compact convex for every τ e J. This is Example 2.3 in Hermes [3]. Hint: Consider the selections r i.\ \y fn(t) = < [z

on J2k-i j on J2k

(k=l,...,n)

2η where [ J J{ is a partition of J into intervals of equal length. i=l

90

§ 7 Topological Properties of Solution Sets

10. Let F(x) = { y e IR2: = (1 - xj)yl, y2 e [ - 1 , 1 ] } for χ 6 IR2. Then P t (0) is not closed, for every τ > 0. This is Example 2.4 in Hermes [3]. 2n Hint: Let τ > 0, [0, τ ] = U J{ as in the previous problem with a = τ, and consider (=1 solutions u„ such that u'n 2 = 1 on J 2 t _ u ' n 2— — 1 on J2k for k = 1 11. Let J = [0, a ] c R, X = IR", U 0. This is Theorem 3.3 in Aubin [3].

§ 8 Comparison of Solutions

We consider again initial value problem (1)

u' e F(t, u)

a.e. on J = [0, a] c= U,

u(0) =

xoeX,

with X = IR" and F\ J χ X -*• 2 X \ 0 , and start with some results about comparison of solutions in terms of the partial ordering given by a cone Κ α X.

8.1 Preliminaries. A cone is a closed convex Κ c= X such that λΚ c= Κ for all λ > 0 and Κ η ( - K ) = {0}. Of couse we only consider cones Κ φ {0}. Then < stands for the partial ordering defined by K, i.e. χ < y iff ^ — x e K, and this yields the partial ordering u < ν for functions u, v. J -* X, defined by u < ν iff u(i) < v(t) on J. Sometimes it will be useful to consider also the dual K* — {y e I : {)», x> > 0 o n K } o f K . The metric projection Ρ κ : X -*• Κ satisfies 0

whenever x* e K* and x*(x) = 0.

ο

Indeed, (2) is obvious if xe Κ since then x* e K* and x*(x) = 0 imply x* = 0. If χ g δΚ, we know that ζ e TK(x) iff x*(z) < 0 whenever x* e X* and x*(x) = supx*(y), κ by Proposition 4.1(c). But the latter is equivalent to x*(z) > 0 whenever x* e X* and x*(x) = infx*(y), and, since λΚ α Κ for λ > 0, we have x*(x) = i n f x * ( j ) iff x* e K* κ κ and x*(x) = 0. Notice also that Κ a TK(x) for all χ e K, and Κ = TK{ 0). Now, f:X -*X is said to be monotone w.r. to Κ if χ < y implies f(x) < f(y), i.e. f{x + y) e f{x) + Κ

for all χ 6 X and y e K,

while / is said to be quasimonotone w.r. to Κ if x*ifiy)

— f(x)) > 0

whenever x* e Κ*, χ < y and x*(x — y) = 0.

By (2), this is evidently the same as (3)

fix + y)e fix) + TKiy)

for all χ e X and y e K,

and therefore the difference between the two definitions is geometrically obvious. For functions / : J χ X X we shall use the same names provided fit, ·) has these properties for every t e J.

92

§ 8 Comparison of Solutions

Example 8.1. Let X = IR" and /(x) = Ax with an η χ η-matrix A. For the standard cone Κ — R" = {x e X: xt > 0 for i = 1, ..., n} we have Κ = Κ*, f is monotone iff fly > 0 for all i,j, but quasimonotone iff αϋ > 0 for all i,j with i φ j. In case η = 3 and Κ = {χ e X: (x? + x^)1/2 < x 3 } we have Κ = K* and x*(x) = 0 for x* e K* and χ e dK\{0}

iff x* = ( —x l5 —x2, x 3 ).

Hence, choosing r A =

α

~ß . y

ß α δ

δ α

we get a quasimonotone / for all (α, β, γ, δ) e IR4, but / is monotone iff a > 0 and β = y = 3 = 0.

8.2 Extremal Solutions I. Given the cone Κ 0).

Let ν and w be solutions of (10) and (11), respectively. Then ζ — ν — w has z(0) e K. Suppose there is a first τ > 0 such that ζ(τ) e dK. Then there exists x* 6 K* with x*(e) > 0 and χ*(ζ(τ)) = 0. Therefore φ(ή = χ*(ζ(ί)) satisfies I ο > -φ(

τ

- h) =

J

X*(/ £ (S, Ms) + Φ))

- fs(s, w(s))) ds +

yx*(e)h.

τ-h

Dividing by h > 0 and letting h-+ 0, we get the contradiction x*(e) < 0, since ft is quasimonotone. Thus w < ν for all solutions ν of (10), and we may let γ -»• 0 + to get νε.

