Solution Sets for Differential Equations and Inclusions 9783110293562, 9783110293449

Table of contents :
Preface
Notations
1 Topological structure of fixed point sets
1.1 Case of single-valued mappings
1.1.1 Fundamental fixed point theorems
1.1.1.1 Banach’s fixed point theorem
1.1.1.2 Brouwer’s fixed point theorem
1.1.1.3 Schauder’s fixed point theorem
1.1.2 Approximation theorems
1.1.3 Browder-Gupta theorems
1.1.4 Acyclicity of the solution sets of operator equations
1.1.5 Nonexpansive maps
1.1.5.1 Existence theory
1.1.5.2 Solution sets
1.2 The case of multi-valued mappings
1.2.1 Approximation of multi-valued maps
1.2.2 Fixed point theorems
1.2.3 Multi-valued contractions
1.2.4 Fixed point sets of multi-valued contractions
1.2.5 Fixed point sets of multi-valued nonexpansive maps
1.2.6 Fixed point sets of multi-valued condensing maps
1.2.6.1 Measure of noncompactness
1.2.6.2 Condensing maps
1.3 Admissible maps
1.3.1 Generalities
1.3.2 Fixed point theorems for admissible multi-valued maps
1.3.3 The general Brouwer fixed point theorem
1.3.4 Browder-Gupta type results for admissible mappings
1.3.5 Topological dimensions of solution sets
1.4 Topological structure of fixed point sets of inverse limit maps
1.4.1 Definition
1.4.2 Basic properties
1.4.3 Multi-maps of inverse systems
2 Existence theory for differential equations and inclusions
2.1 Fundamental theorems
2.1.1 Existence and uniqueness results
2.1.2 Picard-Lindelöf theorem
2.1.2.1 Maximal solutions
2.1.3 Peano and Carathéodory theorems
2.1.3.1 Peano theorem
2.2 The extendability problem
2.2.1 Global existence theorems
2.2.2 Existence results on noncompact intervals
2.2.2.1 The Lipschitz case
2.2.2.2 The Lipschitz-Nagumo case
2.2.2.3 The Nagumo case
2.2.3 A boundary value problem on the half-line
2.3 The case of differential inclusions
2.3.1 Initial value problems
2.3.1.1 A Nagumo type nonlinearity
2.3.1.2 A Lipschitz nonconvex nonlinearity
2.3.2 Boundary value problems
2.3.2.1 The convex case
2.3.2.2 The nonconvex case
3 Solution sets for differential equations and inclusions
3.1 General results
3.1.1 Kneser-Hukuhara theorem
3.1.2 Problems on bounded intervals
3.1.3 Problems on unbounded intervals
3.1.4 Second-order differential equations
3.1.5 Abstract Volterra equations
3.1.6 Aronszajn type results for differential inclusions
3.2 Second-order differential inclusions
3.2.1 The convex case
3.2.2 The nonconvex case
3.2.3 Solution sets
3.3 Higher-order differential inclusions
3.4 Neutral differential inclusions
3.4.1 The convex case
3.4.2 The nonconvex case
3.4.3 Solutions sets
3.5 Nonlocal problems
3.5.1 Main results
3.5.2 A viability problem
3.6 Hyperbolic differential inclusions
3.6.1 Existence results
3.6.1.1 The convex case
3.6.1.2 The nonconvex case
3.6.2 Solution sets
4 Impulsive differential inclusions: existence and solution sets
4.1 Motivation
4.1.1 Ecological model with impulsive control strategy
4.1.2 Leslie predator-prey system
4.1.3 Pulse vaccination model
4.2 Semi-linear impulsive differential inclusions
4.2.1 Existence results
4.2.1.1 The convex case
4.2.1.2 The nonconvex case
4.2.2 Structure of solution sets
4.3 A periodic problem
4.3.1 Existence results: 1 ∊ ρ(T(b))
4.3.2 The convex case: a direct approach
4.3.3 The convex case: an MNC approach
4.3.4 The nonconvex case
4.3.5 The parameter-dependant case
4.3.5.1 The convex case
4.3.5.2 The nonconvex case
4.3.6 Filippov’s Theorem
4.3.7 Existence of solutions: 1 ∉ ρ(T(b))
4.3.7.1 A nonlinear alternative
4.3.7.2 A Poincare translation operator
4.3.7.3 The MNC approach
4.4 Impulsive functional differential inclusions
4.4.1 Introduction
4.4.2 Existence results
4.4.3 Structure of the solution set
4.5 Impulsive differential inclusions on the half-line
4.5.1 Existence results and compactness of solution sets
4.5.1.1 The convex u.s.c. case
4.5.1.2 The nonconvex Lipschitz case
4.5.1.3 The nonconvex l.s.c. case
4.5.2 Topological structure via the projective limit
4.5.2.1 The nonconvex case
4.5.2.2 The convex case
4.5.2.3 The terminal problem
4.5.3 Using solution sets to prove existence results
5 Preliminary notions of topology and homology
5.1 Retracts, extension and embedding properties
5.2 Absolute retracts
5.3 Homotopical properties of spaces
5.4 Čech homology (cohomology) functor
5.5 Maps of spaces of finite type
5.6 Čech homology functor with compact carriers
5.7 Acyclic sets and Vietoris maps
5.8 Homology of open subsets of Euclidean spaces
5.9 Lefschetz number
5.10 The coincidence problem
6 Background in multi-valued analysis
6.1 Continuity of multi-valued mappings
6.1.1 Basic notions
6.1.2 Upper semi-continuity
6.1.2.1 Generalities
6.1.2.2 Ɛ — δ u.s.c. mappings
6.1.2.3 U.s.c. maps and closed graphs
6.1.3 Lower semi-continuity
6.1.3.1 Generalities
6.1.3.2 Ɛ — δ l.s.c. mappings
6.1.4 Hausdorff continuity
6.2 The selection problem
6.2.1 Michael’s selection theorem
6.2.2 Michael’s family of subsets
6.2.3 σ—selectionable mappings
6.2.4 The Kuratowski-Ryll-Nardzewski selection theorem
6.2.5 Aumann and Filippov theorems
6.2.6 Hausdorff measurable multi-valued maps
6.2.7 Product-measurability and the Scorza-Dragoni property
6.3 Decomposable sets
6.3.1 The Bressan-Colombo-Fryszkowski selection theorem
6.3.2 Decomposability in L1(T,E)
6.3.3 Integration of multi-valued maps
6.3.4 Nemytskǐ operators
Appendix
A.1 Axioms of the Cech homology theory
A.2 The Bochner integral
A.3 Absolutely continuous functions
A.4 Compactness criteria in C([a,b], E), Cb([0,∞), E), and PC ([a, b], E)
A.5 Weak-compactness in L1
A.6 Proper maps and vector fields
A.7 Fundamental theorems in functional analysis
A.8 C0-Semigroups
References
Index

Citation preview

De Gruyter Series in Nonlinear Analysis and Applications 18 Editor in Chief Jürgen Appell, Würzburg, Germany Editors Catherine Bandle, Basel, Switzerland Alain Bensoussan, Richardson, Texas, USA Avner Friedman, Columbus, Ohio, USA Karl-Heinz Hoffmann, Munich, Germany Mikio Kato, Kitakyushu, Japan Umberto Mosco, Worcester, Massachusetts, USA Louis Nirenberg, New York, USA Boris N. Sadovsky, Voronezh, Russia Alfonso Vignoli, Rome, Italy Katrin Wendland, Freiburg, Germany

Smaïl Djebali Lech Górniewicz Abdelghani Ouahab

Solution Sets for Differential Equations and Inclusions

De Gruyter

Mathematics Subject Classification 2010: 26E25, 34-01, 34G20, 34A37, 34A60, 34B15, 34B37, 34B40, 45D05, 47D60, 47G10, 47H04, 47H08, 47H10, 54-01, 54C15, 54C60, 54C65, 54H20, 54H25, 55M15, 55N05.

ISBN 978-3-11-029344-9 e-ISBN 978-3-11-029356-2 ISSN 0941-813X Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the internet at http://dnb.dnb.de. © 2013 Walter de Gruyter GmbH, Berlin/Boston Typesetting: P TP-Berlin Protago-TEX-Production GmbH, www.ptp-berlin.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen Printed on acid-free paper Printed in Germany www.degruyter.com

Smaïl Djebali: To the memory of my parents Lech Górniewicz: To Maria, Grzegorz, and Oskar Abdelghani Ouahab: To Ouahab, Nadjmi, Hidaoui, and Baliki families

Preface

In 1890, Peano [392, 393] showed that the Cauchy problem x 0 .t / D f .t; x.t //; x.t0 / D x0 ;

for t 2 Œt0 ; a;

(1)

where f W Œ0; a  Rn ! Rn is continuous, has local solutions although the uniqueness does not hold in general. This observation became a motivation for studying the structure of the set S of solutions to (1). Peano himself had shown that, in the monodimensional case n D 1, all sections S.t / D fx.t / W x 2 Sg are nonempty, compact, and connected (i.e., a continuum) (the Peano funnel) in the standard topology of the real line, for t in some neighbourhood of t0 . H. Kneser generalised this result in 1923 [294] to the case of arbitrary dimension n 2 f1; 2; : : :g. In 1928, Hukuhara [265] proved that S is a continuum in the more general framework of the Banach space of continuous functions endowed with the sup-norm. In 1942, N. Aronszajn [33] improved Kneser’s theorem in finite-dimensional spaces by showing that the Peano funnel is even an Rı -set, i.e., it is homeomorphic to the intersection of a decreasing sequence of compact contractible spaces (or compact absolute retracts), where Rı is a concept introduced by N. Aronszajn himself. In particular, this implies that S is acyclic which means, without a Lipschitzianity of the right-hand side f of (1), that the set S of solutions of (1) may not be a singleton but, from the point of view of algebraic topology, it is equivalent to a point, in the sense that it has the same homology group as one-point space. Since the famous Schauder fixed point theorem or more generally the Lefschetz fixed point theorem yielded the existence of fixed points for some classes of mappings and that uniqueness is not in general guaranteed, then a natural question usually addressed is to characterise the set of fixed points. Following Aronszajn’s theory [33], important progress was made by F. Browder and C. Gupta in 1969 [86] but the theory really received a new impetus after the publication of the Browder–Gupta theorem. In the same year, York [497, Theorem 1.3] proved that the solution set S.F; x0 / of the autonomous Cauchy problem x 0 2 F .x/; x.0/ D x0 is contractible provided F is a Carathéodory bounded multi-valued function with compact, convex values which admits a Lipschitzian selection. This together with some -selectionability properties proved later by Lasry–Robert [319] and Górniewicz [215] have led to several results concerning the topological structure of the set of solutions of many classes of differential equations and inclusions. In particular, it has been shown that the boundedness

viii

Preface

of F may be relaxed and replaced by linear growth or even by some Nagumo type conditions. In case of differential equations, many results follow immediately from the Browder–Gupta theorem and the Szufla type lemma [447–451]. Several generalisations to the case of differential inclusions, integro-differential inclusions, and functional differential inclusions have recently been considered in the following papers and references therein [19,24,26,39,42,71,72,90–92,94,105,110,115,117,123,142, 144, 148, 158, 195, 196, 210, 213, 238, 268, 302, 313, 329, 331–333, 349–351, 369–372, 372–385, 406, 428, 432, 440, 441, 451–454, 471, 505, 506]. Evidently the characterisation of the set of fixed points for some operators implies corresponding results on the structure of solution sets for initial and boundary value problems; regarding this approach, a concise account is given in [156]. In fact, many papers have been precisely concerned with fixed point properties of nonlinear algebraic and differential operators; we quote for instance [132,133,157,309,310,336,347,408, 409, 425, 455, 475]. Earlier, Krasnosel’skiˇı and Perov [292] proved a connectedness principle for single-valued compact mappings in 1959. Then some extensions have been obtained by Górniewicz and Pruszko [223] in 1980 and B. D. Gel’man [195] in 1987 (see also [319]). In this monograph, we develop this theory and use it to address some questions about the solvability and the structure of solution sets of many classes of differential and integral equations and inclusions. Further to AR properties of some solution sets in connection with contraction mappings, we have been mainly concerned with contractibility, Rı -contractibility or merely Rı -structure of solution sets of problems associated with differential inclusions. As we shall see in this monograph, this is rather related to the analytic properties enjoyed by the right-hand side of the equation or the inclusion. This book is an attempt to offer a comprehensive exposition of this theory by giving a systematic presentation of classical and recent results obtained in the last couple of years. We presented a detailed description of methods spread over the literature and concerning the topological structure of fixed point sets and solution sets for differential equations and inclusions. Our main motivation and primary goal are twofold: first, to provide many of the basic techniques and results recently developed about this theory; second, to assemble the literature that is disseminated and scattered in several papers of pioneering researchers who developed the functional analytic framework of this field over the past few decades. We hope to render these results more readily accessible to graduate and post-graduate students and also to more advanced researchers interested in this theory. The methods now called Browder–Gupta method, Banach method, and inverse limit method are presented and most of the advanced results achieved to date and concerning the above three methods are surveyed in this monograph. Moreover, several examples of applications relating to initial and boundary value problems are discussed in detail.

Preface

ix

The presentation of the book is reasonably self-contained since we have assumed familiarity only with basic knowledge of real functional analysis; almost no profound knowledge of topology is required. Prerequisites are standard graduate courses in general topology. The book is intended for instructors active in research areas with interests in topological properties of fixed point mappings and applications; it will also be beneficial to advanced graduate researchers since it aims to provide students with the necessary understanding of the subject with no deep background material needed. We have intentionally included a rich and very extensive bibliography in which the reader can find further results. We hope this monograph will fill the vacuum in the literature regarding the topological structure of fixed point sets and its applications to differential equations and inclusions. Of course, the bibliography is also enriched by some fundamental titles dealing with topology and functional analysis used in this book. Essentially, the book is divided into four main chapters and two supplementary chapters. Chapters 5 and 6 give an overview of the necessary background of topology and elementary multi-valued analysis, respectively; they contain the basic notions for a useful basis for the entire book. Indeed, basic notions and even advanced parts of topology necessary for a good reading of Chapter 1 are outlined in Chapter 5; this chapter could be useful for readers more interested in algebraic topology. In contains the main definitions and properties of common notions of retraction, contractibility, and acyclicity frequently used to describe the topological structure of the sets of solutions. In Chapter 6, we have collected most important results about multi-valued mappings and their topology (measurability, continuity, selection problems, etc). However, auxiliary results from functional analysis are gathered together separately in the Appendix. In 1946, S. Eilenberg and D. Montgomery observed that, by using the Vietoris mapping theorem, the Lefschetz fixed point theorem could be carried over to multi-valued acyclic maps of compact ANR-spaces. Starting from the classical fixed point theory, Chapter 1 focuses on fundamental results recently obtained and which concern the topological structure of single-valued and multi-valued fixed point mappings. The case of nonexpansive maps, which lack strict contractivity, is considered but also the structure of solution sets for multi-valued contractions is investigated. Even the class of admissible maps which contain as particular cases acyclic maps is studied in this chapter; this abstract theory plays a key role in the investigation of solution sets of many initial and boundary value problems. We have however tried to indicate as far as possible the original sources of the various latest results we have learned about. Most of the classical known results on the existence (and uniqueness) theory for ordinary differential equations and inclusions are collected in Chapter 2. This chapter encompasses basic and useful results important to understanding the study of the structure of the solution sets for problems posed either on bounded or unbounded domains. This chapter relies heavily on the fixed point theory developed in Chapter 1 and it could be read independently of the rest of the material. The theory of Aronszajn and Browder–Gupta is applied in Chapter 3 to investigate the topological structure of the solution sets for some classes of differential equations

x

Preface

and inclusions, extending the classical Kneser–Hukuhara theorems. In particular, for a Cauchy problem associated to an ordinary differential equation, it is proved that the solution set is acyclic even when Lipschitzianity of the nonlinearity fails. In Chapter 3, several examples of applications are studied in detail. Further to Rı properties of the set of solutions, some recent results about the contractibility or the acyclicity are presented. Finally, in Chapter 4, our goal was to present a detailed account of the existence theory together with the investigation of structure of solution sets of impulsive differential equations and inclusions. This chapter is designed as a survey of some recent results obtained by the authors and others. Acknowledgments. During the preparation of this monograph we have received much encouragement and help from colleagues and friends we cannot all cite here and thank. Some parts of the monograph were prepared during stays of the first author in the Schauder Center for Nonlinear Studies (Toru´n, Poland) and in J. L. Lions Laboratory (Paris VI University, France), also scientific visits of the second author in Algeria, and visits of the third author in the Mathematics Department of Santiago de Compostela University (Spain). The authors would like to warmly thank these laboratories and institutions for their kind hospitality. Finally, we express our special and sincere thanks to Walter de Gruyter in Berlin (Germany) for accepting publication of this monograph in the “De Gruyter Series in Nonlinear Analysis and Applications”. Toru´n and Algiers, May 2010–May 2012

Smaïl Djebali, Lech Górniewicz, and Abdelghani Ouahab

Notations

The most frequently used notations, symbols, and abbreviations are listed below. 

N n f0g D f1; 2; : : : ; g set of positive natural numbers



Q set of rational numbers



R set of real numbers



Rn n-dimensional real Euclidean space



Bn (or Kn ) unit ball in Rn



Sn1 D @Bn boundary of Bn unit sphere in Rn



.A/ Lebesgue measure of A



dim X dimension of the space X



diam A D supfd.x; y/ j .x; y/ 2 Ag diameter of the set A, where A is a subset of a metric space .E; d /



d.x; A/ D inffd.x; y/ j y 2 Ag



dist.A; B/ D d.A; B/ D inffd.x; y/ j x 2 A; y 2 Bg



Hd.A; B/ D supfd.a; B/ j a 2 Ag distance from A to B



Hd .A; B/ D max.Hd .A; B/; Hd .B; A// Hausdorff distance between the sets A and B



.X; k:k/ real Banach space with norm k:k



B.x0 ; r / open ball with radius r centred at x0



N 0 ; r / closed ball B.x



B.A; "/ D O".A/ D fx 2 E j d.x; A/ < "g "-neighbourhood of the subset A  E



cl A D A closure of the set A



co A D conv .A/ closure of the convex hull of the set A

xii

Notations



C.K; E/ space of continuous functions on the compact space K with values in a space E, endowed with the norm kf k D supfkf .x/kE j x 2 Kg



Lp .; ; E/ (Banach) Lebesgue space of Bochner -measurable functions with R 1=p p-th summable power with norm kf kp D  kf kp E d



f W X ! Y single-valued map



F W X ! P.Y / (or F W X ( Y ) multi-valued map (or set-valued map) (or multi-map)



Pp .E/ D fY 2 P.E/ j Y has property pg



kF .x/kP D supfkyk j y 2 F .x/g



L ˝ B product-measurable



F Nemytskiˇı operator associated with a single-valued map f



SF .y/ selection set of the multi-valued map F



SF .:/ superposition operator associated with F



S.f; x0 / set of solution to x 0 D f .t; x/; x.t0 / D x0



' 1 .B/ small counter image of B under '



1 'C .B/ large counter image of B under '



Gr .F / graph of the multi-function F



Pp .X/ set of subsets of X having property p



fjA restriction of f to A



A characteristic function of the set A



FPP fixed point property



Fix .f / set of fixed points of f



AR absolute retract



ANR neighbourhood absolute retract



ES extension property



SP selection property



LC locally contractible

Notations 

ˇ H (H  ) the Cech homology (cohomology) functor



.X; X0 / (X0  X) pair



pW .X; X0 / ! .Y; Y0 / map of pairs



Deg.f; ; y0 / topological degree of f on  with respect to y0



ind .p; q/ fixed point index for admissible mappings



.f / Lefschetz number of f



˛.A/ Kuratowski measure of noncompactness of the set A



.A/ Hausdorff measure of noncompactness of the set A



' homotopic to



n set-theoretic difference

xiii

Contents

Preface

vii

Notations

xi

1

1

Topological structure of fixed point sets 1.1 Case of single-valued mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Fundamental fixed point theorems . . . . . . . . . . . . . . . . . . . . . . 1.1.1.1 Banach’s fixed point theorem . . . . . . . . . . . . . . . . . 1.1.1.2 Brouwer’s fixed point theorem . . . . . . . . . . . . . . . . 1.1.1.3 Schauder’s fixed point theorem . . . . . . . . . . . . . . . . 1.1.2 Approximation theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Browder–Gupta theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Acyclicity of the solution sets of operator equations . . . . . . . . 1.1.5 Nonexpansive maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5.1 Existence theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5.2 Solution sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 5 7 11 13 20 23 23 26

1.2 The case of multi-valued mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Approximation of multi-valued maps . . . . . . . . . . . . . . . . . . . . 1.2.2 Fixed point theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Multi-valued contractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Fixed point sets of multi-valued contractions . . . . . . . . . . . . . . 1.2.5 Fixed point sets of multi-valued nonexpansive maps . . . . . . . . 1.2.6 Fixed point sets of multi-valued condensing maps . . . . . . . . . . 1.2.6.1 Measure of noncompactness . . . . . . . . . . . . . . . . . . 1.2.6.2 Condensing maps . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 27 30 33 35 38 39 39 43

1.3 Admissible maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Fixed point theorems for admissible multi-valued maps . . . . . 1.3.3 The general Brouwer fixed point theorem . . . . . . . . . . . . . . . . . 1.3.4 Browder–Gupta type results for admissible mappings . . . . . . . 1.3.5 Topological dimensions of solution sets . . . . . . . . . . . . . . . . . .

44 44 53 58 60 62

xvi

2

3

Contents

1.4 Topological structure of fixed point sets of inverse limit maps . . . . . . . 1.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Multi-maps of inverse systems . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 66 67

Existence theory for differential equations and inclusions

72

2.1 Fundamental theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Existence and uniqueness results . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Picard–Lindelöf theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2.1 Maximal solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Peano and Carathéodory theorems . . . . . . . . . . . . . . . . . . . . . . 2.1.3.1 Peano theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72 72 73 75 77 77

2.2 The extendability problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Global existence theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Existence results on noncompact intervals . . . . . . . . . . . . . . . . 2.2.2.1 The Lipschitz case . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2.2 The Lipschitz–Nagumo case . . . . . . . . . . . . . . . . . . 2.2.2.3 The Nagumo case . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 A boundary value problem on the half-line . . . . . . . . . . . . . . .

79 79 82 82 83 86 88

2.3 The case of differential inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Initial value problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1.1 A Nagumo type nonlinearity . . . . . . . . . . . . . . . . . . 2.3.1.2 A Lipschitz nonconvex nonlinearity . . . . . . . . . . . . 2.3.2 Boundary value problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2.1 The convex case . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2.2 The nonconvex case . . . . . . . . . . . . . . . . . . . . . . . .

94 94 94 97 99 100 103

Solution sets for differential equations and inclusions

105

3.1 General results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Kneser–Hukuhara theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Problems on bounded intervals . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Problems on unbounded intervals . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Second-order differential equations . . . . . . . . . . . . . . . . . . . . . 3.1.5 Abstract Volterra equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.6 Aronszajn type results for differential inclusions . . . . . . . . . . .

105 105 108 109 111 113 114

3.2 Second-order differential inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 The convex case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 The nonconvex case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Solution sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

122 122 127 130

Contents

xvii

3.3 Higher-order differential inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 3.4 Neutral differential inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 The convex case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 The nonconvex case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Solutions sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

135 136 142 146

3.5 Nonlocal problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 3.5.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 3.5.2 A viability problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

4

3.6 Hyperbolic differential inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1.1 The convex case . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1.2 The nonconvex case . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Solution sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

154 155 155 159 160

Impulsive differential inclusions: existence and solution sets

163

4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Ecological model with impulsive control strategy . . . . . . . . . . 4.1.2 Leslie predator-prey system . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Pulse vaccination model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 163 164 165

4.2 Semi-linear impulsive differential inclusions . . . . . . . . . . . . . . . . . . . . 4.2.1 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1.1 The convex case . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1.2 The nonconvex case . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Structure of solution sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

166 166 167 181 186

4.3 A periodic problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Existence results: 1 2 .T .b// . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 The convex case: a direct approach . . . . . . . . . . . . . . . . . . . . . . 4.3.3 The convex case: an MNC approach . . . . . . . . . . . . . . . . . . . . . 4.3.4 The nonconvex case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 The parameter-dependant case . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5.1 The convex case . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5.2 The nonconvex case . . . . . . . . . . . . . . . . . . . . . . . . 4.3.6 Filippov’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.7 Existence of solutions: 1 62 .T .b// . . . . . . . . . . . . . . . . . . . . . 4.3.7.1 A nonlinear alternative . . . . . . . . . . . . . . . . . . . . . . 4.3.7.2 A Poincaré translation operator . . . . . . . . . . . . . . . . 4.3.7.3 The MNC approach . . . . . . . . . . . . . . . . . . . . . . . . .

197 198 199 207 212 215 215 217 220 230 230 233 233

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5

Contents

4.4 Impulsive functional differential inclusions . . . . . . . . . . . . . . . . . . . . . 4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Structure of the solution set . . . . . . . . . . . . . . . . . . . . . . . . . . . .

236 236 237 246

4.5 Impulsive differential inclusions on the half-line . . . . . . . . . . . . . . . . . 4.5.1 Existence results and compactness of solution sets . . . . . . . . . . 4.5.1.1 The convex u.s.c. case . . . . . . . . . . . . . . . . . . . . . . . 4.5.1.2 The nonconvex Lipschitz case . . . . . . . . . . . . . . . . . 4.5.1.3 The nonconvex l.s.c. case . . . . . . . . . . . . . . . . . . . . 4.5.2 Topological structure via the projective limit . . . . . . . . . . . . . . 4.5.2.1 The nonconvex case . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2.2 The convex case . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2.3 The terminal problem . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Using solution sets to prove existence results . . . . . . . . . . . . . .

250 250 251 258 262 265 266 271 274 283

Preliminary notions of topology and homology

288

5.1 Retracts, extension and embedding properties . . . . . . . . . . . . . . . . . . . 288 5.2 Absolute retracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 5.3 Homotopical properties of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 ˇ 5.4 Cech homology (cohomology) functor . . . . . . . . . . . . . . . . . . . . . . . . . 304 5.5 Maps of spaces of finite type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 ˇ 5.6 Cech homology functor with compact carriers . . . . . . . . . . . . . . . . . . . 313 5.7 Acyclic sets and Vietoris maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 5.8 Homology of open subsets of Euclidean spaces . . . . . . . . . . . . . . . . . . 319 5.9 Lefschetz number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 5.10 The coincidence problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 6

Background in multi-valued analysis

337

6.1 Continuity of multi-valued mappings . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Upper semi-continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2.2 "  ı u.s.c. mappings . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2.3 U.s.c. maps and closed graphs . . . . . . . . . . . . . . . . . 6.1.3 Lower semi-continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3.2 "  ı l.s.c. mappings . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4 Hausdorff continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

339 339 341 341 344 345 346 346 349 350

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xix

6.2 The selection problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Michael’s selection theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Michael’s family of subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 selectionable mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 The Kuratowski–Ryll-Nardzewski selection theorem . . . . . . . 6.2.5 Aumann and Filippov theorems . . . . . . . . . . . . . . . . . . . . . . . . 6.2.6 Hausdorff measurable multi-valued maps . . . . . . . . . . . . . . . . . 6.2.7 Product-measurability and the Scorza–Dragoni property . . . . .

354 355 358 362 366 378 382 383

6.3 Decomposable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 The Bressan–Colombo–Fryszkowski selection theorem . . . . . 6.3.2 Decomposability in L1 .T; E/ . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Integration of multi-valued maps . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Nemytskiˇı operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

390 390 390 392 393

Appendix

399

ˇ A.1 Axioms of the Cech homology theory . . . . . . . . . . . . . . . . . . . . . . . . . . 399 A.2 The Bochner integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 A.3 Absolutely continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 A.4 Compactness criteria in C.Œa; b; E/, Cb .Œ0; 1/; E/, and P C.Œa; b; E/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 A.5 Weak-compactness in L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 A.6 Proper maps and vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 A.7 Fundamental theorems in functional analysis . . . . . . . . . . . . . . . . . . . . 411 A.8 C0 -Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 References

415

Index

451

Chapter 1

Topological structure of fixed point sets

In this chapter, we shall present a survey of the fundamental fixed point theory both for single-valued and multi-valued mappings. Moreover, we shall formulate all wellknown results concerning topological structure of fixed point sets for mappings. Since the respective results for multi-valued mappings are generalisations of the singlevalued case, we will present proofs mainly in the multi-valued case. Note that several notions from topology and related topics (such as AR-spaces) are gathered together in Chapter 5. So we often refer the reader to this supplementary chapter and the references contained therein.

1.1 Case of single-valued mappings 1.1.1 Fundamental fixed point theorems 1.1.1.1 Banach’s fixed point theorem Let .X; d / and .Y; d1 / be two metric spaces. A mapping f W X ! Y is called contractive (or Banach contraction) provided there exists ˛ 2 Œ0; 1/ such that: 8 x; y 2 X; d1 .f .x/; f .y//  ˛d.x; y/: Assume that A is a subset of X and f W A ! A is a contraction. Then f has at most one fixed point, i.e., a point x 2 A such that f .x/ D x. Indeed, if f .x/ D x and f .y/ D y, then we have d.f .x/; f .y// D d.x; y/  ˛d.x; y/ and it contradicts our assumption that ˛ 2 Œ0; 1/. First, we recall the famous Banach contraction principle (see, e.g., [4, Theorem 1.1], [231, Theorem 1.1]: Theorem 1.1. Let C be a (nonempty) closed subset of a complete metric space .E; d / and let N W C ! C be a K-contraction, then N has unique fixed point and we have d.N n .x/; x/ 

Kn d.N.x/; x/; 1K

8 x 2 C:

Proof. Let x 2 C and define the sequence xn D N n .x/; where N n D N ı    ı N . Using the fact that N is K-contraction, we get d.xnC1 ; xn /  K n d.N.x/; x/;

n 2 N:

2

Chapter 1 Topological structure of fixed point sets

By the triangle inequality, we deduce that for m  n m X

d.xn; xm /  d.xn; xnC1 / C    C d.xm1 ; xm / 

K i 1 d.N.x/; x/:

i DnC1

Thus, d.xn; xm / 

K n .1  K mn / jK m  K n j d.N.x/; x/ D d.N.x/; x/: 1K 1K

Since K 2 .0; 1/, K n is a Cauchy sequence in R hence a Cauchy sequence with respect to d and thus converges to some limit x 2 C . Also the continuity of N guarantees that d.N.xn1 /; N.x// ! 0; as n ! 1: Moreover, d.N.xn1 /; N.x // D d.xn ; N.x // and we have d.x ; N.x //  d.x ; xn / C d.xn; N.x // D d.x; xn / C d.N.xn1 /; N.x //: Then d.x; N.x // D 0 whence N.x / D x . For the uniqueness of the fixed point, assume that there are y1 ; y2 2 C such that y1 D N.y1 /; y2 D N.y2 /; then d.y1 ; y2 / D d.N.y1 /; N.y2 /  Kd.y1 ; y2 /: It follows that d.y1 ; y2 / D 0, that is y1 D y2 . Now, we show that d.N n.x/; x/ 

Kn d.N.x/; x/: 1K

Using the definition of xn , we get d.N n.x/; x/ D lim d.xn1 ; xm / m!1

 lim

m!1

Hence, d.N n.x/; x/ 

K n .1  K mn / d.N.x/; x/: 1K

Kn d.N.x/; x/; 8 x 2 C: 1K

We prove the following local version of the Banach contraction principle.

3

Section 1.1 Case of single-valued mappings

Theorem 1.2. Let .X; d / be a complete metric space and B.x0 ; r0 / D fx 2 X j d.x; x0 / < r0 g be the open ball in X with radius r0 and centred at some point x0 2 X. Assume that f W B.x0 ; r0 / ! X is a contractive map such that: d.x0 ; f .x0 // < .1  ˛/r0 ; where ˛ 2 Œ0; 1/ is the contraction constant for f . Then f has exactly one fixed point. Proof. Let 0 < r1 < r0 be a constant such that d.f .x0 /; x0 /  .1˛/r1 < .1˛/r0 . We let K.x0 ; r1 / D B.x0 ; r1 /: Then K.x0 ; r1 / is a complete metric space. Let us define a map e.x/ D f .x/; f

for every x 2 K.x0 ; r1 /:

In view of Theorem 1.1, for the proof it is sufficient to show that e.K.x0 ; r1 //  K.x0 ; r1 /: f Let x 2 K.x0 ; r1 /; then we have: e.x//  d.x0 ; f .x0 // C d.f .x0 /; f .x// d.x0 ; f  .1  ˛/r1 C ˛d.x; x0 /  .1  ˛/r1 C ˛r1 D r1 and the proof is completed. Now, let us consider a Banach space .E; k  k/ and its subset U  E. A map f W U ! E is called a contractive field provided there exists a contractive map F W U ! E such that f .x/ D x  F .x/;

for every x 2 U:

As a consequence of Theorem 1.2, we now prove the domain invariance theorem: Theorem 1.3. Let U be an open subset of E and f W U ! E be a contractive field, then (1) f .U / is an open subset of E, (2) the map F W U ! f .U / is a homeomorphism. Proof. First, we shall show that f .U / is an open subset of E. Let y0 2 f .U /, i.e., f .x0 / D x0  F .x0 / D y0 , for some x0 2 U . Since U is open, there is some r > 0 such that B.x0 ; r /  U . Let ˛ 2 Œ0; 1/ be the constant of contraction given for E,

4

Chapter 1 Topological structure of fixed point sets

where f .x/ D x  F .x/ for every x 2 U . We claim that B.y0 ; .1  ˛/r /  f .U /. In this order we will show that f .B.x0 ; r // D B.y0 ; .1  ˛/r /: In fact, let y 2 B.y0 ; .1  ˛/r / and define a map G W B.x0 ; r / ! E by G.x/ D y C F .x/: Evidently, G is a contractive map with constant ˛. Moreover, we have: kx0  G.x0 /k D kx0  y  F .x0 /k D kf .x0 /  yk D ky0  yk < .1  ˛/r: Consequently, in view of Theorem 1.2, there is a point x 2 B.x0 ; r / such that G.x/ D x. Hence F .x/ C y D x and y D x  F .x/ D f .x/, so the proof is completed. Theorem 1.4. The conclusion of Theorem 1.1 remains valid if there exists m 2 f1; 2; : : :g such that f m is a contraction. Sketch of the proof. For x 2 X, define '.x/ D d.x; f m .x//. Then .'.f .n/ .x//n is a Cauchy sequence, hence converges to some limit x0 . In addition, for all x 2 X, and all n  1, we have '.f .n/ .x//  k rn max '.f .k/ .x//; kn1

where rn is the largest integer greater or equal to n=m. Hence x0 D 0 and so .f .n/ .x//n is a Cauchy sequence, hence converges to some limit y. We have d.y; f .m/ .y// D 0 which implies that f .y/ D f .m/ .y/. Since f .m/ has only one fixed point, we conclude that y D f .y/. Remark 1.5. Let X D C.Œ0; b; R/ and F W X ! X be defined by Z

t

F .x/.t / D

x.s/ds: 0

Then F is not a contraction if b > 1; however F .n/ .x/.t / D

1 .n  1/Š

is a contraction for a large enough n.

Z

t 0

.t  s/n1 x.s/ds

5

Section 1.1 Case of single-valued mappings

1.1.1.2 Brouwer’s fixed point theorem We start with some general definitions and observations. Definition 1.6. We say that a Banach space X has the fixed point property (FPP for short) if each continuous map f W X ! X has a fixed point. As an easy consequence, we have Proposition 1.7. (a) If X and Y are homeomorphic and X has the FPP, then Y has also the FPP. (b) If C is a retract of X and X is a Hausdorff space with the FPP, then C has the FPP. Now, we prove the Brouwer fixed point theorem (1912). Theorem 1.8. Let X be a finite-dimensional normed space, C X a nonempty closed bounded convex subset, and f W C ! C a continuous map. Then f has at least one fixed point. Proof. (a) First, consider the particular case C D B1 . If f .x0 / D x0 ; for some x0 2 @C , then we are done; otherwise f .x/ ¤ x; 8x 2 @C . Then consider the continuous deformation ft .x/ D x  tf .x/. For t 2 Œ0; 1/ and x 2 @C , we have the estimates: kft .x/k  jkxk  t kf .x/kj D jR  t kf .x/kj  R  t kf .x/k  .1  t /R > 0: Indeed, since f .C /  C , then t kf .x/k < kf .x/k  R; 8 t 2 Œ0; 1Œ. The Leray– ı

Schauder topological degree (see, e.g., [326]) Deg .Id  tf; C ; 0/ is then well defined and equals, by homotopy, ı

ı

Deg .Id; C ; 0/ D 1 D Deg .Id  f; C ; 0/ ¤ 0: ı

By the existence property of the degree, there exists x 2C such that .Id  f /.x/ D 0 , f .x/ D x. (b) Let R > 0 be large enough so that C  B R . Since C is a retract of BR and BR is homeomorphic to B1 . Then, from part (a) and Proposition 1.7, we conclude that C has the FPP. In fact, the result below allows us to only consider closed balls; for a proof of the lemma, we refer to [387, Proposition 3.5.1]. Lemma 1.9. Let C  Rn be a nonempty bounded closed convex subset with nonempty interior. Then C is homeomorphic to the closed unit ball.

6

Chapter 1 Topological structure of fixed point sets

For the sake of completeness, we mention that it is easy to prove that the Brouwer fixed point theorem is equivalent to [Corollary 5.15, Chapter 5]. Indeed, to sum up, we have Proposition 1.10. If X be a finite-dimensional Banach space, then the following statements are equivalent: (a) B has the FPP. (b) S is not a retract of B (c) S is not contractible. Proof. .a/ H) .b/. Arguing by contradiction, assume there exists a continuous function f from B1 .0/ on @B1 .0/ such that f .x/ D x; 8 x 2 @B1 .0/. Applying Brouwer’s fixed point theorem to the map g D f; we obtain some x0 2 B1 .0/ such that g.x0 / D x0 , i.e., f .x0 / D x0 . Now f .B1 .0//  @B1 .0/, then x0 2 @B1 .0/ and kx0 k D 1. Since f self-maps the boundary, we obtain that f .x0 / D x0 hence x0 D x0 I then we have proved that x0 D 0; leading to a contradiction. .b/ H) .a/. Again by contradiction, if the Brouwer fixed point theorem does not hold, then there would exist a continuous function f W B1 .0/ ! B1 .0/ which is fixed point free. Then consider the map gW B1 .0/ ! B1 .0/ such that g.x/ is the point where the half-line Œf .x/; x/ meets @B1 .0/; more precisely g.x/ D f .x/ C t .x/.x  f .x//, where t .x/ is the positive root of the second-order algebraic equation t 2 kx  f .x/k2 C 2t hf .x/; x  f .x/i C kf .x/k2 D 1: By definition, g self-maps the boundary @B1 .0/, proving our claim. .b/ H) .c/. This follows from [Proposition 5.34, Chapter 5]. .c/ H) .b/. By contradiction, assume that S is a retract of B1 .0/; then there is some retraction r W B1 .0/ ! S. Now B1 .0/ being contractible, there exists a homotopy h.t; x/ W B1 .0/ ! B1 .0/ such that h.0; :/ D IjB1 .0/ and h.1; :/ is constant for all t 2 Œ0; 1. The composition H D r ı h ı i is a homotopy H.0; :/ D IjB1 .0/ and H.1; :/ is constant for all t 2 Œ0; 1, which is a contradiction. The following nonlinear alternative, due to Böhl, is an immediate consequence (it is even equivalent to Brouwer’s fixed point theorem) (see [214, Theorem 2.5]: Corollary 1.11 (Nonlinear alternative). Every continuous function f W B ! Rn has at least one of the following properties: (i) the equation x D f .x/ has a solution. (ii) there exist x 2 Sn1 and t 2 .0; 1/ such that x D tf .x/.

7

Section 1.1 Case of single-valued mappings

1.1.1.3 Schauder’s fixed point theorem A fundamental difference between finite and infinite-dimensional spaces is given by: Proposition 1.12. Let B be the unit ball in a normed space X. Then, we have dim X < C1 , every continuous map f W B ! B has a fixed point. Proof. One implication is nothing but the Brouwer fixed point theorem. To prove the converse, consider the space of sequences .xn /n2N normed by k xk D supn j xn j and the subset of vanishing sequences X D f.xn/n2N j limn!C1 xn D 0g. Finally, define the map f W X ! X by .f .x//1 D

1 C k xk 2

and .f .x//n D xn1 ; n > 1:

Then f is fixed point free for otherwise 8n  1; xnC1 D xn D    D x1 D contradicting 1Ck2 xk 62 X.

1Ck xk 2

;

Remark 1.13. Another counter-example is given by the space X D l2 D f.xn /n2N j  P P 2 2 1=2 . n2N xn < 1g of summable-square sequences with norm kxk D n2N xn If B is the closed unit ball in X, then define the mapping f W B ! B by q  f .x/ D 1  kxk2 ; x1 ; x2 ; : : : : The function f is continuous but is fixed point free for otherwise there p would exist some x 2 X such that x D f .x/; hence kxk D kf .x/k D 1; x1 D 1  kxk2 D 0, and x2 D x1 D    D 0, contradicting the fact that kxk D 1. From above, we deduce that in infinite-dimensional spaces, continuity of mappings is not enough to get fixed point theorems. We need some stronger assumptions, namely that the range f ./ is compact. Now, we are ready to prove the Schauder fixed point theorem and some of its variants. Theorem 1.14 (Schauder’s fixed point theorem, 1930). (see, e.g., [430, Theorem 4.1.1]) Let C be a nonempty convex closed bounded subset of a Banach space X and K W C ! C a compact map. Then K has a fixed point. Remark 1.15. In fact, we only need X to be a normed space. In addition, C is only required to be convex (not necessarily closed and bounded). Proof. First proof. (a) Let C D B.0; 1/ be the unit ball. If there exists x0 2 @C such that K.x0 / D x0 ; then we are done. Otherwise for all t 2 Œ0; 1, the degree Deg .Kt ; C; 0/; where Kt D I  tK; is well defined. Indeed, if

8

Chapter 1 Topological structure of fixed point sets

there exists some x 2 @C such that tK.x/ D x; then R D kxk D t kK.x/k  Rt for K.C /  C ; hence t D 1, contradicting kK.x/k D R D kxk. By the homotopy property, we deduce that Deg .K; C; 0/ D 1; yielding our claim. (b) C is a nonempty convex closed bounded subset. Consider the continuous retraction R W X ! C , let B be a ball containing C , and R

R

let the diagram B ! C ! K.B/. The map K ı R is compact for K is compact and R bounded. From the first step, the map K ı R has a fixed point x0 2 B; x0 D .K ı R/.x0 /. But R.x0 / 2 C and by assumption K.C /  C I then K.R.x0 // 2 C hence x0 2 C . Of course, the result still holds if C is homeomorphic to a convex closed set. Second proof. The Schauder fixed point theorem may also be deduced from the Brouwer fixed point theorem and the approximation theorem of compact mappings (Theorem 1.30). Now, let Kn be such a compact approximation of K and Nn the subspace spanned by the image of Kn . The set C is convex and the image Kn.C / is contained in the convex hull of K.C /; hence Kn sends C into C \ Nn . Consequently, Kn sends the closed bounded C \Nn into itself. The Brouwer fixed point theorem implies that, for each n, the map Kn has a fixed point xn in C \Nn . As n ! C1, the sequence Kn .xn/ admits by compactness a subsequence, still denoted Kn .xn /, which converges uniformly. Hence, the sequence xn D Kn .xn / converges to some limit x0 as n ! 1. However, kxn  K.xn /k D kKn .xn /  K.xn /kI then lim Kn .xn / D K.x0 / and finally K.x0 / D x0 .

n!1

An immediate consequence is given by Corollary 1.16. Let C be a nonempty compact convex subset of a Banach space X and f W C ! C be a continuous map. Then f has a fixed point. Corollary 1.17. Let C be a nonempty closed convex subset (non-necessarily bounded) of a Banach space X and f W C ! C be a continuous map such that f .C / lies in a compact subset of C . Then f has a fixed point. Proof. There exists a compact subset A  C such that f .C /  A  C . Setting A0 D conv.A/, we obtain some fixed point in A0 , hence in C (notice that A0 is convex, compact and f .A0 /  A  A0  C /. When the closed bounded set is not self-mapped, we have some interesting results from the application point of view. Corollary 1.18 (Nonlinear alternative). Let  be a bounded open subset of a Banach space X with 0 2  and let f W  ! X be a compact map. Then either (i) f has a fixed point in , or (ii) there exist x 2 @ and t 2 .0; 1 such that x D tf .x/.

Section 1.1 Case of single-valued mappings

9

Proof. If condition (ii) does not hold, then 8 x 2 @; 8 t 2 Œ0; 1; .I  tf /.x/ ¤ 0: This implies that the degree Deg .I  tf; ; 0/ is well defined and is equal, by homotopy, to Deg .I; ; 0/ D 1. Letting t D 1, we obtain that f has a fixed point in . Corollary 1.19. Let X be a Banach space and K W X ! X be a compact map satisfying the hypothesis: .H/ 9 r > 0 j 8 t 2 Œ0; 1 .tK.x/ D x ) x 2 B.0; r //: Then K has a fixed point in B D B.0; r /. Remark 1.20. (a) .H/ is an a priori estimate assumption. (b) Corollary is equivalent to the Schauder fixed point theorem. (c) This corollary has also the following version: Theorem 1.21 (Schaefer’s theorem, 1955). Let X be a Banach space and K W X ! X be a compact map. Then we have the alternative: either the equation tK.x/ D x has a solution for all t 2 Œ0; 1, or the set S D fx 2 Xj 9 t 2 Œ0; 1; tK.x/ D xg is unbounded. Corollary 1.22 (Rothe’s theorem, 1957). Let B be an open ball in a Banach space X and K W X ! X be a compact map such that K.@/  B. Then K has a fixed point in B. Similarly to Theorem 1.3, we also have Corollary 1.23 (Schauder’s domain invariance). Let U be an open subset of a normed space E and f W U ! E be an injective completely continuous field. Then (a) f .U / is an open subset of E, (b) the map F W U ! f .U / is a homeomorphism. We omit the proof but we give a useful consequence Corollary 1.24 (Fredholm alternative). Let E be a normed space E and F W E ! E be a completely continuous linear operator. Then either (a) the equation x  F .x/ D 0 has a solution, or (b) the equation y D x  F .x/ has a unique solution for each y 2 E.

10

Chapter 1 Topological structure of fixed point sets

Next, we formulate some generalisation of the Schauder fixed point theorem. The first one is concerned with locally convex spaces instead of normed spaces; for the proof we refer, e.g., to [231, Theorem 1.10]: Theorem 1.25 (Tychonoff, 1935). Let X be a locally convex space, C  X a nonempty compact convex subset, and f W X ! X be a continuous map. Then f has a fixed point. The next fundamental fixed point theorem is the Lefschetz fixed point theorem. If in the coincidence Theorem 5.119, we put q.x/ D F .x/ and p D IdX then we get: Theorem 1.26 (Lefschetz fixed point theorem). Let X be an ANR-space and let F W X ! Y be a compact map, then: (1) F is a Lefschetz map, i.e., F W H .X/ ! H .Y / is a Leray endomorphism and (2) if ƒ.F / ¤ 0, then there exists a point x 2 X such that F .x/ D x. For the definitions of AR and ANR-spaces and the basic notions about topology and homology, we refer the reader to Chapter 5. Let us remark that the above version of Theorem 1.26 was proved by A. Granas (see [231]). When X is a compact ANR and F is an acyclic map with ƒ.F / ¤ 0, Theorem 1.26 is known as Eilenberg–Montgomery fixed point theorem (see, e.g., [Corollary 7.4, [231]]). Observe that if X is an AR-space, then for any compact map F W X ! X the Lefschetz number ƒ.F / of F is equal to 1. Therefore, from Theorem 1.26, we get (see also [Corollary 5.124, Chapter 5]): Theorem 1.27 (Granas–Schauder fixed point theorem). If X 2 AR and F W X ! X is a compact map, then F has a fixed point. We shall end this section by proving a Leray–Schauder nonlinear alternative using retract theory instead of degree theory. Theorem 1.28 (Nonlinear alternative). Let E be a normed space and K be the closed ball in E with centre 0 and radius . Then every compact map F W K ! E has at least one of the following two properties: (1) F has a fixed point, (2) there exists x 2 @K (i.e., kxk D ) and  2 Œ0; 1/ such that x D F .x/. Proof.

8 x <  ; kxk r .x/ D : x;

if kxk  , if kxk < :

11

Section 1.1 Case of single-valued mappings

Then the map r ı F W K ! K is compact. By using Theorem 1.27, we get a point x 2 K such that r .F .x// D x. If F .x/ 2 K , then r .F .x// D F .x/ D x, i.e., x is a fixed point of F . If F .x/ 2 E n K , then we have r .F .x// D  This implies that kxk D  and  D proof is completed.

F .x/ D x: kF .x/k

 kF .x/k

< 1. Consequently, x D F .x/ and the

1.1.2 Approximation theorems Let us recall the well-known Lasota–Yorke approximation lemma (for more information, see [147, 215]). Lemma 1.29. Let E be a normed space, X a metric space and f W X ! E be a continuous map. Then, for each " > 0; there is a locally Lipschitz map f" W X ! E such that kf .x/  f" .x/k < "; for every x 2 X: Proof. Let " > 0 be fixed. For every x 2 X; consider the open sets V".x/ D fy 2 Xj d.y; x/ < "=2g: We have X D [x2X V" .x/: The metric space X is paracompact by Stone’s theorem; hence there exists fU j  2 ƒg a locally finite refinement of fV".x/ j x 2 Xg; i.e., an open cover of X such that x 2 X has a neighbourhood V .x/ with V .x/ \ U 6D ; only for finitely many  2 ƒ; and such that to each  2 ƒ, there exists x 2 X with U  V" .x/. Define the map k W X ! R by ( 0; for x 62 U ; k .x/ D d.x; @U /; for x 2 U and let k .x/ ; 2ƒ k .x/

 .x/ D P

for x 2 X:

Therefore, k is Lipschitz on X; fU j  2 ƒg is locally finite, and  is locally Lipschitz on X. For every  2 ƒ; we may choose some b 2 U and define f" .x/ D

X 2ƒ

 .x/f .b /;

for x 2 X:

12

Chapter 1 Topological structure of fixed point sets

Then f" is locally Lipschitz in X and satisfies kf".x/  f .x/k D k  .x/Œf" .b /  f .x/k 

X

 .x/kf .b /  f .x/k:

2ƒ

Suppose that for some x0 ; we have  .x0 / 6D 0. Then, x 2 U  U" .x0 /. Hence kf .b /  f .x/k  " and therefore X  .x/" D ": kf".x/  f .x/k  2ƒ

Also we have the following Schauder approximation theorem for compact mappings: Theorem 1.30 (Schauder’s approximation theorem). Let U be an open subset of a normed space E and let f W X ! U be a compact map. Then for every " > 0, there exists a finite-dimensional subspace E n."/ of E and a compact map f" W X ! U such that: (1) kf .x/  f" .x/k < ", for every x 2 X, (2) f" .X/  E n."/ , (3) the maps f" and f W X ! U are homotopic. Proof. Given " > 0 (which we can assume to be sufficiently small) f .X/ is contained in the union of open balls B.yi ; "/ with B.yi ; 2"/  U , i D 1; : : : ; k. For every i D 1; : : : ; k, define i W X ! RC , i .x/ D maxf0; "  kf .x/  yi kg and i .x/ i W X ! Œ0; 1; i .x/ D Pk : j D1 j .x/ Now, define f" W X ! U by putting f" .x/ D

k X

i .x/  yi :

i D1

Let E n."/ be a subspace of E spanned by vectors y1 ; : : : ; yn , i.e., E n."/ D spanfy1 ; : : : ; yk g: Then f" .X/  convfy1 ; : : : ; yn g; so f" is a compact map. We have: kf .x/  f" .x/k 

k X i D1

i .x/kf .x/  yi k < ":

Section 1.1 Case of single-valued mappings

13

Moreover, the map hW X  Œ0; 1 ! U , h.x; t / D tf .x/ C .1  t /f" .x/ is a good homotopy joining f and f" and the proof is completed. In fact, we can prove that K" is a finite polyhedron with vertices y1 ; : : : ; yn . Recall that a polyhedron is a union of a finite number of simplices, where the intersection of any two simplices is either a common face or is empty. A simplex is the convex hull of an affinely independent set of points.

1.1.3 Browder–Gupta theorems The famous Schauder fixed point theorem or more generally the Lefschetz fixed point theorem says that there exists at least one fixed point for the class of compact mappings of ANR-s. So, a natural question is to characterise the set of all fixed points for such mappings. The first result, which is still a main one, was proved by Aronszajn in 1942 [33] and later improved by Browder–Gupta in 1969 [86]. Below we shall present a slight generalisation of the above mentioned result (see [215] for more details). Definition 1.31. Let f W X ! Y be a continuous function and let y 2 Y . (a) We shall say that f is proper at the point y provided that there exists " > 0 such that for any compact set K  B.y; "/ the set f 1 .K/ is compact, where B.y; "/ is the open ball in Y with the centre at y 2 Y and radius ". (b) f W X ! Y is called proper provided that for any compact K  Y the set f 1 .K/ is compact. Of course any proper map f W X ! Y is proper at every point y 2 Y . Now we are able to reformulate the Browder–Gupta theorem: Theorem 1.32. Let E be a Banach space and f W X ! E be a continuous map such that the following conditions are satisfied: (1) f is proper at 0 2 E, (2) for every " > 0 there exists a continuous map f" W X ! E such that: (2a) kf .x/  f" .x/k < " for every x 2 X, e " W f 1 .B.0; "// ! B.0; "/ defined by (2b) the map f " e " .x/ D f" .x/; f is a homeomorphism.

for every x 2 f"1 .B.0; "//

14

Chapter 1 Topological structure of fixed point sets

Then the set f 1 .f0g/ is an Rı -set. Sketch of proof of Theorem 1.32. First, we have to prove that f 1 .f0g/ is nonempty. For every " D 1=n, n 2 f1; 2; : : :g, we take a map fn W X ! E which satisfies 1.32 (2). In view of 1.32 (2b), for every n, we can find a point xn 2 X such that fn .xn / D 0. It follows that: kf .xn /k D kf .xn/  fn .xn /k < 1=n: So the sequence ff .xn/gn2N is convergent to the point 0 2 E. Since f is proper at 0 2 E, we can assume without loss of generality that the sequence fxng is convergent to a point x 2 E. Now, from the continuity of f , it follows that f .x/ D 0 and consequently f 1 .f0g/ ¤ ;. Now let us denote by S the set f 1 .f0g/. It follows that S is nonempty and compact. For every " D 1=n, n 2 f1; 2; : : :g let An D fn .S/ where fn are chosen according to 1.32 (2). Then from 1.32 (2a) we deduce that An  B.0; 1=n/. Note that fAn gn is a sequence of compact sets. Let Cn D co.An /; where co .An / refers to the closure of the convex hull of An : It follows from Mazur’s lemma A.46 that Cn is a compact convex subset of B.0; 1=n/. Now, by using 1.32 (2b) we deduce that set Dn D fn1 .Cn / is an absolute retract (because it is homeomorphic to the convex set Cn ). Therefore, we can proceed in the same way as in the proof of [Theorem 7, [86]] and our theorem follows from [Lemma 5, [86]]. Note that the assumptions in Theorem 1.32 are analogous to [86, Theorem 7]. Let us remark also that Theorem 1.32 has exactly the same proof if we replace the Banach space E by an arbitrary Fréchet space and open balls by convex symmetric open neighbourhoods of the zero point 0 2 E. We shall show this in the multi-valued case. Now, we are going to explain the scope of fixed point interpretation of Theorem 1.32. Assume that X  E and F W X ! E is a given mapping. We let f W X ! E, f .x/ D x  F .x/. Then f is called the vector field associated with F . We have: f 1 .f0g/ D Fix .F / D fx 2 X j F .x/ D xg: Observe that if F" W X ! E is an "-approximation of F then f" .f" .x/ D x  F" .x// is an "-approximation of f . It is well known that if F is a compact map or k-set contraction or condensing map which has "-approximation of the same type, then all assumptions of Theorem 1.32 are satisfied for the field f . We would like to conclude that Theorem 1.32 contains as a special case many results, the so-called generalisations of the Browder–Gupta theorem (see [86, 131, 132, 156, 319, 320, 395, 474, 475], and [450]).

Section 1.1 Case of single-valued mappings

15

Let K be a convex subset of a normed space .Z; j : j/ and .Y; k  k/ a Banach space. Let X be the space of all continuous locally bounded maps y W K ! Y (i.e., bounded on each bounded subset of K) equipped with the topology of locally uniform convergence. Let t0 2 K; for " > 0 denote by K" the set ft 2 K j jt  t0 j  "g; for x 2 X denote by xjK" the restriction of the map x to the set K". In the spirit of the Browder–Gupta theorem, we have the following results which turn out to be useful for applications (for the proofs see, e.g., [476] for K bounded and [310, 311] for K unbounded). Theorem 1.33. Let M D fy 2 X j ky.t /  r .t /k  p.t /g; where p W K ! RC is a non-negative locally bounded continuous function and r 2 X. Let a continuous map N W M ! M satisfy (1) there exist t0 2 Œ0; 1/ and y0 2 RN such that ky0  r .t0 /k  p.t0 /; for all y 2 M satisfying N.y/.t0 / D y0 . (2) N.M/ is a locally equi-uniformly continuous map in the sense that .8 " > 0/ .8 > 0/ .9 ı > 0/ .8 y 2 M/ we have .8 t1 ; t2 2 K / .jt1  t2 j < ı ) kN.y/.t1 /  N.y/.t2 /k < "/: (3) .8 " > 0/ .8 x; y 2 M/ .xjK" D yjK" / ) .N x/jK" D .Ny/jK" and the Palais–Smale condition holds (i.e., each sequence fyn gn2N  X such that yn  Nyn D 0; contains a convergent subsequence). Then F ix.N / is a compact Rı -set. Note that Dubois and Morales recently obtained a characterisation of the Palais– Smale condition in terms of 0-closed operators (we say that an operator F W X ! Y is 0-closed if for every closed subset V  Y , the inclusion 0 2 F .V / implies that 0 2 F .V /). More precisely, it is proved that: Proposition 1.34 ([158, Theorem 3.5]). (a) If F is continuous and satisfies the Palais–Smale condition, then I F is 0-closed. (b) Conversely, if I  F is 0-closed and Fix.F / is compact, then F satisfies the Palais–Smale condition. For the particular case when Y is a Hilbert space, the following theorem is a generalisation of Theorem 1.33.

16

Chapter 1 Topological structure of fixed point sets

Theorem 1.35. Let Y be a real Hilbert space, F W K ! Pcl;cv .Y / be a continuous locally bounded map, and let M D fx 2 X j x.t / 2 F .t /; t 2 Kg: If a continuous map N W M ! M satisfies the conditions 1.33 (2), 1.33 (3), and (1) there exist t0 2 Œ0; 1/ and y0 2 F .t0 / such that for each x 2 X we have N.x.t0 // D y0 , then Fix.N / is a compact Rı -set. The following result is useful in applications since it is more adapted to Volterra integral equations. Theorem 1.36. Let E D C.Œ0; a; Rm / be the Banach space of continuous maps with the usual max-norm and let X D K.0; r / D fu 2 E j kuk  r g be the closed ball in E. Assume that F W X ! E is a compact map and f W X ! E is a compact vector field associated with F such that (1) there exists an x0 2 Rm such that F .u/.0/ D x0 , for every u 2 K.0; r /; (2) for every " 2 .0; a and for every u; v 2 X, if u.t / D v.t / for each t 2 Œ0; ", then F .u/.t / D F .v/.t / for each t 2 Œ0; ". Then there exists a sequence fn W X ! E of continuous proper mappings satisfying conditions 1.32 (1)–1.32 (2) with respect to f . Sketch of proof. For the proof it is sufficient to define a sequence Fn W X ! E of compact maps such that: F .x/ D lim Fn .x/; n!1

uniformly in x 2 X

(1.1)

and fn W X ! E;

fn .x/ D x  Fn .x/; is a one-to-one map:

(1.2)

To do this, we additionally define the mappings rn W Œ0; a ! Œ0; a by putting: h ai 8 ˆ t 2 0; ; < 0; n rn .t / D a i a ˆ : t ; t2 ;a : n n Now we are able to define the sequence fFn gn2f1;2;:::g as follows: Fn .x/.t / D F .x/.rn .t //;

for x 2 X; n 2 f1; 2; : : :g:

(1.3)

17

Section 1.1 Case of single-valued mappings

It is easily seen that Fn is a continuous and compact mapping for n D 1; 2; : : :. Since jrn .t /  t j  a=n, we deduce from compactness of F and (1.3) that lim Fn .x/ D F .x/;

n!1

uniformly in x 2 X:

Now we shall prove that fn is a one-to-one map. Assume that for some x; y 2 X we have fn .x/ D fn .y/: Then x  y D Fn .x/  Fn .y/: If t 2 Œ0; a=n, then we have x.t /  y.t / D F .x/.rn .t //  F .y/.rn.t // D F .x/.0/  F .y/.0/: Thus, in view of 1.36 (1), we obtain x.t / D y.t /;

for every t 2 Œ0; a=n.

Finally, by successively repeating the above procedure n times we obtain that x.t / D y.t /;

for every t 2 Œ0; a.

Therefore, fn is a one-to-one map and the proof is complete. From Theorem 1.32 and Theorem 1.36, we deduce a Szufla type result [449–452]. Corollary 1.37. Assume that f and F are as in Theorem 1.36. Then f 1 .0/ D Fix .F / is an Rı -set. Now from Theorem 1.32 we deduce: Proposition 1.38. Let X be a space, .E; k  k/ be a Banach space, and f W X ! E be a proper map. Assume further that, for each " > 0, a continuous map f" W X ! E is given and the following two conditions are satisfied: (a) kf".x/  f .x/k < "; for every x 2 X; (b) for every " > 0 and u 2 E in a neighbourhood of the origin such that kuk  "; the equation f" .x/ D u has exactly one solution x. Then the set S D f 1 .0/ is an Rı -set. Regarding the set of fixed points of continuous operators, we also mention the following version by Vidossich [476]; for the proof, we refer to [156, Theorem 4.3]:

18

Chapter 1 Topological structure of fixed point sets

Theorem 1.39. Let X be a Banach space and f W X ! X be a continuous and 0closed map such that f is the uniform limit of a sequence ffn gn2N of homeomorphisms from X to X. Then (a) the set S D f 1 .0/ is nonempty, (b) if S is compact, then it is an Rı -set. With the Palais–Smale condition satisfied, a similar result is proved in [319, Theorem 2.4]; we reproduce the proof. Theorem 1.40. Let X be a Banach space,   X an open subset (not necessarily bounded), and F W  ! X be a continuous satisfying the Palais–Smale condition and such that I  F is the uniform limit of a sequence I  fn of homeomorphisms from  to X. If the set Fix.F / is nonempty, then it is an Rı . Proof. Since F is continuous, then S D Fix.F / is closed; in addition the Palais– Smale condition implies that S is compact. Let Sn D fx 2  j kx  Fn .x/k  "n g for some sequence of real numbers ."n /n2N converging to 0. Since Sn is homeomorphic to a closed ball, it is an AR-set. We prove that lim d.Sn ; S/ D 0. On the conn!1

trary, assume that there exist ı > 0, a sequence of integers nk increasing to infinity, and a subsequence xnk 2 Snk such that d.xnk ; S/  ı for all k 2 N. Now, since lim kxnk  F .xnk /k D 0, then there exists some subsequence converging to x0 and k!1

then, by continuity, F .x0 / D x0 contradicting d.x0; S/  ı > 0. For the sake of completeness, we recall that the original version of Aronszajn’s theorem [86, Lemma 5] can be formulated in terms of limit sets as follows: Theorem 1.41 ([86, 158]). Let X be a metric space, fRn gn2N be a sequence of compact absolute retracts in X, and R0  X be a subset satisfying the following conditions: (a) R0  Rn , 8 n 2 N. (b) lim Rn D R0 (in the sense of Hausdorff distance). n!1

(c) Each neighbourhood of R0 contains a subsequence of fRn gn2N. Then R0 is a compact Rı -set. Observe that without condition (c), such a subset R0 is called a topologically simple subset of X by Górniewicz and Pruszko (see [223, Definition 1.1]) who obtained a generalisation of the Krasnosel’skiˇı–Perov–Rabinowitz theorem (see [319, p. 114]) for a fixed point set of a compact map to be topologically simple (see [223], Theorem 1.9). Also, the following version can be found in [475, Theorem 2.2, pp. 606–607]; it has been improved by S. Szufla in [450, p. 972]:

19

Section 1.1 Case of single-valued mappings

Theorem 1.42. Let X be a metric space, .E; k:k/ a Banach space, and f W X ! E be a proper map such that there exists a sequence of positive numbers ."n /n2N converging to 0, a positive number r , and a sequence of maps ffn gn2N W X ! E satisfying the conditions: (a) kfn.x/  f .x/k < "n ; for every x 2 X, (b) for any x 2 E with kxk < r , the set of all solutions of the equation x D fn .x/ is connected. Then the set S D f 1 .0/ is compact and connected. Next, we present the original Krasnosel’skiˇı–Perov–Rabinowitz theorem. Let X be a Banach space,   X a bounded open subset, f W  ! X a compact map and F D I  f the associated compact vector field. Theorem 1.43. Assume that f satisfies the condition: (1) for every " > 0, there exists a compact map f" such that kf .x/  f" .x/k  "; 8 x 2  and, for kuk  ", the equation .I  f" /.x/ D u has at most one solution x. If, further, the topological degree Deg .F; ; 0/ is well defined and nonzero, then the set S D F 1 .0/, that is the set of fixed points of f , is compact and connected. Proof. Since Deg .F; ; 0/ 6D 0, S is nonempty. The compact perturbation vector field F is proper, hence S is compact. To prove it is connected, we argue by contradiction and assume that there exist two disjoint nonempty open subsets U and V such that U \ V D ;; S \ U 6D ;; S \ V 6D ;; and S  U [ V: By the additivity and excision properties of the Schauder degree (see [177, 326, 405]), we have Deg .F; ; 0/ D Deg .F; U; 0/ C Deg .F; V; 0/: (1.4) A contradiction will be reached as soon as we prove that the two degrees on the righthand side vanish. Now, let x0 2 S \ U , i.e., f .x0 / D x0 and define the homotopy by the convex combination H.; x/ D G" .x/ C .1  /F .x/, where  2 Œ0; 1, G" .x/ D F" .x/  F" .x0 /, and F" D I  f" . First, we check that, for " small enough, the degree Deg .H; V; 0/ is well defined, i.e., there is no zero of H on the boundary of V . Indeed, for all x 2 @V , we have the lower bounds kH.; x/k  kF .x/k  kf .x/  f" .x/k  kx0  f" .x0 /k  inf kF .x/k  2" > 0 x2@V

20

Chapter 1 Topological structure of fixed point sets

whenever 0 < " < infx2@V kF .x/k. By the invariance property of the degree, we have Deg .F; V; 0/ D Deg .G" ; V; 0/. Moreover, G".x0 / D 0 and then condition 1.43 (1) implies that G" .x/ 6D 0 for all x 2 V which means that Deg .F; V; 0/ D 0. Finally, S \ V 6D ; implies that Deg .F; U; 0/ D 0, leading to a contradiction, whence our claim. We also mention that a generalisation of the Krasnosel’skiˇı–Perov–Rabinowitz theorem was proved by B. D. Gel’man in 1987 (see [195, 196]), proving connectedness of the set of fixed points of multi-valued maps.

1.1.4 Acyclicity of the solution sets of operator equations We consider a parameter-dependent completely continuous map and establish the acyclicity of the set of fixed points of the map for a fixed value of the parameter. The next lemma is a very recent result due to B. D. Gel’man. Lemma 1.44 (Lemma 1, [197]). Let X be a metric space and A be a compact subset of X. Suppose that there exists a sequence of continuous maps fhn gn2N, hn W Œ0; 1A ! X satisfying the following conditions: (1) given any " > 0 one can find a number n0 such that for any n > n0 the set hn .A  Œ0; 1/  O".A/; where O".A/ D fx 2 A j d.x; A/ < "g; (2) for any x 2 A and n 2 N; the equality x D hn .x; 0/ is valid; (3) for any n 2 N there is a point an 2 X such that an D hn .x; 1/ for all x 2 A. Then the set A is acyclic. Proof. Fix an arbitrary positive number n such that hn .A  Œ0; 1/  O".A/ Let in W A ! A  O".A/ be an inclusion map such that in .x/ D x; for every x 2 A. By 1.44 (2) and 1.44 (3) this map is homotopic to the constant map jn .x/ D an , the homotopy equivalence being established by the map hn .x; /. Consequently, the induced homomorphism in W HL n.O" .A// ! HL n.A/ is the zero homomorphism. Since the system of neighbourhoods O".A/ induces the direct limit of cohomology groups lim HL n.O" .A// D HL n.A/; !

21

Section 1.1 Case of single-valued mappings

then the identity map i W A ! A induces the zero homomorphism (see, e.g., [Theorem 8.4, [338]]), i  W fHM n.A/g ! fHM n.A/g: Hence,

HM n.A/ D 0 for any n > 0;

which means that the set A is acyclic. Let E be a Banach space, C a bounded closed subset of E; and g W C ! E a completely continuous map. In the next result, we assume that F ix.g/ 6D ;. Theorem 1.45. Assume that there is a sequence gn W C ! E (n 2 N) of completely continuous mappings such that (1) there exists a compact convex subset K  E that 1 [

gn .C /  K;

nD1

(2) there exists a continuous map g W E ! E such that lim gn .x/ D g.x/;

n!1

for every x 2 X,

(3) the sequence of maps fgn gn2N is equicontinuous on any compact set A  (i.e., for every " > 0 one can find a ı > 0 such that for any n 2 N and any x1 ; x2 2 A satisfying the inequality kx1  x2 k < ı the condition jgn .x1 /  gn .x2 /j < ") holds true; (4) the equation x D gn .x/ C .1  /.g.y/  gn .y// has exactly one solution for every n and for every x 2 A and  2 Œ0; 1. Then the set Fix.g/ is acyclic. ı

Proof. From 1.45 (1) and 1.45 (2), we have g.C /  K and F ix.g/ is compact. Denote K0 D fa C b  c j a; b; c 2 Kg: It easily seen that K0 is compact and F ix.g/  K1 where K1 D K0 \ C . Since the sequence fgn gn2N is equicontinuous and converges point-wise on the compact set K1 to the continuous map g; it has a subsequence that converges uniformly to g on K1 . Without loss of generality, we may assume that the sequence fgn gn2N itself converges uniformly to g on the set K1 . Consider the sequence of

22

Chapter 1 Topological structure of fixed point sets

maps hn W F ix.g/  Œ0; 1 ! E; hn .y; / D x; where x is the solution to the equation x D gn .x/ C .1  /.g.y/  gn .y//. Let us verify that the sequence of maps fhn gn2N satisfies the hypothesis of Lemma 1.44. It is clear that, by condition 1.45 (3), the map hn is well defined and continuous. It is also evident that for any point y 2 F ix.g/ and any n, we have hn .y; 0/ D y since y is the unique solution to the equation x D gn .x/ C .g.y/  gn .y// and hn .y; 1/ does not depend on the choice of y and equals an , where an is the unique fixed point of the map gn . Now, we prove condition 1.44 (1) in Lemma 1.44. Fix an arbitrary positive number > 0 and let us show that there exists n0 such that for any n > n0 , we have the inclusion hn .F ix.g/  Œ0; 1/  O .F ix.g//: We shall prove this by contradiction. Suppose that there is a number 0 > 0; a sequence f.yn ; n /g  F ix.g/  Œ0; 1, and a sequence fxn gn2N; xn D hn .yn ; n /; such that xn 62 O0 .F ix.g//. Since F ix.g/  Œ0; 1 is a compact set, it may be assumed without loss of generality that n ! ; yn ! y. By 1.45 (1), we may assume that gn .xn / ! z. Also, we note that xn D gn .xn / C .1  n /.g.yn /  gn .yn // D gn .xn / C Œ.1  n/g.yn / C gn .yn /  gn .yn / 2 K: Then the sequence xn ! x; where x D z C .1  /.g.y/  gn .y//: Using the uniform convergence of the maps gn to g on the set K1 , we obtain lim gn .xn / D g.x/ D z. Then n!1

kg.x/  xk D .1  /kg.y/  gn .y/k  kg.y/  gn .y/k:

(1.5)

Let D D C nO0 .F ix.g//. It is clear that this set is closed and g.x/ D x for any x 2 K1 . Since the map g is completely continuous, there exists a number "0 > 0 such that kg.x/  xk > "0 for any point x 2 D. Take a number n1 so large that the inequality kg.y/gn .y/k < " is satisfied for any n > n1 . Then by (1.5) the inequality kg.x/  xk < " holds true, which means that x 2 O0 .F ix.g//. As a result, for a sufficiently large n, we have xn 2 O0 .F ix.g//, which contradicts the assumption made above. The contradiction we have arrived at proves the inclusion hn .F ix.g/  Œ0; 1/  O0 .F ix.g//: Thus, all conditions of Lemma 1.44 are satisfied and, consequently, the set F ix.g/ is acyclic. Our next characterisation theorem can be formulated as follows:

23

Section 1.1 Case of single-valued mappings

Theorem 1.46. Assume that the conditions 1.45 (1), 1.45 (2), 1.45 (4) and (1) on every compact subset A  C we have fgnjA g ! gjA , i.e., fgnjA g is uniformly convergent to gjA . Then the set Fix.g/ is acyclic. Sketch of proof. Let us put B D Fix.g/. Then from 1.45 (1) and 1.46 (1), we obtain that B is compact. We consider a sequence hn W B  Œ0; 1 ! E of mappings defined as follows: hn .; y/ D x; if and only if x is a solution of the equation x D gn .x/ C .1  /.g.y/  gn .y//. Now, it is easy to verify that the sequence fhn gn2N satisfies all conditions of Lemma 1.44.

1.1.5 Nonexpansive maps 1.1.5.1 Existence theory Definition 1.47. Let X be a Banach space and F W X ! X a map. We say that F is a nonexpansive map provided kF .x/  F .y/k  kx  yk;

for every x; y 2 X:

Remark 1.48. A nonexpansive map on a Banach space need not have a fixed point as shown by the translation map f .x/ D x C x0 for some x0 2 X n f0g. Moreover, the identity map shows that in general uniqueness does not hold for nonexpansive maps. This property justifies the study of fixed point sets of nonexpansive maps. Next, we present some classical fixed point theorems for nonexpansive maps. Theorem 1.49. Let .X; k:k/ be a Banach space, C  X a nonempty closed convex and f W C ! C a nonexpansive mapping. Then for any ı > 0, f has a ı-fixed point in C , that is xı 2 X such that kxı  f .xı /k < ı (delta fixed point). Proof in case C D B.0; R/. For any r 2 .0; 1/; the mapping rf is a contraction and then admits a fixed point xr 2 C . We have 0  kf .xr /  xr k D kf .xr /  rf .xr /k D .1  r /kf .xr /k  .1  r /R: The claim then follows on passing to the limit as r ! 1 . Theorem 1.50. Let C  E be a closed subset of a Banach space and f W C ! E a continuous mapping. Assume that (a) f .C / is compact.

24

Chapter 1 Topological structure of fixed point sets

(b) f has a ı-fixed point in C for every ı > 0. Then f has a fixed point in C . Proof. From (b), consider a sequence .xn /n2N 2 C such that lim kf .xn /xn k D 0. n!1

Since f .C / is compact, there exists a subsequence .xnk /k2N such that lim f .xnk / D k!1

y 2 f .C /. Therefore, y D lim xnk and f .y/ D y. k!1

Corollary 1.51. Let .X; k:k/ be a Banach space, C  X a nonempty compact convex subset, and f W C ! C a nonexpansive mapping. Then f has a fixed point. Remark 1.52. The compactness of C cannot be relaxed to weak-compactness as shown by the following counter-example provided by Alspach in 1981 [9]. Let X D L1 Œ0; 1, Z 1 h.t /dt D 1g; C D fh 2 Xj 0  h.t /  2 for a.e. t and 0

and f W C ! C be defined by ( minf2; 2h.2t /g; f .t / D maxf0; 2h.2t  1/  2g;

if 0  t  1=2 if 1=2 < t  1:

Then C is a nonempty convex, weakly compact and f is an isometry which is fixed point free. Now, we prove Theorem 1.53. Let C be a nonempty closed bounded subset in a Banach space X and f W C ! C be a nonexpansive map such that .I  f /.C / is closed in X. Then f has a fixed point. Proof. Without loss of generality, we may assume 0 2 C . Since C is bounded, there is some large R > 0 such that C  BR .0/. Let .n /n  .0; 1/ be an increasing sequence tending to 1 and let fn D n f . Then, for every n, fn is a contraction hence admits a unique fixed point xn 2 X. We have: kf .xn /  xnk D kf .xn /  nf .xn /k  R.1  n/ which tends to 0 as n ! 1. Since .I f /.C / is closed, we have that 0 2 .I f /.C /; hence f has a fixed point in X. Definition 1.54. Let X be a Banach space and let C  X be a nonempty bounded closed convex set. A point c 2 C is said to be diametral, if sup kx  ck D diam C: x2X

Section 1.1 Case of single-valued mappings

25

We say that C has normal structure if for any given bounded, closed, convex set K  C containing more than one point there exists a nondiametral c 2 K. Proposition 1.55. (a) Every nonempty compact convex set C in a Banach space X has normal structure. (b) Every nonempty bounded closed convex set C of a uniformly convex Banach space X has normal structure. Recall Definition 1.56. A space X is said to be uniformly convex if it satisfies the following geometric condition: 8 " > 0; 9 ı > 0; 8 .x; y/ 2 X 2 ; x C y    kx  yk  "; kxk  1; kyk  1 )    1  ı: 2 Remark 1.57. (a) Hilbert spaces and Lebesgue spaces Lp ./ (for 1 < p < 1) are uniformly convex spaces (see e.g., [165]). (b) Any uniformly convex Banach space is reflexive and has the fixed point property (see [7, 118, 430]). The uniform convexity is a geometric property of the unit ball: if we slide a rule of length " > 0 in the unit ball, then its midpoint must stay within a ball of radius 1  ı for some ı > 0. In particular, the unit sphere must be “round” and cannot include any line segment. The following result describes the situation for nonexpansive maps in the framework of uniformly convex spaces. A proof may be found in [4], Theorem 2.1. This is a result proved by Browder [85], Göhde [203] and Kirk [283] in 1965 in the case of a Hilbert space and then extended to uniformly convex Banach spaces by Goebel in 1969 (for the proof, see also [231, Theorem 1.3] or [201, 202]). Theorem 1.58. Let C be a nonempty closed bounded convex subset in a uniformly convex space. Then each nonexpansive map f W C ! C has a fixed point. When the geometry of the Banach space is replaced by some properties of the set C , we obtain the following result: Theorem 1.59 (Theorem 7.1.23, [193]). Let C be a nonempty convex weakly-compact with normal structure subset of a Banach space. Then each nonexpansive map f W C ! C has a fixed point.

26

Chapter 1 Topological structure of fixed point sets

Remark 1.60. The weak-compactness of C cannot be dispensed as the following counter-example shows. Let X D C Œ0; 1, C D fx 2 Xj 0  h.t /  1 for all t and x.0/ D 0; x.1/ D 1g; and f W C ! C be defined by f .x/.t / D tx.t /; t 2 Œ0; 1. Then C is a nonempty convex, weakly compact but has no normal structure (indeed r .C / WD infx2C sup y2C kx  yk D diam C D 1) and f is fixed point free nonexpansive map. However, in case of reflexive spaces, normal structure of sets is sufficient. We have: Theorem 1.61 (Theorem 3.4.31, [387]). Let C be a nonempty, convex with normal structure subset of a reflexive Banach space. Then each nonexpansive map f W C ! C has a fixed point. 1.1.5.2 Solution sets Applying Theorem 1.40 with the Palais–Smale condition satisfied, we first prove Theorem 1.62 ( [319, Theorem 2.7]). Let X be a Banach space and F W X ! X be a nonexpansive map satisfying the Palais–Smale condition. If S D Fix.F / is nonempty, then it is an Rı -set. Proof. Without loss of generality, we may assume that 0 2 Fix.F /. Then kF .x/k  kxk and as in the proof of Theorem 1.40, we have that S is compact, hence bounded; let S  B.0; R/ D . For 0 < " < R, let F" D .1  "=R/F . For every x 2 , we have kF .x/  F" .x/k D k"FR.x/k  R" kxk  " and the approximation map F" is a contraction. Hence, I  F" is a homeomorphism on X. Let y 2 B.0; "/ and x D .I  F" /1 .y/. We have kxk  .1  "=R/kF .x/k C .I  F" /1 kyk  .1  "=R/kxk C "; proving that x 2 . Therefore, .I  F" /1 is a homeomorphism from B.0; "/ to the approximate set of fixed points S" D fx 2  j kx  F" xk  "g. Finally, as an application of Theorem 1.46, we prove (see also [Theorem 3, [197]]): Theorem 1.63. Let C be a convex closed bounded subset of a normed space E and let g W C ! C be a completely continuous and nonexpansive map. Then Fix.g/ is an acyclic set. Sketch of proof. For the proof we consider a sequence of completely continuous mappings gn W C ! C defined as follows:  1 1 g.x/ C x0 ; gn .x/ D 1  n n

Section 1.2 The case of multi-valued mappings

27

where x0 is a fixed point in C and n D 1; 2; : : :. Then, it is not difficult to verify that all assumptions of Theorem 1.46 are satisfied. Without compactness of the nonexpansive map, we have a weaker result owed to G. Vidossich [Theorem 3.1, [475]]: Theorem 1.64. Let X be a Banach space and f W X ! X be a nonexpansive map such that I f sends bounded sets into closed sets and one of the following conditions hold: .x/k (a) inf">0 supkxk" kfkxk < 1,

(b) f is a contraction for points sufficiently far, i.e., there are " > 0 and k 2 Œ0; 1/ such that kf .x/  f .y/k  kkx  yk, for kxk  ", (c) f .X/ is bounded. Then the set of fixed points of f is a nonempty connected set.

1.2 The case of multi-valued mappings In this section we shall present needed classes of multi-valued mappings. Next, we shall deal with fundamental fixed point theorems for the respective classes of multivalued mappings. Finally, we shall discuss Browder–Gupta type results for multivalued mappings. For details we recommend [215]. However, as we did for singlevalued maps, we will start with some approximation theorems.

1.2.1 Approximation of multi-valued maps It is well known that methods of algebraic topology started by S. Eilenberg and D. Montgomery [166] in 1946 and developed, for example, in [207, 209, 216, 228–230, 272] provide probably the most powerful tool in the fixed point theory of multi-valued maps. However, in order to build the fixed point theory for these maps, one must use quite a complex homological apparatus. In addition, there is another useful technique available in fixed point theory of multi-valued maps, namely that of a singlevalued approximation which is much simpler than the first one. Apparently, the idea of constructing such an approximation seems to be an old one and goes back to the works of J. von Neumann (cf. [348]) in 1963; later it was studied by many authors (cf. [27, 28, 36, 55, 75, 115,116, 139, 217, 218, 319, 337]). Below we shall present most general approximation results proved recently in [218] and [217] (see also [36, 76] and [305]). Let .X; dX / and .Y; dY / be two metric spaces and in the Cartesian product X  Y , consider the max-metric dXY : dXY ..x; y/; .u; v// D maxfdX .x; u/; dY .y; v/g; for x; u 2 X and y; v 2 Y: Secondly, we shall use the following result on uniform continuity of maps.

28

Chapter 1 Topological structure of fixed point sets

Lemma 1.65. Let K be a compact subset of X and let f W X ! Y be a continuous map. Then for each " > 0, there exists > 0 such that d2 .f .x/; f .u// < ", provided d1 .u; x/ < and x; u 2 O .K/. Proof. Assume on the contrary that there exists " > 0 such that for every n D 1; 2; : : : ; there are xn ; un 2 O1=n.K/ such that for every n 2 f1; 2; : : :g d.xn ; un / < 1=n and d.f .xn /; f .un //  ": xn; e un 2 K such that Since xn ; un 2 O1=n.K/, we can find e x n / < 1=n d1 .xn ;e

and d1 .un ;e un / < 1=n:

(1.6)

and d1 .un ;e un / < 1=n:

(1.7)

Then we deduce that un ;e x n / < 3=n d1 .e

un are convergent. Now, since K is compact, we can assume that the sequences e x n and e So, in view of (1.7), we have: lim e x n D lim e un D x:

n!1

n!1

(1.8)

Consequently, from (1.6), we get lim xn D lim un D x:

n!1

n!1

Then lim f .un / D lim f .xn / D f .x/;

n!1

n!1

and this contradicts the fact that: d2 .f .un /; f .xn //  ";

for every n:

Definition 1.66. Let F W X ! Pcp .Y / be a multi-valued mapping, Z  X, and " > 0. A mapping f W Z ! Y is called "-approximation (on the graph) of F if Gr .f /  O" .Gr .F //: If Z D X and f is an approximation (on the graph of G), then we write f 2 a.G; "/. Some important properties about approximation of multi-valued maps are summarised in the following; for the proof, we refer to [215]. Proposition 1.67. (1) A mapping f W Z ! Y is an "-approximation of a multi-valued map G W X ! Pcp .Y / if and only if f .x/ 2 O" .G.O" .x/// for each x 2 Z; where Z  X.

29

Section 1.2 The case of multi-valued mappings

(2) Let P be a compact space, r W P ! X a continuous map, and let G W X ! Pcp .Y / be u.s.c. Then, for each  > 0 there exists "0 > 0 such that for any " (0 < " < "0 ) and any "-approximation f W X ! Y of G, the map f ı r W P ! Y is a -approximation of G ı r . (3) Let C be a compact subset of X and G W X ! Pcp .X/ is an u.s.c. mapping such that C \ F ix.G/ D ;. Then there exists " > 0 such that, for every f 2 a.G; "/, we have F ixf \ C D ;. (4) Let C be a compact subset of X. Then, for every " > 0, there is ı > 0 such that the restriction fjC of f to C is an "-approximation of the restriction fjC of G W X ! P.Y / to C , whenever f 2 a.G; ı/. (5) Let X be compact and  W X  Œ0; 1 ! Pcp .Y / be a multi-valued map. Then, for every t 2 Œ0; 1 and for every " > 0, there exists ı > 0 such that ht 2 a.t ; ı/; whenever h 2 a.; ı/; where ht W X ! Pcp .Y / and t W X ! Y are defined as follows: t .x/ D .x; t /; ht .x/ D h.x; t /;

for every x 2 X

and t 2 Œ0; 1:

(6) Let G W X ! Pcp .Y / and g W Y ! Z be two mappings (with G u.s.c. and g continuous). Then, for every " > 0, there exists ı > 0 such that gıf 2 a.gıG; "/; whenever f 2 a.G; ı/. (7) Let G W X ! Pcp .Y / and F W Z ! Pcp .T / be two multi-valued mappings. Then, for every " > 0 there exists ı > 0 such that, if f 2 a.G; ı/ and g 2 a.F; "/; then f  g W X  Z ! Y  T is an "-approximation of G  F W X  Z ! Pcp .Y  T /. Theorem 1.68 (Cellina’s approximation selection). Let .X; d / be a metric space and Y a normed space. Then every u.s.c. multi-map F W X ! Pcv;cp .Y / has an "approximation selection f" 2 a.F; "/, for every " > 0. Proof. Fix " > 0 and x 2 X; let ı D ı.x/; ı < " be such that F .B.x; ı.x///  O" .F .x//: The family of balls fB.x; .x//gx2X ; where .x/ D 14 ı.x/ covers the paracompact space X. Using Stone’s theorem, this cover has a locally finite refinement fVi gi 2I and f gi 2I a corresponding partition of unity. Now, choosing for each i 2 I an arbitrary point yi 2 F .Vi /; define the map f" W X ! Y by f" .x/ D

X i 2I

i .x/yi :

30

Chapter 1 Topological structure of fixed point sets

The map f" is the desired one. Let x 2 X belong to all members of the family fVi gniD1 from the covering fvi gi 2I . Every Vi ; i D 1; : : : ; n is contained in some ball B.xi ; .xi //; therefore x 2 \niD1 B.xi ; .xi //. Let k 2 f1; : : : ; ng be such that k D max1i n .xi /. Take x 0 D xk ; then we have xi 2 B.x; k /, hence xi 2 B.x 0 ; 2 k / for all i D 1; : : : ; n. Thus, B.xi ; .xi //  B.x 0 ; 4 k /;

i D 1; : : : ; n:

Then yi 2 F .xi /  F .B.xi ; .xi //  F .B.x 0 ; 4 k //  O" .F .x 0 //; for all i D 1; : : : ; n: Using the fact that O" .F .x 0 // is convex, then f" .x/ 2 O".F .x 0 //. Since x 2 Vi ; i D 1; : : : ; n; we have also F .x/  O" .F .x 0 //: For the continuity of f" , we can use Lemma 1.29.