• The last two results evidently imply

8.3 Extremal Solutions II

95

Theorem 8.1. Let the conditions of Proposition 8.1 be satisfied and Κ φ ψ. Then (1) has a minimal solution uand u^ is the minimal solution for problem (7). The same kind of reasoning leads to a natural condition in case we are looking for a maximal solution u* of (1). Since T_K{—y) = — TK(y) on K, (6) has to be changed to (12)

F has a quasimonotone selection / such that F{t, x) c: f(t, χ) — Κ on J χ X

and we have ο

,

Corollary 8.1. Let the conditions of Proposition 8.1 be satisfied with Κ φ 0 and (12) instead of (6). Then (1) has a maximal solution u*, and u* is the maximal solution of (7) with f from (12) instead o f f . Let us remark that in case η = 1 conditions (6) and (12) are always satisfied since F{t, x), as a compact subset of R, has a minimal f{t, x) and a maximal element f(t, x) and quasimonotonicity is no condition in case Κ = U+. However, for η > 2 it is easy to see that u^ may not exist if the geometric condition (6) is violated. Consider Example 8.2. Let η = 2, Κ = R i and F(x) = {y e R 2 : |>>| < 1 and p{y, Κ) < 5} on IR2 for some δ < 1. Then F has no selection / such that F(x) c f(x) + Κ on IR2 and u' e F(u) a.e., u(0) = 0 has no minimal solution; notice that ( — St, 0) and (0, — St) are solutions, hence w+(i) < ( — St, —St), but also F(x) cz ( — S, —S) + K, hence > ( - S , —S) a.e. and therefore u^(t) = ( — St, — St), which yields the contradiction N f l S = ρ(«;(ί), K) < S a.e.

8.3 Extremal Solutions II. While finding extremal solutions was relatively easy in case F(t, ·) is continuous, the same problem becomes harder if F(t, ·) is only use. Of course we need not consider the case where F(t, ·) is lsc since Example 7.1 tells us that u* may not exist even if η = 1 and F(t, x) is convex. The following example shows that u^ may now have nothing to do with u for (7). Example 8.3. Let X = U and Κ = U+. The general use F: X 2 * \ 0 with compact convex values is given by F(x) = [

t

/Γ

1

j

gm{s,

u"(s))

ds

t

for h > 0 and η > m evidently implies v'0 > f{t, t;0) a.e. on J. Therefore u 1, \e\ = 1) with closed convex values. Hence, by Corollary 2.1, Gm has a sequence of continuous selections which at every ω e Ω is dense in G (co). Therefore it is obvious that G = ( / — Κ ) η 5 2 (0) has a sequence of continuous selections f" such that m

G(co)

=

{ / " ( ω ) : η > 1}

on Ω,

f

n

( ( o ) < f ( ( o ) - ß „ e

on Ω

for some

ßn

> 0.

98

§ 8 Comparison of Solutions

By means of ( / " ) we get (3") inductively such that also / ' < for i = 1, . . . , η and g"{a>) < / ( ω ) — y„e on Ω for some yn > 0. Indeed, let g1 = / 1 and, given g1, ..., g", consider Η{ω) = (g»(a>) + Κ) η ( / where ε > 0 is such that g", fn+l ο

η +

» + Κ) η (/(ω) - ee-Κ)

η B2(0),

< f — 3ee. Since gn and / " + 1 are continuous and

/ ( ω ) — 2ee e Η(ω), we have Η lsc with closed convex values, hence get a continuous selection gn+1 of Η(ω), which completes the induction step. Since / is a selection of G, given ω e Ω, we find a subsequence ( / " ( ω ) ) such that fm{d) -* /(ω), and therefore gm(co) - » / ( ω ) since fm(co) < gm(co) < /(ω). But the entire sequence (^"(ω)) is convergent, since Κ is fully regular. Therefore, gn((o) / ( ω ) on Ω, which completes the proof. • By means of obvious modifications we also obtain Corollary 8.2. Problem (1) has a maximal solution if the conditions of Theorem 8.2 are satisfied with (6) replaced by (12).