1.2.2 Fixed point theorems In connection with Proposition 1.67, we start by giving a consequence of Schauder’s fixed point theorem: for given two spaces X; Y , we let: A0 .X; Y / D fG W X ! Pcp .Y /j G

is u.s.c. and for every " > 0, there exists f 2 a.G; "/g:

a.G; "/ is as defined in Definition 1.66. The class A0 is adequate for obtaining global fixed point theorems. Now, we shall describe properties of A0 . Theorem 1.69. Let G 2 A0 .X; X/ be a multi-map and X a compact AR-space, then F ix.G/ 6D ;. Proof. Let " D 1=n; n D 1; 2; : : : ; and let fn 2 a.G; 1=n/. Then from the Schauder fixed point theorem we obtain that fn .xn / D xn ; for some xn 2 X. Without loss of generality we can assume that lim fn .xn / D lim xn D x. Then we can choose a n!1

n!1

sequence .un ; vn / 2 Gr .G/ such that: d1 .xn ; un / < 1=n

and d1 .xn ; vn / < 1=n;

n D 1; 2; : : : :

and hence we obtain: lim un D lim vn D lim fn .xn / D lim xn D x:

n!1

n!1

n!1

n!1

Since G is u.s.c., the graph of G is closed in X X and consequently x 2 G.x/, which completes the proof.

Section 1.2 The case of multi-valued mappings

31

Now, we recall some classical fixed point theorems needed to prove existence results. The first one is the so-called nonlinear alternative of Leray and Schauder (see [215, 231]). First, we present a multi-valued version of the Schauder fixed point theorem. It is also known as the Kakutani or the Bohnenblust–Karlin fixed point theorem. Indeed, Kakutani [274] proved this theorem in 1941 in Rn and Bohnenblust and Karlin [74] extended the theorem to Banach spaces in 1950. Theorem 1.70 ([74]). Let E be a normed linear space, X 2 Pcv;cp .E/; and F W X ! Pcv;cl .X/ be an upper semi-continuous multi-valued map. Then F has a fixed point in X. Proof. From Theorem 1.68, there exists a 1=n-approximation selection for every n 2 N; i.e., there exists fn W X ! X a family of continuous maps such that fn 2 a.F; 1=n/; since F is u.s.c. then F 2 A0 .X; X/. Using the fact that X is convex, then X 2 AR. By Theorem 1.69, there exists x 2 X such that x 2 F .x/. Equivalently, it is also possible to use Cellina’s approximation selection (Theorem 1.68), to get a sequence of continuous maps fn W X ! X such that Gr fn  O"n .Gr f /. Then the Schauder fixed point theorem yields some xn 2 X, such that xn D f .xn /. By compactness of X, .xn /n has some subsequence converging to a limit x 2 F .x/. Remark 1.71. For this theorem, Fan and Glicksberg obtained in 1952 some generalisation to locally convex spaces (leading by the way to Tikhonov’s fixed point theorem, 1935) Now, we give the following equivalence result of Theorem 1.70. Theorem 1.72 (Kakutani’s theorem). Let E be a normed linear space, X 2 Pcv;b .E/; and F W X ! Pcv;cp .X/ be an upper semi-continuous multi-valued map with F .X/ compact. Then F has a fixed point in X. Proof. Let C D co F .X/ satisfy the hypotheses of Theorem 1.72 and C 2 Pcp;cv .E/. From Mazur’s lemma A.46 and the compactness of the multi-valued operator F; we deduce that the multi-valued map F W C ! Pcp;cv .C / is u.s.c. By Theorem 1.70, there exists y 2 C such that y 2 F .y/. Now, we present the classical multi-valued version of the nonlinear alternative of Leray and Schauder.

32

Chapter 1 Topological structure of fixed point sets

Theorem 1.73. Let E be a normed linear space, F W E ! Pcv;cp .E/ an upper semicontinuous multi-valued map and completely continuous multi-valued map. Suppose that there exists r > 0 such that x 2 F .x/

and

0    1 ) kxk  r:

Then F has a fixed point in B.0; r /. Proof. Let M D fx 2 E j x 2 F .x/;  2 .0; 1/gI then M is a bounded set in E. Hence, there exists r > 0 such that F .M/  B.0; r / D fx 2 E j kxk  rg: Let K D supfkyk j y 2 F .B.0; 2r//g;

and k D max.K; 2r C 1/

and consider the multi-valued operator G W E ! P.E/ defined by 8 ˆ < F .x/ \ B.0; 2r /; if F .x/ \ B.0; 2r / 6D ;; G.x/ D ˆ : 2r F .x/; if F .x/ \ B.0; 2r / D ;: k We can prove that G.B.0; 2r //  B.0; 2r/; G.:/ 2 Pcp;cv .B.0; 2r // and G is u.s.c. Then using Theorem 1.72, there exists x 2 B.0; 2r/ such that x 2 G.x /. Assume that x 2 2r F .x / with F .x / \ B.0; 2r / D ;; then there exists y 2 k F .x / such that x D

2r k 2r x ) 2r < ky k  k ) y ) y D < 1: k 2r k

Then x 2 M ) y 2 F .M/ ) ky k  r ) 2r < r : This is a contradiction with F .x / \ B.0; 2r /; hence x 2 F .x /, as claimed. We end this section with a nonlinear alternative. Corollary 1.74. Let .X; j  j/ be a normed space and F W X ! Pcl;cv .X/ a compact, u.s.c. multi-valued map. Then either one of the following conditions holds: (a) F has at least one fixed point, (b) the set M WD fx 2 X; x 2 F .x/;  2 .0; 1/g is unbounded. The single-valued version may be stated as follows and is known as the Schaefer nonlinear alternative:

33

Section 1.2 The case of multi-valued mappings

Corollary 1.75. Let X be a Banach space and C  X a nonempty bounded closed convex subset. Assume that U is an open subset of C with 0 2 U and let G W UN ! C be a a continuous compact map. Then (a) either there is a point u 2 @U and  2 .0; 1/ with u D G.u/, (b) or G has a fixed point in U .

1.2.3 Multi-valued contractions Let .X; d / and .Y; d 0 / be two metric spaces. Definition 1.76. A multi-valued operator F W X ! Pcl .Y / is called (a) a contraction if it is k-Hd -Lipschitz with 0  k < 1; (b) locally Lipschitz, if for every x 2 X there is an open neighbourhood Vx of x in X and kx > 0 such that: Hd 0 .F .u1 /; F .u2 //  kx d.u1 ; u2 /;

for each u1 ; u2 2 Vx :

In connection with Lemmas 6.24 and 6.23 from Chapter 6, we have the following result without compactness. Lemma 1.77. Let .Y; d 0 / be a complete metric space and F W X ! Pcl .Y / be a Lipschitz (or locally Lipschitz) with closed graph, then F is u.s.c. Proof. Let V be open in Y and x 2 X be such that F .x/  V . We show that there exist open neighbourhoods Wx of x in X and F .Wx /  V . Assume that for every open neighbourhood Wx there exist xw 2 Wx and yw 2 F .xw / such that yw 62 V . Then for every " > 0 there exist x" 2 B.x; "/ and y" 2 F .x" / such that y" 62 V . Take "n D 1=n; n 2 f1; 2; : : :g; thus d.xn; x/ < 1=n

and d 0 .yn ; ym /  d 0 .yn ; F .x// C Hd 0 .F .x/; F .xm //:

Then d.xn; x/ < 1=n and d 0 .yn ; ym /  Hd 0 .F .xn /; F .x// C Hd 0 .F .x/; F .xm //: Since F is a Lipschitz multi-valued map, we obtain d.xn; x/ < 1=n and d 0 .yn ; ym /  L.d.xn; x/ C d.x; xm //: Hence ,fxngn2N converges to x in X and fyn gn2N is a Cauchy sequence in Y . Using the fact that F has a closed graph, we obtain that y 2 F .x/; where y D lim yn . But n!1

fyn gn2N  Y nV ; this implies that y 2 Y nV , which is contradiction with F .x/  V .

34

Chapter 1 Topological structure of fixed point sets

It easy to see that the following proposition holds. Proposition 1.78. Let K be a compact convex subset of a normed space E and F W E ! P.Y / be a locally Lipschitz multi-valued map. Then the restriction F jK W K ! P.Y / of F to K is a Lipschitz map. Now, we present the classical fixed point theorem for contraction of multi-valued operators proved by H. Covitz and S. B. Nadler in 1970 [127] (see also Deimling, [146] Theorem 11.1). Theorem 1.79. Let .X; d / be a complete a metric space. If F W X ! Pcl .X/ is a contraction, then F ixF 6D ;. Proof. Assume that Hd .F .x/; F .y//  kd.x; y/ for every x; y 2 X; where k 2 Œ0; 1/. Let x 2 X and D.x/ D fy 2 Xj d.y; x/  d.x; F .x//g: Since F .x/ is closed, then D.x/ \ F .x/ 6D ;: So we can select x1 2 F .x/ such that d.x; x1 /  d.x; F .x//. If D.x1 / D fy 2 X j d.y; x1 /  d.x1 ; F .x1 /g; then we can select x2 2 F .x1 /; we have d.x1 ; x2 /  d.x1 ; F .x1 // ) d.x1 ; x2 /  Hd .F .x/; F .x1 //: This implies that d.x1 ; x2 /  kd.x; x1 /  kd.x; F .x//: Continuing this procedure, we can find a sequence fxn j n 2 Ng  X such that d.xn; xnC1 /  d.xn; F .xn //: It follows that d.xn; xnC1 /  d.xn ; F .xn //  Hd .F .xn1 /; xn /  kd.xn1 ; xn /  k n d.x; F .x//: So, it is easy to verify that fxngn2N is a Cauchy sequence. Let x0 D lim xn . Then n!1

we have fxng ! u, xnC1 2 F .xn / for every n 2 N, and the following estimates hold: 0  d.x0 ; F .x0 //  d.xnC1; x0 / C d.xnC1; F .x0 //  d.xnC1; x/ C kd.xn ; x0 /: Passing to the limit, as n ! 1, we get x0 2 F .x0 /, as claimed.

Section 1.2 The case of multi-valued mappings

35

There are many generalisations of Theorem 1.79. We recommend [286] and [490] for further details. Next, we present a multi-valued version of the nonlinear alternative of contractive type in Fréchet spaces. Let E be a Fréchet space with the topology generated by a family of semi-norms j  jn and corresponding distances dn .x; y/ D jx  yjn .n 2 N/. First, we start with Definition 1.80. A multi-valued map F W E ! P.E/ is called an admissible contraction with constant fkngn2N if for each n 2 N; there exists kn 2 .0; 1/ such that (a) Hdn .F .x/; F .y//  kn jx  yjn for all x; y 2 E; where Hd is the Hausdorff distance. (b) for every x 2 E and every " > 0; there exists y 2 F .x/ such that jx  yjn  dn .x; F .x// C "; for every n 2 N: A subset A  E is bounded if for every n 2 N; there exists Mn > 0 such that jxjn  Mn ; for every x 2 A. The following nonlinear alternative for multi-valued contraction is owed to Frigon [182]: Theorem 1.81. Let E be a Fréchet space, U  E an open neighbourhood of the origin, and let N W U ! P.E/ be a bounded admissible multi-valued contraction. Then either one of the following statements holds: (C1) N has a fixed point, (C2) there exist  2 Œ0; 1/ and x 2 @U such that x 2 N.x/.

1.2.4 Fixed point sets of multi-valued contractions Below we shall concentrate our considerations on the topological structure of the set of fixed points of contraction mappings. First, observe that multi-valued contractions can possess not necessarily a unique fixed point (see also Theorem 1.153). Example 1.82. Let F W R ! Pb .R/ be a map defined as follows: F .x/ D A; for every x 2 R; where A  R is a nonempty set. Then F as a constant map is a contraction. Of course we have: Fix.F / D fx 2 R j x 2 F .x/g D A: Since, contrary to the single-valued case, the set Fix.F / of a contraction F may have many elements, it is interesting to look for its topological properties. In this framework, the following result is classical (see [215, 417]):

36

Chapter 1 Topological structure of fixed point sets

Theorem 1.83. Let X be a complete metric space and F W X ! Pcp .X/ be a multivalued contraction. Then the set Fix.F / is compact. Proof. Since F is multi-valued contraction, then there exists  2 Œ0; 1/ such that Hd .F .x/; F .y//  d.x; y/;

for all x; y 2 X:

Firstly, we prove that Fix.F / is closed. Let .xn /n2N  F ix.F / be a sequence converging to x; we show that x 2 F ix.F /. d.x; F .x//  d.x; xn / C Hd .F .xn /; F .x//  .1 C /d.xn ; x/ ! 0;

as n ! 1:

Then x 2 F .x/, which implies that Fix.F / is closed, hence complete. Assume Fix.F / is not compact. Since it is complete it cannot be paracompact. Thus, there exist some ı > 0 and some sequence fxngn2N in Fix.F / such that d.xn ; xm /  ı; for any two different integers n and m: Let

 D inffr j 9 a 2 X B.a; r / contains infinitely xn0 g:

Since for every a 2 X; the ball B.a; 2ı / contains at most xn; one has   ı2 . Fix " > 0 1 such that 0 < " <  1C and choose a 2 X such that the set J D fn j xn 2 B.a;  C "/g is infinite: For each n 2 J , we have d.xn; F .a//  Hd .F .xn /; F .a//  d.xn; a/ < . C "/; and we can choose some yn 2 F .a/ such that d.xn ; yn / < . C "/: By compactness of F .a/, there is a b 2 F .a/ such that J 0 D fn 2 J j d.yn ; b/ < "g is infinite: Then for each n 2 J 0 ; we have d.xn; b/ < . C "/   C ".1 C / < I this is a contradiction with the definition of  since B.b; r / contains infinitely many xn where r D  C ".1 C /.

37

Section 1.2 The case of multi-valued mappings

Regarding the topological structure of the solution set, we state two fundamental results. The first one is due to B. Ricceri [408] in 1987 and the second one was proved by A. Bressan, A. Cellina and A. Fryszkowski [82] in 1991. Theorem 1.84. Let E be a Banach space and let X be a nonempty convex closed subset of E. Suppose F W X ! Pcl;cv .X/ is a contraction. Then the set Fix.F / is an absolute retract. Theorem 1.85. If X D L1 .T / for some measure space T and F W X ! Pb .X/ is a contraction with decomposable values then Fix.F / is an absolute retract. We are now in a position to prove our main result. Theorem 1.86. Let X be a complete absolute retract and ˆW X ! P.X/ [ f;g be a multi-valued contraction such that ˆ 2 SP.X/. Then Fix.ˆ/ is a complete ARspace. Recall that ˆ 2 SP.X/ means that ˆ has the selection property (see [Definition 6.71, Chapter 6]). Proof. Since Fix.ˆ/ is nonempty and closed in X, we only have to show that if Y 2 M, Y  is a nonempty closed subset of Y , and f  W Y  ! Fix.ˆ/ is a continuous function, then there exists a continuous extension f W Y ! Fix.ˆ/ of f  over Y . Let d be the metric of X, L 2 0; 1Œ be such that Hd .ˆ.x 0 /; ˆ.x 00 //  Ld.x 0; x 00 / for all x 0 ; x 00 2 X, and M 2 1; L1 Œ. The assumption X 2 AR yields a continuous function f0 W Y ! X fulfilling f0 .y/ D f  .y/ in Y  . We claim that there is a sequence ffngn0 of continuous functions from Y into X with the following properties: (1) fn jY  D f  for every n 2 N, (2) fn .y/ 2 ˆ.fn1 .y// for all y 2 Y , n 2 N, (3) d.fn .y/; fn1 .y//  Ln1 d.f1 .y/; f0 .y// C M 1n ; for every y 2 Y , n 2 N. To see this, we proceed by induction on n. From Proposition 6.66, it follows that the function h0 W Y ! 0; C1Œ defined by h0 .y/ D dist.f0 .y/; ˆ.f0 .y/// C 1;

for y 2 Y;

is continuous; moreover, one clearly has ˆ.f0 .y// \ B.f0 .y/; h0 .y// 6D ; for all y 2 Y . Bearing in mind that ˆ 2 SP.X/, we obtain a continuous function f1 W Y ! X satisfying f1 .y/ D f  .y/ in Y  and f1 .y/ 2 ˆ.f0 .y// in Y . Hence, conditions 1.86 (1)–1.86 (3) are true for f1 . Suppose now we have constructed p continuous functions f1 ; : : : ; fp from Y into X in such a way that 1.86 (1)–1.86 (3) hold whenever

38

Chapter 1 Topological structure of fixed point sets

n D 1; : : : ; p. Since ˆ is Lipschitzian with constant L, 1.86 (2) and 1.86 (3) apply if n D p, and LM < 1, for every y 2 Y; we achieve dist.fp .y/; ˆ.fp .y///  Hd .ˆ.fp1 .y//; ˆ.fp .y///  Ld.fp1 .y/; fp .y//  Lp d.f1 .y/; f0 .y// C LM 1p < Lp d.f1 .y/; f0 .y// C M p so that

ˆ.fp .y// \ B.fp .y/; Lp d.f1 .y/; f0 .y// C M p / 6D ;:

Because of the assumption ˆ 2 SP.X/, this procedure yields a continuous function fpC1 W Y ! X with the properties: fpC1 jY  D f  ;

fpC1 .y/ 2 ˆ.fp .y//;

dist.fpC1 .y/; fp .y//  Lp d.f1 .y/; f0 .y// C M p ;

for every y 2 Y; for all y 2 Y:

Thus, the existence of the sequence ffng is established. We next define, for any a > 0, the set Ya D fy 2 Y j d.f1 .y/; f0 .y// < ag. Obviously, the family of sets fYa j a > 0g is an open covering of Y . Moreover, due to 1.86 (3) and the completeness of X, the sequence ffngn2N converges uniformly on each Ya . Let f W Y ! X be the point-wise limit of ffngn2N. It is easy to see that the function f is continuous. Further, due to 1.86 (1) one has f jY  D f  . Finally, the range of f is a subset of Fix.ˆ/ since, by 1.86 (1), f .y/ 2 ˆ.f .y// for all y 2 Y . This completes the proof. The same arguments used to prove Theorem 1.86 actually produce the following more general result. Theorem 1.87. Let D  M, X be a complete absolute retract, and ˆW X ! P.X/ [ f;g be a multi-valued contraction having the selection property with respect to D. Then, for any Y 2 D and any nonempty closed set Y0  Y , every continuous function f0 W Y0 ! Fix.ˆ/ admits a continuous extension over Y . Finally, note that Theorems 1.84 and 1.85 are special cases of Theorem 1.87.

1.2.5 Fixed point sets of multi-valued nonexpansive maps For general multi-valued nonexpansive maps, we first mention the following fixed point theorem (see [323]). Theorem 1.88. Let C be a convex closed bounded subset of a uniformly convex Banach space E and let G W C ! Pcl .C / be nonexpansive. Then Fix.G/ is nonempty. Regarding the structure of the fixed point set, we have the following result due to Gel’man.

Section 1.2 The case of multi-valued mappings

39

Theorem 1.89 ([196, Theorem 2.5.5]). Let   X be a bounded open subset of a Banach space, F W  ! Pcp X be a compact multi-valued mapping such that F is nonexpansive and the topological index (rotation of the vector field F ) i.I F; @/ 6D 0. Then Fix.F / is nonempty and connected. We end this section with another result of Gel’man; this result turns out to be useful when dealing with -selectionable maps: Theorem 1.90 ([196, Theorem 2.5.3]). Let   X be a bounded open subset of a Banach space, F W  ! Pcp X be a compact multi-valued mapping such that there exists a decreasing sequence of compact multi-valued maps Fn W  ! Pcp X with F .x/ D \1 nD1 Fn .x/ and such that Fix.Fn / is connected for each n 2 N. Then Fix.F / is a nonempty compact and connected set. Proof. It is clear that the sets Fix.Fn / and Fix.F / are compact. We argue by contradiction assuming that Fix.F / is not connected. There there exist two open disjoint sets U and V such that Fix.F /  U [V , Fix.F /\U 6D ;, and F ix.F /\V 6D ;. By connectedness of Fix.Fn /, for each n, there exists some xn 2 Fix.Fn / such that xn 2 U [ V . By compactness, the sequence .xn /n has a subsequence still denoted .xn /n converging to some limit x0 . It follows that x0 2 \1 nD1 Fix.Fn /, i.e., x0 2 Fix.F /, contradicting the fact that Fix.F /  U [ V and proving our claim.

1.2.6 Fixed point sets of multi-valued condensing maps 1.2.6.1 Measure of noncompactness We shall define the measure of noncompactness on Pb .E/. Recall that a subset A  E is relatively compact provided the closure cl A is compact. Definition 1.91. Let E be a Banach space and Pb .E/ the family of all bounded subsets of E. Then the function: ˛W Pb .E/ ! RC defined by: ˛.A/ D inff" > 0 j A admits a finite cover by sets of diameter  "g is called the Kuratowski measure of noncompactness, (the ˛-MNC for short). Another function ˇW Pb .E/ ! RC defined by: ˇ.A/ D inffr > 0 j A can be covered by finitely many balls of radius r g is called the Hausdorff measure of noncompactness. Definition 1.91 is very useful since ˛ and ˇ have interesting properties, some of which are listed in the following Proposition 1.92. Let E be a Banach space with dim E D 1 and W Pb .E/ ! RC be either ˛ or ˇ. Then:

40

Chapter 1 Topological structure of fixed point sets

(1) .A/ D 0 if and only if A is relatively compact, (2) .A/ D jj.A/ and .A1 C A2 /  .A1 / C .A2 /, for every  2 R and A; A1 ; A2 2 Pb .E/, (3) A1  A2 implies .A1 /  .A2 /, (4) .A1 [ A2 / D maxf.A1 /; .A2 /g, (5) .A/ D .conv.A//, (6) the function W Pb .E/ ! RC is continuous with respect to the metric Hd on Pb .E/. Proof. You will have no difficulty in checking 1.92 (1)–1.92 (4) and 1.92 (6) by means of Definition 1.91. Concerning 1.92 (5), we only have to show that .conv.A//  .A/, Sm since A  conv.A/ and therefore .A/  .conv.A//. Let  > .A/ and A  i D1 Mi with ı.Mi /   if  D ˛ and Mi D B.xi ; / if  D ˇ. Since ı.conv.i //   and B.xi ; / are convex, we may assume that Mi are convex. Since 

conv.A/  conv M1 [ conv

m [

 Mi

i D2 m   [  :::; Mi  conv M1 [ conv M2 [ conv



i D3

it suffices to show that .conv.C1 [ C2 //  maxf.C1 /; .C2 /g; Now, we have

[

conv.C1 [ C2 / 

for convex subsets C1 and C2 .

ŒC1 C .1  /C2 ;

01

and since C1 [ C2 is bounded there exists an r > 0 such that kxk  r for all x 2 .C1 [ C2 /. Finally, given " > 0, we find 1 ; : : : ; p such that Œ0; 1 

p  [ i D1

" " i  ; i C r r

and therefore conv.C1 [ C2 / 

p [

Œi C1 C .1  i /C2 C cl B.0; "/:

i D1

Section 1.2 The case of multi-valued mappings

41

Hence, 1.92 (2)–1.92 (4) and the obvious estimate .cl B.0; "//  2" imply .conv.C1 [ C2 //  maxf.C1 /; .C2 /g C 2"; for every " > 0. Consequently, the proof is completed. Now, let us state the following obvious observation: Remark 1.93. For every A 2 Pb .E/, we have ˇ.A/  ˛.A/  2ˇ.A/: We shall end this section by considering two examples and by formulating a generalisation of the Cantor theorem. Example 1.94. Assume that dim E D 1. Now, let us compute the measures of a ball B.x0 ; r / D fx0 g C r  B.0; 1/. Evidently, .B.x0 ; r // D r .cl B.0; 1// D r .S/; where S D @B.0; 1/ D fx 2 E j kxk D 1g:

S Furthermore, ˛.S/  2 and ˇ.S/  1. Suppose ˛.S/ < 2. Then S D niD1 Mi with the closed sets Mi and ı.Mi / < 2. Let E n be an n-dimensional subspace of E. Then n [ n S \E D Mi \ E n i D1

and in view of the Lusternik–Schnirelman–Borsuk theorem (see [146]), there exists i such that the set Mi \ E n contains a pair of antipodal points, x and x. Hence @.Mi /  2 for this i, a contradiction. Thus, ˛.S/ D 2 and 1D

˛.S/  ˇ.S/  1; 2

i.e., we have ˛.B.x0 ; r // D 2r and ˇ.B.x0 ; r // D r provided dim E D 1. Example 1.95. Let r W E ! cl B.0; 1/ be the ball retraction defined as follows: 8 < x; if kxk  1; x r .x/ D ; if kxk > 1: : kxk Let A 2 Pb .E/. Since r .A/  conv.A [ f0g/, we obtain .r .A//  .A/: In other words, we can say that r is a nonexpansive map with respect to the Kuratowski or Hausdorff measure of noncompactness. Now we note that the following version of the Cantor theorem holds true:

42

Chapter 1 Topological structure of fixed point sets

Theorem 1.96. If  D ˛ or  D ˇ and fAn gn2N is a decreasing T sequence of closed nonempty subsets in Pb .E/ such that limn .An / D 0, then A D 1 nD1 An is a nonempty compact subset of E. The following characterisation of Rı -sets, which develops the well-known Hyman theorem [267], was shown by D. Bothe. Theorem 1.97 ([78,397]). Let X be a complete metric space,  denotes the measure of noncompactness in X, and let B 6D ;. Then the following statements are equivalent: (1) B is an Rı -set; (2) B is an intersection of a decreasing sequence fBng of closed contractible spaces with .Bn / ! as n ! 1I (3) B is compact and absolutely neighbourhood contractible. Now, we present the abstract definition of MNC. For more details, we refer to [6, 276, 463] and some references therein. Definition 1.98. Let E be a Banach space and .A; / a partially ordered set. A map ˇW P.E/ ! A is called a measure of noncompactness on E, MNC for short, if ˇ.co / D ˇ./ for every bounded  2 P.E/. Notice that if D is dense in ; then co  D co D and hence ˇ./ D ˇ.D/. Definition 1.99. A measure of noncompactness ˇ is called (a) Monotone if 0 ; 1 2 P.E/; 0  1 implies ˇ.0 /  ˇ.1 /. (b) Nonsingular if ˇ.fag [ / D ˇ./ for every a 2 E;  2 P.E/. (c) Invariant with respect to the union with compact sets if ˇ.K [ / D ˇ./ for every relatively compact set K  E; and  2 P.E/. (d) Real if A D RC D Œ0; 1 and ˇ./ < 1 for every bounded . (e) Semi-additive if ˇ.0 [ 1 / D max.ˇ.0 /; ˇ.1 // for every 0 ; 1 2 P.E/. (f) Lower-additive if ˇ is real and ˇ.0 C1 /  ˇ.0 /Cˇ.1 / for every 0 ; 1 2 P.E/. (g) Regular if the condition ˇ./ D 0 is equivalent to the relative compactness of .

43

Section 1.2 The case of multi-valued mappings

1.2.6.2 Condensing maps Definition 1.100. Let M be a closed subset of a Banach space E and ˇW P.E/ ! .A; / an MNC on E. A multi-valued map FW M ! Pcp .E/ is said to be ˇ-condensing if for every   M, the relation ˇ./  ˇ.F.// implies the relative compactness of . For condensing a multi-valued map, we present the following result: Lemma 1.101. Let FW M ! Pcp .E/ be a ˇ-condensing multi-valued map, where ˇ is a monotone MNC in E. If the fixed points set F ix.F/ is bounded, then it is compact. Proof. It is clear that F ix.F/  F .F ix.F/; then ˇ.F ix.F/  ˇ.F .F ix.F/// which implies that F ix.F/ is compact. Some important results on fixed point theory with MNCs are recalled hereafter (see, e.g., [276] for the proofs and further details). The first one is a compactness criterion. Lemma 1.102 ( [276], Theorem 5.1.1). Let N W L1 .Œa; b; E/ ! C.Œa; b; E/ be an abstract operator satisfying the following conditions: (1) N is -Lipschitz: there exists > 0 such that for every f; g 2 L1 .Œa; b; E/ Z

b

jf .s/  g.s/jds;

jNf .t /  Ng.t /j 

for all t 2 Œa; b:

a

(2) N is weakly-strongly sequentially continuous on compact subsets: for any compact K  E and any sequence ffngn2N  L1 .Œa; b; E/ such that ffn .t /gn2N  K for a.e. t 2 Œa; b; the weak convergence fn * f0 implies the strong convergence N.fn / ! N.f0 / as n ! 1. Then for every semi-compact sequence ffn gN  L1 .Œ0; b; E/; the image sequence N.ffn gN/ is relatively compact in C.Œa; b; E/. Lemma 1.103 ([276], Theorem 5.2.2). Let an operator N W L1 .Œa; b; E/ ! C.Œa; b; E/ satisfy conditions (1.102.1)–(1.102.2) together with (1) There exists 2 L1 .Œa; b/ such that for every integrably bounded sequence ffn gn2N; we have .ffn .t / j n 2 Ng/  .t /; where  is the Hausdorff MNC.

for a.e. t 2 Œa; b;

44

Chapter 1 Topological structure of fixed point sets

Then

Z .fN.fn /.t / j n 2 Ng/  2

b

.s/ds;

for all t 2 Œa; b;

a

where is the constant in .1.102:1/. The next results are concerned with the structure of solution sets for ˇ-condensing u.s.c. multi-valued maps. Proposition 1.104 ([276]). Let V  E be a bounded open neighbourhood of zero and N W V ! Pcp;cv .E/ be a ˇ-condensing u.s.c. multi-valued map, where ˇ is a nonsingular measure of noncompactness defined on subsets of E, satisfying the boundary condition x 62 N.x/ for all x 2 @V and 0 <  < 1. Then F ix N 6D ;. Proposition 1.105 ([276]). Let W be a closed subset of a Banach space E and FW W ! Pcp .E/ be a closed ˇ-condensing multi-valued map where ˇ is a monotone MNC on E. If the fixed point set Fix F is bounded, then it is compact. Proposition 1.106 ([276], Theorem 3.3.4). Let K  E be a convex closed set, U  K a nonempty bounded relatively open set, a 2 U be an interior point, and N W U ! Pcp;cv .K/ be an u.s.c. ˇ-condensing multi-map satisfying the boundary condition x  a 62 .N.x/  a/ for all x 2 @U and  2 .0; 1/. Then F ix.N / is a nonempty compact set.

1.3 Admissible maps 1.3.1 Generalities Now we shall define a class of admissible mappings (see [207] and [215]) which is important from the point of view of fixed point theory. Definition 1.107. A multi-valued map ' W X ! P.Y / is called admissible if there exists a diagram: p

q

X (H !Y; in which p is Vietoris, q is continuous and such that, for every x 2 X, we have: '.x/ D q.p 1 .x//: Since p is closed, we get that any admissible map is u.s.c. An u.s.c. map ' W X ! P.Y / is called acyclic if '.x/ is a compact acyclic set, for every x 2 X.

45

Section 1.3 Admissible maps

Proposition 1.108. Any acyclic map is admissible. Proof. Assume that ' W X ! P.Y / is an acyclic map. Let ' be the graph of ' and p' W ' ! X, p' .x; y/ D x, q' W ' ! Y , q' .x; y/ D y be the natural projection. Since p'1 .x/ is homeomorphic to '.x/, we get that p'1 .x/ is compact acyclic. Therefore, for the proof it is sufficient to show that p' is closed. Let C be a closed subset of ' . If x0 62 p' .C /, then fx0 g'.x0 /\C D ;, and so fx0g'.x0 /  .X  Y / n C . Observe that C , as a closed subset of a closed subset ' of X  Y , is closed in X  Y , too. Now, ' is u.s.c. and, therefore, there are two open sets V and U such that x0 2 V , '.x0 /  U , V  U  X  Y n C and V D ' 1 .U /. Thus, V \ p' .C / D ;, which proves the proposition. The following example shows that the composition of two acyclic mappings need not be acyclic. Example 1.109. Let S1 denote the unit sphere in R2 . Let ' W S1 ! P.S1 / be defined as follows: p '.x/ D fy 2 S1 j jx  yj  3g: Obviously ' is acyclic, but '.'.x// D S1 , for every x, and consequently ' ı ' is not acyclic. However, we have Proposition 1.110. Let ' W X ! P.Y / and W Y ! P.Z/ be two admissible mappings; then the composition ı ' W X ! P.Z/ of ' and is also admissible. In particular, the composition of two acyclic mappings is admissible. In order to prove Proposition 1.110, it is sufficient to consider the following diagram: X ks

p

q / Y ks

g_ FFF O FFFF FFFF FFF f p

FFFF

p1

1 w; ww w w ww ww q

q1

/Z

 1

in which the pair .p; q/ determines ', .p1 ; q1 / determines , f .y; z/ D y, q.y; z/ D z and p is the fibre product of p1 and q. Then p ı p is a Vietoris map and, therefore, we get: . ı '/.x/ D .q1 ı q/..p ı p/1 .x//: In what follows, the pair .p; q/ determining an admissible map ' will be called a selected pair for ' (written .p; q/  '). Definition 1.111. Let ' W X ! P.Y / be an admissible map and .p; q/ its selected pair. We define the induced linear map: q ı p1 W H .X/ ! H.Y /

46

Chapter 1 Topological structure of fixed point sets

and the induced set f' g of linear maps by putting: f' g D fq ı p1 j .p; q/  'g: It easily follows from the example 1.109 that f' g can be even an infinite set. Nevertheless, we have Proposition 1.112. If ' W X ! P.Y / is an acyclic map, then f' g D ' is a singleton. Proof. Consider the graph ' of ' and the natural projections: p' W ' ) X, q' W ' ! Y . Assume, furthermore, that .p; q/  '. Then the following diagram is commutative:

' } O @@@ q' }}}}} @@ } } @@ }}}} } } z

p'

f

X d\ B BBB BB

BBB p BBBBB

>Y || | | || q ||

where f .u/ D .p.u/; q.u//. Now, the commutativity of the above diagram implies that q ı p1 D .q' / ı .p' /1  , and the proof is completed. Definition 1.113. Let '; W X ! P.Y / be two admissible maps. We shall say that ' is homotopic to (written ' ) if there are .p; q/  ', .p1 ; q1 /  , a pair .p; q/, and two continuous maps f and g such that the following diagram is commutative: p

X ks i0



@ @

X  Œ0; 1 ks O

p

i1

X ks

@@ q @@ @@  q /Y

O ~? ~ ~~ g ~~ q1 ~ ~

f

p1

1

where i0 .x/ D .x; 0/, i1 .x/ D .x; 1/. Since .i0 / and .i1 / are equal isomorphisms by applying the functor H to the above diagram, we obtain: Proposition 1.114. If '

, then f' g \ f

g

¤ ;.

Starting from now, we shall assume that all multi-valued mappings are admissible and all topological spaces are regular. Recall the

47

Section 1.3 Admissible maps

Definition S1.115. A multi-valued map ' W X ! P.Y / is called compact if the set '.X/ D x2X '.x/ is contained in a compact subset of Y . Observe that if .p; q/  ', then ' is compact if and only if (a single-valued map) q is compact. We let: K.X; Y / D f' j X ! P.Y / j ' is compactgI if X D Y , then we write K.X/ for K.X; X/. Definition 1.116. A map ' W X ! P.Y / is called locally compact if, for every x 2 X, there exists an open neighbourhood Ux of x in X such that the restriction: 'jUx W Ux ! P.Y / of ' to Ux is a compact map. Observe that if X is a locally compact space, then any admissible map ' W X ! P.Y / is locally compact. Obviously, any compact map is locally compact. We let: Kloc .X; Y / D f' j X ! P.Y / j ' is locally compactg and Kloc .X/ D Kloc.X; X/. We have: K.X; Y /  Kloc.X; Y /: Definition 1.117. A map ' W X ! P.X/ is called eventually compact if there exists n such that the nth iteration ' n j X ! P.X/ of ' is a compact map. Let E.X/ D f' j X ! P.X/ j ' is eventually compactg: We have: K.X/  E.X/: Definition 1.118. A map ' W X ! P.X/ is called asymptotically compact if S n (1) for every x 2 X, the orbit 1 nD1 ' .x/ is contained in a compact subset of X, and T n (2) the centre, sometimes also called the core, C' D 1 nD1 ' .X/ of ' is nonempty and contained in a compact subset of X. Let ASC.X/ D f' j X ! P.X/ j ' is asymptotically compactg: Definition 1.119. A map ' W X ! P.X/ is called a compact attraction if there exists a compact set K  X such that, for every open neighbourhood U of K in X and, for every x 2 X, there exists n D nx such that ' m .x/  U , for every m  n; then K is called the attractor of '.

48

Chapter 1 Topological structure of fixed point sets

Let CA.X/ D f' W X ! P.X/ j ' is a compact attractiong: Definition 1.120. A map ' W X ! P.X/ is called a compact absorbing contraction if there exists an open set U  X such that: (1) '.U /  U and the map e ' W U ! P.U /, e ' .x/ D '.x/, for every x 2 U , is compact, and (2) for every x 2 X, there exists n D nx such that ' n .x/  U . Let CAC.X/ D f' W X ! P.X/ j ' is a compact absorbing contractiong: Evidently, we have: K.X/  CAC.X/  CA.X/:

(1.9)

At first, we prove: Proposition 1.121. EC.X/  ASC.X/: Proof. Let ' 2 EC.X/ and assume that K D ' n0 .X/ is a compact set. Then, for every x 2 X, we have: 1 [

' i .x/  fxg [ f'.x/g [ : : : [ f' n0 1 .x/g [ ' n0 .X/:

i D1

Thus, the set fxg [ f'.x/g [ : : : [ f' n0 1 .x/g [ ' n0 .X/ is compact, i.e., every orbit is a relatively compact set. Moreover, we have: 8i  0 ' n0 Ci .K/ D ' n0 .' i .K//  ' n0 .X/  K: Consequently, ;¤

1 \ i D0

'

n0 Ci

.K/ 

1 \ i D0

'

n0 Ci

.X/ D

1 \

' i .X/

i D0

which implies that the core C' of ' is nonempty and relatively compact. So the proof of Proposition 1.121 is completed. Proposition 1.122. ASC.X/  CA.X/.

49

Section 1.3 Admissible maps

Proof. Let ' 2 ASC.X/. Then the set C' is nonempty and compact. It is enough to show that C' is an attractor of '. Let V be an open neighbourhood of C' in X and let S i x 2 X. We put L D 1 i D1 ' .x/. Then, for 0  j  n and for arbitrary n we get: 1 [

' 2n .x/ 

' i Cj .x/ D ' j

1 [

i D0

 ' i .x/  ' j .L/;

i D0

and so: '

2n

.x/ 

n \

' j .L/:

j D0

Therefore, it is enough to show that there is a natural number nx such that: nx \

' j .L/  V:

j D0

In fact, we obtain:

Tn

1 \

An 

nD0

1 \

' n .X/ D C'  V;

nD0

where An D i D0 ' .L/. Since An is a decreasing sequence of compact sets, this implies that there are natural numbers n1 < n2 <    < nk such that i

An1 \ An2 \ : : : \ Ank  V; but An1 \ An2 \ : : : \ Ank D Ank ; and so nx D nk is the required natural number. The proof of Proposition 1.122 is completed. Summing up the above, we get: EC.X/  ASC.X/  CA.X/ [ [ K.X/  CAC.X/ In what follows, we introduce the following notations: EC0 .X/ D EC.X/ \ Kloc .X/; ASC0 .X/ D ASC.X/ \ Kloc .X/; CA0 .X/ D CA.X/ \ Kloc .X/; and start by the following: Lemma 1.123. EC0 .X/  CAC.X/.

(1.10)

50

Chapter 1 Topological structure of fixed point sets

Proof. Let ' 2 EC0 .X/, M D ' n0 .X/ be compact, and let K D Then K is compact and '.K/ D

n0 [

Sn0 1 i D0

' i .M /.

' i .M /  K [ M D K:

i D1

Since ' is locally compact, there exists an open neighbourhood W of K in X such that L D '.W / is compact. Now, we define open sets V0 ; : : : ; Vn0 such that L \ '.Vi /  Vi 1 and K [ ' n0 i .L/  Vi , i D 1; : : : n0 . Namely, we put V0 D W . If V0 ; : : : ; Vi are needed sets then .K [ ' n0 i .L// \ .L n Vi / D ;. Therefore, there is an open set V such that K [ ' ni .L/  V  V  Vi [ .X n L/. Letting Vi C1 D ' 1 .V /, we obtain '.K [ ' n0 .i C1/ .L// D '.K/ [ '.' n0 .i C1/.L//  K [ ' n0 i .L/  V; and so

K [ ' n0 .i C1/.L/  Vi C1 :

Moreover, we have '.Vi C1 /  V  Vi [ .X n L/; so L \ '.Vi C1 /  Vi : Letting U D V0 \ : : : \ Vn0 , we have M  K  U and '.U /  '.V0 / \    \ '.Vn0 /  L \ '.V1 / \    \ '.Vn0 /: Consequently, we have '.U /  .L \ '.V1 // \    \ .L \ '.Vn0 / \ L  V0 \    \ Vn0 D U; and '.U / is compact. Since M  U , for every x 2 X, we deduce that ' n0 .x/  U , and the proof is completed. The following example shows that EC.X/ 6 CAC.X/. Example 1.124. Let C D ffxng  R j fxng is a bounded sequenceg be the space of bounded sequences with the usual supremum norm. We define F W C ! C as follows: F .fxn g/ D f0; x1 ; 0; x3 ; 0; x5 ; 0; : : :g: Then F 2 D 0, so F 2 EC.C /, but F … CAC.C /. Now, we prove

51

Section 1.3 Admissible maps

Proposition 1.125. CA0 .X/  CAC.X/. Proof. Let ' 2 CA0 .X/ and let K be a compact attractor for '. Since ' is locally compact, there exists an open neighbourhood W of K in X such that L D '.W / is a compact set. We have 1 [ ' i .W /: LX  i D0

This implies that f' i .W /gi 2N is an open covering of L in X. Therefore, there is a finite subcovering ' i1 .W /; : : : ; ' ij .W /: S Let n D maxfi1 ; : : : ; ij g and V D niD0 ' i .W /. Then we have LV

and W  V

Consequently, X

1 [

and ' i .W /  ' i .V /:

' i .W / 

i D0

1 [

' i .V /:

i D0

We get '.V / D

n [

' i C1 .W / D '.W / [

i D0

n1 [

' i .W /  '.W / [ V  L [ V D V

i D1

and, moreover, we have '

nC1

.V / 

n [

'

ni C1

.W / D

i D0

n [

i

' .'.W // 

i D0

n [

' i .L/:

i D0

S Consequently, the set niD0 ' i .L/ is a compact subset of V , and so we have shown that e ' 2 EC.X/, where e ' D 'jV . In view of Lemma 1.123, we infer e ' 2 CAC.V /, but this immediately implies that ' 2 CAC.X/. It fact, if U is an open subset of V such that 1 1 [ [ V  e ' i .U /  ' i .U /; i D0

then X

1 [ i D0

i D0

'

i

.V / 

1 [

' i .U /;

i D0

' .U / D '.U / is a compact subset of V . The proof is completed. and e

52

Chapter 1 Topological structure of fixed point sets

Summing up the above, we get K.X/  EC0 .X/  ASC0 .X/  CA0 .X/  CAC.X/:

(1.11)

Finally, let us remark that all the above inclusions are proper. Let E be a Banach space and let  be a measure of noncompactness in E. Definition 1.126. Assume that X and Y are subsets of E. An admissible map ' W X ! P.Y / is called: (1) k-set contraction, k 2 Œ0; 1/, if for every bounded A  X we have: .'.A//  k.A/I (2) condensing if for every bounded A  X such that .A/ > 0 we have: .'.A// > .A/: We let C.X; Y / D f' W X ! P.Y / j ' is condensingg; Ck .X; Y / D f' W X ! P.Y / j ' is a k-set contractiong and also we put C.X/ D C.X; X/;

Ck .X/ D Ck .X; X/:

Evidently, we have: K.X; Y /  Ck .X; Y /  C.X; Y /: The following proposition is evident: Proposition 1.127. If X  Y  E and ' 2 C.X; Y /, then Fix.'/ D fx 2 X j x 2 '.x/g is relatively compact in E; in particular if X is a closed subset of E, then Fix.'/ is compact. Theorem 1.128. If X is a closed bounded subset of E and ' 2 C.X/, then ' 2 CA.X/. Proof. Let X1 D '.X/; : : : ; Xn D '.Xn1 /; : : : and let X1 D cl .X1 /; : : : ; Xn D T cl .Xn /; : : :. Let M D 1 nD1 Xn . Evidently M is an attractor for '. Moreover, we get: .M / D lim .Xn / D 0 n!1

and hence M is a compact attractor for '.

53

Section 1.3 Admissible maps

1.3.2 Fixed point theorems for admissible multi-valued maps In this section, we shall present the Lefschetz and in particular Schauder’s fixed point theorems for admissible CAC -mappings. Then we shall inform about the fixed point index for such a class of admissible mappings. Finally, the case of admissible condensing mappings will be discussed. We shall start from the following lemma. Lemma 1.129. Let U be an open subset of a normed space E and let ' W U ! P.U / be an admissible compact map. Then the Lefschetz set ƒ.'/ D f.q ıp1 / j .p; q  'g is well defined and if ƒ.'/ ¤ f0g, then ' has a fixed point. Proof. Let .p; q/  ' be a selected pair. Since ' is compact, coq is compact, too. By Theorem 5.119, q ı p1 is a Leray endomorphism and consequently ' is a Lefschetz map, i.e., the Lefschetz set ƒ.'/ of ' is well defined. Consequently, if .q ı p1 / ¤ 0, then p and q have a coincidence u 2 U , i.e., p.u/ D q.u/ (see again Theorem 5.119). Thus, z D p.u/ is a fixed point of ' and the proof is completed. In what follows, we shall say that ' W X ! P.X/ is a Lefschetz map provided the Lefschetz set ƒ.'/ D f.q ı p1 / j .p; q/  'g is well defined. Now, we prove the Lefschetz fixed point theorem for ANR-s. Theorem 1.130. If X 2 ANR and ' W X ! P.X/ is an admissible compact map, then: (1) ' is a Lefschetz map and (2) ƒ.'/ ¤ f0g implies that Fix.'/ ¤ ;. Proof. Without loss of generality, we can assume that X is a retract of an open set U  E, where E is a normed space. Let r W U ! X be the retraction map and i W X ! U the inclusion mapping. We define: W U ! P.U / by the formula D i ı'ır . Since ' is a compact admissible map, by using Proposition Theorem 1.110 we deduce that is compact admissible, too. Now, let .p; q/  ' be a selected pair. We can choose a selected pair .e p ;e q /  such that (see [215]) 1 e q ı e p 1  D i ı q ı p ı r :

Therefore, the following diagram is commutative: r / H.X/ O PPP 1 iPPıq ıp 1 P P e q  ıe p1 q ıp PPP  P / H.X/ H.U / r

H.U / hP O



54

Chapter 1 Topological structure of fixed point sets

and, consequently, from Proposition 5.108 and Lemma 1.129, we deduce that: .q ı p1 / D .e q ı e p 1  /: If we assume that .q ı p1 / ¤ 0, then .e q ı e p 1  / ¤ 0 and we get a point u 2 U such that u 2 .u/. Finally, we have r .u/ D x 2 '.x/ and the proof is completed. Observe that if X is an acyclic ANR or in particular if X 2 AR, then for every compact admissible map, we have: Corollary 1.131 (Schauder’s fixed point theorem). If X 2 AR and ' W X ! P.X/ is a compact admissible map, then Fix.'/ ¤ ;. Now, we shall generalise Theorem 1.130 to the following one: Theorem 1.132 (Generalised Lefschetz fixed point theorem). Let X 2 ANR and ' W X ! P.X/ be an admissible CAC map, then: (1) the Lefschetz set ƒ.'/ of ' is well defined, and (2) ƒ.'/ ¤ f0g implies that Fix.'/ ¤ ;. Proof. Let U be an open subset of X chosen according to the definition of CACmappings. Then the map '1 W U ! P.U /, '1 .x/ D '.x/ for every x 2 U is compact admissible and e ' W .X; U / ! P..X; U //, e ' .x/ D '.x/ for every x 2 X is an admissible map of pairs. Let .p; q/  ' be a selected pair and let .e p ;e q /, .p1 ; q1 / be induced by .p; q/ selected pairs of e ' and '1 , respectively. From the definition of CAC-maps and the fact that we consider homology with compact carriers, it follows that the linear map: p 1 e q ı e  W H.X; U / ! H.X; U / is weakly nilpotent and hence ƒ.e q ı e p 1  / D 0. Since open subsets of ANRs are ANRs again, it follows that the Lefschetz number ƒ..q1 / ı .p1 /1  / is well defined and we get 1 1 p 1 0 D ƒ.e q ı e  / D ƒ.q ı p /  ƒ..q1 / ı .p1 / /:

Consequently, if ƒ.q ı p1 / ¤ 0, then from Theorem 1.130, Fix.'/ ¤ ;. But Fix.'/ D Fix.'1 / and the proof is completed. In particular, we have: Corollary 1.133 (Generalised Schauder’s fixed point theorem). If X 2 ANR and ' W X ! P.X/ is an admissible CAC-map, then Fix.'/ ¤ ;.

55

Section 1.3 Admissible maps

Below, we shall sketch the fixed point index and topological degree theory for admissible mappings. We shall define the fixed point index for compact admissible mappings on open subsets of ANRs. Consider the diagram: p

q

X (H !Y: The above diagram induces a map '.p; q/ W X ! P.Y / by the formula: '.p; q/.x/ D q.p 1 .x//;

for every x 2 X:

In what follows, we shall identify the map '.p; q/ with the pair .p; q/. Of course '.p; q/ is admissible, but we keep only one selected pair .p; q/ of this map. Moreover, '.p; q/ is compact if and only if q is compact. For a multi-valued map '.p; q/ W X ! P.Y /, we denote: C.p; q/ D fz 2 j p.z/ D q.z/g; Fix.p; q/ D Fix.'.p; q// D fx 2 X j x 2 q.p 1 .x//g: Evidently, p.C.p; q// D Fix.p; q/ and so: Fix.p; q/ ¤ ; ” C.p; q/ ¤ ;: Recall that in the case when .p; q/ W U ! P.RN / is compact, where U is an open subset of Rn we already defined the fixed index I.p; q/ of the pair .p; q/. Now, by using the Schauder approximation theorem and the topological characterisation of ANRs, this index can be taken up to the case when .p; q/ W U ! X is a compact admissible map and we shall denote it ind.p; q/. The properties of “ind” are collected in the following: Theorem 1.134. Assume that .p; q/ W U ! P.X/ is a multi-valued map and Fix .p; q/ is compact. We have (1) (Existence). If ind.p; q/ ¤ 0, then Fix .p; q/ ¤ ;. (2) (Localisation). If V is an open subset of X such that Fix .p; q/  V  U , then ind.p; q/ D ind.p1 ; q1 /; where

p1

q1

V (Hp 1 .V /!X;

p1 .z/ D p.z/; q1 .z/ D q.z/;

i.e., .p1 ; q1 / W V ! P.X/ is the restriction of .p; q/.

56

Chapter 1 Topological structure of fixed point sets

(3) (Additivity). Assume that U D U1 [ U2 , where U1 ; U2 are open in X. Assume, furthermore, that .p1 ; q1 / W U1 ! P.X/, .p2 ; q2 / W U2 ! P.X/ are respective restrictions of .p; q/, Fix .p1 ; q1 /, Fix .p2 ; q2 / are compact and Fix .p1 ; q1 / \ Fix .p2 ; q2 / D ;, then ind.p; q/ D ind.p1 ; q1 / C ind.p2 ; q2 /: (4) (Homotopy). If .p1 ; q1 /, .p2 ; q2 / W U ! P.X/ are homotopic and the joining homotopy of .p1 ; q1 / with .p2 ; q2 / has a compact set of fixed points, then ind.p1 ; q1 / D ind.p2 ; q2 /: e D U \ A, (5) (Contraction). Assume that q.p 1 .U //  A, A 2 ANR and let U e ! P.A/ be the respective contraction of .p; q/. Then: .p1 ; q1 / W U ind.p; q/ D ind.p1 ; q1 /: (6) (Multiplicity). Let U  X, U 0  X 0 be open sets and .p; q/ W U ! P.X/, .p 0 ; q 0 / W U 0 ! P.X 0 / be two maps such that Fix .p; q/ and Fix .p 0 ; q 0 / are compact sets of fixed points and ind.p  p 0 ; q  q 0 / D ind.p; q/  ind.p 0 ; q 0 /: (7) (Normalisation). Assume that .p; q/ W U ! P.U / is compact. Then ind.p;e q / D ƒ.p; q/; where e q W ! X, e q .z/ D q.z/, for every z 2 . Let us add that if U is an open subset of a normed space and .p; q/ D '.p; q/ W U ! P.E/ is a compact admissible vector field. Moreover, if we assume that 0 … ˆ.p; q/.x/ for every x 2 @U , then we can define the topological degree Deg.ˆ.p; q// by putting: Deg.ˆ.p; q// D ind'.p; q/: (1.12) Remark 1.135. Let us notice that all results presented in this section can be formulated for the space X to a retract of an open set in a locally convex space E. To end this section we shall present the Lefschetz fixed point theorem for admissible condensing mappings. First, we prove the following: Proposition 1.136. Let .X; d / be a complete bounded space and let ' W X ! P.X/ be a condensing map. Then ' is an asymptotically compact map, in particular ' has a compact attractor.

57

Section 1.3 Admissible maps

Proof. Since ' is condensing, we get: lim .' n .X// D 0:

n!1

It implies that the core C' D

1 \

' n .X/

nD1

is compact and nonempty. Moreover, let O.x/ D fx; '.x/; ' 2 .x/; : : :g be an orbit of x 2 X with respect to '. Then we have O.x/ D fxg [ '.O.x// and consequently, if we assume that .O.x// > 0, then we get: .O.x// D .'.O.x/// < .O.x//; leading to a contradiction. So ' is asymptotically compact and therefore it has a compact attractor. Corollary 1.137. Let U be an open subset of a Banach space E and let ' W U ! P.U / be a condensing map. If there exists a closed bounded subset B of E such that '.U /  B  U , then ' has a compact attractor. In fact, by applying Proposition 1.136 to e ' W B ! B, e ' .x/ D '.x/ for every x 2 B, we get Corollary 1.137. Consequently, we have: Corollary 1.138. Let U and ' W U ! P.U / be the same as in Corollary 1.137. Then ' is a Lefschetz map. We need the following definition: Definition 1.139. A complete, bounded metric space .X; d / is called a special ANR (written X 2 ANR/ provided there exists an open U of a Banach space E and two continuous mappings r W U ! X and s W X ! U such that: (1) r ı s D idX , (2) r and s are nonexpansive, i.e., .r .B//  .B/ and .s.A//  .A/ for two arbitrary bounded sets A and B. We are able to prove the following version of the Lefschetz fixed point theorem: Theorem 1.140. Let X 2 ANR and let ' W X ! P.X/ be a condensing and admissible map. Then ' is a Lefschetz map and ƒ.'/ ¤ f0g implies Fix.'/ ¤ ;.

58

Chapter 1 Topological structure of fixed point sets

Proof. From Proposition 1.136 we deduce that ' has a compact attractor. Let U; r W U ! X and s W X ! U be as in Definition 1.139. We define the map e ' W U ! U by putting: e ' D s ı ' ı r: In view of 1.139 (2), we deduce that e ' is a condensing map. Observe that if A is a compact attractor of ', then s.A/ is a compact attractor e ' (see 1.139 (1)). Consequently, the map e ' W U ! U is condensing with a compact attractor. Furthermore, we have the following commutative diagram: U o

s

ppp  pppp U o

' e

X pp8

'ır pppp s



'

X

Thus, ƒ.'/ ¤ f0g if and only if ƒ.e ' / ¤ f0g and our theorem follows from Theorem 1.132. The following lemma is obvious. Lemma 1.141. Let ' W X ! P.X/ be a map. Assume further that A is a compact attractor for ' and V is an open neighbourhood of A in X. Then there exists an open neighbourhood U of A in X such that (1) '.U /  U , (2) A  U  V . Finally, by standard arguments, we obtain: Theorem 1.142. Assume that X is a nonexpansive retract of some open set W in a Banach space E. Assume further that ' W X ! P.X/ is an admissible CA-mapping with a compact attractor A. If there exists an open neighbourhood V of A in X such that the restriction 'jV W V ! P.X/ of ' to V is a condensing map, then ' is a Lefschetz map and ƒ.'/ ¤ f0g implies Fix.'/ ¤ ;.

1.3.3 The general Brouwer fixed point theorem Now, we can summarise the Brouwer fixed point theorem and its generalisation for multi-valued mappings (more details may be found in [212, 214]. First, we introduce the following classes of functions: (1) (Convex valued maps). When X  Rn is compact convex and nonempty, let Co.X/ D fF W X ! P.X/ j F is u.s.c. with convex valuesg:

59

Section 1.3 Admissible maps

(2) (Acyclic mappings). For a space X, let A.X/ D fF W X ! P.X/ j F is u.s.c. with acyclic valuesg: (3) (n-Acyclic maps). For a space X, let An .X/ D fF W X ! P.X/ j F is u.s.c. and rd X M i .F /  n  2  i; i D 0; 1; : : : ; n  2g; n  1: (4) (Maps with acyclic components). For a space X and a fixed natural number m, let A1;m .X/ D fF W X ! P.X/ j F is continuous and F .x/ consists of one or m acyclic componentsg: (5) (Admissible maps). For a space X, let A.X/ D fF W X ! P.X/ j F is admissibleg: (6) (n-Admissible maps). For a space X, let An.X/ D fF W X ! P.X/ j F is n-admissibleg: (7) (Composition of maps with acyclic components). For a space X, let Ac .X/ D fF W X ! P.X/ j F D Fk ı    ı F1 ^ Fi W Xi ! Xi C1 ; X1 D XkC1 D X ^ Fi 2 A1;mi .Xi ; Xi C1 /g: Recall the relative dimension of A in X: rdX .A/ D supfdim C j C  A and C closed in Xg and M i .F / D fx 2 X j H i .F .x// 6D 0g 0

.i > 0/;

0

M .F / D fx 2 X j H .F .x// 6D Qg: If we let F.X/ D fF W X ! P.X/ j F is u.s.c. with compact values and F has a fixed pointg; where X is an Euclidean absolute retract, then we get (see [212, Theorem 2.8]) Theorem 1.143. (a) (Kakutani, 1941) Co.X/  F.X/ provided X is compact, convex, and nonempty. (b) (Eilenberg-Montgomery, 1946) [166] A.X/  F.X/.

60

Chapter 1 Topological structure of fixed point sets

(c) (Górniewicz, 1979) [208] An .X/  F.X/ provided X D Bn is the closed unit ball. (d) (Górniewicz, 1976) [207] A.X/  F.X/. (e) (Kryszewski, 1994) [306] An .X/  F.X/ provided X D Bn is the closed unit ball. (f) (O’Neill, 1947) [362] A1;m .X/  F.X/. (g) (Dzedzej, 1985) [163] Ac .X/  F.X/.

1.3.4 Browder–Gupta type results for admissible mappings There is a natural and essential problem to formulate an appropriate multi-valued version of the Browder–Gupta theorem. In this order, see [19, 39, 73, 119, 123, 156, 190, 195–197, 213, 238, 319, 372, 373]. The most general results are obtained by G. Gabor (see [190]) and B. D. Gel’man (see [197]). Theorem 1.144. Let X be a metric space, E a Fréchet space, fUk gk2N a base of open convex symmetric neighbourhoods of the origin in E, and let ' W X ! P.E/ be an u.s.c. proper map with compact values. Assume that there is a sequence of compact convex valued u.s.c. proper maps 'k W X ! E such that (a) 'k .x/  '.N1=k .x// C Uk , for every x 2 X, (b) If 0 2 '.x/, then 'k .x/ \ Uk ¤ 0, (c) for every k 2 N and every u 2 E with u 2 Uk , the inclusion u 2 'k .x/ has an acyclic set of solutions. Then the set S D ' 1 .0/ is compact and acyclic. Proof. Step 1. We show that S is nonempty. To this end, notice that for every k 2 N we can find xk 2 X such that 0 2 'k .xk /. Assumption (i) implies that there are zk 2 N1=k .xk /, yk 2 'k .zk / and uk 2 Uk such that 0 D yk C uk . Thus yk ! 0. Consider 1 the compact set K D fyk g [ f0g. Since ' is proper, the set 'C .K/ is compact. 1 Moreover, fzk gk2N  'C .K/. Thus we can assume, without loss of generality, that fzk gk2N converges to some point x 2 X. By the upper semi-continuity of ', we have 0 2 '.x/ and hence S ¤ ;. Step 2. Since ' is proper, the set S is compact. We show that it is acyclic. By assump1 tion (ii), the set Ak D 'kC .Uk / is nonempty. Consider the map W Ak ! P.Uk / defined by k .x/ D 'k .x/ \ Uk . Since Uk is contractible and k is u.s.c. convex

Section 1.3 Admissible maps

61

valued surjection (see (c)), we can apply Corollary 3.12 in [190] to obtain that Ak is acyclic. Now we show that for every open neighbourhood U of S in X, there exists k  1 such that Ak  U . Indeed, assume on the contrary that there is an open neighbourhood U of S in X such that Ak 6 U for every k  1. It means that there are xk 2 Ak with xk 62 U and, consequently, there are yk 2 'k .xk / such that yk 2 Uk . Assumption (i) implies that there are zk 2 B.xk ; 1=k/, vk 2 '.zk /, and uk 2 Uk such that yk D vk C uk . Therefore, vk D yk  uk 2 2Uk which implies that vk ! 0. Consider the compact set K0 D fvk g [f0g. Since ' is proper, we can assume that fzk g and, consequently, fxk g converges to some point x 2 X. Thus x 2 S. On the other hand, x 62 U , leading to a contradiction and our theorem follows from Lemma 3.10 in [190]. Remark 1.145. It is easy to see that in the above result we can assume that X is a subset of a Fréchet space. Then, instead of neighbourhoods, we can consider sets x C Vk , where fVk gk2N is the base of open convex symmetric neighbourhoods of the origin. As a consequence of Theorem 1.129 and properties of a topological degree of u.s.c. compact convex valued maps (see, e.g., [215] or [327]), one can obtain the following theorem generalising the result of Czarnowski in [132]. Theorem 1.146. Let  be an open subset of a Fréchet space E, fUk gN be the base of open convex symmetric neighbourhoods of the origin in E, and ˆ W  ! P.E/ be a compact u.s.c. map with compact convex values. Suppose that x 62 ˆ.x/ for every x 2 @, and the topological degree Deg.j  ˆ; ; 0/ of .j  ˆ/ is different from zero, where j W  ! E is an inclusion. Assume further that there exists a sequence fˆk W  ! P.E/gk2N of compact u.s.c. maps with compact convex values such that (i)

ˆk .x/  ˆ.x C Uk / C Uk , for every x 2 ,

(ii) if x 2 ˆ.x/, then x 2 ˆk .x/ C Uk , (iii) for every u 2 Uk the set Suk of all solutions to the inclusion x  ˆk .x/ 3 u is acyclic or empty, for every n > 0. Then the fixed point set Fix .ˆ/ of ˆ is compact and acyclic. Proof. Define the maps ', 'k W  ! P.E/, ' D j ˆ, 'k D j ˆk . One can check that ', 'k are proper maps. To apply Theorem 1.144, it is sufficient to show that, for sufficiently large k and for every u 2 Uk , the set Suk is nonempty. For each k 2 N, define the map ‰ W  ! P.E/, ‰.x/ D ˆk .x/ C u, for every x 2 . We prove that, for sufficiently large k, Deg.j  ‰k ; ; 0/ ¤ 0 which implies, by the existence property of a degree, nonemptiness of Suk .

62

Chapter 1 Topological structure of fixed point sets

Since ' is a closed map (see, e.g., [215]), we can find, for sufficiently large k, a neighbourhood Uk of the origin such that '.@/ \ Uk D ;. Consider the following homotopy Hk W   Œ0; 1 ! P.E/, H.x; t / D .1  t /ˆ.x/ C t ‰k .x/. We show that Zk D fx 2 @ j x 2 Hk .x; t / for some t 2 Œ0; 1g D ; for sufficiently large k. Suppose, on the contrary, that there is a subsequence of fHk gk2N (we denote it also by fHk gk2N), points xk 2 @, and numbers tk 2 Œ0; 1 such that xk 2 Hk .xk ; tk /, that is xk D .1  tk /yk C tk sk C tk u, for some yk 2 ˆ.xk / and sk 2 ˆ.xk /. Assumption (i) implies that there are zk 2 xk C Uk and vk 2 ˆ.zk / such that sk 2 vk C Uk . By the compactness of ˆ, we can assume that yk ! y and vk ! v. Therefore, sk ! v. Moreover, we can assume that tk ! t 2 Œ0; 1. This implies that xk ! x0 D .1  t /y C t v C t u or, equivalently, that 0 D .1  t /.x0  y/ C t .x0  v/  t u. But by the upper semi-continuity of ', we obtain that x0  y 2 '.x0 / and x0  v 2 '.x0 /. Since ' is convex valued, 0 2 .1  t /'.x0 / C t '.x0 /  t u  '.x0 /  t u. This implies that '.x0 / \ Uk ¤ ;, a contradiction.

1.3.5 Topological dimensions of solution sets We define the covering dimension dim .A/ of a set A (see [170, p. 385]) as follows: Definition 1.147. (a) dim .A/  n; n 2 N if for all finite open cover U of A, there exists a finite open refinement V such that V1 \ : : : VnC2 D ; for all V1 ; : : : ; VnC2 2 V which are pairwise distinct. (b) If dim .A/  n but not dim .A/  n  1, then dim .A/ D n. (c) if dim .A/  n does not hold for any n 2 N, then dim .A/ D 1. The covering dimension provides some information about the connectedness of the components of metric spaces. The proof of the following result can be found in Engelking [170]: Lemma 1.148. Let E be a locally compact metric space. Then dim .A/ D 0 if and only if the connected components of E are singletons. Regarding applications, an important result has been proved by B. Ricceri in 1997: Theorem 1.149 ([409, Theorem 1]). Let X; Y be Banach spaces, ˆ W X ! Y be a continuous linear surjective operator, and ‰ W X ! Y be a continuous operator with relatively compact range. Then dim .fx 2 X j ˆ.x/ D ‰.x/g/  dim1 .ˆ1 .0//: We need the following lemma owed to Saint-Raymond [416].

63

Section 1.3 Admissible maps

Lemma 1.150. Let .X; d / be a compact metric space and let .E; k  k/ be a Banach space. Assume further that ' W X ! Pcp;cv .E/ is a l.s.c. multi-valued map such that the topological dimension dim '.x/  n, x 2 X and x 62 '.x/. Then there exists a continuous selection f W X ! E of ' such that x ¤ f .x/ 2 '.x/ for every x 2 X. Note that Lemma 1.150 provides additional information to that given in Michael’s selection theorem. Theorem 1.151. Let U be an open subset of E and let ' W U ! P.E/ be a condensing map such that: (1) ; ¤ Fix.'/  U , (2) there exists an open V  U and a l.s.c. map W V ! Pcp;cv .E/ satisfying the following condition: .x/  '.x/; Fix. / D Fix.'/

and

dim .x/  n

for every x 2 V:

Then dim.Fix.'//  n. Proof. Assume on the contrary that dim.Fix.'//  n  1; then we get a contradiction with Lemma 1.150. The following result is due to Z. Dzedzej and B. D. Gel’man [164] (see also [22]). Theorem 1.152. Let E be a Banach space and F W E ! B.E/ be a contraction with convex values and a constant ˛ < 1=2. Assume, furthermore, that the topological dimension dim F .x/ of F .x/ is greater or equal to n for some n and every x 2 E. If Fix .F / is compact, then dim Fix .F /  n: In fact, J. Saint-Raymond [Theorem 8, [417]] proved the following result, which states that the set of fixed points of a multi-valued contraction can be unbounded. Theorem 1.153. Let C be a closed convex subset of a Banach space and F W C ! Pcl .C / a multi-valued k-contraction with a fixed point x0 . Then diam Fix .F / 

1k diamf.F /.x0 /g: 2

The proof relies on the following technical lemma (see [Theorem 7, [417]]): Lemma 1.154. Let C be a closed convex subset of a Banach space and F W C ! Pcl .C / a multi-valued k-contraction such that 0 2 F .0/, y0 2 F .0/, and y0 6D 0. Then supfkxk j x 2 Fix .F /g  .1  k/ky0 k > 0:

64

Chapter 1 Topological structure of fixed point sets

Proof of Theorem 1.153. Let the multi-valued map G be defined by G.x/ D F .x C x0 /  x0 for all x 2 C 0 D C  x0 . Then G is a k-contraction, G.x/  C 0 , and 0 2 G.0/. For r < diam .F .x0 // D diam .G.0//, let y0 2 G.0/ be such that ky0 k > r=2. Then by Lemma 1.154, G has a fixed point xN such that kxk N > .1  q/r=2. Thus, x1 D x0 C xN is a fixed point of F and kx1  x0 k > r .1  q/=2 and the conclusion of the theorem follows. Theorem 1.153 was generalised to estimating the topological dimension of Fix .F / by the same author in case of Hilbertian space [Theorem 9, [416]]. Open problem. Is it possible to prove Theorem 1.152 for a complete AR-space E D X and F W X ! Pb .X/ with values belonging to a Michael family M.X/‹ Following D. Miklaszewski, we would like to discuss some generalisations of Theorem 1.152 (see also [21]). Theorem 1.155. Let X be a retract of a Banach space E and F W X ! Pb .X/ be a compact continuous multi-valued map with values being elements of the Michael family M.X/ such that F .x/ n fxg 2 C k2 , for every x 2 Fix .F /. Then the set Fix .F / has the dimension greater or equal to k. Proof. Suppose on the contrary that dim.Fix .F // < k. Let us consider the maps W Fix .F / ! Pb;cv .E/ and ' W Fix .F / ! E n f0g defined by the formulas .x/ D F .x/  x D fy  x j y 2 F .x/g and '.x/ D .x/ n f0g D .F .x/ n fxg/  x. We are going to prove that the family f'.x/ j x 2 Fix g is equi-LC 1 . Let y 2 '.x0 / and r be a positive number such that 0 62 BE .y; 3r /. Suppose that the set BE .y; r / \ '.x/ is nonempty for a fixed point x of F . Then BE .y; r / \ '.x/ D Œ.BE .y C x; r / \ F .x//  x. Let z 2 BE .y C x; r / \ F .x/. It is easy to show that BE .y C x; r / \ F .x/  BE .y C x; 3r / \ F .x/. But the second set of these three sets being in the Michael family M.X/ is C 1 as well as its translation, so the inclusion of BE .y; r / \ '.x/ into the set BE .y; 3r / \ '.x/ is homotopically trivial, and the family f'.x/ j x 2 Fix .F /g is equi-LC 1 . Then ' has a selection f (see [220]). Then the map g W Fix .F / ! X defined by the formula: g.x/ D f .x/ C x is a selection of F . We conclude that there exists a selection h of F being an extension of g. But h has a fixed point x 0 2 Fix .F /, h.x 0 / D g.x 0 / D f .x 0 / C x 0 D x 0 , f .x 0 / D 0 2 '.x/, which is a contradiction. In the case when dim X < 1, by analogous considerations as in the proof of Theorem 1.155, we obtain: Theorem 1.156. Let X be a retract of a Banach space E and F W X ! B.X/ be a S continuous (i.e., both l.s.c. and u.s.c.) map such that F .X/ D fF .x/ j x 2 Xg is a compact set. Assume that the values of F satisfy the following conditions: (i)

F .x/ n fxg is C k2 , for every x 2 Fix .F /,

65

Section 1.4 Topological structure of fixed point sets of inverse limit maps

(ii) F .x/ is C k , for every x 2 X, (iii) fF .x/ j x 2 Fix .F /g is equi-LC k2 in E, (iv) fF .x/ j x 2 Xg is equi-LC k in X. Then dim.Fix .F //  k. The proof of Theorem 1.156 is quite analogous to that of Theorem 1.155. Finally, note that one can show an example of a continuous (i.e., both l.s.c. and u.s.c.) map with contractible values of the local dimension 2 such that (iii) and (iv) are satisfied, but the dimension of the set of fixed points equals 1.

1.4 Topological structure of fixed point sets of inverse limit maps 1.4.1 Definition Let us recall that an inverse system of Hausdorff topological spaces is a family S D ˇ .X˛ ; ˛ ; J /; where J is a poset directed by the relation ; X˛ is a Hausdorff topoˇ logical space, for every ˛ 2 J; and ˛ W Xˇ ! X˛ is a continuous mapping, for each ˇ

two elements ˛; ˇ 2 J such that ˛  ˇ. Moreover, for each ˛  ˇ  ; ˛ satisfies ˛˛ D IdX˛ and ˛ˇ ˇ D ˛ . By ˛ˇ ˇ , it is meant the composite ˛ˇ ı ˇ . The following subset of the product …˛2J X˛ n o lim S D .x˛ / 2 …˛2J X˛ j ˛ˇ .xˇ / D x˛ ; for all ˛  ˇ is called a limit (or projective limit) of the inverse system S. The inverse limit of the corresponding inverse system is just the product. The limit projective lim S is also called the generalised intersection \˛2J X˛ (see, e.g., [170] or [277], p. 439). An element of lim S is called thread or fibre of the system S. One can see that if we denote by ˛ W lim S ! X˛ a restriction of the projection p˛ W …˛2J X˛ ! X˛ onto the ˇ

˛th axis, then we obtain that ˛ ˇ D ˛ ; for each ˛  ˇ. Note that in general the inverse limit may be empty. Let us give an important example of inverse systems. Example 1.157. For every m 2 N, let Cm D C.Œ0; m; Rn / be the Banach space of all continuous functions on the closed interval Œ0; m into Rn and C D C.Œ0; 1/; Rn / the Fréchet space of continuous functions. For p  m, consider the restriction maps

66

Chapter 1 Topological structure of fixed point sets

p

p

m j Cp ! Cm defined by m .x/ D xjŒ0;m . It is easy to see that C is isometrically ˚  p homeomorphic to the limit of the inverse system Cm ; m ; N . The maps m W C ! Cm defined by m .x/ D xjŒ0;m correspond to suitable projections.

1.4.2 Basic properties Now, we summarise some useful properties of limits of inverse systems (see [190]). ˇ

Proposition 1.158. Let S D fX˛ ; ˛ ; J g be an inverse system. Q (1) The limit lim S is a closed subset of ˛2J X˛ .  (2)

If, for every ˛ 2 J , X˛ is

(2i)

compact, then lim S is compact; 

(2ii) compact and nonempty, then lim S is compact and nonempty;  (2iii) a continuum, then lim S is a continuum;  (2iv) compact and acyclic, then lim S is compact and acyclic;  (2v) metrisable, J is countable, and lim S is nonempty, then lim S is metrisable.   Proofs of Proposition 1.158 can be found in [18,19,22,190]. In case J is countable, we have (see [190], Proposition 3.2): Proposition 1.159. Let S D fXn; np ; Ng be an inverse system. If each Xn is a compact Rı -set, then lim S is Rı , too.  Proof. We let: 1 n o Y ˇ Qn D .xi / 2 Xi ˇ xi D in .xn/; for i  n : i D1

It is easy to see that each Qn is homeomorphic to the Rı -set 1 \ nD1

Q1

i Dn Xi .

Notice that

1 n o Y ˇ Qn D .xi / 2 Xn ˇ xi D in.xn / for every n  1 and i  n D lim S:  i D1

This implies that lim S D Lim Qn , and consequently, it is an Rı -set, as required.  The following example shows that a limit of an inverse system of absolute retracts does not have to be an absolute retract.

Section 1.4 Topological structure of fixed point sets of inverse limit maps

67

Example 1.160. Consider a family fXn gn2N of subsets of R2 defined as follows: Xn D .Œ0; 1=n  Œ1; 1/ [ f.x; y/ j y D sin 1=x and 1=n < x  1g: One can see that for each m; n  1 such that m  n we have Xm  Xn . Define the maps nm W Xm ! Xn and nm .x/ D x. Therefore, S D fXn; nm ; Ng is an inverse system of compact absolute retracts. It is evident that lim S is homeomor phic to the intersection of all Xn . On the other hand, XD

1 \

Xn D f.0; y/ j y 2 Œ1; 1g [ f.x; y/ j y D sin 1=x and 0 < x  1g;

nD1

and X is not an absolute retract since, for instance, X is not locally connected. Note that in [73] the following information on the limit of an inverse system of absolute retracts has been given. p

Proposition 1.161. Let S D fXn ; n ; Ng be an inverse system of compact absolute p retracts such that Xn  Xp and n be a retraction for all n  p. Then lim S has the  fixed point property, i.e., every continuous map f W lim S ! lim S has a fixed point.   p

Example 1.162. Consider the inverse system S D fXn ; n ; Ng such that Xn D p Œn; 1/ and n W Xp ,! Xn are inclusion maps for n  p. It is obvious that lim S is  homeomorphic to the intersection of all Xn which is an empty set. This shows that the compactness assumption in 1.158 (2) is important in obtaining a nonemptiness of the limit lim S. 

1.4.3 Multi-maps of inverse systems Next, we introduce the notion of multi-maps of inverse systems. Suppose that two o n o n ˇ ˇ0 0 0 systems S D X˛ ; ˛ ; J and S D Y˛0 ; ˛0 ; J are given. Definition 1.163. By a multi-map from the system S into the system S 0 ; we mean a family f; ' .˛0 / g consisting of a monotone function  W J 0 ! J; that is .˛ 0 /  .ˇ 0 /; for ˛ 0  ˇ 0 ; and of multi-maps ' .˛0 / W X .˛0 / ! P.Y˛0 / defined for every ˛ 0 2 J 0 and such that for each ˛ 0  ˇ 0 ˇ0

.ˇ 0 /

˛0 ' .ˇ 0 / D ' .˛0 /  .˛0 / : A map of systems f; ' .˛0 / g induces a limit map ' W lim S ! P.lim S 0 / defined by '.x/ D …˛0 2J 0 ' .˛0 / .x .˛0 / / \ lim S 0 :

68

Chapter 1 Topological structure of fixed point sets

In other words, a limit map is a map such that ˛0 ' D ' .˛0 /  .˛0 / ; for every ˛ 0 2 J . 0 In terms of countable inverse systems, '..xn /n2N/ D …1 nD1 'n .xn / \ lim S . Since a topology of the limit of an inverse system is the one generated by the base consisting of all sets of the form ˛ .U˛ /; where ˛ runs over an arbitrary set cofinal in J (given a directed set .J; /, K  J is called cofinal if for every ˛ 2 J , there exists ˇ 2 K such that ˇ ˛) and U˛ are open subsets of the space X˛ ; it is easy to prove the following continuity property for limit maps. 0

Proposition 1.164. Let S D fX˛ ; ˛ˇ ; J g and S 0 D fY˛0 ; ˛ˇ0 ; J 0 g be two inverse systems, and ' W lim S ! P. lim S 0 / [ f;g be a limit map induced by the map   f; ' .˛0 / g. If, for every ˛ 0 2 J 0 , ' .˛0 / is (i)

u.s.c. with compact values, then ' is u.s.c.;

(ii) l.s.c., then ' is l.s.c.; (iii) continuous, then ' is continuous. Regarding the structure of the fixed point sets of limit maps, we have (see [19], Theorem 2.8): ˇ

Theorem 1.165. Let S D fX˛ ; ˛ ; J g be an inverse system, and ' W lim S !  P. lim S/[f;g be a limit map induced by a map fid; '˛ g, where '˛ W X˛ ! P.X˛ /[  f;g. If the fixed point sets of '˛ are compact acyclic, then the fixed point set of ' is compact acyclic, too. Proof. Denote by F˛ the fixed point set of '˛ , for every ˛ 2 J , and by F the fixed ˇ point set of '. We will show that ˛ .Fˇ /  F˛ . ˇ

ˇ

ˇ

Let xˇ 2 Fˇ . Then xˇ 2 'ˇ .xˇ / and ˛ .xˇ / 2 ˛ 'ˇ .xˇ /  '˛ ˛ .xˇ /, which ˇ

implies that ˛ .xˇ / 2 F˛ . ˇ

Similarly, we show that ˛ .F/  F˛ . Denote by  ˛ W Fˇ ! F˛ the restriction ˇ

ˇ

of ˛ . One can see that S D fF˛ ;  ˛ ; J g is an inverse system. By Proposition 1.158, the set F is acyclic and the proof is complete. From the above proof, we obtain: p

Theorem 1.166. Let S D fXn; n ; Ng be an inverse system and ' W lim S !  P. lim S/ be a limit map induced by a map fid; 'n g, where 'n W Xn ! P.Xn /. If  fixed point sets of 'n are compact Rı , then the fixed point set of ' is Rı , too.

Section 1.4 Topological structure of fixed point sets of inverse limit maps

69

Recall that a lower semi-continuous map ' W X ! P.X/, where X is a metric space, has the selection property with respect to a subclass D of the class of metric spaces, if, for any Y 2 D, any pair of continuous functions f W Y ! X and h W Y ! .0; 1/ such that .y/ D cl Œ'.f .y// \ Nh.y/ .f .y// ¤ ;;

y 2 Y;

and any nonempty closed set Y0  Y , every continuous selection of jY0 admits a continuous extension g over Y fulfilling g.y/ 2 .y/ for all y 2 Y . If D is a class of all metric spaces, then we recall that ' has the selection property (' 2 SP .X/). Note that, for example, every closed convex valued l.s.c. map from a Fréchet space E into itself (and more generally, with values in any Michael family of subsets of E) has the selection property. Moreover, if X is a closed subset of L1 .T; E/, where E is a Banach space and ' W X ! P.X/ is a l.s.c. map with closed decomposable values, then ' 2 SP .X/. Now, we state a result that extends and unifies the results obtained by B. Ricceri and A. Bressan. A. Cellina and A. Fryszkowski about the fixed point set of multi-valued contractive maps (see [219] or [220, Theorem 2.3]]). Theorem 1.167. Let X be a complete absolute retract and let ' W X ! P.X/ be a multi-valued contraction map with a constant 0  k < 1. Suppose that ' 2 SP .X/. Then the set Fix.'/ is a complete absolute retract. The above result gives us the following applications. p

Corollary 1.168. Let S D fXn; n ; Ng be an inverse system, and ' W lim S !  P. lim S/ be a limit map induced by a map fid; 'n g, where 'n W Xn ! P.Xn /. If  all Xn are complete absolute retracts and all 'n are compact-valued contractions having the selection property, then Fix.'/ is compact Rı . Proof. By Theorem 1.167, all the fixed point sets Fn of 'n are absolute retracts. Since every map 'n has compact values, Theorem 1 in [408] implies the compactness of Fn . Therefore, our assertion follows from Proposition 1.159. p

Corollary 1.169. Let S D fXn; n ; Ng be an inverse system, and ' W lim S !  P. lim S/ be a limit map induced by a map fid; 'n g, where 'n W Xn ! P.Xn /. If  all Xn are Fréchet spaces and all 'n are contractions with convex compact values, then Fix.'/ is compact Rı . Below we shall present some applications of Theorem 1.167 to function spaces. We start with the following important examples of inverse systems. Remark 1.170. In the same manner as in Example 1.157, we can show that the Fréchet spaces C.Œ0; 1/; RN /, L1 .Œ0; 1/; RN / of all locally integrable functions,

70

Chapter 1 Topological structure of fixed point sets

AC.Œ0; 1/; RN / of all locally absolutely continuous functions and C k .Œ0; 1/; RN / of all continuously differentiable functions up to the order k can be considered as limits of suitable inverse systems. More generally, every Fréchet space is a limit of some inverse system of Banach spaces. Using the inverse system described in Example 1.157, we can give an example of a limit map induced by a map (of an inverse system) consisting of contractions (even with the same constant of contraction) which is not a contraction with respect to the metric in a limit of this system. Example 1.171. Consider the map f W C.Œ0; 1/; R/ ! C.Œ0; 1/; R/, f .x/ D x=2. This map is a contraction (with 1=2 as a constant of contractivity) with respect to each semi-norm pm . Suppose that there is k; 0  k < 1; such that d.f .x/; f .y//  k d.x; y/;

for any x; y 2 C.Œ0; 1/; R/:

Take L such that maxf1=2; kg < L < 1. We show that there are functions x; y 2 C.Œ0;1/; R/ such that d.f .x/; f .y//  Ld.x; y/. Indeed, let y  0 and x  2L=.1  L/. Then pm .x  y/ D 2L=.1  L/, for every m  1. One can easily check that pm .x  y/ 2L2 L D 1 C pm .x  y/ 1CL and

pm .f .x/  f .y// pm .x  y/=2 2L2 D DL> : 1 C pm .f .x/  f .y// 1 C pm .x  y/=2 1CL

Hence, 1 X 1 pm .x  y/ Ld.x; y/ D L 2m 1 C pm .x  y/ mD1



1 X mD1

pm .f .x/  f .y// D d.f .x/; f .y//: 1 C pm .f .x/  f .y//

This implies that f is not a contraction with respect to the metric in C.Œ0; 1/; R/. Now we formulate the results obtained in the previous section in a special case of function spaces to some examples described in Example 1.157 and Remark 1.170. Corollary 1.172. Let 'm W C.Œ0; m; RN / ! P.C.Œ0; m; RN // (respectively, 'm W L1 .Œ0; m; RN / ! P.L1 .Œ0; m; RN //, m 2 N be compact-valued contractions having the selection property, and such that 'p .x/jŒ0;m D 'm .xjŒ0;m /, for every x 2 C.Œ0; p; RN / (respectively, L1 .Œ0; p; RN //, p  m. Define the multi-valued map ' W C.Œ0; 1/; RN / ! P.C.Œ0; 1/; RN // (respectively, ' W L1 .Œ0; 1/; RN / !