8.4 Related Problems. In standard references about differential inequalities for the single-valued case, see e.g. Lakshmikantham/Leela [1], Szarski [1] and Walter [1], one has also considered questions such as: Under which conditions is v: J -* X such that ν < u for at least one u s S(x 0 ) or all u e S(x0)7 Given another w: J -* X with ν < w, under which conditions does the order interval 0 , w] = {u e C(J): y(i) < w(i) < w(f) on J) contain at least one, or all u e S(x 0 )? Evidently, existence of extremal solutions is the most difficult question of this type, since then ν and/or w have to be solutions too. So far one has tried to extend the satisfactory answers for the simplest case η — 1 by means of componentwise imitation for Κ = IR" or, in a few cases, by use of functionals from K* instead of coordinates for more general K. The picture becomes clearer if one considers such problems also in the singlevalued case as what they are, namely special cases of the existence theory for (multivalued) differential equations on closed sets. By the results in the previous sections one knows what one has to do, i.e. the almost necessary conditions d o not fall from heaven but are dictated by this point of view. Let us illustrate this by means of a few examples, always under the assumption that F:J χ X 2 * \ 0 has closed values and satisfies (8), ο

,

and < refers to a cone Κ 0 ,

v'2< f2(t,

+ x t , v2)

for Xj > 0.

However, if / is quasimonotone w.r. to K, the latter two follow from the first one. T o have ν < u for all u e S(x 0 ) means, essentially, that Κ should be forward invariant for (15), hence §7.2 allows some (unsatisfactory) answers; see Problems 1 and 2. (ii) Given two functions v, w 6 C{J) such that ν < w and i>(0) < XQ < vv(0), finding a solution u e [t;, w] means to find u e S(x0) such that graph(m) C~G = graph(K 0 )

where

c

K0(t) = |>(i), w(i)]

on J.

So, by §5.4(iii), an almost necessary condition is (16)

({1} χ F(t, x ) ) n TG{t, χ) φ 0

for (t, x) e G - graph(K 0 )

with t < a,

i.e. to t e [0, a) and χ e [u(i), w(0] there exist y e F(t, χ), λ„ -> 0 + , τη -*• 0 and >>"-»• 0 such that χ + X„{y - y") e [i>(sB), w ( s j ] for sn = t 4- λ„(ί - τη). In case F: G -»· 2 X \0 is use with convex values (and c(t) = c in (8)) this is also sufficient, by Corollary 5.3, and in case F is lsc (and c{t) = c in (8)) it is sufficient to have this condition for all y e F(t, x), i.e. {1} χ F(t, x) + αλ (J-7442/HD57/P99/WSL/28-2-92)

for some α > 0 and all λ e (0, 0), (s, z) e G

§ 8 Comparison of Solutions

100

if δ > 0 is sufficiently small. Choosing s = t + λ, this implies ~~~~—- + y > α Λ

for A e (0, ,(f) implies D+Vi(t) < fi(t, x) and Xj = Wj(t) implies fj(t, x) < D + w/t)". This is just M. Müller's classical condition; see e.g. the books mentioned before. F o r differentiable v, w and (t, x) e G with t < a this can be expressed in terms of functionals from K*\ x*(x - v(t)) = 0 for some x* e K* implies x*(p'(f) - f{t, x)) < 0 and x*(x - w(r)) = 0 for some x* e K* implies x*(w'(i) - f(t, x)) > 0. Example 8.7. Let Κ = {χ e IR3: (x? + x\)m < x 3 }. By Example 8.1, given that υ and w are differentiable, (16) implies x*(v'(t) - f(t,v(t)))< 0 if x* e K*, xeK0(t) and x*(x — y(i)) = 0, and similarly if x*(x — w(i)) = 0 instead. In other words, (16) implies Müller's condition (in terms of K*) from the previous example. But the converse is not true; consider e.g. ν = 0 and w(i) = w e δ Κ \ {0}; for y = f(t, x) and χ = w the functional condition only means w ^ -I- w2y2 > w3>>3, while (16) requires w + Xn(y — y") e [0, w], i.e. y = Aw with A < 0.

8.5 Gronwall's Lemma. In the single-valued case a crude way of comparing solutions consists in the use of norm estimates. Suppose, for example, that / : J χ X X is measurable in t with / ( · , 0) e Lx(J) and \f(t,x)-f(t,y)\(0) e B a (x 0 )· Then φ ( ή — |u(i) — y(r)| satisfies

for t e J

(17) ο

ο

8.5 Gronwall's Lemma

101

and talk of " G r o n w a l l L e m m a " means that this implies φ(ί)

< φ(ή,

where φ is the

solution of φ' = k(t)q> + p(t) a.e., 0.

Mn-^·)), (18) implies

t (19)

o n

t ds,

φ0(t)

< δ + j p{s) ds 0

for φ„(ί) — |wn+i(i) — u„(t)\.