Section 1.4 Topological structure of fixed point sets of inverse limit maps

71

P.L1 .Œ0; 1/; RN //, '.x/jŒ0;m D 'm .xjŒ0;m /, for every x 2 C.Œ0; 1/; RN / (respectively, L1 .Œ0; 1/; RN //. Then Fix.'/ is a compact Rı . Remark 1.173. From the above corollary one can infer that, if each 'm has convex values (or decomposable in the case of L1 spaces), then Fix.'/ is compact Rı . The inverse system approach described above gives us an easy way to study the topological structure of solution sets of differential problems on noncompact intervals. Namely, the suitable operator with solutions as fixed points can be often considered as a limit map induced by maps of Banach spaces of functions defined on compact intervals. For more details, we refer to [18, 19, 22], and [190].

Chapter 2

Existence theory for differential equations and inclusions This chapter deals with the classical existence theory for Cauchy problems and boundary value problems for differential equations and inclusions. Such problems are considered on compact and noncompact intervals of the real line. Basic notions are also presented with various kinds of right-hand nonlinearities.

2.1 Fundamental theorems The general form of nt h -order differential equation is a relation F .t; y; y .1/; : : : ; y .n/ / D 0;

(2.1)

where F is a function defined on a subset D.F /  R  RnC1 with unknown function y. Under some regularity on the function F , we can express problem (2.1) in the form y .n/ .t / D f .t; y.t /; y .1/ .t /; : : : ; y .n1/ .t //;

(2.2)

where f W D.f /  R  Rn ! Rn . This equation explicitly defines y .n/ as a function of t; y; y 0 ; : : : ; y .n1/ by means of the relation F .t; y; y 0 ; : : : ; y .n/ / D 0.

2.1.1 Existence and uniqueness results In this section, we study the existence of local and maximal solutions of the following Cauchy problem for first-order differential equations: y 0 .t / D f .t; y.t //;

y.t0 / D y0 ;

(2.3)

where f W R  Rn ! Rn is a given function and y0 2 Rn . Definition 2.1. We say that problem (2.3) has a local solution if there exist r > 0; h > 0 and a function y 2 C.Œt0  h; t0 C h; B .y0 ; r // such that y.t0 / D y0 y 0 .t / D f .t; y.t //;

for every t 2 Œt0  h; t0 C h

and .t; y.t // 2 Œt0  h; t0 C h  B.y0 ; r /; The following result is easily checked.

for every t 2 Œt0  h; t0 C h:

73

Section 2.1 Fundamental theorems

Lemma 2.2. Let f W Œt0 ; b  Rn ! Rn be a continuous function. Then y 2 C.Œt0 ; b; Rn / is solution of problem y .n/ .t / D f .t; y.t //;

y.t0 / D y0 ;

if and only if 1 y.t / D y0 C .n  1/Š

Z

t

.t  s/n1 f .s; y.s//ds;

t 2 Œt0 ; b:

0

2.1.2 Picard–Lindelöf theorem We start with a uniqueness result for problem (2.3). Theorem 2.3. Let f W Œt0 ; b  Rn ! Rn be a continuous function such that there exists K > 0 satisfying for all x; y 2 Rn ; t 2 Œt0 ; b:

kf .t; x/  f .t; y/k  Kkx  yk;

Then the initial value problem (2.3) has a unique solution on Œt0 ; b. Proof. We first transform problem (2.3) into a fixed point problem by considering the operator N1 W C.Œt0 ; b; Rn / ! C.Œt0 ; b; Rn / defined by Z t N1 .y/.t / D y0 C f .s; y.s//ds; t 2 Œt0 ; b: t0

We show that there exists n0 2 N such that N n0 is a contraction. Indeed, consider y1 ; y2 2 C.Œt0 ; b; Rn /. Then we have for each t 2 Œt0 ; b Z t kf .s; y1 .s//  f .s; y2 .s//kds kN1 .y1 /.t /  N1 .y2 /.t /k  t0

Z

t

K

ky1 .s/  y2 .s/kds: t0

Thus, kN1 .y1 /.t /  N1 .y2 /.t /k  K.t  t0 /ky1  y2 k1 : From the definition of N1 ; we have Z

t

N1 .N1 .y1 //.t / D y0 C

f .s; N1 .y1 .s///ds;

t 2 Œt0 ; b

t0

and

Z

t

N1 .N1 y2 //.t / D y0 C

f .s; N1 .y2 .s///ds; t0

t 2 Œt0 ; b:

(2.4)

74

Chapter 2 Existence theory for differential equations and inclusions

Using the estimate (2.4), we get kN12 .y1 /.t /  N12 .y2 /.t /k 

..t  t0 /K/2 ky1  y2 k1 : 2Š

Now, assume that for n 2 N kN1n.y1 /.t /  N1n .y2 /.t /k 

..t  t0 /K/n ky1  y2 k1 ; nŠ

then, by induction, we have: Z kN1nC1 .y1 /.t /  N1nC1 .y2 /.t /k 

t

t0 Z t

 t0 Z t

kf .s; N1n .y1 /.s//  f .s; N1n .y2 /.s//kds KkN1n.y1 /.s/  N1n .y2 /.s/kds

 t0

..s  t0 /K/n dsky1  y2 k1 : nŠ

Therefore, kN1nC1.y1 /.t /  N1nC1 .y2 /.t /k 

..t  t0 /K/nC1 ky1  y2 k1 : .n C 1/Š

Then, for every n 2 N and 8 y1 ; y2 2 C.Œt0 ; b; Rn /, we have kN1n.y1 /  N1n .y2 /k1 

..b  t0 /K/n ky1  y2 k1 : nŠ

P n0 ..bt0 /K/i < 1; then there exists n0 2 N such that ..btn00/K/ < 1. Since 1 i D0 iŠ Š Hence, N1 satisfies the assumptions of [Theorem 1.4, Chapter 1], ending the proof of the theorem. Theorem 2.4 (Picard–Lindelöf). Let f 2 C.U; Rn /, where U is an open subset of RN C1 , and .t0 ; y0 / 2 U . If f is locally Lipschitz continuous in the second argument, then there exists a unique local solution of problem (2.3). Proof. The function f D f .t; y/ is continuous at .t0 ; y0 / and is locally Lipschitz with respect to y. Then for every " > 0 there exists ı > 0 such that for every .t; y/ 2 U with k.t  t0 ; y  y0 /k  ı we have kf .t; y/  f .t0 ; y0 /k  " and kf .t; x/  f .t; y/k  ıkx  yk; for all .t; x/; .t; y/ 2 Œt0  ; t0 C   B.y0 ; /. Let C D Œt0  ı; t0 C ı  B.y0 ; ı/; since C is compact, there exists M > 0 such that kf .t; y/k  M;

for all .t; y/ 2 C:

75

Section 2.1 Fundamental theorems

 ı  and consider the operator N defined Let I0 D Œt0  h; t0 C h with h  min ı; M on C.I0 ; B.y0 ; ı// by Z

t

N1 .y/.t / D y0 C

f .s; y.s//ds;

t 2 I0 :

t0

We have Z

t

kN1 .y/.t /  y0 k 

kf .s; y.s//kds  M.t  t0 /  ı: t0

Then N1 .y/.t / 2 B.y0 ; ı/;

for all t 2 I0 :

Arguing as in the proof of Theorem 2.3, we obtain that problem (2.3) has a unique local solution on I0 . 2.1.2.1 Maximal solutions Definition 2.5. Let y1 W J1 ! Rn be a solution to (2.3). We say that y2 W J2 ! Rn is a continuation of y1 if y2 is a solution to (2.3) with J1  J2 and y1 .t / D y2 .t /; for every t 2 J1 where Ji ; i D 1; 2 are two intervals in R. Definition 2.6. A solution to Cauchy problem (2.3) is a maximal solution if there is no continuation of it. Lemma 2.7. Let f W R  Rn ! Rn be a continuous, locally Lipschitz map and let y W J ! Rn be a solution of problem (2.3). If y is maximal, then there exist ! ; !C 2 R [ f1; 1g such that J D .! ; !C /. J is called the maximal interval of y. Proof. Let J D Œ! ; !C  and consider the Cauchy problem z 0 D f .t; z.t //; z.!C / D y.!C /:

(2.5)

Since f is locally Lipschitz, then by Theorem 2.3 there exist h > 0 and z W .!C  h; !C C h/ ! Rn solution of problem (2.5). Let ( y.t /; for t 2 .! ; !C ; e y .t / D z.t /; for t 2 .!C ; !C C h/: It is clear that e y is a solution of problem (2.3) which is a contradiction with the fact that y is maximal. Hence J D Œ! ; !C /. By the same method, we can prove that J is open in ! .

76

Chapter 2 Existence theory for differential equations and inclusions

Theorem 2.8 (Cauchy–Lipschitz). Let f W RRn ! Rn be a continuous and locally Lipschitz map with respect to the second variable. Then there exists at least a maximal solution to problem (2.3). Proof. Set I D f[J j there exists a unique solution of problem .2.3/ defined on J g. From Theorem 2.3, the set I is nonempty. Since for all such J; we have t0 2 J; then I is connected in R. Moreover the solution defined by e y D yjJ

y is a solution of problem (2.3) on I

is a maximal solution on I . Now, we give a characterisation of the behaviour of maximal solution in terms of the behaviour near the right end of the time interval. This is related to extendability of local solutions. Proposition 2.9. Let us suppose that the conditions of Theorem 2.8 are satisfied. Let y be a maximal solution to (2.3) corresponding to the initial value y0 ; defined in the maximal interval Œt0 ; !C /. If !C < 1 then lim sup ky.t /k D 1. t !!C

Proof. Suppose that !C < 1 and lim sup ky.t /k  K < 1. Then for each " > 0; t !!C

there exists ı > 0 such that ky.t /k < 1;

for every t 2 Œt0 ; !C /:

Using the fact that f is a continuous function, there exists M > 0 such that kf .t; u/k  M;

for all .t; u/ 2 Œt0 ; !C   B.0; K/:

Now, given t1 ; t2 2 Œt0 ; !C /; the following estimate holds: ky.t2 /  y.t1 /k  M jt2  t1 j: By the Cauchy criterion for limits, lim y.t / WD l < 1; which permits the extension t !!C

by continuity of y to the compact interval Œt0 ; !C ; consider the following problem z 0 D f .t; z.t //;

z.!C / D l:

(2.6)

Appealing to Theorem 2.3, we obtain a solution to (2.6) defined on some interval I D .!C ; !C C h/. Then there exists y1 solution of problem (2.3), which is a continuation of yI but this is contradiction with the fact that y is a maximal solution on Œt0 ; !C /.

77

Section 2.1 Fundamental theorems

2.1.3 Peano and Carathéodory theorems 2.1.3.1 Peano theorem Theorem 2.10 (Peano theorem). Let f W R  Rn ! Rn be a continuous function. Then problem (2.3) has at least one local solution. Proof. Let " > 0; then by continuity of f there exists > 0 such that for all t 2 R; y 2 Rn , we have jt  t0 j  ; ky  y0 k  : This implies that kf .t; y/  f .t0 ; y0 /k  ": Since the set C D Œt0  ; t0 C   B.y0 ; / is compact in R  Rn , then there exists M > 0 such that kf .t; y/k  M; for all .t; y/ 2 C: Consider the Cauchy problem y 0 .t / D f .t; y.t //;

t 2 J WD Œt0  h; t0 C h; y.t0 / D y0 :

(2.7)

  . It is clear that every solution of problem (2.7) is a fixed point where h  min ; M of the operator N W C.J; B.y0 ; // ! C.J; B.y0 ; // defined by Z

t

.Ny/.t / D y0 C

f .s; y.s//ds;

t 2 J;

t0

and conversely. Let the bounded set K D fy 2 C.J; B.y0 ; // j ky  y0 k1  g: We first prove that N.K/  K. Given y 2 K, we have Z t    f .s; y.s//ds  k.Ny/.t /  y0 k D  Z

t0 t

kf .s; y.s//kds

 t0

 M jt  t0 j: Thus, kN.y/  y0 k1  M h  : We shall use the Schauder fixed point theorem to prove that N has at least one fixed point.

78

Chapter 2 Existence theory for differential equations and inclusions

(i) N is continuous. Let fyng be a sequence such that yn ! y in C.J; B.y0 ; //. Then Z t kf .s; yn .s//  f .s; y.s//kds: k.Nyn /.t /  .Ny/.t /k  t0

Since f is a continuous function, then we have kN.yn /  N.y/k1  kf .; yn .//  f .; y.//k1 ! 0; as n ! 1: (ii) N.K/ is equicontinuous in C.J; B.y0 ; //. Let l1 ; l2 2 J; l1 < l2 and y 2 K. Then for each t 2 Œt0  h; t0 C h; we have Z l2 kf .s; y.s//kds  M jl2  l1 j: k.Ny/.l2 /  .Ny/.l1 /k  l1

We can see that k.Ny/.l2 /  .Ny/.l1 /k tends to zero independently of y 2 K, as l2  l1 ! 0. Since K is a bounded set in C.J; B.y0 ; //, as a consequence of the Arzelá– Ascoli theorem we can conclude that N W K ! K is continuous and completely continuous. By the Schauder fixed point theorem, we deduce that N has a fixed point y in B.y0 ; / which is a local solution to problem (2.3). In fact, we can even prove a more precise result of local solutions. Theorem 2.11 (Carathéodory theorem). Let f W R  Rn ! Rn be a Carathéodory function. Assume that for every compact K D Œa; b  C  R  Rn , there exists p 2 L1 .Œa; b; RC / such that kf .t; x/k  pK .t /;

for all x 2 C and almost all t 2 Œa; b:

Then the Cauchy problem (2.3) has at least one local solution. Proof. Let h; r > 0 and .t0 ; y0 / 2 K D Œt0  h; t0 C h  B.y0 ; r /. Then there exists pK 2 L1 .Œt0  h; t0 C h; RC / such that and for almost all t 2 Œt0  h; t0 C h:

kf .t; x/k  p.t /; for all x 2 B.y0 ; r /;

Consider the function m W Œt0  h; t0 C h ! RC defined by Z t p.s/ds: m.t / D t0

It is clear that m is a continuous function; then for given " > 0; there exists > 0 such Rt that for every t 2 Œt0  h; t0 C h, jt0  t j  implies that t0 p.s/ds < ". Hence, Z

t0 C

p.s/ds < ": t0

79

Section 2.2 The extendability problem

Let J D Œt0  h ; t0 C h , where h  min.h; / and W D fy 2 C.J; B.y0 ; r // j ky  y0 k1  "g; where r D min.r; "/. Finally, consider the operator N W W ! W defined by Z t f .s; y.s//ds; t 2 J: .Ny/.t / D y0 C t0

For y 2 W; we have Z

Z

t

k.Ny/.t /  y0 k 

kf .s; y.s//kds 

t0 C

p.s/ds

t0

t0

Hence, kN.y/  y0 k1  "; that is N.W /  W . Using the same reasoning used in Theorem 2.10, we obtain that the operator N has at least one fixed point which is a local solution of problem (2.3).

2.2 The extendability problem 2.2.1 Global existence theorems First we need a result known as Gronwall–Bihari theorem. Lemma 2.12 ([48]). Let I D Œa; b and u; gW I ! R be positive real continuous functions. Assume there exist c > 0 and a continuous nondecreasing function hW R ! .0; C1/ such that Z t g.s/h.u.s// ds; 8 t 2 I: u.t /  c C a

Then u.t /  H

1

Z

t

 g.s/ ds ;

8t 2I

a

provided

Z

C1 c

dy > h.y/

Z

b

g.s/ ds: a

Here H 1 refers to the inverse of the function H.u/ D

Ru

dy c h.y/ ;

Definition 2.13. (a) A map f W R  Rn ! Rn is said to be Carathéodory if (i) t 7! f .t; x/ is measurable for each x 2 Rn I (ii) x  7 ! f .t; x/ is continuous for almost all t 2 RI

for u  c.

80

Chapter 2 Existence theory for differential equations and inclusions

(b) if further, for each q > 0; there exists hq 2 L1loc .R; RC / such that kf .t; x/k  hq .t /; for all kxk  q and for almost all t 2 R then f is said L1 -Carathéodory. (c) f is L1loc -Carathéodory if there exists h 2 L1 .R; RC / such that kf .t; x/k  h.t /; for all x 2 Rn and for almost all t 2 R: We now present a global existence result under a nonlinearity f satisfying a Nagumo type growth condition. Theorem 2.14. Assume that (1) f W J  Rn ! Rn is a Carathéodory function where J D Œt0 ; b. (2) There exist a function p 2 L1 .Œt0 ; b; RC / and a continuous nondecreasing function W Œ0; 1/ ! .0; 1/ such that kf .t; x/k  p.t / .kxk/; with

Z

for a.e. t 2 Œt0 ; b and each x 2 Rn Z

b

1

p.s/ds < ky0 k

t0

du  .u/

Then problem (2.3) has at least one solution defined on Œt0 ; b. Remark 2.15. This theorem shows that if f satisfies the Lipschitz condition and t 7! kf .t; 0/k is integrable, then problem (2.3) has a solution but, in contrast to Theorem 2.4, we have no uniqueness result. The proof uses the nonlinear alternative of Leray and Schauder (see [Corollary 1.18, Chapter 1]): Proof. Define the map N W C.J; Rn / ! C.J; Rn / by Z t f .s; y.s//ds; t 2 Œt0 ; b: N.y/.t / D y0 C t0

Clearly, the fixed points of N are solutions to (2.3). In order to apply [Corollary 1.74, Chapter 1], we first show that N is completely continuous. The proof will be given in several steps. Step 1. N maps bounded sets into bounded sets in C.J; Rn /. Indeed, it is enough to show that there exists a positive constant ` such that for each y 2 Bq D fy 2 C.J; Rn / j kyk1  qg one has kN.y/k1  `. Here kyk1 D

81

Section 2.2 The extendability problem

sup ky.t /k refers to the sup-norm in C.J; Rn /. Let y 2 Bq , then by 2.14 (2) for each t 2J

t 2 J; we have Z

Z

t

kN.y/.t /k  ky0 k C

kf .s; y.s//kds  ky0 k C

t

.q/

t0

p.s/ds: t0

Thus,

Z kN.y/k1  ky0 k C

b

p.s/ds WD `:

.q/ t0

Step 2. N maps bounded sets into equicontinuous sets of C.J; Rn /. Let r1 ; r2 2 Œt0 ; b; r1 < r2 and Bq D fy 2 C.Œt0 ; b; Rn /jkyk1  qg be a bounded set of C.J; Rn /. For y 2 Bq , we have Z r2 p.s/ds: kN.y/.r2 /  N.y/.r1 /k  .q/ r1

As r2  r1 ! 0, the right-hand side of the above inequality tends to zero for p 2 L1 . Step 3. N W C.J; Rn / ! C.J; Rn / is continuous. Let fyn g be a sequence such that yn ! y in C.J; Rn /. Then there is an integer q such that kyn k1  q for all n 2 N and kyk1  q; hence yn 2 Bq and y 2 Bq . By the Lebesgue dominated convergence theorem, we get Z

b

kN.yn /  N.y/k1 

kf .s; yn .s//  f .s; y.s//kds ! 0; as n ! 1: t0

Thus N is continuous. Step 4. A priori estimates. Let y 2 C.J; Rn / be a solution of y D N.y/ for some  2 .0; 1/. We have Z t p.s/ .ky.s/k/ds: ky.t /k  ky0 k C t0

By the Gronwall–Bihari inequality (Lemma 2.12), we can easily prove that there exists M > 0 such that kyk1  M: Set U D fy 2 C.J; Rn /j kyk1 < M C 1g: As a consequence of Steps 1 to 4 together with the Ascoli–Arzelá theorem, we conclude that the map N W U ! C.J; Rn / is compact. From the choice of U there is no y 2 @U such that y D Ny for any  2 .0; 1/. As a consequence of [Corollary 1.74, Chapter 1], we deduce that N has a fixed point y 2 U ; solution of (2.3) on J .

82

Chapter 2 Existence theory for differential equations and inclusions

2.2.2 Existence results on noncompact intervals 2.2.2.1 The Lipschitz case Theorem 2.16. Let the function f W Œt0 ; 1/Rn ! Rn be L1 -Carathéodory. Assume that there exists p 2 L1 .Œt0 ; 1/; RC / such that kf .t; x/  f .t; y/k  p.t /kx  yk;

for all x; y 2 Rn and a.e. t 2 Œt0 ; C1/:

Then problem (2.3) has a unique global solution on Œt0 ; 1/ Proof. It is clear that all solutions of problem (2.3) are fixed points of the operator N W C.Œt0 ; 1/; Rn / ! C.Œt0 ; 1/; Rn / defined by Z t .Ny/.t / D y0 C f .s; y.s//ds; t 2 Œt0 ; 1/: t0

Set

Z t n n   o o n p.s/ds ; t 2 Œt0 ; 1/ < 1 : Cp D y 2 C.Œt0 ; 1/; R /j j sup ky.t /k exp   t0

Equipped with the norm

Z t n   o p.s/ds W t 2 Œt0 ; 1/ : kykp D sup ky.t /k exp   t0

Cp is a Banach space. Choose  > jpj1 . Then for y1 ; y2 2 Cp ; we have Z t kf .s; y1 .s//  f .s; y2 .s//kds k.Ny1 /.t /  .Ny1 /.t /k  Z

t0 t



p.s/ky1 .s/  y2 .s/kds t0 Z t

D

 Z p.s/ exp 

t0

s

  Z p.r /dr exp 

t0

ky1 .s/  y2 .s/kds Z t  Z p.s/ exp   ky1  y2 kp t0



s

p.r /dr t0

s

 p.r /dr ds

t0

  Z t jpj1 p.s/ds ky1  y2 kp :  exp   t0 Then

jpj1 ky1  y2 kp :  As a consequence of the Banach fixed point theorem, the operator N has a unique fixed point, solution of problem (2.3) and defined on Œt0 ; 1/. kN.y1 /  N.y2 /kp 

83

Section 2.2 The extendability problem

2.2.2.2 The Lipschitz–Nagumo case In order to prove existence and uniqueness of a solution to problem (2.3), we are going to appeal to a fixed point theorem for contraction mappings on Fréchet spaces. To start with, we recall some basic facts. Let X be a Fréchet space with a family of semi-norms fk : kn; n 2 Ng. Given Y  X, we say that Y is bounded if for every n 2 N; there exists Mn > 0 such that kykn  Mn ;

for all y 2 Y:

To the Fréchet space X, we associate a sequence of Banach spaces f.X n; k:kn /g as follows. For every n 2 N; consider the equivalence relation n defined by x n y if and only if kx ykn D 0. We denote by X n D .X= n; k : kn / the quotient space, the completion of X n with respect to k : kn. To every Y  X; we associate a sequence of subsets Y n  X n as follows. For every x 2 X; we denote Œxn the equivalence class n of x of subset X n , and we define Y n D fŒxn j x 2 Y g. We denote Y ; intn .Y n / and @n Y n ; respectively, the closure, the interior and the boundary of Y n with respect to k : kn in X n . We assume that the family of semi-norms fk : kng satisfies kxk1  kxk2  kxk3     ;

for every x 2 X:

Definition 2.17. A function f W X ! X is said to be a contraction if for each n 2 N, there exists kn 2 .0; 1/ such that kf .x/  f .y/kn  kn kx  ykn;

for all x; y 2 X:

Lemma 2.18 (Nonlinear alternative, [185]). Let X be a Fréchet space, Y  X be a closed subset, and let N W Y ! X be a contraction such that N.Y / is bounded. Then one of the following statements holds: (1) N has a unique fixed point; (2) there exist  2 Œ0; 1/; n 2 N; and x 2 @nY n such that kx  N.x/kn D 0. Theorem 2.19 (The Nagumo–Lipschitz case). Assume that f W Œt0 ; 1/  Rn ! Rn is an L1loc -Carathéodory function which satisfies the following assumptions (1) There exist a continuous nondecreasing function p 2 L1loc .Œ0; 1/; RC / such that

for a.e. t 2 Œt0 ; 1/ and each u 2 Rn

kf .t; u/k  p.t / .kuk/; with

Z

W Œ0; 1/ ! .0; 1/ and

1

ky0 k

ds D 1: .s/

84

Chapter 2 Existence theory for differential equations and inclusions

(2) For all R > 0, there exists lR 2 L1loc .Œt0 ; 1/; RC / such that kf .t; u/  f .t; u/k  lR .t /ku  uk;

for a.e. t  t0

and all u; u 2 Rn with kuk; kuk  R. Then problem (2.3) has a unique solution defined on Œt0 ; 1/. Remark 2.20. Clearly, if the mapping t 7! kf .t; 0/k is L1loc , then 2.19 (2) implies 2.19 (1) and also the main hypotheses in Theorem 2.16. However, notice that Theorem 2.19 also provides uniqueness. Proof. The proof will be given in several steps. In order to transform problem (2.3) into a fixed point problem, consider the operator N W C.Œt0 ; 1/; Rn / ! C.Œt0 ; 1/; Rn / defined by: Z

t

N.y/.t / D y0 C

f .s; y.s//ds;

t 2 Œt0 ; 1/:

t0

Let y be a solution of problem (2.3). Then for t 2 Œt0 ; n WD Jn , n 2 N; n > t0 , we have Z t f .s; y.s//ds: y.t / D y0 C t0

This implies by 2.19 (2) that for each t 2 Jn ; we have Z

t

ky.t /k  ky0 k C

p.s/ .ky.s/k/ds: t0

From the Gronwall–Bihari inequality (Lemma 2.12), we get Z n  1 p.s/ds ; 8 t 2 Jn ; ky.t /k  t0

Z

where

.u/ D In particular, kykn 

1

Z

u ky0 k

n

ds : .s/

 p.s/ds WD Mn ;

t0

where

kykn D supfe  Pn .t /ky.t /k W t0  t  ng; Z t pn .s/ds Pn .t / D t0

85

Section 2.2 The extendability problem

T n and ln is the function from 2.19 (2). Then C.Œt0 ; 1/; Rn / D 1 nD1 C.Œt0 ; n; R / is a Fréchet space with the family of semi-norms fk  kn j n 2 N; n  t0 g. Let Y D fy 2 C.Œt0 ; 1/; Rn / j kykn  Mn C 1; for all n 2 N; n > t0 g: Clearly, Y is a closed subset of C.Œt0 ; 1/; Rn /. We shall show that N is a contraction operator on Y . Indeed, consider y; y 2 Y . Then for each t 2 Œt0 ; n and n 2 N, we have the estimates Z t kf .s; y.s//  f .s; y.s//k ds kN.y/.t /  N.y/.t /k  Z 

t0 t

t0 Z t

 Z

t0 t



p.s/ky.s/  y.s/kds pn .s/e Pn .s/ e  Pn .s/ ky.s/  y.s/kds pn .s/e Pn .s/ dsky  ykn

t0 Z t

1 Pn .s/ 0 .e / dsky  ykn t0  1  e Pn.t / ky  ykn:  

Thus, e  Pn .t / kN.y/.t /  N.y/.t /k 

1 ky  ykn : 

Therefore, 1 ky  ykn ; 8 y; y 2 Y;  showing that for  > 1; the operator N is a contraction for all n 2 N. From the choice of Y there is no y 2 @Y n such that y D N.y/ for some  2 .0; 1/. As a consequence of the nonlinear alternative type (Lemma 2.18), we deduce that N has a unique fixed point, solution of problem (2.3). kN.y/  N.y/kn 

The following example shows that the condition (2.19) is important for the global existence: Example 2.21. Consider the following Cauchy problem y 0 D 1 C y 2;

y.0/ D 0:

(2.8)

It is clear that for every R > 0, there exists MR > 0 such that jf .t; x/  f .t; y/j  M jx  yj;

for each x; y 2 R;

where f .t; u/ D 1 Cu2 and problem (2.8) has a unique maximal solution y.t / D  arctan.t /; t 2  2 ; 2 but has no global solution.

86

Chapter 2 Existence theory for differential equations and inclusions

Now, we show that, with only a Nagumo–Bernstein growth condition, we can still have existence but not uniqueness. 2.2.2.3 The Nagumo case Let Cb be the Banach space of all bounded continuous real-valued functions on the interval Œ0; 1/; endowed with the sup-norm k  k1 defined by kyk1 D

sup ky.t /k; t 2Œ0;1/

for y 2 Cb :

We present an existence result for problem (2.3) on J D Œ0; 1/. Theorem 2.22 (The Nagumo case). Suppose the following hypotheses hold: (1) The function f W J  Rn ! Rn is a Carathéodory function. (2) There exist p 2 L1 .Œ0; 1/; RC / and a continuous nondecreasing function Œ0; 1/ ! Œ0; 1/ such that for a.e. t 2 J and each x 2 Rn

W

kf .t; x/k  p.t / .kxk/ with

Z

1

ds D 1: .s/

ky0 k

Then problem (2.3) has at least one solution in Cb .Œ0; 1/; Rn /. Proof. Consider the operator N W Cb ! Cb defined by Z

t

.Ny/.t / D y0 C

f .s; y.s//ds;

t 2 J:

0

In order to apply the Leray–Schauder nonlinear alternative ([Corollary 1.74, Chapter 1]), we first show that N is completely continuous. Step 1. N maps bounded sets into bounded sets in Cb . For this, it is enough to show that there exists a positive constant ` such that for each y 2 Bq D fy 2 Cb jkyk1  qg one has kN.y/k1  `. Let y 2 Bq ; then for each t 2 Œ0; 1/; we have Z

t

N.y/.t / D y0 C

f .s; y.s//ds; 0

87

Section 2.2 The extendability problem

By 2.22 (2), for each t 2 J , we have Z

 k.Ny/.t /k  y0 C

   y0  C

 f .s; y.s//ds 

t 0

Z

1

p.s/ .ky.s/k/ds Z 1 p.s/ds: .kyk1/

0

   y0  C

0

Thus N is well defined. In addition, for y 2 Bq , we have Z

t

.Ny/.t / D y0 C

f .s; y.s//ds;

t 2 J:

0

Then

Z

1

kN.y/.t /k  ky0 k C

.ky.s/k/ds  ky0 k C

t

.q/kpkL1 :

Hence, kN.y/k1  ky0 k C

.q/kpkL1 WD `:

Step 2. N.Bq / is equicontinuous in Cb . Let r1 ; r2 2 J; r1 < r2 < 1. Then Z kN.y/.r2 /  N.y/.r1 /k 

r2

.q/

p.s/ds: r1

As r2  r1 ! 0, the right-hand side of the above inequality tends to zero. Step 3. N.Bq / is equiconvergent at 1. Let y 2 N.Bq /; then Z

t

.Ny/.t / D y0 C

f .s; y.s//ds 0

implies that

Z

1

lim N.y/.t / D y0 C

t !1

f .s; y.s//ds: 0

Using the condition 2.22 (1), for every " > 0; there exists T > 0 such that for every t  T; we have Z 1 .q/ p.s/ds  " ) k.Ny/.t /  lim .Ny/.r /k  ": T

Hence, N.Bq / is equiconvergent in Cb .

r !1

88

Chapter 2 Existence theory for differential equations and inclusions

Step 4. N is continuous. Let fyng be a sequence such that yn ! y in C . Then there is an integer q such that kyn k1  q for all n 2 N and kyk1  q. Since f is a Carathéodory function, the sequence f .t; yn .t // converges to f .t; y.t //. Moreover, 2.22 (1) guarantees that kf .t; yn .t //k  p.t / .q/;

for every t  0 and for all n 2 N:

By the dominated convergence theorem, we deduce Z 1 kf .s; yn .s//  f .s; y.s//kds ! 0; kN.yn /  N.y/k1 

as n ! C1:

0

Thus, N is continuous. Step 5. A priori estimates. Let y 2 Cb be a solution of y D N.y/ for some  2 .0; 1. Then Z

t

ky.t /k  ky0 k C

p.s/ .ky.s/k/ds: 0

By the Gronwall–Bihari inequality (Lemma 2.12), we get the bound kyk1  1 .kpkL1 / WD M; Z

where

.z/ D

z ky0 k

du : .u/

If we let U D fy 2 Cb j kyk1 < M C 1g; then there is no y 2 @U such that y D N.y/ for some  2 .0; 1/. Since N W U ! Cb is compact, the nonlinear alternative of Leray–Schauder ([Corollary 1.74, Chapter 1]) implies that N has a fixed point y in U , solution of problem (2.3) on Œ0; 1/.

2.2.3 A boundary value problem on the half-line The history of boundary value problems (BVPs for short) on infinite intervals starts at the end of the last century with the pioneering work of A. Kneser [293] about monotone solutions and their derivatives on Œ0; 1/ for second-order ordinary differential equations (ODEs). The Kneser type works were followed by A. Mambriani [328] in 1929 and others from the beginning of the fifties until now (see, e.g., [53, 232, 244, 282, 406, 407, 423, 437, 493] and the references therein). At the beginning of the fifties, the study of bounded solutions via BVPs was initiated by C. Corduneanu [124, 125] who considered second-order BVPs on the positive ray as well as

89

Section 2.2 The extendability problem

on the whole real line. Since the sixties, similar problems have been studied, using mostly the lower and upper solutions technique (see, e.g., [25, 53, 56, 178, 419–421]). Since the beginning of the seventies, BVPs on infinite intervals have been studied extensively, and we quote at least four powerful techniques. The first approach, called the sequential method, consists in investigating the limit process for the family of BVPs on infinitely increasing compact intervals. Then the associated function spaces for the related fixed point problems are Banach spaces. This idea has been elaborated in [297–299] for problems on the whole line. For some applications and further interesting results, we refer the reader to [1, 11–16, 23, 25, 322, 400, 403]. If, however, we work directly on the noncompact intervals, then the associated function spaces for the fixed point problems are not Banach, but rather Fréchet spaces, which raises some difficulties (see [122, 325, 335] and the references therein). Moreover, this approach can bring very strong results (see, e.g., [29,108,109,111–114,188,189,250,279,280,425]). Recently, the Conley index approach has been alternatively applied for the same goal, mainly by J. R. Ward, Jr. (see [341, 481–486] and R. Srzednicki [437, 438]) where the link with the Lefschetz index has been employed. Another remarkable recent approach consists in the application of the so-called A-mapping theory (the A-class means the approximation admissible maps); for details and some results, we refer the reader to [304, 402]. In addition to the studies of BVPs for ODEs in Euclidean spaces, there are also some contributions to the study of ODEs in some function spaces (Banach spaces, Hilbert spaces, etc.); see, e.g., [133, 135, 278, 404, 414, 504]. Further generalisations are related to functional problems (see [445, 446]) and especially those for differential inclusions (see, e.g., [30, 114, 400, 422, 504]). Consider the second-order boundary value problem (BVP, for short) on the half-line. (

y 00 D f .t; y.t /; y 0 .t //;

t 2 J WD Œ0; 1/ (2.9)

0

y.0/ D a; lim y .t / D ; t !1

where f W Œ0; 1/  Rn  Rn ! Rn is a continuous function and a; 2 Rn . Notice that if y has a limit in R [ f1g, then the condition lim y 0 .t / D implies that !1

y 0 .t / D : t !C1 t lim

Definition 2.23. A function y 2 C 2 .J; Rn / (or y 2 AC 1 .J; Rn /) is called solution of problem (2.9) if y satisfies y 00 .t / D f .t; y.t /; y 0 .t //; t 2 Œ0; 1/; y.0/ D a; and lim!1 y 0 .t / D . The following proposition provides a useful integral formulation of problem (2.9), which will be used in the main result of the subsection.

90

Chapter 2 Existence theory for differential equations and inclusions

Proposition 2.24. Let f W Œ0; 1/  Rn  Rn ! Rn be a continuous function. A function y 2 C 1 .Œ0; 1/; Rn / is a solution of problem (2.9) if and only if Z t Z 1 0 y.t / D aC t C sf .s; y.s/; y .s//dsC tf .s; y.s/; y 0 .s//ds; t 2 J: (2.10) 0

t

Proof. Let y be a function in C 1 .Œ0; 1/; Rn /, solution of problem (2.10). Then Z 1 y 0 .t / D C f .s; y.s/; y 0 .s//ds; for every t 2 J; t

which yields lim y 0 .t / D . Then the second condition of (2.9) is fulfilled. Moreover, t !1

y.0/ D a and

y 00 .t / D f .t; y.t /; y 0 .t //;

for all t 2 J;

which means that y is a solution on Œ0; 1/ of (2.9). Conversely, let us suppose that y is a solution of problem (2.9). Then Z t .t  s/f .s; y.s/; y 0 .s//ds; t 2 J y.t / D a C y 0 .0/t  0

and 0

Z

0

t

y .t / D y .0/ 

f .s; y.s/; y 0 .s//ds;

t 2 J:

0

Using the fact that lim y 0 .t / D ; we get t !1

Z

0

1

y .0/ D C

f .s; y.s/; y 0 .s//ds:

0

Hence,

Z

t

y.t / D a C t C

sf .s; y.s/; y 0 .s//ds C

0

Z

1

tf .s; y.s/; y 0 .s//ds;

t 2 J:

t

The proof of the proposition is complete. Now, we present an existence result for problem (2.9). Theorem 2.25. Assume that (1) there exists a non-negative continuous real-valued function W Œ0; 1/  RC  RC ! Œ0; 1/ and for every t  0 .t; :; :/ is increasing on Rn  Rn such that for all .t; x; y/ 2 Œ0; 1/  .Rn /2 kf .t; x; y/k  with

Z

1 0

.t; kxk; kyk/

.s; kxk; kyk/ds < 1:

91

Section 2.2 The extendability problem

(2) There exists a real number c > k k such that Z 1 .s; sc C kak; c/ds  c  k k: 0

Then problem (2.9) has at least one solution y such that .c  kak C 2k k/t  ky 0 .t /k  c;

for every t  0:

Proof. Let Cb1 D fy 2 C 1 .Œ0; 1/; Rn / j sup ky 0 .s/k < 1g. .Cb1 ; k  k1 / is a Banach s0

space endowed with the norm defined by kyk1 D max.ky.0/k; ky 0 k1 / with ky 0 k1 D sup ky 0 .s/k. Let s0

ky 0 .t /k  c;

C D fy 2 Cb1 j y.0/ D a;

for all t  0g:

It is clear that C is a closed convex bounded subset of Cb1 . Consider the operator N W Cb1 ! Cb1 defined by Z 1 Z t sf .s; y.s/; y 0 .s//ds C tf .s; y.s/; y 0 .s//ds; t 2 J: .Ny/.t / D a C t C 0

t

In order to apply the Schauder fixed point theorem, we first show that N is completely continuous. The proof will be given in several steps. Step 1. N.C / is a bounded set in Cb1 . It is enough to show that there exists a positive constant ` such that for each y 2 C , one has kN.y/k1  `. Let y 2 C , then for each t 2 Œ0; 1/, we have Z 1 Z t 0 sf .s; y.s/; y .s//ds C tf .s; y.s/; y 0 .s//ds: N.y/.t / D a C t C 0

t

Then .Ny/.0/ D a and .Ny/0 .t / D C

Z

1

f .s; y.s/; y 0 .s//ds:

t

By 2.25 (1), we have for each t 2 J  k.Ny/ .t /k   C 0

Z

     C      C

1 0

Z

Z

1 0

0

Thus, N is well defined.

 f .s; y.s/; y 0 .s//ds 

1

.s; ky.s/k; ky 0 .s/k/ds .s; sc C kak; c/ds:

(2.11)

92

Chapter 2 Existence theory for differential equations and inclusions

Step 2. N.C / is equicontinuous in Cb1 . For some r1 ; r2 2 J; r1 < r2 < 1; we have Z r2 kf .s; y.s/; y 0 .s//kds kN.y/.r2 /  N.y/.r1 /k  k kjr1  r2 j C r Z 1 1 kf .s; y.s/; y 0 .s//kds C jr2  r1 j r2 Z r2 C r1 kf .s; y.s/; y 0 .s//kds r1 Z r2  k kjr1  r2 j C .s; sc C kak; c/ds r Z 1 1 C jr2  r1 j .s; sc C kak; c/ds r2 Z r2 C r1 .s; sc C kak; c/ds r1

and 0

Z

0

r2

k.Ny/ .r2 /  .Ny/ .r1 /k 

.s; sc C kak; c/ds:

r1

As jr2  r1 j ! 0, the right-hand side of the above inequality tends to zero. Step 3. N.C / is equiconvergent at 1. Given y 2 N.C /; we have Z

0

.Ny/ .t / D C

1

f .s; y.s/; y 0 .s//ds:

t

Hence, lim .Ny/0 .t / D . Using the condition 2.25 (1), for every " > 0, there exists t !1

T > 0 such that for every t  T; we have Z

1

.s; sc C kak; s/ds  ":

T

Then k.Ny/0 .t /  lim .Ny/0 .r /k  " and .N 0 .C // is equiconvergent in Cb1 . r !1

Step 4. N is continuous. Let fyn g be a sequence such that yn ! y in C . Then kyk1  q; .Nyn /.0/ D .Ny/.0/ D a and there is an integer q such that kyn0 k1  q for all n 2 N. Since yn0 converges uniformly to y 0 , then Z lim

n!1 0

t

yn0 .s/ds

Z

t

D 0

y 0 .s/ds:

93

Section 2.2 The extendability problem

Hence, lim yn .t / D y.t /. Using the fact that f is a continuous function, the sen!1

quence f .t; yn .t /; yn0 .t // converges to f .t; y.t /; y 0 .t // Moreover, 2.25 (1) guarantees that kf .t; yn .t /; yn0 .t //k 

.t; ct C kak; c/;

for every t  0 and for all n 2 N:

By the dominated convergence theorem, we conclude that kN 0 .yn /  N 0 .y/k1 

Z

1 0

kf .s; yn .s/; yn0 .s//  f .s; y.s/; y 0 .s//kds ! 0;

as n ! 1. Thus, N is continuous. As a consequence of Steps 1 to 4 together with the compactness criterion ([Corollary A.26, Appendix]), we deduce that N.C / is relatively compact in Cb1 . Step 5. N.C /  C . Given y 2 C; we have from (2.11) Z

0

1

k.Ny/ .t /  k 

.s; ky.s/k; c/ds:

t

Since ky 0 .t /k  c ) ky.t /k  ct C kak; we obtain by 2.25 (2) 0

Z

1

k.Ny/ .t /  k 

.s; cs C kak; c/ds  c C k k  k k:

t

Hence, k.Ny/0 .t /k  c and N.C /  C . From the Schauder fixed point theorem, the mapping N has a fixed point y 2 C which is solution of (2.3) on Œ0; 1/. Moreover, for every t 2 Œ0; 1/, we have ky 0 .t /  k  c C kak  k k: Then .c  kak C 2k k/  ky 0 .t /k  c C kak and .c  kak C 2k k/t C kak  ky.t /k  .c C kak/t C kak:

94

Chapter 2 Existence theory for differential equations and inclusions

2.3 The case of differential inclusions 2.3.1 Initial value problems Consider the first-order differential inclusion:  0 y 2 F .t; y.t //; a.e . t 2 J WD Œt0 ; b; y.0/ D y0 ;

(2.12)

where F W J  Rn ! P.Rn / is a multi-valued map and y0 2 Rn . Let us start by defining what we mean by a solution to problem (2.12). Definition 2.26. A function y 2 AC.Œt0 ; b; Rn / is said to be a solution of (2.12) if y satisfies the differential inclusion y 0 .t / 2 F .t; y.t // a.e. on J and the condition y.t0 / D y0 . 2.3.1.1 A Nagumo type nonlinearity The first result of this section is concerned with the existence of solutions to problem (2.12) when the nonlinearity satisfies some growth condition including a Lipschitz one. Lemma 2.27. Let .X; k  k/ be a normed space and F W X ! Pcl;cv .X/ a compact, u.s.c. multi-valued map. Then either one of the following conditions holds: (a) F has at least one fixed point, (b) the set M WD fx 2 X; x 2 F .x/;  2 .0; 1/g is unbounded. Theorem 2.28. Suppose that (1) F W J  Rn ! Pcp;cv .Rn / is a Carathéodory multi-function. (2) There exist a continuous nondecreasing function W Œt0 ; 1/ ! .0; 1/ and p 2 L1 .J; RC / such that for a.e. t 2 J and each u 2 Rn kF .t; u/kP WD supfkvk j v 2 F .t; u/g  p.t / .kuk/ with

Z

Z

b

1

p.s/ds < t0

Then problem (2.12) has at least one solution.

ky0 k

du  .u/

95

Section 2.3 The case of differential inclusions

Proof. Consider the multi-valued map N W C.J; Rn / ! P.C.J; Rn // defined by: Z t o n g.s/ds; t 2 Œt0 ; b ; (2.13) N.y/ WD h 2 C.J; Rn / j h.t / D y0 C t0

where

n g 2 SF ;y D g 2 L1 .J; Rn / j g.t / 2 F .t; y.t //;

o for a.e. t 2 J :

(2.14)

Since F is a Carathéodory function, the set SF ;y is nonempty (see [478]). We shall show that N is a completely continuous multi-valued map, u.s.c. with convex closed values. The proof will be given in several steps. First, notice that N.y/ is convex for each y 2 C.J; Rn /. This follows from the convexity of SF ;y , since F has convex values. Step 1. N maps bounded sets into bounded sets in C.J; Rn /. Indeed, it is enough to show that there exists a positive constant ` such that for each h 2 N.y/; y 2 Bq D fy 2 C.J; Rn / j kyk1  qg one has khk1  `. If h 2 N.y/, then there exists g 2 SF ;y such that for each t 2 J; we have Z t g.s/ds: h.t / D y0 C t0

By 2.28 (2), we have for each t 2 J Z t kh.t /k  ky0 k C kg.s/kds  ky0 k C

Z

t

.q/

t0

p.s/ds: t0

Then for each h 2 N.Bq /; we have Z khk1  ky0 k C

b

p.s/ds WD `:

.q/ 0

Step 2. N maps bounded sets into equicontinuous sets of C.J; Rn /. Let 1 ; 2 2 J; 1 < 2 and Bq D fy 2 C.J; Rn / j kyk1  qg be a bounded set of C.J; Rn /. For each y 2 Bq and h 2 N.y/, there exists g 2 SF ;y such that Z t g.s/ds; t 2 J: h.t / D y0 C t0

Thus,

Z kh.2 /  h.1 /k 

2

1

Z kg.s/kds 

2

.q/

p.s/ds:

1

As j2  1 j ! 0, the right-hand side of the above inequality tends to zero. As a consequence of Step 1 and 2 together with the Ascoli–Arzéla theorem, we conclude

96

Chapter 2 Existence theory for differential equations and inclusions

that N W C.J; Rn / ! Pcp;cv .C.J; Rn // is a completely continuous multi-valued map. Step 3. N is u.s.c. Since N is completely continuous, we prove that N has a closed graph. Let hn 2 N.yn / be such that yn ! y and hn ! h . We shall prove that h 2 N.y /. hn 2 N.yn / means that there exists gn 2 SF ;yn such that Z

t

hn .t / D y0 C

gn .s/ds; t 2 J: t0

Consider the linear continuous operator

W L1 .J; Rn / ! C.J; Rn / Z t g.s/ds: g 7! .g/.t / D t0

Since hn .:/  y0 2 .SF ;yn /;

ıSF has, by Lemma 6.155, a closed graph, then .y ; h y0 / 2 Gr . ıSF /. Hence, there exists g 2 SF ;y such that Z

t

h .t / D y0 C

g .s/ds;

t2J

(2.15)

t0

for some g 2 SF ;y ; proving our claim. Steps 1 and 2 show that N is locally compact; hence, N is u.s.c. Step 4. A priori estimates. Let y 2 N.y/ for some  2 Œ0; 1I then there exists g 2 SF ;y such that Z

t

y.t / D y0 C 

g.s/ds: 0

From 2.28 (2), we get Z

t

ky.t /k  v.t / WD ky0 k C

p.s/ .ky.s/k/ds;

t 2 J:

t0

Then

v 0 .t / D p.t / .ky.t /k/;

t 2 J and v.t0 / D ky0 k:

Using the nondecreasing character of ; we find that v 0 .t /  p.t / .v.t //;

t 2 J:

97

Section 2.3 The case of differential inclusions

Integrating over t 2 Œt0 ; b; we get Z

v.t / v.t0 /

du  .u/

Z

b

p.s/ds;

8 t 2 Œt0 ; b:

0

In view of 2.28 (2), we finally obtain the bound sup jy.t /j  v.t /  sup t 2Œt0 ;b

t 2Œt0 ;b

11

Z

b

 p.s/ds WD M:

t0

If U D fy 2 C.J; Rn / j kyk1 < M C 1g; then there is no y 2 @U such that y 2 N.y/ for some  2 .0; 1/. The nonlinear alternative of Leray–Schauder (Theorem 1.73, Chapter 1) implies that N has a fixed point y in U solution of (2.12). 2.3.1.2 A Lipschitz nonconvex nonlinearity Theorem 2.29. Assume that (1) F W Œt0 ; b  Rn ! Pcp .Rn / has the property that F .; u/ W Œt0 ; b ! Pcp .Rn / is measurable for each u 2 Rn . (2) Hd .F .t; u/; F .t; u//  l.t /ku  uk, for each t 2 J and u; u 2 Rn where l 2 L1 .J; RC / and F .t; 0/  l.t /B.0; 1/ for a.e. t 2 J . Then problem (2.12) has at least one solution. Remark 2.30. Note that 2.29 (2) implies that F has at most linear growth. Hence 2.28 (2) is satisfied with .s/ D 1 C s. Moreover, F is u.s.c. and l.s.c. with respect to the second variable. However, Theorem 2.28 cannot be applied here since F has not necessarily convex values. Proof. We shall show that N defined by (2.13) satisfies the assumptions of Lemma 1.79. The proof will be given in two steps. Step 1. N.y/ 2 Pcl .C.J; Rn //; for each y 2 C.J; Rn /. Indeed, let .hn /n0 2 N.y/ be such that hn ! hQ in C.J; Rn /. Then hQ 2 C.J; Rn / and for each t 2 J Z t hn .t / D y0 C gn .s/ds; where gn 2 SF ;y : t0

98

Chapter 2 Existence theory for differential equations and inclusions

Using the fact that F has compact values together with 2.29 (2), we may pass to a subsequence if necessary to get that gn converges to g in L1 .J; Rn / and hence g 2 SF ;y . Then, for each t 2 J , as n ! 1 Q / D y0 C hn .t / ! h.t

Z

t

g.s/ds: 0

So hQ 2 N.y/. Step 2. We claim that there exists  < 1 such that Hd .N.y1 /; N.y2 //  ky1  y2 k1 for each y1 ; y2 2 C.J; Rn / Let y1 ; y2 2 C.J; Rn / and h1 2 N.y1 /. Then there exists g1 2 SF ;y1 such that for each t 2 Œt0 ; b Z t g1 .s/ds: h1 .t / D y0 C t0

From 2.29 (2), we know that Hd .F .t; y1 .t //; F .t; y2 .t ///  l.t /ky1 .t /  y2 .t /k;

p.p. t 2 Œt0 ; b:

Hence, there is some w.t / 2 F .t; y2 .t // such that kg1 .t /  w.t /k  l.t /ky1 .t /  y2 .t /k;

t 2 Œt0 ; b:

Now consider the multi-valued map U W Œt0 ; b ! P.Rn / defined by U.t / D fw 2 Rn j kg1 .t /  wk  l.t /ky1 .t /  y2 .t /kg: Since g1 is measurable, U is also. Hence, the multi-valued operator V .t / D U.t / \ F .t; y2 .t // is measurable (see [107]). By the Kuratowski–Ryll-Nardzewski theorem, V has a measurable selection g2 ./. Hence, g2 .t / 2 F .t; y2 .t // and kg1 .t /  g2 .t /k  l.t /ky1 .t /  y2 .t /k; Let us define for each t 2 Œt0 ; b Z

t

h2 .t / D y0 C

g2 .s/ds: t0

t 2 Œt0 ; b:

99

Section 2.3 The case of differential inclusions

We have the estimates Z

t

kh1 .t /  h2 .t /k 

kg1 .s/  g2 .s/k ds t0 Z t

 

t0 Z t

Z

t0 t



l.s/e L.s/ e  L.s/ ky1 .s/  y2 k.s/ ds l.s/e L.s/ ky1  y2 k1 ds l.s/e L.s/ ky1  y2 k1 ds

t0

Z t 0 1 e L.s/ ds  ky1  y2 k1  0 1 L.t / ky1  y2 k1 ;  e  where L.t / D

Rt 0

l.s/ds; t 2 Œt0 ; b; and kyk1 D supfe  L.t /jy.t /j j t 2 Œt0 ; bg;

 > 1:

Then kh1  h2 k1 

1 ky1  y2 k1 : 

By an analogous relation, obtained by interchanging the roles of y1 and y2 ; we arrive at 1 Hd .N.y1 /; N.y2 //  ky1  y2 k1 :  So, N is a contraction and thus, by Lemma 1.79, N has a fixed point y, solution of problem (2.12).

2.3.2 Boundary value problems Consider the two-point boundary value problem: (

x 00 2 F .t; x.t //; x.0/ D x.1/ D 0;

a.e . t 2 J WD Œt0 ; b;

(2.16)

100

Chapter 2 Existence theory for differential equations and inclusions

2.3.2.1 The convex case Theorem 2.31. Assume F W J RC ! Pcp;cv .RC / is a multi-valued L1loc -Carathéodory mapping such that 0 62 F .:; :/ and 8 There exist a continuous nondecreasing function ˆ ˆ < W Œ0; 1/ 7! .0; 1/ and p 2 L1 .J; RC / such that .H1 / kF .t; x/kP  p.t / .jxj/ for a.e. t 2 J; all x 2 R; and ˆ ˆ : R0 9 R0 > 0; .R0 /  jpj  1 Then problem (2.16) has at least one positive solution. Remark 2.32. It is obvious that any integrably bounded multi-function satisfies .H1 /. Remark 2.33. When F is an L1loc -Carathéodory multi-valued mapping, we know from a result due to Lasota and Opial [317] that for each x 2 C.J; R/; the set SF ;x is nonempty. Thus, we can define a multi-operator SF W C.J; RC / ! P.C.J; RC // x 7! SF .x/ D SF ;x : Proof. Consider the convex subset of C.J; RC /  w 2 C.J; RC /; w.0/ D 0; w is nondecreasing; and Rt : KD 0  w.t /  w.s/  .R0 / s p./d ; for all 0  s  t  1 It is clear that for every w 2 K; kwk1  R1 WD jpj1 .R0 / and K is compact by Ascoli–Arzéla lemma. Furthermore, any element w 2 K is absolutely continuous. Thus, we can define S K ! C.J; R/ such that x D S.w/ is a unique solution of the problem ( x 00 .t / D w 0 .t /; t 2 Œ0; 1 x.0/ D x.1/ D 0: Now define the multi-valued map G W C.J; RC / ! P.C.J; RC // by G.x/ D fy 2 C.J; RC /; y.t / D

Z

t

v.s/ ds for some v 2 SF ;x g: 0

e D G ı S are studied. Next, the properties of the mapping G

101

Section 2.3 The case of differential inclusions

e Step 1. G.K/  K. Let w 2 K and y 2 G.w/I then there exist x 2 C.J; R/ and v 2 SF ;x such that Z

t

v.s/ ds; t 2 Œ0; 1:

y.t / D 0

It is clear that y is a nondecreasing function and, for 0  s  t  1; Z t Z t Z t y.t /  y.s/ D v./ d   kF .; x.//kP d   p./ .jx./j/ d ; s

s

s

kxk1  R1 ; and

Z 0  y.t /  y.s/ 

t

.R1 /

p./d : s

From the definition of R0 and R1 ; it follows that .R1 / 

.R0 /;

showing that G.K/  K. e Step 2. G.w/ is convex for each w 2 K. e then there exist x 2 C.J; R/ and v1 ; v2 2 SF ;x such that, Indeed, if y1 ; y2 2 G.w/, for each t 2 Œ0; 1, we have Z t vi .s/ ds; i D 1; 2: yi .t / D 0

Let 0  ˛  1. Then for each t 2 Œ0; 1; we have Z t Œ˛v1 .s/ C .1  ˛/v2 .s/ ds: .˛y1 C .1  ˛/y2 /.t / D 0

Since SF ;x is convex (because F has convex values), we get e ˛y1 C .1  ˛/y2 2 G.w/: e maps bounded sets into bounded sets in C.J; RC /. Step 3. G Indeed, it is enough to show that there exists a positive constant ` such that for each e w 2 Br D fw 2 C.J; RC / j kwk1  r g; one has kG.w/k 1  `. Let w 2 Br and e y 2 G.w/I then there exist x 2 C.J; R/ and v 2 SF ;x such that for each t 2 J; we have Z t y.t / D v.s/ ds; t 2 Œ0; 1: (2.17) 0

102

Chapter 2 Existence theory for differential equations and inclusions

Using .H1 / and noting that is nondecreasing, we obtain that kxk1  r and then for each t 2 J Z 1 Z t jv.s/j ds  p.s/ .jx.s/j/ ds  jpj1 .r /: jy.t /j  0

0

e maps bounded sets into equicontinuous sets of C.J; RC /. Step 4. G Let Br be the ball centred at the origin and of radius r in C.J; RC /; we prove that the e family set fGw; w 2 Br g is relatively compact. As in Step 3, it is clear that this set is bounded. To check that it is equicontinuous, let t1 ; t2 2 J be such that t1 < t2 . From .H1 /, we have Z t2 p.s/ ds jy.t2 /  y.t1 /j  .r / t1

where the right-hand side tends to zero as t2  t1 ! 0. e is u.s.c. Step 5. G e has a closed graph. Let By [Lemma 6.155, Chapter 6], it suffices to prove that G e e  /. wn ! w ; yn 2 G.wn / and yn ! y as n ! 1. We claim that y 2 G.w e n / means that there exist xn 2 C.J; R/ and vn 2 SF ;xn such that Indeed, yn 2 G.w for each t 2 J; Z t

yn .t / D

vn .s/ ds;

t 2 Œ0; 1:

0

We shall prove that there exists v 2 SF ;x such that for each t 2 J Z t v .s/ ds; t 2 Œ0; 1: y .t / D 0

Consider the continuous linear operator

W L1 .J; R/ ! C.J; R/ u 7! u defined by

Z

t

. u/.t / D

u.s/ds;

t 2 Œ0; 1:

0

By [Lemma 6.155, Chapter 6], the operator ıSF has a closed graph and the definition e yields that of G yn 2 .SF ;xn / D . ı SF /.xn /: Moreover, it is easy to see that the operator S is continuous. Then the sequence .xn /n2N is convergent and so there exists an M  0 such that kxnk1  M;

8 n 2 N:

103

Section 2.3 The case of differential inclusions

Hence, jvn .t /j  p.t / .M /;

for a.e. t 2 J and all n 2 N

and vn ! v a.e. in R; as n ! C1. By the Lebesgue dominated convergence theorem, lim yn .t / D y .t /; t 2 J . Since xn ! x ; we finally deduce from the n!1

continuity of F and that y 2 .SF ;x / D . ı SF /.x /; ending our claim. e is completely continuous and hence has nonempty compact valFrom Steps 3–5, G e W K ! Pcl;cv .K/ satisfies all conditions ues. To sum up, the multi-valued map G of [Theorem 1.70, Chapter 1] and therefore has a fixed point w in K. It follows that x D S.w/ is a fixed point of N; whence a solution to problem (2.16) in S.K/. ConRt versely, if x is solution to problem (2.16), then w defined by w.t / D 0 x.s/ds is a e and lies in K. Since K is compact and S is continuous, fixed point of the mapping G the set S.K/ is compact and the last statement of the theorem follows. 2.3.2.2 The nonconvex case Our final existence theorem in this chapter is Theorem 2.34. Assume that the multi-valued map F W J  RC ! Pcp .RC / is integrably bounded, satisfies 0 62 F .:; :/ and ( .H2 /

.a/ .t; x/ 7! F .t; x/ is L ˝ B measurableI .b/ x 7! F .t; x/ is lower semi-continuous for a.e. t 2 J:

Then problem (2.16) has at least one positive solution. We need the following auxiliary results. Lemma 2.35 (see [146, 181]). Let F W J  R ! Pcp .R/ be an integrably bounded multi-valued function satisfying .H2 /. Then F is of lower semi-continuous type. Proof. From 2.35 and [Theorem 6.138, Chapter 6], there exists a continuous selection function f W C.J; RC / ! L1 .J; RC / such that f .x/.t / 2 F .t; x/ for every x 2 C.J; RC / and a.e. t 2 J . Next, consider the boundary value problem for an autonomous ordinary differential equation: 

x 00 .t / D f .x/.t /; x.0/ D x.1/ D 0:

a.e. t 2 J

(2.18)

104

Chapter 2 Existence theory for differential equations and inclusions

Clearly, if x 2 C.J; RC / is a solution of problem (2.18), then x is a solution to problem (2.16). Problem (2.18) is then reformulated as a fixed point problem for the operator A W C.J; RC / ! C.J; RC / defined by Z

1

.Ax/.t / D

G.t; s/f .x/.s/ds;

(2.19)

0

where G is the Green’s function for G 00 D 0 with homogeneous Dirichlet boundary conditions. We first check that A is completely continuous. From continuity of G, A is continuous. (a) A maps bounded sets into bounded sets. Let B be a bounded subset of C.J; RC / and u 2 B. Then kAxk  M D jpj1 where jf .x.t //j  jp.t /jI this implies the boundedness of A.B/. (b) The set fAx j x 2 Bg is equicontinuous. For t1 ; t2 2 J , we have ˇZ 1 ˇ ˇ ˇ ŒG.t1 ; s/  G.t2 ; s/f .x.s//ds ˇ j.Ax/.t1 /  .Ax/.t2 /j D ˇ 0

ˇZ 1 ˇ ˇ ˇ ŒG.t1 ; s/  G.t2 ; s/ds ˇ:  jpj1 ˇ 0

Letting jt1 t2 j ! 0; the claim follows. With Steps 1–2, the Arzéla–Ascoli lemma implies that A is completely continuous. (c) Uniform a priori bounds. For every fixed point x D A.x/ with  2 .0; 1/, we have, as in (b), that kxk1  M D jpj1 : Let

U D fx 2 C.J; RC / j kuk1 < M C 1g:

From the choice of U; there is no solution x 2 @U such that x D A.x/ for some  2 .0; 1/. As a consequence of the nonlinear alternative of Leray–Schauder type for single-value mappings ([Corollary 1.74, Chapter 1]), we deduce that A has a fixed point x in U; which is a solution for problem (2.16).

Chapter 3

Solution sets for differential equations and inclusions Some of the results obtained in Chapter 1 are now used to investigate the topological structure of solution sets for initial and boundary value problems associated with differential equations and inclusions. As for existence theorems, they are mainly based on results of Chapter 2. Several results of Aronszajn–Browder–Gupta type are given, extending Kneser–Hukuhara classical results on the topological structure of the solution sets. The cases of differential equations and inclusions are discussed separately.

3.1 General results We start this section with a very classical result.

3.1.1 Kneser–Hukuhara theorem Consider the Cauchy problem ( y 0 .t / D f .t; y.t //; y.0/ D y0

t 2 J D Œ0; a

(3.1)

where f W J  B.0; b/ ! Rn is single-valued continuous function. Let ˛ D min.a; b=M /; where M D

sup

kf .t; x/k and I D Œ0; ˛:

.t;x/2J B.0;b/

For problem 3.1, we prove a Kneser type result (see [145], Theorem 2.3 for the case of Banach spaces with an additional compactness condition and [431], Corollary 4.6 for the case of differential inclusions). Theorem 3.1. The set S.y0 / of all solutions defined on I is a continuum in C.I; B.0; b// (i.e., S.y0 / is closed and connected). In particular, the section S.t / D fy.t / j y 2 S.y0 /g is a continuum in Rn for all t 2 Œ0; b. Proof. From Peano’s existence theorem [Theorem 2.10, Chapter 2], S.y0 / 6D ; and it clear that this set is compact by Ascoli–Arzéla lemma. We claim that S.y0 / is connected. On the contrary, suppose that two nonempty compact sets S1 and S2 exist and satisfy S.y0 / D S1 [ S2 and S1 \ S2 D ;:

106

Chapter 3 Solution sets for differential equations and inclusions

Hence, 0 D Hd .S1 ; S2 / > 0, where Hd is the Hausdorff distance (see Chapter 6 for main properties of multi-valued maps and their continuity). Define the function W C.J; Rn / ! R by .y/ D Hd.y; S1 /  Hd .y; S2 /; where

Hd .x; Si / D supfd.x; y/ j y 2 Si g;

i D 1; 2:

The function is continuous and we have .y/  0 for y 2 S1

and .y/  0

for y 2 S2 :

Let " > 0 be a real number. By Lemma 1.29, we can find an approximation function f" W J Rn ! Rn such that kf" .x/f" .y/k  k" .t /kx yk and kf" .x/f .x/k  " where the function k" .:/ is locally integrable on J . Let y1 2 S1 and y2 2 S2 be fixed. Consider the functions f1 ; f2 ; f  defined by fi .t; y/ D f" .t; y/ C f .t; yi .t //  f" .t; yi .t //;

for i D 1; 2

and f  .t; y/ D f1 .t; y/ C .f2 .t; y/  f1 .t; y//;

for  2 Œ0; 1:

Given .t; x/; .t; y/ 2 J  Rn ; we have: kfi .t; x/  fi .t; y/k  kf".t; x/  f" .t; y/k  k" .t /kx  yk;

for i D 1; 2:

Consequently, the functions fi are locally Lipschitzian as linear combinations of f . Moreover, we have kfi .t; y/  f .t; y/k D kf" .t; y/ C f .t; yi /  f" .t; yi /  f .t; y/k  kf" .t; y/  f .t; y/k C kf".t; yi /  f .t; yi /k  2" and kf  .t; y/  f .t; y/k   kf .t; y2 /  f" .t; y2 / C f" .t; y1 /  f .t; y1 /k C kf1 .t; y/  f .t; y/k   .kf .t; y2 /  f" .t; y2 /k C kf" .t; y1 /  f .t; y1 /k/ C kf1 .t; y/  f .t; y/k  ." C "/ C 2"  4": From the Picard existence theorem (Theorem 2.3, Chapter 2), the Cauchy problem ( y 0 D f  .t; y.t //; a.e. t 2 J; (3.2) y.0/ D y0

107

Section 3.1 General results

has a unique solution y  in J for all  2 Œ0; 1. Given 1 ; 2 2 Œ0; 1; we have Z t    2 1 2 2 1 1  f .s; y .s//  f .s; y .s//ds  ky .t /  y .t /k D   0

Z

t

 j2  1 j

kf .s; y2 .s//  f" .s; y2 .s// 0

 f .s; y1 .s// C f" .s; y1 .s//kds

Z

t

C

kf".s; y 2 .s//  f" .s; y 1 .s//kds:

0

From Gronwall’s inequality, we get ky 2  y 1 k1  2bj2  1 j".e

Rt 0

k" .s/ds

 1/:

Therefore, the function ' W Œ0; 1 ! C.J; Rn / defined by './ D y  is continuous on Œ0; 1. Also the function W Œ0; 1 ! R defined by

./ D .'.// D .y  / is continuous on the same interval. Since f 0 .t; y1 .t // D f .t; y1 .t // D y10 .t /

and f 1 .t; y2 .t // D f .t; y2 .t // D y20 .t /;

from the uniqueness of solution to problem (3.2), we infer that y 0 D y1 2 S1 and y 1 D y2 2 S2 , where y 0 ; y 1 are solutions of problem (3.2) in cases  D 0 and  D 1 respectively. Hence,

.0/ D H .y1 ; S2 /  0

and .1/ D H .y2 ; S1 /  0 :

By the intermediate value theorem applied to the continuous function '; there is a " 2 .0; 1/ such that ."/ D 0. Now let f"ngn2N be a sequence with limit lim "n D "; n!1

let yn D y "n ; and consider the sequence fyngn2N. All the functions yn are solutions of (3.2), so that yn .0/ D y0 ; for all n 2 N. Let zn .t / be the function representing the error made when approximating the derivatives of solutions of (3.3) by yn0 .t /; that is yn0 .t / D f .t; yn .t // C zn .t /; where kzn .t /k D ky 0"n .t /  f .t; yn .t /k D kf " .t; yn .t //  f .t; yn .t //k  4"n: Hence,

lim kyn0 .t /  f .t; yn .t //k D 0:

n!1

108

Chapter 3 Solution sets for differential equations and inclusions

It follows that the sequence fyn gn2N admits a subsequence converging uniformly to a solution y of problem (3.3), that is y 2 S. But .y/ D lim .y "n / D .y " / ."/ D 0; n!1

leading to a contradiction. Therefore, S.y0 / is a continuum and, by the continuity of the projection y ! y.t /; the section S.t / is a continuum in Rn .

3.1.2 Problems on bounded intervals Consider the first-order initial value problem: (

y 0 .t / D f .t; y.t //; y.t0 / D y0 ;

a.e. t 2 J D Œt0 ; b;

(3.3)

where f W J  Rn ! Rn is a given single-valued function y0 2 Rn . Denote by S.f; y0 / the set of all solutions of problem (3.3). We prove an Aronszajn type result for this problem. Theorem 3.2. Assume that f W J  Rn ! Rn is a Carathéodory function satisfying the Nagumo condition: (1) There exist a function p 2 L1 .J; RC / and a continuous nondecreasing function W Œ0; 1/ ! Œ0; 1/ such that kf .t; y/k  p.t / .kyk/; with

Z

for a.e. t 2 J and each y 2 Rn ; Z

b

1

p.s/ds < ky0 k

t0

du  .u/

Then the set S.f; y0 / is Rı . Proof. Let F W C.Œt0 ; b; Rn / ! C.Œt0 ; b; Rn / be defined by Z

t

F .y/.t / D y0 C

f .s; y.s//ds;

t 2 Œt0 ; b:

t0

Thus, Fix F D S.f; y0 /. From Theorem 2.14 we know that S.f; y0 / 6D ; and there exists M > 0 such that kyk1  M;

for every y 2 S.f; y0 /;

109

Section 3.1 General results

where kyk1 D sup ky.t /k. Define t 2Œt0 ;b

8 ˆ < f .t; y/; 

e.t; y/ D f My ˆ ; : f t; kyk

if ky.t /k  M if ky.t /k  M;

e .t; y/ D f .t; r .y// where r is the radial retraction of the closed ball B.0; M /. that is f e is Carathéodory too and is integrably bounded Since f is Carathéodory, the function f 1 by 3.2 (1). So there exists h 2 L .J; RC / such that e.t; y/k  p.t / .M / WD h.t /; kf Consider the modified problem (

a.e. t and all y 2 Rn :

(3.4)

e.t; y.t //; y 0 .t / D f y.t0 / D y0 :

e; y0 / D FixF e where F e W C.J; Rn / ! We can easily prove that S.f; y0 / D S.f C.J; Rn / is defined analogously by Z t e .s; y.s//ds; t 2 J: e .y/.t / D y0 C F f t0

We deduce that e.y/k1  ky0 k C .b  t0 /khkL1 WD R; kF

8 y 2 Rn :

e is uniformly bounded. By the Ascoli–Arzéla lemma, we can prove that F e Then F e is compact; this allows us to define the compact perturbation of the identity G.y/ D e .y/ which is a proper map. From the compactness of F e and using [Corollary 1.36, yF Chapter 1], we can easily prove that all conditions of [Theorem 1.32, Chapter 1] are e; y0 / D G e 1 .0/ is an Rı -set, hence acyclic. met. Therefore, the solution set S.f

3.1.3 Problems on unbounded intervals Consider the Cauchy problem:  0 y .t / D f .t; y.t //; y.0/ D y0 ;

a. e. t 2 J D Œ0; 1/;

(3.5)

where f W Œ0; 1/  Rn ! Rn is a given function. Let Cb be the Banach space of bounded continuous vector-valued functions on the interval Œ0; 1/; endowed with the sup-norm k  k1 defined by kyk1 D

sup ky.t /k; t 2Œ0;1/

for y 2 Cb :

Now we prove the following result extending Theorem 3.2:

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Chapter 3 Solution sets for differential equations and inclusions

Theorem 3.3. Assume that the following conditions hold: (1) The function f W J  Rn ! Rn is a Carathéodory function. (2) There exist p 2 L1 .J; RC / and a continuous nondecreasing function Œ0; 1/ such that for a.e. t 2 J and each x 2 Rn

kf .t; x/k  p.t / .kxk/; Z

with

WJ !

1 ky0 k

ds D 1: .s/

Then the solution set of problem (3.5) is a nonempty Rı in Cb .J; Rn /. Proof. Let S D fy 2 Cb .J; Rn / j y is a solution of problem.3.5g. From Theorem 2.22 we know that S 6D ;. Now, we prove that S is compact. Let fyn j n 2 Ng  S; then Z t yn .t / D y0 C f .s; yn .s//ds; t 2 J: 0

Using Gronwall’s lemma, we can prove that kynkb  ‰ 1 .kpkL1 /; where

Z ‰.t / D

t ky0 k

for all n 2 N; ds : .s/

Step 1. Using Ascoli–Arzéla lemma, we can prove, as in Theorem 2.22, that the sequence fyn j n 2 Ng is compact in Cb .Œ0; 1/; Rn /; then there exists a subsequence still denoted fyn j n 2 Ng which converges to y 2 Cb .J; Rn /. Using the fact that f is an L1 -Carathéodory function, by the Lebesgue dominated convergence theorem, we find that Z t

y.t / D y0 C

f .s; y.s//ds;

t 2 J;

0

proving that S is compact. Step 2. S is an Rı -set. Consider the operator N W Cb .J; Rn / ! Cb .J; Rn / defined by Z t f .s; y.s//ds; t 2 J; .Ny/.t / D y0 C 0

i.e., S D Fix.N /. Let M D fy 2 Cb .Œ0; 1/; Rn / j ky.t /  r .t /k  p .t /; t 2 J g;

111

Section 3.1 General results

Rt  where r .t / D y0 and p .t / D ‰ 1 0 p.s/ds . It is clear that p is non-negative continuous function and is locally bounded. Let 0 2 K D Œ0; 1/ and y0 2 Rn ; then ky0  r .0/k  p .0/ for all y 2 M with .Ny/.0/ D y0 . Step 3. N.M/ is a uniformly continuous map. Let " > 0 and > 0 be such that t1 ; t2 2 K , then Z t2 p.s/ .ky.s/k/ds: k.Ny/.t2 /  .Ny/.t1 /k  t1

From the definition of M; we get ky.t /k  ky0 k C p .t /; Then

for all y 2 M:

ky.t /k  ky0 k C ‰ 1 .kpkL1 / WD M :

Hence,

Z

k.Ny/.t2 /  .Ny/.t1 /k 

t2

.M /

p.s/ds ! 0;

as jt1  t2 j ! 0:

t1

and .8 " > 0/ .8 x; y 2 M/ .xjK" D yjK" / ) .N x/jK" D .Ny/jK" . Since N is compact then the Palais–Smale condition holds. By Theorem 1.33, Fix.N / is an Rı -set.

3.1.4 Second-order differential equations Consider the second-order Cauchy problem, ( y 00 .t / D f .t; y.t /; y 0 .t //; y.t0 / D a; y 0 .t0 / D c;

a.e. t 2 J D Œt0 ; b;

(3.6)

where f W J  Rn  Rn ! Rn is a given function. Denote by S.f; a; c/ the set of all solutions of problem (3.6). We prove an Aronsajn type result for this problem. Theorem 3.4. Assume that f W J  Rn  Rn ! Rn is a Carathéodory function such that (1) there exist a function p 2 L1 .J; RC / and a continuous nondecreasing function  W Œ0; 1/  Œ0; 1 ! .0; 1/ satisfying kf .t; x; y/k  p.t /.kxk; kyk/; with

Z

Z

b

for a.e. t 2 J and each x 2 Rn

1

p.s/ds < t0

Then the solution set S.f; a; c/ is Rı .

kakC.bC1/kck

du  .u; u/

112

Chapter 3 Solution sets for differential equations and inclusions

Proof. Let F W C 1 .J; Rn / ! C 1 .J; Rn / be defined by Z

t

F .y/.t / D a C t c C

.t  s/f .s; y.s/; y 0 .s//ds;

t 2 J:

t0

Thus, Fix F D S.f; a; c/. From Theorem 3.2 we know that S.f; a; c/ 6D ;, and there exists M > 0 such that kyk1  M ; Define

for every y 2 S.f; a; c/:

8 ˆ < f .t; x; y/; e  Mx My  f .t; x; y/ D ˆ : f t; ; ; kxk kyk

if kxk C kyk  M ; if kxk C kyk  M :

e is Carathéodory too and it is integrably Since f is L1 -Carathéodory, the function f 1 bounded by 3.4 (1). So there exists m 2 L .J; RC / such that e.t; x; y/k  m.t /; kf

for a.e. t and all x; y 2 Rn :

(3.7)

Consider the modified problem 

e.t; y.t /; y 0 .t //; y 00 .t / D f y.t0 / D a; y 0 .t0 / D c:

a.e. t 2 J;

e ; a; c/ D Fix F e , where F e W C 1 .J; Rn / ! P.C 1 .J; Rn // It is clear that S.f; a; c/ D S.f is defined by e .y/.t / D a C t c C F

Z

t

e .s; y.s/; y 0 .s//ds; f

t 2 J:

t0

Hence, e.y/k1  kak C ..b  t0 / C 1/kck C .b C 1/khkL1 : kF e is uniformly bounded. By the Ascoli–Arzéla lemma, we can prove that F e W Then F C 1 .J; Rn / ! C 1 .J; Rn / is compact which allows us to define the compact perturbae e .y/ which is a proper map. From the compactness tion of the identity G.y/ D y F e of F and the Lasota–Yorke approximation theorem, we can easily prove that all cone; a; c/ D G e 1 .0/ is ditions of Corollary 1.37 are met. Therefore, the solution set S.f an Rı -set, hence an acyclic space.

113

Section 3.1 General results

3.1.5 Abstract Volterra equations Consider the abstract Volterra equation (

y 0 .t / D V .y/.t /; a.e. t 2 Œ0; T  y.0/ D y0 2 Rn ;

(3.8)

where V W C.Œ0; T ; Rn / ! L1 .Œ0; T ; Rn / is a continuous operator of Volterra type, i.e., if x.t / D y.t / for t 2 Œ0; "; ."  T /, then V .x/.t / D V .y/.t / for a.e. t 2 Œ0; ". A first result is given by the following one (see [359, Theorem 2.2]) Theorem 3.5. Assume that there exists  2 L1 .Œ0; T ; Rn / such that for any y 2 C.Œ0; T ; Rn /, we have kV .y/.t /k  .t /, for a.e. t 2 Œ0; T . Then the solution set for (3.9) is Rı in C.Œ0; T ; Rn /. Remark 3.6. The condition that the Volterra operator is integrably bounded may be easily relaxed to a Nagumo type growth condition: there exist ˛ 2 L1 Œ0; T  and g W Œ0; 1/ ! .0; 1/ such that for a.e. t 2 Œ0; T  and all y 2 C.Œ0; T ; Rn /, hy.t /; V .y.t //i  ˛.t /ky.t /kg.ky.t /k/ with

Z

Z

T

1

˛.s/ds < 0

ky0 k

ds ; g.s/

where h; :i denotes the Euclidian scalar product. If we rather assume a boundedness condition on the operator V then we can study the topological structure of SV in the space A.Œ0; T ; Rn / D ff 2 C.Œ0; T ; Rn / j f 0 2 L1 .Œ0; T ; Rn /g, L1 .Œ0; T ; Rn / being supplied with the weak topology instead of SV  C.Œ0; T ; Rn /. We have (see [Theorem 2.2, [2]]) Theorem 3.7. Assume that there exists a constant M > 0 such that for any y 2 C.Œ0; T ; Rn / we have kV .y/.t /k  M for a.e. t 2 Œ0; T . Then the solution set SV for (3.9) is a continuum (nonempty compact connected set) in A.Œ0; T ; Rn /. Proof. (a) SV is compact in A.Œ0; T ; Rn /. Let fy˛ g˛2ƒ be a Moore–Smith sequence in a compact set in A.Œ0; T ; Rn / with fy˛ g˛2ƒ  SV . By assumption, wa .t / D y˛0 .t /=M lies in the unit ball of L1 .Œ0; T ; Rn /. By the Banach–Alaoglu compactness 0 =M converges criterion (see [83,499]), there is some subsequence N of ƒ such that ym  1 n weakly to some limit w 2 L .Œ0; T ; R /, as m ! 1 in N . Since ym 2 SV , then by Theorem 3.5, there exists y 2 SV \ C.Œ0; T ; Rn / and a subsequence, still denoted 0 converges weakly to M w as M ! 1. Now, [37, p. 14] ym , converging to y and ym

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Chapter 3 Solution sets for differential equations and inclusions

0 implies that ym converges weakly in L1 .Œ0; T ; Rn /g as m ! 1. Passing to the limit in the integral equation

Z

t

ym .t / D ym .0/ C 0

we obtain at the limit

Z

t

y.t / D y.0/ C

0 ym .s/ds;

y 0 .s/ds;

0 0 so that y 0 D M w almost everywhere. Hence, ym ! y in C.Œ0; T ; Rn / and ym  0 converges weakly to y as m ! 1, which proves compactness of the solution set. (b) SV is connected in A.Œ0; T ; Rn /. We argue by contradiction, assuming that SV D A[B where A and B are two nonempty closed disjoint subsets of A.Œ0; T ; Rn /. In particular, A and B are nonempty disjoint subsets of C.Œ0; T ; Rn /. Since we already know that SV is connected in C.Œ0; T ; Rn / by Theorem 3.5, then it suffices to prove that A and B are closed subsets of C.Œ0; T ; Rn /. Let fym gm1  A be a converging sequence to y in C.Œ0; T ; Rn /; then there exists a subsequence wM D y 0 m=M converging weak to some limit w 2 L1 .Œ0; T ; Rn /; then again [37, p. 14] guarantees 0 converges weakly to M w in L1 .Œ0; T ; Rn / and y 0 D M w a.e. Then y 2 A that ym for A 2 A.Œ0; T ; Rn / is closed, proving our claim.

Remark 3.8. The result of Theorem 3.7 remains valid in the space A1 .Œ0; T ; Rn / D ff 2 C.Œ0; T ; Rn / j f 0 2 L1 .Œ0; T ; Rn /g (see [Theorem 2.3, [2]]). Regarding the differential inclusion ( y 0 .t / 2 F .t; y.t //; y.0/ D y0 2 Rn ;

a.e. t 2 Œ0; T 

(3.9)

we have (see [Theorem 3.3, [2]]): Theorem 3.9. If F W Œ0; T   Rn ! Pcv;cp is L1 -Carathéodory, then S.F; y0 / is a nonempty compact connected set in A1 .Œ0; T ; Rn /. Finally, we mention that in the paper [2], we can also find further results about the solution sets for a class of differential and integral inclusions.