π Consequently, pn(t) = Σ φ((ή satisfies ί=0 t

t

Pn(t) < Pn+1(0 ^ 1,

and therefore p„(t) < 0, we have to show that there exists u e S 2 0 satisfying \u — t>|0 < η. Due to Example 6.3 we also know that to every ε > 0 there exists a closed J £ c J with < ε such that FoUx* continuous. 2. Given ε > 0, we find a closed A 3, limu(i) does not exist; nevertheless we have |u(i)| < 3 on [0, 1). This is t-»l Example 1.1 from Deimling [2], a modification of an example given in Dieudonne [1] for X = (c)0, the space of real sequences tending to zero. Evidently these phenomena cannot occur if we assume that / maps bounded sets into relatively compact sets. However, it is not necessary to be so restrictive. We can also allow certain perturbations of such "compact" / . This improvement will be achieved by means of

9.2 Measures of Noncompactness

113

9.2 Measures of Noncompactness. Let S be the family of bounded subsets of X . Then a: © R + , defined by (1) o c ( B ) = inf{ 0: Β admits a finite cover by sets of diameter < d } , is called (Kuratowski-) measure of noncompactness, and ß : S -» R+, defined by (2) ß ( B ) = inf{r > 0: Β can be covered by finitely many balls of radius < r}, is called ball measure of noncompactness. These functions have some useful properties. Proposition 9.1. L e t X be a B a n a c h s p a c e o f dim X — oo, © t h e f a m i l y o f b o u n d e d s u b s e t s o f X , a n d y : © -+ IR+ be e i t h e r o c ( - ) o r β ( · ) . T h e n (a) = 0 i f f Β is c o m p a c t . (b) y is a s e m i n o r m , i . e . γ ( λ Β ) = a n d y ( B t + B 2 ) < 7(#i) + y ( B 2 ) . (c) Β ! c B 2 i m p l i e s < γ(B2);y(Bx υ B2) = m a x ^ ) , γ(Β2)}. (d) y(conv B ) - y ( B ) , y ( B ) = y ( B ) . (e) y is c o n t i n u o u s w . r . t o d H . (f) α(Β Γ (χ 0 )) = 2r a n d ß ( B r ( x 0 ) ) = r . A proof can be found e.g. in § 7.3 of Deimling [3]. In general one has β ( Β ) < o l ( B ) < 2 ß { B ) and both inequalities may be strict. Notice also that in (1) the covering sets can be chosen as subsets of B, and ß(B) depends on the set from which the centers of the covering balls are chosen, i.e. if Β t2 if |x(s)| < t2 '

It is easy to check that / is uniformly continuous on J χ X, f(t, ·) is Lipschitz of constant k{t) = 1 jt, and (4) with k(t) = l/t for t > 0 is best possible, since for Β = {χ ε Χ: 0 < x(s) < t2 on J} we have tfit, Β) = Β and oc(B) = t2 for t > 0. This example is from Mönch [1]. In the sequel we shall avoid such cases by making the assumption keLl(J). Finally, since we shall have to deal with sequences, it is useful to know that ß( ·) has a simple representation for separable X, i.e. we have Proposition 9.2. Let X be a separable Banach space and (X„) be an increasing sequence of subspaces with dim Xn< ao and X = [J Xn. Then n> ι (5)

ß(B) = lim sup p(x, X„) rt-*oo Β

for Be ©

and (6)

Proof

ß(B) = lim Hm p(xk, X„) n->oo oo

for bounded

Β = {xk: k > 1}.

1. Let X = {ζ,: i > 1} and Xn = s p a n { z ! , . . . , z„}. Then dim X„ < oo and X =

(J X„. Let S(B) be the right hand side of (5) and ε > 0. Then supp(x, Xm) < S(B) + ε η>1 Β for some m. Let C — [y e Xm: p(x, Xm) = \x — y\

for some χ e B}.

ρ Since Β is bounded, we have C bounded and dim Xm < oo, hence C k 0 , hence relative compactness of C = {y 6 Xm: p(xk,

= \xk-y\

for some k > k0}

yields ß{B) = ß{{xk: k > k0})

J

=

[0,

a]

c

IR

sup |υ„(ί)| < 1 a.e. π

I } ) ds

for 0


ο)(0

o n Js

implies

|u„(t) - /(ω„)(ί)| < φ(ω„){ή

for all large n, and therefore we have v„ = ν„χ^ + wnxJxJi s Η(ωπ)

if w„ e Η(ω„).

on J6 Now,

l ^ - W o l i = J b , ( t ) - f o ( 0 M t + J |w„(t) — I70(t)| Λ Ji J\Jt ^

- t?oll +

< \ t>n -

yoli

φ(ω„)(ή J\Js + l