3.1.6 Aronszajn type results for differential inclusions In this section, we will consider some differential inclusions. The first result is given by ( y 0 2 F .t; y.t //; a.e. t 2 J D Œ0; b; (3.10) y.0/ D y0 ;

115

Section 3.1 General results

where F W J  Rn ! P.Rn / is a multi-valued map and y0 2 Rn . Let S.F; y0 / denote the set of all solutions of problem (3.10). We present some results about the topological structure of S.F; y0 / (see [210, 213, 215]). Theorem 3.10. Let F W J Rn ! Pcp;cv .Rn / be an mLL-selectionable multi-valued map (see Chapter 6) satisfying the Nagumo condition. (1) There exist a continuous nondecreasing function W Œ0; 1/ ! .0; 1/ and p 2 L1 .J; RC / such that for a.e. t 2 J and each u 2 Rn kF .t; u/kP WD supfkvk j v 2 F .t; u/g  p.t / .kuk/; with

Z

Z

b

1

p.s/ds < 0

ky0 k

du  .u/

Then, for every y0 2 Rn ; the set S.F; y0 / is contractible. Proof. Since F is a Carathéodory multi-valued function satisfying 3.10 (1), Theorem 2.28 implies that S.F; y0 / is nonempty. Now F is mLL-selectionable implies the existence of a measurable, locally Lipschitz, and integrably bounded selection f  F . Let the single-valued problem ( y 0 .t / D f .t; y.t //; t 2 J; (3.11) y.0/ D y0 : By the Cauchy–Lipschitz theorem (Theorem 2.3), problem (3.11) has exactly one solution for every y0 2 Rn . Define the function h W S.F; y0 /  Œ0; 1 ! S.F; y0 / by ( y.t /; for 0  t  ˛b; h.y; ˛/.t / D x.t /; for ˛b < t  b; where x D S.f; y0 / is the unique solution of problem (3.11). In particular, ( y; for ˛ D 1; h.y; ˛/ D x; for ˛ D 0: We prove that h is a continuous homotopy. Let .yn ; ˛n / 2 S.F; y0 /  Œ0; 1 be such that .yn ; ˛n / ! .y; ˛/; as n ! 1. We shall prove that h.yn ; ˛n / ! h.y; ˛/. We have ( yn .t /; for t 2 Œ0; ˛n b; h.yn ; ˛n /.t / D x.t /; for t 2 .˛n b; b: Since ˛n 2 Œ0; 1, we distinguish between three cases:

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Chapter 3 Solution sets for differential equations and inclusions

(a) If lim ˛n D 0; then h.y; 0/.t / D x.t /; for t 2 Œ0; b and n!1

kh.yn ; ˛n /  h.y; 0/k1  kyn  xk N Œ0;˛n b ! 0;

as n ! 1;

where N Œ0;˛n b D kyn  xk

sup t 2Œ0;˛n b

kyn .t /  x.t N /k:

(b) The case when lim ˛n D 1 is treated similarly. n!1

(c) If ˛n 6D 0 and 0 < lim ˛n D ˛ < 1; then we may distinguish between two n!1

sub-cases: (i) Assume t 2 Œ0; ˛b. yn 2 S.F; y0 / implies the existence of some vn 2 SF ;yn such that for t 2 Œ0; ˛n b Z t yn .t / D y0 C vn .s/ds; (3.12) 0

where SF ;y is as defined in (2.14). Since fyn gn2N converges to y in C.J; Rn , then some R > 0 exists and satisfies kyn k1  R: From 3.10 (1), we get kvn .t /k  p.t / .R/;

for all n 2 N:

Hence, the sequence fvn gn2N  L1 .Œ0; ˛b; Rn / is bounded and uniformly integrable in L1 .Œ0; ˛b; Rn . By the Dunfor–Pettis compactness criterion (see Appendix), we may assume, without loss of generality, that vn converges weakly to v in L1 .Œ0; ˛b; Rn / as n ! 1. Mazur’s lemma (see Appendix) implies the existence of a double sequence .˛n;k /k;n2N such that 8 n 2 N, P 9 k0 .n/ 2 N W ˛n;k D 0; 8 k  k0 .n/, 1 ˛n;k D 1; 8 n 2 N and kDn P1 the sequence of convex combinations gn .:/ D kDn ˛n;k vk .:/ converges strongly to v in L1 . Since F takes convex values, using Lemma 6.51, we obtain that for a.e. t 2 Œ0; ˛b \ v.t / 2 fgk .t /; k  ng; n1



\

cofvk .t /; k  ng

n1



\

n1

cof

[

(3.13) F .t; yk .t //g

kn

D co.lim sup F .t; yk .t ///: k!1

117

Section 3.1 General results

Moreover, F is u.s.c. with compact values; then by Lemma 6.48, we have lim sup F .t; yk .t // D F .t; y.t //;

for a.e. t 2 Œ0; ˛b:

k!1

This with (3.13) implies that v.t / 2 co F .t; y.t //. In addition F .; / has compact convex values; hence, v.t / 2 F .t; y.t //;

a.e. t 2 Œ0; ˛b:

By the Lebesgue dominated convergence theorem and passing at the limit in (3.12), we deduce that for t 2 Œ0; ˛b; we have Z t v.s/ds: y.t / D y0 C 0

We deduce that sup t 2 Œ0; ˛bkh.yn ; ˛n /.t /  h.y; ˛/.t / ! 0, as n ! 1. (ii) If t 2 .˛b; b, then h.y; ˛/.t / D x.t N / and again kh.yn ; ˛n /  h.y; ˛/k1 ! 0;

as n ! 1

for ˛n ! ˛. Therefore, h is a continuous homotopy, proving that S.F; y0 / is contractible to the point x D S.f; y0 /. A second result is given by: Theorem 3.11. Let F W J  Rn ! Pcp;cv .Rn / be a Carathéodory, C a-selectionable multi-valued map which satisfies (3.10.1). Then the solution set S.F; y0 / is Rı -contractible. Proof. From Theorem 2.28, S.F; y0 / 6D ;. Now we replace the single-valued homotopy h W S.F; y0 /  Œ0; 1 ! S.F; y0 / in Theorem 3.10 by the multi-valued homotopy … W S.F; y0 /  Œ0; 1 ! P.S.F; y0 // defined by n x.t /; for 0  t  ˛b; o n z 2 S.f; ˛b; x/ ….x; ˛/ D y 2 C.Œ0; b; Rn / j y.t / D z.t /; for ˛b < t  b; where f  F is a Carathéodory selection of F and S.f; ˛b; x/ is the solution set of the problem ( y 0 .t / D f .t; y.t //; t 2 Œ˛b; b; y.t / D x.˛b/: In other words, ( ….x; ˛/.t / D

x.t /; for 0  t  ˛b; S.f; ˛b; x/.t /; for ˛b < t  b;

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Chapter 3 Solution sets for differential equations and inclusions

Note that Theorem 3.2 implies that the set S.f; ˛b; x/ is Rı . In addition, from the definition of …; ….x; 0/ D S.f; 0; x/ and x 2 ….x; 1/ for every x 2 S.F; y0 /. It remains to prove that ….; / is u.s.c. Since ….; / has nonempty compact values, we only check that … is locally compact and has a closed graph. Finally, we show that for each x; ˛, ….x; ˛/ is an Rı -set. This will be performed in three steps. Step 1. To prove that … is locally compact, consider two sub-steps. (a) The multi-valued map e S W Œ0; b  Rn ! Pcp .C.Œ0; b; Rn // defined by e S.u; y0 / D S.f; u; y0 / is u.s.c. Here S.f; u; y0 / refers to the solution set of the problem ( y 0 .t / D f .t; y.t //; a.e. t 2 Œu; b; y.u/ D y0 : On the contrary, assume that e S is not u.s.c. at some point .t0 ; y0 /. Then there exists an open neighbourhood U of e S.t0 ; y0 / in C.Œ0; b; Rn /, such that for every open neighbourhoodV at .t0 ; y0 / in the metric space Œ0; bRn there exists .t1 ; y1 / 2 V such that e S.t1 ; y1 / 6 U . Let Vn D f.t; y/ 2 Œ0; b  Rn j d..t; y/; .t0 ; y0 // < 1=ng; for each n 2 N where d denotes the product metric in Œ0; b  Rn . Then for S.tn ; yn / such that yn 62 U . each n 2 N; we get some .tn ; yn / 2 Vn and yn 2 e Define the maps Gt0 ;y0 ; Ft0 ;y0 W C.Œ0; b; Rn / ! C.Œ0; b; Rn / by

Z Ft0 ;y0 .y/.t / D y.t0 / C

t

f .s; y.s//ds;

t 2 Œt0 ; b;

t0

and the compact perturbation of the identity Gt0 ;y0 .y/ D y  Ft0 ;y0 .y/;

for t 2 Œ0; b and y 2 C.Œ0; b; Rn /:

By a simple calculation, for y 2 C.Œ0; b; Rn /; t; t0 2 Œ0; b, and y0 2 Rn ; we have Ft0 ;y0 .y/.t / D y0  F0;y0 .y/.t0 / C F0;y0 .y/.t /: Consequently, Gt0 ;y0 .y/.t / D y0 C F0;y0 .y/.t0 / C G0;y0 .y/.t /: From the definition of e S, we have e S.t0 ; y0 / D Gt1 .0/; for each .t0 ; y0 / 2 Œ0; b  Rn : 0 ;y0

119

Section 3.1 General results

Since Ft0 ;y0 is a compact map, the compact perturbation of the identity Gt0 ;y0 is S.tn ; yn /. Hence, proper. Moreover, yn 2 e 0 D Gtn ;yn .yn /.t / D y.tn /  F0;y0 .yn /.tn /  G0;y0 .yn /.t / and Gt0 ;y0 .yn /.t / D y0 C F0;y0 .yn /.t0 / C G0;y0 .yn /.t /: Then, we have the successive estimates kGt0 ;y0 yn .t /k  kyn .tn /  y0 k C kF0;yn .t0 / .yn /.t /  F0;yn .t0 / .yn /.t0 /k Z t0 Z tn     kyn .tn /  y0 k C  f .s; yn .s//ds  f .s; y.s//ds  0 0 Z tn kf .s; yn .s//  f .s; y.s//kds  kyn .tn /  y0 k C 0 Z tn kf .s; yn .s//kds: C t0

In addition, we have kGt0 ; 0 yn .t /k  2kyn.tn /  yn .t0 /k1 C ky0 .t0 /  yn .t0 /k Z b Z tn C kf .s; yn .s//  f .s; y.s//kds C p.s/ds: t0

0

Now lim yn D y0 and lim tn D t0 imply that lim Gt0 ; 0 .yn / D 0. Since n!1

n!1

n!1

.A/. Moreover, fyng  the set A D fGt0 ;y0 .yn /g is compact, then so is Gt1 0 ; 0 A. Without loss of generality, we may assume that lim yn D y0 ; hence y0 2 n!1

e S.t0 ; y0 /  U but this is a contradiction to the assumption that yn 62 U for each n. This proves that e S is u.s.sc. (b) … is locally compact. For some r > 0; let B  I D f.x; ˛/ 2 S.F; y0 /  Œ0; 1 j kxk1 C j˛j  r g and let fyn gn2N 2 ….B  I /I then there exists .xn ; ˛n / 2 B  I such that ( xn.t /; for 0  t  ˛n b yn .t / D zn .t /; for ˛n b < t  b; zn 2 S.f; ˛nb; xn .˛n b//: Since S.F; y0 / is compact, there exist subsequences of fyngn2N and f˛ngn2N which converge to some limit y 2 S.F; y0 / and ˛ respectively. Since e S is u.s.c. then e S.Œ0; b  fxn.˛n b/gn2N [ fx.˛b/g/ is compact in C.Œ0; b; Rn /. S.Œ0; b  fxn.˛n b/gn2N [ fx.˛b/g/, then there exists a subseSince zn .t / 2 e quence of fzn .:/gn2N which converges to z. Since e S has a closed graph, then z2e S .˛b; x/ D S.f; ˛b; x/. Therefore, … is locally compact.

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Chapter 3 Solution sets for differential equations and inclusions

Step 2. … has a closed graph. Let .xn ; ˛n / ! .x ; ˛/; hn 2 ….xn ; ˛n / and hn ! h as n ! C1. hn 2 ….yn ; ˛n / means that there exists zn 2 S.f; ˛n b; n / such that for each t 2 J ( xn .t /; for 0  t  ˛n b hn .t / D zn .t /; for ˛nb < t  b: We must prove that there exists z 2 S.f; ˛b; x/ such that for each t 2 J ( x .t /; for 0  t  ˛b h .t / D z .t /; for ˛b < t  b: Clearly .˛n b; xn / ! .˛; x / as n ! 1 and we can easily show that there exists a subsequence fzn gn2N converging to some limit z . The cases ˛ D 0 and ˛ D 1 can be treated similarly. From the above arguing, we find that z 2 S.f; ˛b; x/; proving that h 2 ….x ; ˛/. Step 3. We claim that ….x; ˛/ is an Rı -set for each fixed ˛ 2 Œ0; 1 and x 2 S.:/. Since F is Carathéodory, [Theorem 6.125, Chapter 6] implies that F is -Caselectionable, i.e., there exists a decreasing sequence of multi-valued maps Fk W Œ0; b  Rn ! P.Rn / .k 2 N/ which have Carathéodory selections and satisfy FkC1 .t; u/  Fk .t; u/; and F .t; u/ D

for a.e. t 2 Œ0; b; all u 2 Rn

1 \

Fk .t; u/; u 2 Rn :

kD0

Then ….x; ˛/ D

1 \

S.Fk ; x/:

kD0

Notice further that the Nagumo type condition [(3.10.1)] implies the existence of some constant M > 0 such that ky.t /k  M;

for each t 2 J:

Define the multi-valued map 8 ˆ if kyk  M and t 2 J < F .t; y/;

 FM .t; y/ D My ˆ ; if kyk  M and t 2 J : F t; kyk which is obviously integrably bounded, Carathédory, and satisfies S.FM ; y0 / D S.F; y0 /. Again [Theorem 6.125, Chapter 6] implies that FM is in fact -mLLselectionable which means that for every k D 0; 1; : : : ; Fk has an m  LL-selection.

121

Section 3.1 General results

From Theorem 3.10, we know that the sets S.Fk ; x/ are contractible. Therefore, ….x; ˛/ is an Rı -set, as claimed. As a consequence of Steps 1–3, all properties in Definition 5.72 are satisfied. Therefore, the set S.F; y0 / is Rı -contractible, ending the proof of the theorem. Remark 3.12. Notice that in Step 1 of the proof of Theorem 3.11, we have proved u.s.c. dependence of the solution set upon initial data. From Theorems 3.10 and 3.11, we obtain: Theorem 3.13. Let F W Œ0; b  RN ! Pcp;cv .RN / be a multi-valued map. (1) If F is   mLL-selectionable, then the set S.F; y0 / is an intersection of a decreasing sequence of contractible sets. (2) If F is   C a-selectionable, then the set S.F; y0 / is an intersection of a decreasing sequence of Rı -contractible spaces. Another result regarding the topological structure of the solution sets is given by Theorem 3.14. Let F W J  Rn ! Pcp;cv .Rn / be a Carathéodory multi-valued map which satisfies the following condition: (1) There exist a function p 2 L1 .J; RC / and a continuous nondecreasing function W Œ0; 1/ ! Œ0; 1/ such that kF .t; z/kP  p.t / .kzk/; with

Z

for a.e. t 2 J and each z 2 Rn ; Z

b

1

p.s/ds < 0

ky0 k

du  .u/

Then the solution set S.F; y0 / is a Rı . Proof. It is clear that from Theorem 2.28 S.F; y0 / 6D ; and there exists M > 0 such that ky.t /k  M; for each t 2 J: Let

8 ˆ if kyk  M and t 2 J < F .t; y/; 

e F .t; y/ D My ˆ ; if kyk  M and t 2 J: : F t; kyk

We can see that FM is an integrably bounded Carathéodory multi-valued map and e S.FM ; y0 / D S.F; y0 /. Now we apply Theorem 6.125 to the multi-valued map F e to deduce that F is   C a-selectionable, i.e., there exists a decreasing sequence of

122

Chapter 3 Solution sets for differential equations and inclusions

e k W Œ0; b  Rn ! P.Rn / .k 2 N/ which have Carathéodory multi-valued maps F selections and satisfy e kC1 .t; u/  F e k .t; x/ F and e .t; x/ D F

for almost all t 2 Œ0; b; x 2 RN 1 \

e k .t; x/; x 2 Rn : F

kD0

Then e; y0 / D S.F

1 \

ek ; y0 / D S.F; y0 /: S.F

kD0

eF e is integrably bounded, then by Theorem 6.125, for every k D 0; 1; : : :, F ek Since F e has m  LL-selection. Hence, from Theorem 3.11, the set S.F k ; y0 / is contractible for each k 2 N. Hence, S.F; y0 / is an Rı -set.

3.2 Second-order differential inclusions Consider the problem: (

y 00 .t / 2 F .t; y.t /; y 0 .t //; y.0/ D a; y 0 .0/ D c;

t 2 J WD Œ0; b;

(3.14)

where F W J  Rn ! P.Rn / is a multi-valued map, and a; c 2 Rn . We first define what we mean by a solution of problem (3.14). Definition 3.15. A function y 2 AC 1 .Œ0; b; Rn / is said to be a solution of (3.14) if there exists v 2 L1 Œ0; b such that v.t / 2 F .t; y.t /; y 0 .t // a.e. on J and y is the solution of the integral equation Z

t

y.t / D a C t c C

.t  s/v.s/ds;

t 2 J:

0

3.2.1 The convex case Theorem 3.16. Let F W J  Rn  Rn ! Pcp;cv .Rn / be a multi-valued map such that (1) the multi-map t ! F .t; :; :/ is measurable, for every t 2 J; the multi-map .x; y/ ! F .t; x; y/ is u.s.c. from Rn  Rn into Rn ;

123

Section 3.2 Second-order differential inclusions

W Œ0; 1/  Œ0; 1/ ! .0; 1/

(2) there exist a nondecreasing continuous function and p 2 L1 .J; RC / such that kF .t; x; y/kD  p.t / .kxk; kyk/; with

Z

for a.e. t 2 J and each .x; y/ 2 Rn  Rn Z

b

1

p.t /ds < 0

A

du .u; u/

and A D kak C bkck: Then problem (3.14) has at least one solution and the solution set is compact in C 1 .J; Rn /. Remark 3.17. Since F is Carathéodory, the multi-map t 7! F .t; y.t /; y 0 .t // is measurable whenever y 2 C 1 .J; Rn /. Proof. Part 1. Existence of solution. The proof will be given in several steps. Step 1. We first transform the problem (3.14) into a fixed point problem by considering the operator N W C 1 .J; Rn / ! C 1 .J; Rn / defined by  Z t N.y/ D h 2 C 1 .J; Rn / j h.t / D a C t c C .t  s/v.s/ds; t 2 J 0

where v 2 SF ;y D fv 2 L1 .J; Rn / j v.t / 2 F .t; y.t /; y 0 .t // a.e. t 2 J g: We shall use Lemma 1.74 to prove that N has a fixed point. First, notice that N.y/ is convex for each y 2 C 1 .J; Rn / for F has convex values. Claim 1. N maps bounded sets into bounded sets in C 1 .J; Rn /. Indeed, it is enough to show that there exists a positive constant ` such that, for each y 2 Bq D fy 2 C 1 .J; Rn / j kyk1  qg .q > 0/, one has kN.y/kP  `. Let y 2 Bq ; h 2 N.y/; then there exists v 2 SF ;y such that Z

t

h.t / D a C t c C

.t  s/v.s/ds;

for t 2 J

0

and 0

Z

t

h .t / D c C

v.s/ds; 0

for t 2 J:

124

Chapter 3 Solution sets for differential equations and inclusions

Hence, Z t kh.t /k  kak C t kck C .t  s/kv.s/kds 0 Z t  kAk C .t  s/p.s/ .ky.s/k; ky 0 .s/k/ds: 0

Therefore,

Z

b

khk0  kAk C b .q; q/

p.s/ds WD `; 0

and

Z

kh0 k0  kck C b .q; q/

b

p.s/ds WD `1 : 0

Hence, kN.y/kP  ` C `1 ; where kN.y/kP D supfkhk0; h 2 N.y/g. Claim 2. N maps bounded sets into equicontinuous sets of C 1 .J; Rn /. Let l1 ; l2 2 J; 0  l1 < l2 and Bq be a closed ball of C 1 .J; Rn / as in Claim 2. Let y 2 Bq ; then for each t 2 J , we have Z

l2

kh.l2 /  h.l1 /k  jl1  l2 jkck C l2

p.s/ .ky.s/k; ky 0 .s/k/ds

l1

Z

l1

Cjl2  l1 j .q; q/

p.s/ds 0

 jl1  l2 jkck C jl2  l1 j .q; q/kpkL1 Z l1 Cjl2  l1 j .q; q/ p.s/ds 0

and 0

Z

0

kh .l2 /  h .l1 /k 

l2

.q; q/

p.s/ds: l1

Since p 2 L1 , then kh.l2 /  h.l1 /k and kh0 .l2 /  h0 .l1 /k tend to zero, as l2  l1 ! 0. Consequently, N.Bq / is compact in C 1 .J; Rn /. Claim 3. N has a closed graph. Let yn ! y , hn 2 N.yn /, and hn ! h . We prove that h 2 N.y /. hn 2 N.yn / means that there exists vn 2 SF ;yn such that for each t 2 J Z

t

hn .t / D a C t c C

.t  s/vn .s/ds 0

125

Section 3.2 Second-order differential inclusions

and h0n .t / D c C

Z

t

vn .s/ds: 0

We must prove that there exists v 2 SF ;y such that for each t 2 J Z t .t  s/v .s/ds: h .t / D a C t c C 0

Note that khn  h k1 ! 0; as n ! 1: Now, consider the linear continuous operator W L1 .J; Rn / ! C 1 .J; Rn / defined by Z t .t  s/v.s/ds: . v/.t / D 0

From the definition of ; we know that hn .t / 2 .SF ;yn /: Since yn ! y and ı SF is a closed graph operator by Lemma 6.155, then there exists f 2 SF ;y such that Z t .t  s/f .s/ds; t 2 J: h .t / D a C t c C 0

Hence, h 2 N.y /; proving our claim. Finally, [Lemma 6.23, Chapter 6] implies that N is u.s.c. Claim 4. A priori bounds. Let y 2 C 1 .J; Rn / and y 2 N.y/. Then there exists v 2 SF ;y such that for each t 2 J we have Z t y.t / D a C t c C .t  s/v.s/ds 0

and y 0 .t / D c C

Z

t

v.s/ds: 0

This implies by 3.16 (2) that for each t 2 J we have Z t p.s/ .ky.s/k; ky 0 .s/k/ds ky.t /k  kak C bkck C b

(3.15)

0

and 0

Z

t

ky .t /k  kck C 0

p.s/ .ky.s/k; ky 0 .s/k/ds:

(3.16)

126

Chapter 3 Solution sets for differential equations and inclusions

Consider the function  defined by .t / WD supfky.s/k C ky 0 .s/k j 0  s  t g;

0  t  b:

Let t  2 J be such that .t / D ky.t  /k C ky 0 .t  /k; by the inequalities (3.15) and (3.16), we have for t 2 Œ0; b Z t p.s/ ..s/; .s//ds: .t /  kak C b.kck C 1/ C .1 C b/ 0

Consequently,

Z  .t /  01 b C 1/

b

 p.s/ds D M0

0

which implies that kykC 1  M0 Z z du ; where 0 .z/ D .u; u/ A

z  A:

Let U WD fy 2 C 1 .J; Rn / j kyk1 < M0 C 1g and consider the operator N W U ! Pcv;cp .C 1 .J; Rn //. From the choice of U , there is no y 2 @U such that y 2 N.y/ for some  2 .0; 1/. As a consequence of the Leray–Schauder nonlinear alternative ([Corollary 1.74, Chapter 1]), we deduce that N has a fixed point y in U which is a solution of problem (3.14). Part 2. Compactness of the solution set. Consider the set S.a; c/ D fy 2 C 1 .J; Rn / j y is a solution of problem .3.14/g: M. From Part 1, S.a; c/ 6D ; and there exists f M such that for every y 2 SF ; kyk1  f 1 Since N is completely continuous, N.S.a; c// is relatively compact in C .Œ0; b; Rn /. Let y 2 S.a; c/I then y 2 N.y/ and SF  N.S.a; c//. It remains to prove that S.a; c/ is a closed set in C 1 .J; Rn /. Let yn 2 S.a; c/ such that yn converges to y in C 1 .J; Rn /. For every n 2 N; there exists vn .t / 2 F .t; yn .t /; yn0 .t //; a.e. t 2 J such that Z t

yn .t / D a C t c C

.t  s/vn .s/ds

(3.17)

vn .s/ds:

(3.18)

0

and yn0 .t / D c C

Z

t 0

f/B.0; 1/; hence fvngn2N is inteM; M Theorem 3.16 (2) implies that vn .t / 2 p.t / .f grably bounded. As a consequence, there exists a subsequence, still denoted fvn gn2N,

127

Section 3.2 Second-order differential inclusions

which converges weakly to some limit v.:/ 2 L1 . Mazur’s lemma implies the existence of a double sequence f˛n;k gk;n2N such that 8 n 2 N, 9 k0 .n/ 2 N W ˛n;k D P 0; 8 k  k0 .n/, 1 ˛ D 1; 8 n 2 N, and the sequence of convex combinaP1 kDn n;k tions gn .:/ D kDn ˛n;k vk .:/ converges strongly to v in L1 . Since F takes convex values, using Lemma 6.51 we obtain that v.t / 2

\

fgk .t /; k  ng; a.e. t 2 J

n1



\

cofvk .t /; k  ng

n1



\

cof

n1

[

(3.19)

F .t; yk .t /; yk0 .t //g

kn

D co.lim sup F .t; yk .t /; yk0 .t ///: k!1

Moreover, F is u.s.c. with compact values; then by Lemma 6.48 we have lim sup F .t; yn .t /; y 0 .t // D F .t; y.t /; y 0 .t //;

for a.e. t 2 J:

n!1

This with (3.19) implies that v.t / 2 co F .t; y.t /; y 0 .t //. Since F .:; :/ has closed, convex values, we deduce that v.t / 2 F .t; y.t /; y 0 .t //; for a.e. t 2 J . Let Z

t

z.t / D a C t c C

.t  s/v.s/ds;

t 2 J:

0

The Lebesgue dominated convergence theorem implies that Z

b

kyn  zk1  .b C 1/

kgn .s/  v.s/kds ! 0;

as n ! 1:

0

Hence, y.t / D z.t /; t 2 J , proving that S.:; :/ 2 Pcp .C 1 .J; Rn /.

3.2.2 The nonconvex case In this part, we present a second existence result to problem (3.14) with a nonconvex valued right-hand side. Theorem 3.18. Assume that the multi-map F W J  Rn  Rn ! Pcp .Rn / is such that (1) t 7! F .t; x; y/ is measurable for each x; y 2 Rn .

128

Chapter 3 Solution sets for differential equations and inclusions

(2) There exists a function l 2 L1 .J; RC / such that for a.e. t 2 J and all x1 ; y1 ; x2 ; y2 2 Rn , Hd .F .t; x1 ; y1 /; F .t; x2 ; y2 //  l.t /.kx1  y1 k C jx2  y2 k/; with Hd .0; F .t; 0; 0//  l.t /;

for a.e. t 2 J:

Then problem (3.14) has at least one solution. Proof. Let N W C 1 .J; Rn / ! P.C 1 .J; Rn // be defined as in Theorem 3.16. We shall show that N satisfies the assumptions of Lemma 1.79. The proof will be given in two steps. Step 1. N.y/ 2 Pcl .C 1 .J; Rn // for each u 2 C 1 .J; Rn /. Indeed, let fyngn2N 2 N.y/ be such that yn ! y in C 1 .J; Rn /. Then there exists vn 2 SF ;y such that for each t 2 J Z

t

yn .t / D a C t c C

.t  s/vn .s/ds and

0

yn0 .t /

Z

t

DcC

vn .s/ds: 0

Let vn be fixed; from Lemma 6.91 there exists wn .:/ 2 F .:; .y .:/; y0 .:// such that kvn.t /  wn .t /k  d.vn .t /; F .t; y .t /; y0 .t ///: Since F .; / has compact values, there exists a subsequence wn ./ such that wn ./ ! w./;

as n ! 1;

wn .t / 2 F .t; y .t /; y0 .t //;

a.e. t 2 J;

and kvn.t /  wn .t /k  d.vn .t /; F .t; y .t /; y0 .t ///  p.t /kyn  y kC 1 : This implies that vn .:/ converges to w.:/ 2 F .:; y .:/; y0 .://. Now, we prove that w 2 L1 .J; Rn /: Using the fact that F .; / is p-Lipschitz and Lemma 6.91, we get kvn.t /k  d.vn .t /; F .t; 0; 0// C 2l.t /  2l.t / C l.t /kyn .t /k; Thus, kvn .t /k  .2 C M /l.t /; Hence,

Z

a.e. t 2 J:

b

kyn.t /  z.t /k 

kvn.s/  v.s/kds; 0

a.e. t 2 J:

129

Section 3.2 Second-order differential inclusions

where Z

t

.t  s/v.s/ds;

z.t / D a C t c C

t 2 J:

0

From the Lebesgue dominated convergence theorem we conclude that y D z. This implies that v.t / 2 F .t; y.t /; y 0 .t //; a.e. t 2 J . Finally, Z

t

y .t / D a C t c C

.t  s/v.s/ds; 0

that is y 2 N.y/. Step 2. There exists  < 1 such that for each y; y 2 C 1 .J; Rn /:

Hd .N.y/; N.y //  ky  y k1 ;

Let y; y 2 C 1 .J; Rn / and h 2 N.y/. Then there exists v.:/ 2 F .:; y.:/; y 0 .:// such that for each t 2 J Z t .t  s/v.s/ds: h.t / D a C t c C 0

From 3.18 (2) we have Hd .F .t; y.t /; y 0 .t //; F .t; y .t /; y0 .t ///  l.t /Œky.t /  y .t /k C ky 0 .t /  y0 .t /k: Hence, there is u 2 F .t; y.t /; y 0 .t // such that kv.t /  uk  l.t /Œky.t /  y .t /k C ky 0 .t /  y0 .t /k;

t 2 J:

Consider the multi-map V W J ! P.Rn / given by V .t / D fu 2 Rn j kv.t /  uk  l.t /Œky.t /  y .t /k C ky 0 .t /  y0 .t /kg: Since the multi-valued operator V .t / D V .t / \ F .t; y .t /; y0 .t // is measurable (see Proposition III.4 in [107]), there exists v.t / a measurable selection for V . Then v.t / 2 F .t; y .t /; y0 .t // and kv.t /  v.t /k  l.t /Œky.t /  y .t /k C ky 0.t /  y0 .t /k; Let us define for each t 2 J Z

t

h.t / D a C t c C

.t  s/v.s/ds: 0

for each t 2 J:

130

Chapter 3 Solution sets for differential equations and inclusions

For  > 0, we have Z

t

.t  s/l.s/Œky.s/  y .s/k C ky 0 .s/  y0 .s/k ds

kh.t /  h.t /k  0

b   

Z

b e 

t

.e

Rs 0

l.u/du 0

0 Rt

0 l.s/ds

/ dsky  y kB1

ky  y kB1

and 0

kh0 .t /  h .t /k 

Z

t

0

1   

l.s/Œky.s/  y .s/k C ky 0 .s/  y0 .s/k ds

Z

1 e 

t

.e

Rs 0

l.u/du 0

0 Rt

0 l.s/ds

Then kh  hkB1 

/ dsky  y kB1

ky  y kB1 :

1Cb ky  y kB1 ; 

where k  kB1 is the Bielecki type norm on C 1 .J; Rn / defined by kykB1 D supfky.t /ke  l.t /; t 2 Œ0; bg C supfky 0.t /ke 

Rt 0

l.s/ ds

; t 2 J g:

By an analogous relation, obtained by interchanging the roles of y and y ; we obtain that 1Cb Hd .N.y/; N.y //  ky  y kB1 :  For  > 1 C a, N is a contraction and thus, by Lemma 1.79, N has a fixed point y which is a solution of problem (3.14).

3.2.3 Solution sets Let S.a; c/ denote the set of all solutions of problem (3.14). We are in a position to state and prove another characterisation of the geometric structure of S.a; c/. Theorem 3.19. Let F W J  Rn  Rn ! Pcp;cv .Rn / be a Carathéodory and an mLL-selectionable multi-valued map which satisfies condition 3.16 (2). Then, for every a; c 2 Rn ; the set S.a; c/ is contractible.

131

Section 3.2 Second-order differential inclusions

Proof. Let f  F be a measurable, locally Lipschitz selection and consider the single-valued problem ( y 00 D f .t; y.t /; y 0 .t //; a.e. t 2 J; (3.20) y.0/ D a; y 0 .0/ D c: As in [249], we can prove that the Cauchy problem (3.20) has exactly one solution for every a; c 2 Rn . Define the homotopy h W S.a; c/  Œ0; 1 ! S.a; c/ by ( y.t /; for 0  t  ˛b; h.y; ˛/.t / D x.t /; for ˛b < t  b; where x D S.f; a; c/ is the unique solution of problem (3.20). In particular, ( y; for ˛ D 1; h.y; ˛/ D x; for ˛ D 0: To prove that h is a continuous homotopy, let .yn ; ˛n / 2 S.a; b/  Œ0; 1 be such that .yn ; ˛n / ! .y; ˛/; as n ! 1. We shall prove that h.yn ; ˛n / ! h.y; ˛/. We have ( yn .t /; for t 2 Œ0; ˛n b; h.yn ; ˛n /.t / D x.t /; for t 2 .˛n b; b: (a) If lim ˛n D 0; then n!1

for t 2 Œ0; b:

h.y; 0/.t / D x.t /; Hence,

N Œ0;˛n b kh.yn ; ˛n /  h.y; ˛/k1  kyn  yk1 C kyn  xk which tends to 0 as n ! C1. The case when lim ˛n D 1 is treated similarly. n!1

(b) If ˛n 6D 0 and 0 < lim ˛n D ˛ < 1,] then we may distinguish between two n!1

sub-cases: (i) yn 2 S.a; c/ implies the existence of vn 2 SF ;yn such that for t 2 Œ0; ˛n b Z

t

yn .t / D a C t c C

.t  s/vn .s/ds: 0

As a consequence, there exists a subsequence, still denoted fvngn2N , which converges weakly to some limit v.:/ 2 L1 . Mazur’s lemma implies the existence of a double sequence .˛n;k /k;n2N such that 8 n 2 N , 9 k0 .n/ 2 N W ˛n;k D

132

Chapter 3 Solution sets for differential equations and inclusions

P 0; 8 k  k0 .n/, 1 ˛n;k D 1; 8 n 2 N , and the sequence of convex comPkDn 1 binations gn .:/ D 1 kDn ˛n;k vk .:/ converges strongly to v in L . Since F takes convex values, using Lemma 6.51, we obtain that v.t / 2

\

fgk .t /; k  ng; a.e. t 2 J

n1



\

cofvk .t /; k  ng

n1



\

cof

n1

[

F .t; yk .t /; yk0 .t //g

(3.21)

kn

D co.lim sup F .t; yk .t /; yk0 .t ///: k!1

Moreover, F is u.s.c. with compact values; then Lemma 6.48 implies that lim sup F .t; yn .t /; y 0 .t // D F .t; y.t /; y 0 .t //;

for a.e. t 2 J:

n!1

This with (3.19-12) implies that v.t / 2 co F .t; y.t /; y 0 .t //. Since F .:; :/ has closed convex values, we deduce that v.t / 2 F .t; y.t /; y 0 .t //;

for a.e. t 2 J:

By the Lebesgue dominated convergence theorem, we deduce that Z

t

y.t / D a C t c C

.t  s/v.s/ds;

t 2 J:

0

(ii) If t 2 .˛n b; b, then N /: h.yn ; ˛n /.t / D h.y; ˛/.t / D x.t Thus, kh.yn ; ˛n /  h.y; ˛/k1 ! 0;

as n ! 1:

Therefore, h is a continuous function, proving that S.a; c/ is contractible to the point x D S.f; a; c/. Another result is given by: Theorem 3.20. Let F W J  Rn  Rn ! Pcp;cv .Rn / be a Carathéodory, a C aselectionable multi-valued map which is integrably bounded. Then the solution set S.a; c/ is Rı -contractible.

133

Section 3.2 Second-order differential inclusions

Proof. Step 1. Replace the single-valued homotopy h W S.a; c/  Œ0; 1 ! S.a; c/ in Theorem 3.19 by the multi-valued homotopy … W S.a; c/  Œ0; 1 ! P.S.a; c// defined by n n x.t /; ….x; ˛/ D y 2 S.a; c/ j y.t / D z.t /;

if 0  t  ˛b o ; if ˛b  t  b

where f  F and z 2 S.f; ˛b; x/ is the solution set of the problem: (

y 00 .t / D f .t; y.t /; y 0 .t //; y.˛b/ D x.˛b/;

a.e. t 2 Œ˛b; b; y 0 .˛b/ D x.˛b/:

From the definition of …; we have ….x; 0/ D S.f; 0; x/ and x 2 ….x; 1/ for every x 2 S.a; c/. By the same argument used in Theorem 3.11, we can prove that ….; / is u.s.c. Step 2. We claim that ….x; ˛/ is an Rı -set for each fixed ˛ 2 Œ0; 1 and x 2 S.a; c/. Clearly, ….x; ˛/ D S.x˛b/; x 0 .˛b//. Since F is integrably bounded, then from Theorem 6.125, F is -Ca-selectionable; hence there the exists a decreasing sequence of multi-valued maps Fk W Œ0; bRn Rn ! P.Rn / .k 2 N/ which have Carathéodory selections and satisfy FkC1 .t; u; v/  Fk .t; u; v/ and F .t; u; v/ D

1 \

for a.e. t 2 Œ0; b and all u; v 2 Rn

Fk .t; u; v/; t 2 Œ0; b; u; v 2 Rn :

kD0

Then ….x; ˛/ D

1 \

S.Fk ; x/:

kD0

e is integrably bounded, then by Theorem 6.125, for every k D 0; 1; : : : ; F ek Since F has an m  LL-selection. Using Theorem 3.19, we deduce that the sets S.Fk ; x/ are contractible. Therefore, ….x; ˛/ is an Rı -set. As a consequence, all properties in Definition 5.72 are met which implies that the set S.a; c/ is Rı -contractible, ending the proof of the theorem. Finally, we have Theorem 3.21. Let F W J  Rn  Rn ! Pcp;cv .Rn / be a multi-valued map. Assume that all conditions of Theorem 3.18 are satisfied. Then the solution set S.a; c/ is AR.

134

Chapter 3 Solution sets for differential equations and inclusions

Proof. Consider the operator N W C 1 .J; Rn / ! P.C 1 .J; Rn // defined for y 2 C 1 .J; Rn / by Z t o n 1 n .t  s/v.s/ds; t 2 J ; N.y/ D h 2 C .J; R / j h.t / D a C t c C 0

where v 2 SF ;y D fv 2 L1 .J; Rn / j v.t / 2 F .t; y.t /; y 0 .t //; a.e. t 2 J g. Using the fact that F .:; :; :/ has a convex and compact values and by 3.18 (1)–3.18 (2), for every y 2 C 1 .J; Rn /, we have N.y/ 2 Pcv;cp .C 1 .J; Rn //. Using some Bielecki type norm on C 1 .J; Rn /, we can prove that N is a contraction. Hence, from Theorem 1.84, the solution set S.a; c/ D Fix.N / is a nonempty, compact AR-space.

3.3 Higher-order differential inclusions Consider the problem (

y .k/ .t / 2 F .t; y.t /; y 0 .t /; : : : ; y .k1/.t //; 0

y.0/ D y0 ; y .0/ D y1 ; : : : y

.k1/

a.e. t 2 J WD Œ0; b;

.0/ D yk1 ;

(3.22)

where F W J  E k ! Pcp .E/ is a multi-valued map with compact values, .E; j:j/ is a separable Banach space, y0 ; y1 ; : : : ; yk1 2 E, and k 2 f1; 2; : : : g. We state our main result about the topological structure of solution sets in case F satisfies a Lipschitz condition. Theorem 3.22. Assume that (1) F is bounded, (2) for every x 2 E, F .; x/ is measurable, (3) F is Lipschitzian with respect to the second argument, i.e., 9 L > 0 j Hd .F .t; x/; F .t; y//  L

n X

jxi  yi j;

i D1

for all x D .x1 ; : : : ; xk /; y D .y1 ; : : : ; yk / 2 E k . Then the solution set S.F; y0 ; y1 ; : : : ; yk1 / for problem (3.22) is an AR-space. Proof. Define a family of single-valued maps hj W L1 .Œ0; b; E/ ! AC j .j D 0; : : : ; k  1/ as follows: Z t Z s1 Z sj tj ::: z.s/dsdsj : : : ds1 ; .hj .z//.t / D y0 C ty1 C    C yj C jŠ 0 0 0

135

Section 3.4 Neutral differential inclusions

where AC j is the Banach space of absolutely continuous functions with norm kuk D kukC j C sup essfju.j C1/ .t /jg: t 2J

Now, define a multi-valued map

W L1 .Œ0; b; E/ ( L1 .Œ0; b; E/ by

.x/ D fz 2 L1 .Œ0; b; E/ j z.t / 2 F .t; hk1 .x/.t /; : : : ; h0 .x/.t //g; for t 2 Œ0; a: From the Kuratowski–Ryll-Nardzewski selection theorem and the fact that F is bounded, is well defined and has decomposable values in L1 .Œ0; b; E/. Moreover, hk1 .Fix. // D S.F; y0 ; y1 ; : : : ; yk1 /: Moreover, observe that hk1 is a homeomorphism onto its image. Using [Theorem 1.83, Chapter 1], it is then sufficient to prove that is a multi-valued contraction. For this, notice that for x; y 2 L1 .Œ0; b; E/ and every z 2 .x/, there exists t 2 .y/ such that jz  t j1  ˛jx  yj1 ; where ˛ 2 Œ0; 1/ and jxj1 D supt 2J essfe L˛kt jx.t /jg is the Bielecki type norm in L1 .Œ0; b; E/. Then our claim follows from 3.22 (3).

3.4 Neutral differential inclusions It is well known that systems with after-effect, with time lag or with delay are of great theoretical interest and form an important class with regard to their applications. This class of systems is described by functional differential equations and inclusions also called differential equations and inclusions with deviating arguments. Among functional differential equations, one may distinguish some special classes of equations, retarded functional differential equations, advanced functional differential equations, and neutral functional equations and inclusions. In particular, retarded functional differential equations and inclusions describe those systems or processes whose rate of change of state is determined by their past and present states. Such equations are frequently encountered as mathematical models of most dynamical processes in mechanics, control theory, physics, chemistry, biology, medicine, economics, atomic energy, information theory, etc. Since the 1960s, many books have been published on delay differential equations; we refer the reader, for example, to the books by Burton [95,96], Èl’sgol’ts [167], Èl’sgol’ts and Norkin [168], Gopalsmy [205], Azbelez et al. [44], Hale [241], Hale and Lunel [242], Kolmanovskii and Myshkis [295], Kolmanovaskii and Nosov [296], Krasovskii [301], Yoshizawa [498], and the references listed in these books.

136

Chapter 3 Solution sets for differential equations and inclusions

We will consider in this section the first-order neutral differential inclusion: 8 < d Œy.t /  g.t; y / 2 F .t; y /; a.e. t 2 Œ0; b t t dt (3.23) : y.t / D .t /; t 2 Œr; 0; where 0 < r < 1; F W Œ0; bC.Œr; 0; Rn / ! P.Rn / is a multifunction, g W Œ0; b C.Œr; 0; Rn / ! Rn is a given function, and 2 C.Œr; 0; Rn /. For any function y defined on Œr; b and any t 2 Œ0; b; yt refers to the element of C.Œr; 0; Rn / defined by yt ./ D y.t C /;  2 Œr; 0I this means that the function yt represents the history of the state from time t  r up to the present time t . Denote by kukD D

sup ku./k:

2Œr;0

We shall prove existence of solution in both the convex and nonconvex cases and then investigate the topological structure of solution sets. First, we give Definition 3.23. A function y 2 AC.Œ0; b; Rn / \ C.Œr; 0; Rn / is said to be a solution of (3.23) if there exists v 2 L1 .Œ0; b; Rn / such that v.t / 2 F .t; yt / a.e. on J , Œy.t /  g.t; yt /0 D v.t / a.e. t 2 Œ0; b; and y.t / D .t /; for t 2 Œr; 0.

3.4.1 The convex case We first prove a general existence principle: Theorem 3.24. Suppose that: (1) F W J  C.Œr; 0; Rn / ! Pcp;cv .Rn / is a L1 -upper Carathéodory multi-valued map, i.e., the multi-map t 7! F .t; z/ is measurable for every z and for a.e. t 2 Œ0; b, the multi-map y 7! F .t; z/ is u.s.c. and satisfies 8 r > 0; 9 'r 2 L1 Œ0; b;

such that kF .t; z/k  'r .t /;

for a.e. t 2 Œ0; b and every z 2 C.Œr; 0; Rn / with kzkD  r . (2) The function g is continuous and for any bounded subset B in C.Œr; b; Rn /, the set ft ! g.t; yt / j y 2 Bg is equicontinuous in C.Œ0; b; Rn / and there exist constants 0  d1 < 1; d2  0 such that kg.t; u/k  d1 kukD C d2 ; t 2 Œ0; b; u 2 C.Œr; 0; Rn /I

137

Section 3.4 Neutral differential inclusions

(3) there is a constant M; independent of  2 .0; 1/, such that kyk1  M for any solution y to 8 < d Œy.t /  g.t; y / 2 F .t; y /; t t dt : y.t / D  .t /;

a.e. t 2 Œ0; b

(3.24)

t 2 Œr; 0;

Then, problem (3.23) has at least one solution. Here kyk1 D max.kyk0; kykD /; where kyk0 D sup ky.t /k. t 2Œ0;b

Proof. Consider the multi-valued map N W C.Œr; b; Rn / ! P.C.Œr; b; Rn // defined by N.y/ D fh 2 C.Œr; b; Rn /g with  h.t / D

t 2 Œr; 0 t 2 Œ0; b

.t /; Rt .0/  g.0; / C g.t; yt / C 0 v.s/ds;

where v 2 SF ;y D fv 2 L1 .J; Rn / j v.t / 2 F .t; yt /;

for a.e. t 2 Œ0; bg:

We shall show that N is a completely continuous multi-valued map, u.s.c. with convex closed values. The proof will be given in several steps. First, notice that N.y/ is convex for each y 2 C.Œr; b; Rn / because F has convex values. Step 1. N maps bounded sets into bounded sets in C.Œr; b; Rn /. It is enough to show that there exists a positive constant ` such that for each h 2 N.y/; y 2 Bq D fy 2 C.Œr; b; Rn / j kyk1  qg, one has khk1  `. If h 2 N.y/, then there exists v 2 SF ;y such that for each t 2 Œ0; b, we have Z

t

h.t / D .0/  g.0; / C g.t; yt / C

v.s/ds: 0

By 3.24 (1) and 3.24 (2), we have for each t 2 Œ0; b Z kh.t /k  .1 C d1 /k kD C d2 C d1 kyt k1 C

t

kv.s/kds 0

 .1 C d1 /kk kD C d2 C d1 q C k'q kL1 :

138

Chapter 3 Solution sets for differential equations and inclusions

Then for each h 2 N.Bq / we have khk0  .1 C d1 /k kD C d2 C d1 q C k'q kL1 WD ` and khk1  max.`; k kD /: Step 2. N maps bounded sets into equicontinuous sets of C.Œr; b; Rn /. First, let 1 ; 2 2 Œ0; b; 1 < 2 and Bq D fy 2 C.Œr; b; Rn / j kyk1  qg be a bounded set of C.Œr; b; Rn /. For each y 2 Bq and h 2 N.y/, there exists v 2 SF ;y such that Z t v.s/ds; t 2 Œ0; b: h.t / D .0/  g.0; / C g.t; yt / C 0

Thus,

Z jh.2 /  h.1 /j  kg.1 ; y 1 /  g.2 ; y 2 /k C

2

'q .s/ds:

1

Since ft ! g.t; yt / j y 2 Bq g is equicontinuous and 'q 2 L1 .Œ0; b; Rn /, the righthand side of the above inequality tends to zero independently of y as 2  1 ! 0. The equicontinuity for the cases 1 < 2  0 and 1  0  2 are obvious. As a consequence of 3.24 (2), Steps 1–3 together with the Ascoli–Arzelá theorem, we conclude that N W C.Œr; b; Rn / ! Pcp;cv .C.Œr; b; Rn // is a locally compact multi-valued map. Step 3. N has a closed graph. Let yn ! y ; hn 2 N.yn /; and hn ! h , as n ! 1. hn 2 N.yn / means that there exists vn 2 SF ;yn such that Z

t

hn .t / D .0/  g.0; / C g.t; .yn /t / C

vn .s/ds;

t 2 Œ0; b:

0

We must prove that there exists v 2 SF ;y such that Z

t

h .t / D .0/  g.0; / C g.t; yt / C

v .s/ds;

t 2 Œ0; b:

0

Since g.:; y: / is a continuous function and fg.t; .yn /t /; t 2 Œ0; bg [ fg.t; yt /; t 2 Œ0; bg is compact, we have  .hn  .0/  g.0; /  g.t; .yn /t //   .h  .0/  g.0; /  g.t; .y /t //  ! 0; 0

as n ! 1. Consider the linear continuous operator

W L1 .Œ0; b; Rn / ! C.Œ0; b; Rn /

139

Section 3.4 Neutral differential inclusions

defined by

Z

t

g 7! .v/.t / D

v.s/ds: 0

From Lemma 6.155, ı SF is a closed graph operator. Moreover, hn .t /  .0/  g.0; / 2 .SF ;yn /: Since yn ! y ; it follows that Z

t

h .t / D .0/ C g.0; / C g.t; .y /t / C

g .s/ds; 0

for some g 2 SF ;y . As a consequence of Step 2–4 together with the Ascoli–Arzéla theorem and Lemma 6.23, we conclude that N W C.Œr; b; Rn / ! Pcp;cv .C.Œr; b; Rn // is u.s.c. In addition, from 3.24 (3) we have that for every y 2 N.y/;  2 .0; 1/ ) kyk1  M: The nonlinear alternative of Leray–Schauder type ([Corollary 1.74, Chapter 1]) implies that N has a fixed point y 2 C.Œr; b; Rn / solution of problem (3.23). Notice that fixed points of N are a mild solution of problem (3.23), hence satisfying Definition 3.23. In the following theorem, we prove an existence result when the nonlinearity satisfies a Nagumo type growth. Also, the compactness of the solution set is studied together with the upper continuous dependence of the solution set upon initial data. Theorem 3.25. Assume that Hypotheses 3.24 (1) and 3.24 (2) are satisfied together with the following conditions: (1) There exists a nondecreasing continuous function W Œ0; 1/ ! .0; 1/ and 1 C n p 2 L .J; R / such that for each u 2 C.Œr; 0; R / and a.e. t 2 J , we have kF .t; u/kP  p.t / .kukD / with 1 1  d1

Z

where KD

Z

b

1

p.s/ds < 0

K

du ; .u/

.1 C d1 /k kD C d2 : 1  d1

Then problem (3.23) has at least one solution, the solution set S.F; / is compact, and the operator solution ! S.F; / is u.s.c.

140

Chapter 3 Solution sets for differential equations and inclusions

Proof. Part 1. A priori bounds on solutions. Let y be a possible solution of (3.23). We show that 3.25 (1) implies the conditions 3.24 (1)–3.24 (3) in Theorem 3.24. For each t 2 Œ0; b, we have Z

t

y.t / D .0/  g.0; / C g.t; yt / C

f .s/ds;

where f 2 SF ;y :

0

From 3.24 (2) and 3.25 (1), we obtain the estimate: Z t ky.t /k  .1 C d1 /k k1 C d2 C d1 kykD C  p.s/ .kys k1 /ds;

t 2 Œ0; b:

0

(3.25) Consider the function  defined by .t / D supfky.s/k j r  s  t g; 0  t  b: Let t  2 Œr; t  be such that .t / D ky.t /k. If t  2 Œ0; b, then by (3.25), we have for t 2 Œ0; b .t / 

1  .1 C d1 /k kD C d2 C 1  d1

Z

t

 p.s/ ..s//ds :

0

If t  2 Œr; 0, then .t / D k kD and the previous inequality holds. Let us take the right-hand side of the above inequality as v.t /. Then .t /  v.t /; t 2 Œ0; b K D v.0/ D and v 0 .t / D Since

.1 C d1 /k kD C d2 ; 1  d1

1 p.t / ..t //; 1  d1

t 2 Œ0; b:

is nondecreasing, we get v 0 .t / 

1 p.t / .v.t //; 1  d1

t 2 Œ0; b:

By integration, we obtain that, for t 2 Œ0; b, Z

v.t / v.0/

du 1  .u/ 1  d1

Z

b

p.s/ds: 0

In view of 3.25 (1), we obtain .t /  v.t / 

1



1 1  d1

Z

b 0

 p.s/ds WD M0 ;

141

Section 3.4 Neutral differential inclusions

where

Z

s

.s/ D K

du : .u/

Therefore, sup ky.t /k  max.M0 ; k kD /:

t 2Œr;b

Consequently, for every possible solution y to y 2 N.y/ for some  2 .0; 1/, we have the estimate kyk1  max.M0 ; k kD /: The fact that S. / 6D ; follows from Theorem 3.24. Part 2. The multi-valued map is u.s.c. Step 1. We prove that the set S. / D fy 2 C.Œr; b; Rn / j y solution of the problem .3.23/g is compact. Let yn 2 S. /; then there exists vn 2 SF ;yn ; n 2 N such that y.t / D .t /; t 2 Œr; 0 and Z

t

yn .t / D .0/  g.0; / C g.t; .yn /t / C

vn .s/ds; t 2 Œ0; b: 0

It is clear that kyn k1  K;

for every n 2 N:

Arguing as in Theorem 3.24, we can prove that the set fyn j n 2 Ng is equicontinuous. Then, from the Ascoli–Arzéla theorem, we conclude that fyn j n 2 Ng is compact in C.Œr; b; Rn /. Hence, there exists a subsequence converging to y in C.Œr; b; Rn /. 3.25 (1) implies that vn .t / 2 p.t / .K/B.0; 1/. Hence, there is a subsequence, still denoted .vn /n2N , which converges weakly to v./ 2 L1 . Mazur’s lemma implies the Pk.n/ existence of ˛in  0; i D n; : : : ; k.n/ such that i D1 ˛in D 1 and the sequence of Pk.n/ convex combinations fn ./ D i D1 ˛in vi ./ converges strongly to v in L1 . Since F takes convex values, using [Lemma 6.51, Chapter 6], we obtain that \ v.t / 2 ffk .t /; k  ng; a.e. t 2 Œ0; b n1



\

cofvk .t /; k  ng

n1



\

n1

cof

[

(3.26) F .t; .yt /k /g

kn

D co.lim sup F .t; .yt /k //: k!1

142

Chapter 3 Solution sets for differential equations and inclusions

Moreover, F is u.s.c. with compact values; then by [Lemma 6.48, Chapter 6], we have lim sup F .t; .yt /n / D F .t; yt /;

for a.e. t 2 Œ0; b:

n!1

This with (3.26) implies that v.t / 2 co F .t; yt /. Since F .:; :/ has closed, convex values, we deduce that v 2 F .t; yt /; for a.e. t 2 J . Using the fact that g is a continuous function and by the Lebesgue dominated convergence theorem, we conclude that Z

t

v.s/ds; t 2 Œ0; b; y.t / D .t /;

y.t / D .0/g.0; /Cg.t; yt /C

t 2 Œr; 0:

0

Hence, y 2 S. / and thus S. / is compact in C.Œr; b; Rn /. Step 2. S is quasi-compact. Let B  C.Œr; 0; Rn / be a compact set; we show that S.B/ is a relatively compact set in C.Œr; b; Rn /. Let fyn j n 2 Ng 2 S.B/; then there exist n 2 B and n 2 SF ;yn such that yn .t / D n .0/  g.0; n / C g.t; .yn /t / C yn .t / D n .t /; t 2 Œr; 0;

Rt 0

vn .s/ds; t 2 Œ0; b;

where vn 2 SF ;yn . Since B is compact, there exists a subsequence of . n / which converges to . As in Part 1, there exists M > 0 such that kyn k1  M;

for every n 2 N:

From 3.25 (1) and the compactness of f n j n 2 Ng in C.Œr; 0; Rn /, we can prove that fyn j n 2 Ng is compact in C.Œr; b; Rn /. Hence, S.B/ is relatively compact, hence S.:/ is quasi-compact. Finally, it is easy to check that Gr .S/ is closed. Steps 1 and 2 together with Theorem 6.24 imply that S.:/ is u.s.c.

3.4.2 The nonconvex case We now study the case where F is not necessarily convex valued. Our approach here is based on a nonlinear alternative of Leray–Schauder type (Corollary 1.18, Chapter 1) combined with a selection theorem due to Bressan, Colombo, and Fryszkowski (Theorem 6.138, Chapter 6) for lower semi-continuous multi-valued operators with decomposable values. In this section, assume that E is a separable Banach space. Theorem 3.26. Suppose that 3.24 (2) and 3.25 (1) hold together with the following conditions: (1) F W Œ0; b  C.Œr; 0; Rn / ! P.Rn / is a nonempty compact-valued multi-map such that:

143

Section 3.4 Neutral differential inclusions

(a) .t; u/ 7! F .t; u/ is L ˝ B measurable; (b) u 7! F .t; u/ is lower semi-continuous, for a.e. t 2 Œ0; b; Then the neutral initial value problem (3.23) has at least one solution. Proof. Assumptions 3.25 (1) and 3.26 (1) imply that F is of lower semi-continuous type. Then from Theorem 6.138, there exists a continuous selection f W C.Œr; b; Rn / ! L1 .Œ0; b; Rn / such that f .y/ 2 F.y/ for all y 2 C.Œr; b; Rn /. Consider the problem 8 < d Œy.t /  g.t; y / D f .y /; t 2 Œ0; b; t t dt : y.t / D .t /; t 2 Œr; 0:

(3.27)

It is clear that if y 2 C.Œr; b; Rn / is a solution of problem (3.27), then y is a solution to problem (3.23). Consider the operator G W C.Œr; b; Rn / ! C.Œr; b; Rn / defined by ( .t /; if t 2 Œr; 0, Rt G.y/.t / WD .0/  g.0; / C g.t; yt / C 0 f .ys /ds; if t 2 Œ0; b: We first check that G is completely continuous. Firstly, we prove that G W C.Œr; b; Rn / ! C.Œr; b; Rn / is continuous. Let fyn gn2N be a sequence such that yn ! y in C.Œr; b; Rn /. Then there is a positive real number q such that kyn k1  q, for all n 2 N and kyk1  q, that is yn 2 B.0; q/ and y 2 B.0; q/. By the dominated convergence theorem and 3.24 (2), we have Z b kG.yn/  G.y/k1  kg.:; .yn //  g.:; y/k1 jf ..yn /s /  f .ys /jds 0

which tends to 0 as n ! 1I thus G is continuous. As in Theorems 3.24 and 3.25, we can show that G is compact, and there exists a positive constant M > 0 such that for all possible solutions y of problem (3.27), we have kyk1  M . Set U D fy 2 C.Œr; b; Rn / j kyk1 < M C 1g: From the choice of U there is no y 2 @U such that y D G.y/ for some  2 .0; 1/. As a consequence of the nonlinear alternative of Leray–Schauder type (Lemma 1.75, Chapter 1), we deduce that G has a fixed point y in U solution of (3.23).

144

Chapter 3 Solution sets for differential equations and inclusions

Using a fixed point theorem for contraction multi-valued operators, we now present another existence result for problem (3.23) with a right-hand side not necessarily convex. Theorem 3.27. Assume that: (1) F W Œ0; b  C.Œr; 0; Rn / ! Pcp .Rn / has the property that F .; u/ W Œ0; b ! Pcp .Rn / is measurable for each u 2 C.Œr; 0; Rn /; (2) Hd .F .t; u/; F .t; u//  l.t /ku  uk1 , for each t 2 Œ0; b and u; u 2 C.Œr; 0; Rn /, where l 2 L1 .Œ0; b; RC / and Hd .0; F .t; 0//  l.t /

for a.e. t 2 Œ0; b:

(3) g is a continuous function and there exists c 2 Œ0; 1/ such that kg.t; u/  g.t; u/k  cku  uk;

for t 2 Œ0; b and u; u 2 C.Œr; 0; Rn /:

Then problem (3.23) has at least one solution on Œr; b. Proof. We transform (3.23) into a fixed point problem. Consider the map N W C.Œr; b; Rn / ! P.C.Œr; b; Rn // defined by N.y/ D fh 2 C.Œr; b; Rn /g with ( .t /; Rt h.t / D .0/  g.0; / C g.t; yt / C 0 v.s/ds;

if t 2 Œr; 0 if t 2 Œ0; b;

where v 2 SF ;y D fv 2 L1 .Œ0; b; Rn / j v.t / 2 F .t; yt / for a.e. t 2 Œ0; bg: We shall show that N satisfies the assumptions of [Theorem 1.79, Chapter 1]. Step 1. N.y/ 2 Pcl .C.Œr; b; Rn //, for each y 2 C.Œr; b; Rn /. Indeed, let fyngn0 2 N.y/ be such that yn ! y in C.Œr; b; Rn /. Then there exists vn 2 SF ;y such that for each t 2 Œ0; b Z t yn .t / D .0/  g.0; / C g.t; yt / C vn .s/ds: 0

Since F has compact values, from 3.27 (2) we may pass to a subsequence, if necessary, to get that .vn / converges to v in L1 .J; Rn / and hence v 2 SF ;y . Then for each t 2 Œ0; b Z t y.t / D .0/  g.0; / C g.t; yt / C v.s/ds; 0

as n ! C1. So y 2 N.y/.

145

Section 3.4 Neutral differential inclusions

Step 2. There exists 0   < 1; such that Hd .N.y/; N.y//  ky  yk1 ; for each y; y 2 C.Œr; b; Rn /. Let y; y 2 C.Œr; b; Rn / and h1 2 N.y/. Then there exists v1 .t / 2 F .t; yt / such that for each t 2 Œ0; b Z t h1 .t / D .0/  g.0; / C g.t; yt / C v1 .s/ds: 0

From 3.27 (2) we have Hd .F .t; yt /; F .t; y t //  l.t /kyt  y t kD ; t 2 Œ0; b: Hence, there is some w 2 F .t; y t / such that kg1 .t /  wk  l.t /kyt  y t kD ;

t 2 Œ0; b:

Consider U W Œ0; b ! P.Rn /; given by U.t / D fw 2 Rn j kv1 .t /  wk  l.t /kyt  y t kD g: Since the multi-valued operator V .t / D U.t / \ F .t; y t / is measurable, there exists a function v2 .t / which is a measurable selection for V (see [107]). Hence, v2 .t / 2 F .t; y t / and kv1 .t /  v2 .t /k  l.t /ky  ykD ;

for each t 2 Œ0; b:

Now, for each t 2 Œ0; b, let us define

Z

h2 .t / D .0/  g.0; / C g.t; y t / C

t

v2 .s/ds: 0

Then, for t 2 Œ0; b, we have kh1 .t /  h2 .t /k  ckyt  y t kD C c

Z

t

kv1 .s/  v2 .s/k ds Z t sup ky.t C /  y.t C /k C l.s/kys  y s kD ds 0

r  0

0

Z

t

D ce L.t / ky  yk1 C Z Z

 sup jy.s C /  y.s C /j ds r  0

0 t

D ce L.t / ky  yk1 C

 l.s/

l.s/ky.s C   /  y.s C   /k ds;   2 Œr; 0

0 t





l.s/e L.sC / e  L.sC / ky.s C   /  y.s C   /k ds 0 Z t  L.s/0 1  ce L.t / ky  yk1 C ky  yk1 ds e  0 1  ce L.t / ky  yk1 C e L.t /ky  yk1 ; 

D ce L.t / ky  yk1 C

146

Chapter 3 Solution sets for differential equations and inclusions

where L.t / D

Rt 0

e l.s/ds, ( e l.t / D

and

0 l.t /;

t 2 Œr; 0 t 2 Œ0; b;

kyk1 D supfe  L.t /ky.t /k j t 2 Œr; bg:

Hence,

 1 ky  yk1 : kh1  h2 k1  c C  By the analogous relation, obtained by interchanging the roles of y and y; we obtain that  1 Hd .N.y/; N.y//  c C ky  yk1 :  If  > 1=1  c, then N is a contraction and thus, by [Theorem 1.79, Chapter], N has a fixed point y solution to (3.23).

3.4.3 Solutions sets Let S.F; / denote the set of all solutions of problem (3.23). Next, we present two results on the structure of S.F; /. The proofs are the same as in Section 3.1.6, hence omitted. Theorem 3.28. Let F W J  C.Œr; b; Rn / ! Pcp;cv .Rn / be a mLL-selectionable multi-valued map which satisfies condition 3.25 (1) both with (1) The function g is continuous and for any bounded set B in C.Œr; b; Rn /, the set ft ! g.t; yt / j y 2 Bg is equicontinuous in C.Œ0; b; Rn / and there exist constants 0  d1 < 1 such that kg.t; u/g.t; u /k  d1 kuu kD ; for t 2 Œ0; b; and u; u 2 C.Œr; 0; Rn /: Then, for every 2 C.Œr; 0; Rn /; the solution set S.F; / is contractible.

3.5 Nonlocal problems In this section, we present some existence results and properties of solution sets for ordinary differential inclusions with nonlocal conditions. The nonlocal condition y.0/C g.y/ D y0 can be more descriptive in physics with better effect than the classical initial condition y.0/ D y0 (see, e.g., [97, 98, 149]). For example, the multi-point condition: p X ci y.ti /; (3.28) g.y/ D kD1

147

Section 3.5 Nonlocal problems

where ci ; i D 1; : : : ; p; are given constants and 0 < t1 < t2 <    < tp ; describes the diffusion phenomenon of a small amount of gas in a transparent tube (see [149]). In this case, equation (3.28) allows the additional measurements at ti ; i D 1; : : : ; p. Nonlocal Cauchy problems for ordinary differential equations have been investigated by several authors, (see for instance [79, 98, 99, 354–358]). Nonlocal Cauchy problems, in the case where F is a multi-valued map, were studied by, e.g., Benchohra and Ntouyas [64–67] and Boucherif [79]. We are concerned with the first-order differential inclusion with nonlocal conditions: ( y 0 .t / 2 F .t; y.t //; a.e. t 2 J; (3.29) y.0/ D g.y/ C y0 ; where J D Œ0; b; y0 2 Rn , F W J  Rn ! P.Rn / is a multifunction, and g W C.J; Rn / ! Rn is a given function.

3.5.1 Main results The following two existence theorems are concerned with the case where F is convexvalued and nonconvex-valued respectively. For the proof, we refer to [225]. Theorem 3.29. Let F W J  Rn ! Pcp;cv .Rn / be a Carathéodory multi-valued map which satisfies some of the following assumptions: (1) There exist a function p 2 L1 .J; RC / and a continuous nondecreasing function W Œ0; 1/ ! Œ0; 1/ such that kF .t; z/k  p.t / .kzk/; with

Z

for a.e. t 2 J and each z 2 Rn ; Z

b

1

p.s/ds < 0

1

du  .u/

(2) The function g W C.J; R / ! R is continuous and either one of the following conditions holds: n

n

(a) there exist ˛ 2 Œ0; 1/ and ; ˇ  0 such that kg.y/k  kyk˛1 C ˇ; (b) there exists > 0 and 0 <  < 1 such that kg.y/k  kyk1 C : Then problem (3.29) has at least one solution. Moreover, the solution set S.y0 / is compact, and the multi-valued map S W Rn ! Pcp .C.J; Rn //: y0 7! S.y0 / is u.s.c. Theorem 3.30. Assume that (1) F W J  Rn ! Pcp .Rn /; t 7! F .t; x/ is measurable for each x 2 Rn ;

148

Chapter 3 Solution sets for differential equations and inclusions

(2) there exists a function l 2 L1 .J; RC / such that for a.e. t 2 J and all x; y 2 Rn ;

Hd .F .t; x/; F .t; y//  l.t /kx  yk; with Hd .0; F .t; 0//  l.t /;

for a.e. t 2 J;

(3) there exists c 2 Œ0; 1/ such that for every x; y 2 C.J; Rn /:

kg.y/  g.x/k  cky  xk1 ; Then problem (3.29) has at least one solution.

Now, we study the solution set for a first-order nonlocal differential equation. ( a.e. t 2 J D Œt0 ; b; y 0 .t / D f .t; y.t //; (3.30) y.t0 / D g.y/ C y0 ; where f W J  Rn ! Rn and g W C.J; Rn / ! Rn are given functions, and y0 2 Rn . Denote by S.f; y0 / the set of all solutions of problem (3.30). The following theorems provide Aronsajn type results for this problem. Theorem 3.31. Assume that (1) f W J  Rn ! Rn is a Carathéodory function. (2) There exist a function p 2 L1 .J; RC / and a continuous nondecreasing function W Œt0 ; 1/ ! Œ0; 1/ such that kf .t; x/k  p.t / .kxk/; with

Z

for a.e. t 2 J and each x 2 Rn ; Z

b

p.s/ds < t0

1 ky0 k

du  .u/

and 3.29 (3) holds. Then the set S.f; y0 / is Rı . Theorem 3.32 ([225]). Let F W J  Rn ! Pcp;cv .Rn / be a Carathéodory and an mLL-selectionable multi-valued map which satisfies condition .3.30:3/. Then, for every y0 2 Rn ; the set S.y0 / is contractible. Theorem 3.33 ([225]). Let F W J  Rn ! Pcp;cv .Rn / be a Carathéodory, C a-selectionable and integrable bounded multi-valued map. Assume the additional condition 3.29 (2) holds. Then the solution set S.y0 / is Rı -contractible and acyclic.

149

Section 3.5 Nonlocal problems

3.5.2 A viability problem Let K be a closed subset of Rn and x 2 Rn . Define the Bouligand tangent cone to K at x as 1 TK .x/ D fy 2 Rn j lim inf d.x C ty; K/ D 0g t t !0C Definition 3.34. A nonempty closed subset K  Rn is called a proximate retract provided there exists an open neighbourhood U of K in Rn and a retraction r W U ! Rn such that the following two conditions are satisfied: (i) r .x/ D x, for all x 2 K; (ii) kx  r .x/k D dist.x; K/, for every x 2 U . It is well known that the class of proximate retracts contains as special cases closed convex sets and C 2 -manifolds. In addition, it is easily seen that, for given K, if r W U ! K exists, then the retraction is unique. Since one can take a sufficiently small U , for example, by restricting U to U \ fx 2 Rn j dist.x; K/ < ıg; ı > 0; we may assume that kr .x/  xk  ı; for a given ı > 0 and x 2 U . The following lemmas play a key role in our considerations. Lemma 3.35 ([221]). Let K be a proximate retract. Then TK .r .x//  fy 2 Rn j hy; x  r .x/i  0g;

for any x 2 U;

where h; i denotes the inner product in Rn . Proof. If x 2 K, then r .x/ D x. Hence, for every y 2 TK .r .x//, we have hy; x  r .x/i D 0: This implies that TK .r .x//  fy 2 Rn j hy; x  r .x/i  0g: Hence, we assume that x 2 U nK and y 2 Rn are such that hy; x  r .x/i > 0; then lim

t !0C

d.r .x/ C ty; Rn nB.x; kx  r .x/k// > 0: t

150

Chapter 3 Solution sets for differential equations and inclusions

In fact, d.r .x/ C ty; @B.x; kx  r .x/k// t kx  r .x/k  kx C ty  r .x/k  lim t t !0C .kx  r .x/k  kx  ty  r .x/k/kx  r .x/  tyk D lim t kx  r .x/  tyk t !0C hty; x  r .x/  tyi  lim C t kx  r .x/  tyk t !0 t 2 kyk C t hy; x  r .x/i D lim t kx  r .x/ C tyk t !0C hy; x  r .x/i D lim t !0C kx  r .x/ C tyk hy; x  r .x/i > 0: D lim t !0C kx  r .x/k

lim

t !0C

Moreover, K  Rn nB.x; kx  r .x/k/, and thus d.r .x/ C ty; K/  d.r .x/ C ty; Rn nB.x; kx  r .x/k/: In conclusion, y 62 TK .r .x//. The following two auxiliary lemmas are stated without proofs. Lemma 3.36 ([221, Lemma 2.2]). Let K be a proximate retract, r W U ! K a metric retraction and s > 0 be such that K \ B.0; s/ 6D ;; where B.0; s/ is the closure of B.0; s/ in Rn . Then there exists "0 > 0 such that, for any 0 < "  "0 ; there exist subsets K  K"  U of Rn ; K" closed and U" open, and a continuous retraction r" W U" ! K" such that the following conditions are satisfied: \ (i) K" D K; 0 0 is chosen in such a way that O 2".K/  U . Then (1) O 2".K/ is an approximate retract;

151

Section 3.5 Nonlocal problems

(2) fy 2 Rn W hy; x  r .x/i  0g  TO " .K/ .x/; for all x 2 O" .K/I (3) TK .r .x//  fy 2 Rn W hy; x  r .x/i  0g; for all x 2 O " .K/. Now, we shall discuss the existence of viable solutions of some classes of differential inclusions with nonlocal conditions. Our approach here is based on [183,198,211]; we give natural generalisations of some of the results contained, therein for ( y 0 .t / 2 F .t; y.t //; a.e. t 2 Œ0; b; (3.31) y.0/ D g.y/ C y0 ; where F W Œ0; b  K ! P.Rn / is a multi-valued map, g W C.Œ0; b; K/ ! y0 C K is a given function, y0 2 Rn , and K  Rn is a proximate retract. Definition 3.38. A function y 2 AC.Œ0; b; Rn / is called a solution of (3.31) (or a viable solution), if there exists v 2 L1 .Œ0; b; K/, v.t / 2 F .t; y.t //; t 2 Œ0; b such that y 0 .t / D v.t /; t 2 Œ0; b, y.0/ D g.y/ C y0 , and y.t / 2 K; t 2 Œ0; b. Consider the modified problem: ( e .t; y.t //; y 0 .t / 2 F

a.e. t 2 Œ0; b;

y.0/ D g.y/ C y0 ;

(3.32)

e W Œ0; b  Rn ! Rn is defined by where F ( ˛.y/F .t; r .y//; if y 2 U and t 2 Œ0; b; e .t; y/ D F 0; if y 62 U and t 2 Œ0; b; where r W U ! K is the metric retraction and ˛ W Rn ! Œ0; 1 is a continuous Urysohn e is unique up to the choice function satisfying ˛jK  1 and ˛jRn nK  0. Obviously, F of the Urysohn function ˛. The following auxiliary result is easily verified. Proposition 3.39. If F W Œ0; b  K ! P.Rn / is a Carathéodory multifunction in K; e W J  Rn ! P.Rn / is a Carathéodory function on Rn . then F Definition 3.40. A map F W Œ0; b  K ! P.Rn / is called weakly tangent (respectively tangent) to K; if F .t; y/ \ TK .y/ 6D ;; (respectively F .t; y/  TK .y//, for y 2 K and a.e. t 2 Œ0; b. Lemma 3.41. Let F W J  Rn ! P.Rn / tangent to K be a proximate retract in Rn . If y 2 AC.Œ0; b; Rn / is a solution of problem (3.32) and y0 C g.y/ 2 K; then y.t / 2 K; for all t 2 J .

152

Chapter 3 Solution sets for differential equations and inclusions

Proof. Let d W J ! RC be defined by d.t / D d.y.t /; K/; t 2 Œ0; b. We show that d.t / D 0 for all t 2 Œ0; b. Since y.0/ D g.y/ C y0 ; we have d.0/ D 0, and from the definition of d , we can see that jd.t C h/  d.t /j  ky.t C h/  y.t /k;

t 2 Œ0; b;

(3.33)

and so d is an absolutely continuous function. Let t0 2 J be such that y 0 .t0 / 2 F .t0 ; y.t0 //. If y.t0 / 2 U; then y 0 .t0 / 2 TK .r .y.t0 ///, and lim inf

h!0C

d.r .y.t0// C hy 0 .t0 /; K/ D 0: h

(3.34)

We have d.y.t0 Ch/; K/d.y.t0 /; K/  ky.t0 Ch/y.t0 /hy 0 .t0 /kCd.y.t0 /Chy 0 .t0 /; K/; and from (3.34), we obtain lim inf

h!0C

d.t0 C h/  d.t0 /  0: h

(3.35)

If y.t0 / 62 U; then y 0 .t0 / D 0, and d.t0 Ch/d.t0 /  ky.t0 Ch/y.t0 /hy 0 .t0 /k ) lim inf h!0C

d.t0 C h/  d.t0 /  0: h

Since d is differentiable almost everywhere with derivative d 0 .t /  0; a.e. t 2 J; then it is nonincreasing. But d.0/ D 0 and d.t / D 0 for every t 2 J . The following example shows that if we remove the assumption that K is a proximate retract, then the set S.y0 / may even be disconnected. Example 3.42. Let S1 D f.x; y/ 2 R2 j .x  1/2 C y 2 D 1g S2 D f.x; y/ 2 R2 j .x C 1/2 C y 2 D 1g K D S1 [ S2 and define f W Œ0; 1  K ! R2 by ( .y; 1  x/; for .x; y/ 2 S1 f .t; x; y/ D .y; 1 C x/; for .x; y/ 2 S2 It is easy to see the set S.f; 0; 0/ is disconnected and hence is not Rı . Now, we examine the structure of the solution set.

153

Section 3.5 Nonlocal problems

Theorem 3.43. Let F W Œ0; b  K ! Pcp;cv .Rn / be a Carathéodory multifunction tangent to K an approximate retract to Rn and let g satisfy 3.30 (3). Assume further the following conditions hold: (1) there exist p 2 L1 .J; RC / and

W Œ0; 1/ ! .0; 1/ such that

kF .t; u/kP  p.t / .kuk/; with

Z

Z

b

1

p.s/ds < 0

1

for all u 2 Rn ; du  .u/

(2) The function g W C.J; Rn / ! Rn is continuous and L.C.J; Rn //  K; where L.y/ D g.y/ C y0 . Then problem (3.31) has at least one viable solution, the solution set is compact and it is an Rı -set. Proof. Consider the modified problem ( e .t; y.t //; y 0 .t / 2 F y.0/ D g.y/ C y0 ;

a.e. t 2 J;

(3.36)

e is as defined in problem (3.32). From Theorem 3.29, we know that problem where F (3.36) has at least one solution y.t / and the set e / D fy j y is a solution of problem .3.32/g S.F is compact. By Lemma 3.41, we have y.t / 2 K for all t 2 J . Hence, e .t; y.t // D F .t; y.t //; F

t 2 J:

This implies that y.:/ is a solution of problem (3.31) and e /: S.F; K/ D fyjy is a solution of the problem .3.31/g D S.F e/ is an Rı -set. As in [225], we can find some constant M > 0 such We show that S.F that for every y solution of problem (3.32), we have kyk1 < M . Let 8 e .t; y/; ˆ if kyk  M and t 2 J; < F   e F M .t; y/ D e t; My ; if kyk  M and t 2 J: ˆ : F kyk e/ D S.F eM /. e M is an integrably bounded Carathéodory map and S.F It is clear that F e From Theorem 6.125, F M is   mLL-selectionable; then there exists a sequence of ek gkD1 such that multi-valued maps fF ek .t; u/; e kC1 .t; u/  F F

for almost every t 2 Œ0; b; u 2 Rn ;

e e e M .t; u/ D \1 F and F kD1 k .t; u/. By Theorem 3.29, S.F k / is compact and contractible, and so S.F / is an Rı -set.

154

Chapter 3 Solution sets for differential equations and inclusions

Theorem 3.44. Let F W Œ0; b  K ! Pcp;cv .Rn / be a Carathéodory multifunction, weakly tangent to K an approximate retract to Rn , and assume that g satisfies 3.29 (2) and 3.30 (3). Then the solution set for problem (3.31) is a nonempty Rı -set. Proof. Let r W U ! K be the metric retraction. We choose " > 0 such that O2".K/  U and O2".K/ is a proximate neighbourhood retract. We consider the multi-valued map T W O".K/ ! Pcl .Rn / defined by T .x/ D fy 2 Rn j hy; x  r .x/i  0g: We can easily prove that T has a closed graph in O".K/  Rn . The multi-valued mapping F" W J  O".K/ ! Pcl .Rn / defined by F" .t; y/ D F .t; r .y// \ T .r .y// is a Carathéodory integrably bounded multi-valued map. From Lemma 3.37, F" satisfies the tangent condition to K. From Theorem 3.43, the set S.F" / is Rı . Finally, we can observe that, for every y0 2 K; we have S.F / D \1 nD1 S.F1=n /: Hence, S.F / is an Rı -set.

3.6 Hyperbolic differential inclusions In this section, we consider a class of hyperbolic differential inclusions, namely 8 2 < @ u.t; x/ 2 F .t; x; u.t; x//; @t @x : u.0; x/ D .x/; x 2 Jb ; u.t; 0/ D '.t /;

a.e. .t; x/ 2 Ja  Jb ;

(3.37)

t 2 Ja

where F W Ja  Jb  Rn ! P.Rn / is a multi-valued map with compact values, Ja  Jb WD Œ0; a  Œ0; b; 2 AC.Ja ; Rn /, ' 2 AC.Jb ; Rn /, and .0/ D '.0/. Partial differential equations have become the object of increasing investigation in many mathematical models of real world phenomena; see, e.g., the books by Evans [171], Lions [324], Wu [494], and the references therein. In the last three decades, several papers have been devoted to the study of hyperbolic partial differential equations with local and nonlocal initial conditions; see, for instance, [100, 316] and the references cited therein. For similar results with set-valued right-hand side, we refer to the papers by Byszewski and Papageorgiou [101], Papageorgiou [374], and Benchohra and Ntouyas [58, 67]. For the topological structure of hyperbolic differential equations and inclusions (see, e.g., [140, 144, 223]. Here we will present some existence results and study the topological structure for problem (3.37) in the cases when F has either convex or nonconvex values.

155

Section 3.6 Hyperbolic differential inclusions

3.6.1 Existence results Definition 3.45. A function u 2 C.Ja  Jb ; Rn / is said to be a solution of (3.37) if there exists v 2 L1 .Ja  Jb ; Rn / such that v.t; x/ 2 F .t; x; u..t; x/// is satisfied a.e 2 u.t;x/ D v.t; x/, a.e on Ja  Jb ; u.0; x/ D .x/ for each x 2 Jb and on Ja  Jb , @ @t @x u.t; 0/ D '.t / for each y 2 Ja . To get a priori estimates of solutions of problem (3.37), we make use of the following auxiliary lemma [368]: Lemma 3.46. Let 2 C.RC ; RC / be a nondecreasing function with .u/ > 0 for u > 0 and u.x; y/; a.x; y/ 2 C.J1  J2 ; RC /; b.x; y; s; t / 2 C.J1  J2  J2  J2 ; RC /; for x0  s  x  X; y0  t  y  Y; ˛ 2 C 1 .J1 ; J1 /; ˇ.y/ 2 C 1 .J2 ; J2 / be nondecreasing with ˛.x/  x on J1 ; ˇ.y/  y on J2 and k  0 be a constant. Assume that for .x; y/ 2 J1  J2 , Z u.x; y/  k C

˛.x/ Z ˇ.y/ h

a.s; t / .u.s; t // ˛.x0 /

ˇ.y0 /

Z

Z

s

i b.s; t; ; / .u.; //d d dt ds:

t

C ˛.x0 /

ˇ.y0 /

Then for x0  x  x1  N; y0  y  y1  R; we have u.x; y/  G 1 .G.k/ C A.x; y// ; where G 1 is the inverse function of Z G.r / D

r

r0

ds ; r > r0 > 0; .s/

and for .x; y/ 2 J1  J2 , Z A.x; y/ D

˛.x/ Z ˇ.y/  ˛.x0 /

Z

s

Z

t

a.s; t / C

ˇ.y0 /

˛.x0 /

 b.s; t; ; /d d dt ds;

ˇ .y0 /

where J1 D Œx0 ; N  and J2 D Œy0 ; R. 3.6.1.1 The convex case Theorem 3.47. Assume that (1) F W Ja  Jb  Rn ! Pcp;cv .Rn / is a Carathéodory multi-valued map,

156

Chapter 3 Solution sets for differential equations and inclusions

(2) there exist functions p 2 C.Ja  Jb ; RC / and a continuous nondecreasing function W Œ0; 1/ ! .0; 1/; such that kF .t; x; u/kP  p.t; x/ .kuk/; with

for a.e. .t; x/ 2 Ja  Jb and each u 2 Rn ; Z

1Z 1

Z

1

p.s; t /dsdt < 0

k k1

0

du  .u/

Then problem (3.37) has at least one solution. Proof. A solution of problem (3.37) is a fixed point of the operator N W C.Ja  Jb ; Rn / ! P.C.Ja  Jb ; Rn // defined by N.u/ D fh 2 C.Ja  Jb ; Rn /g where Z tZ x v.s; y/dsdy; .t; x/ 2 Ja  Jb ; h.t; x/ D .x/ C '.t /  .0/ C 0

0

and v 2 SF ;u D fv 2 L1 .Ja  Jb j v.:; :/ 2 F .:; :; u.:; ://; on Ja  Jb g: The proof will be given in three steps. First, notice that since F has convex values, then N.u/ is convex for each u 2 C.Ja  Jb ; Rn /. Step 1. (a) N maps bounded sets into bounded sets in C.Ja  Jb ; Rn / Indeed, it is enough to show that there exists a positive constant ` such that for each u 2 Bq D fu 2 C.Ja  Jb ; Rn / j kuk1  qg .q > 0/ one has kN.u/k1  `. Let h 2 N.u/; then there exists v 2 SF ;u such that Z tZ x v.s; y/dsdy: h.t; x/ D .x/ C 0

0

From 3.47 (2) we have for each .t; x/ 2 Ja  Jb Z t1 Z b kh.t; x/k  2k kD C k'kD C 0

.ku.t; x//kp.t; x/dt dx

0

 2k kD C k'kD C kpkL1 .q/ WD `: (b) N maps bounded sets into equicontinuous sets of C.Ja  Jb ; Rn /. Let .tN1 ; x1 /; .tN2 ; x2 / 2 Ja  Jb ; tN1 < tN2 ; x1 < x2 and Bq be a bounded set of C.Ja  Jb ; Rn /. Then kh.tN2 ; x2 /  h.tN1 ; x1 /k  k .x1 /  .x2 /k C k'.t1 /  '.t2 /k Z tN2 Z Z tN2 Z x2 p.t; s/dt ds C .q/ C .q/ 0

x1

tN1

x1

p.t; s/dt ds:

0

The right-hand side tends to zero as tN2  tN1 ! 0 and x2  x1 ! 0. As a consequence of Step 1 together with the Ascoli–Arzéla theorem, we conclude that N W C.Ja  Jb ; Rn / ! C.Ja  Jb ; Rn / is completely continuous.

157

Section 3.6 Hyperbolic differential inclusions

Step 2. N has a closed graph. Let un ! u ; hn 2 N.un / and hn ! h . We shall prove that h 2 N.u /. hn 2 N.un / means that there exists vn 2 SF ;un such that, for each .t; x/ 2 Ja  Jb , we have Z Z t

x

hn .t; x/ D .x/ C '.t /  .0/ C

vn .s; y/dsdy: 0

0

We must prove that there exists v 2 SF ;u such that for each .t; x/ 2 Ja  Jb Z tZ

x

h .t; x/ D .x/ C '.t /  .0/ C

v .s; y/dsdy: 0

0

Consider the linear continuous operator ‰ W L1 .Ja  Jb ; Rn / ! C.Ja  Jb ; Rn / Z tZ

x

v 7! ‰.v/.t; x/ D

v.s; /dsd : 0

0

From Lemma 6.155, the operator ‰ ı SF has a closed graph. Moreover, hn .t; x/ 2 ‰.SF ;un /: Since un ! u ; it follows from Lemma 6.155 that Z tZ

x

h .t; x/ D .x/ C '.t /  .0/ C

v .s; y/dsdy 0

0

for some v 2 SF ;u . Step 3. A priori bounds on solutions. Let u 2 C.Ja  Jb ; Rn / be a possible solution to the differential inclusion u 2 N.u/ for some  2 .0; 1/. Then there exists v 2 SF ;u such that for each .t; x/ 2 J Z tZ u.t; x/ D .x/ C

x

v.s; y/dsdy: 0

0

By 3.47 (2), this implies that for each .t; x/ 2 Ja  Jb , we have Z tZ ku.t; x/k  2k kD C k'kD C

x

kp.s; /k .ku.s; /k/dsd  0

From Lemma 3.46, we deduce the estimate Z tZ x   1 G.k/ C p.s; x/dsdx ; .t; x/  G 0

0

(3.38)

0

for each .t; x/ 2 Œ0; a  Œ0; b;

158

Chapter 3 Solution sets for differential equations and inclusions

where k D k k1. Thus, kuk1  G

1

Z

a

Z

b

ŒG.k/ C

p.s; x/dsdx WD M: 0

0

If U D fu 2 C.Ja  Jb ; Rn / j kuk1 < M C 1g; then N W U ! P.C.Ja  Jb ; Rn // is completely continuous and there is no u 2 @U such that w 2 N.u/ for some  2 .0; 1/. As a consequence of the nonlinear alternative of Leray–Schauder type [Corollary 1.74, Chapter 1], we deduce that N has a fixed point u in U which is a solution to problem (3.37). Theorem 3.48. Under the assumptions of Theorem 3.47, the solution set of problem (3.37) is compact. Proof. Let S. ; '/ D fu 2 C.Ja  Jb ; Rn / j y is a solution of Problem .3.37/g: f such that for every u 2 S; From Theorem 3.47, S. ; '/ 6D ; and there exists M f. Let fun j n 2 Ng  S. / be a sequence. Then there exists un .t; x/ 2 kyk1  M F .t; x; un .t; x//; a.e. t 2 J such that Z tZ x un .t; x/ D .x/ C '.t /  .0/ C vn .s; y/dsdt: (3.39) 0

0

We rewrite un .t; x/ 2 N.un .t; x//;

.t; x/ 2 Ja  Jb ;

For .t; x/ 2 Ja  Jb , we have un .t; x/ 2 N.un .t; x// ) un 2 S. /; where S. / D fu 2 C.Ja  Jb ; Rn / j u is a solution of problem .3.37/g: Since N is compact, the set N.S. // is precompact in C.Ja  Jb ; Rn /; this implies that there exists a subsequence of fun gn2N which converges to u. We shall M /B.0; 1/. Then prove that u 2 N.S. //. 3.47 (2) implies that vn .t; x/ 2 p.t; x/ .f 1 n .vn .:; ://n2N is integrably bounded in L .Ja  Jb ; R /. Then there exists a subsequence, still denoted fvn.:; :/gn2N ; which converges weakly to some limit v 2 L1 .Ja  Jb ; Rn /. Mazur’s Lemma (Lemma A.46, Appendix) yields the existence of ˛in 

159

Section 3.6 Hyperbolic differential inclusions

Pk.n/ 0; i D n; : : : ; k.n/ such that i D1 ˛in D 1 and the sequence of convex combinaPk.n/ tions gn .:; :// D i D1 ˛invi .:; :/ converges strongly to v in L1 . Using Lemma 6.51, we obtain that v.t; x/ 2D co.lim sup F .t; uk .t; x///: k!1

However, the fact that the multi-valued y ! F .:; :; u/ is u.s.c. and has compact values together with Lemma 6.48 imply that lim sup F .t; x; un .t; x// D F .t; x; u.t; x//; a.e. .t; x/ 2 Ja  Jb : n!1

This yields that v.t; x/ 2 co F .t; x; u.t; x//. Finally, F .:; :; :/ has closed, convex values, hence, v.t; x/ 2 F .t; x; u.t; x//; a.e. t 2 Ja  Jb . Hence, Z tZ x v.s; x/dsdx .t; x/ 2 Ja  Jb ; u.t; x/ D .x/ C '.t /  .0/ C 0

0

that is u 2 S. ; '/, ending the proof of the theorem. 3.6.1.2 The nonconvex case Theorem 3.49. Suppose that hypothesis 3.47 (2) holds together with the condition (1) F W Ja  Jb  Rn ! P.Rn / is a nonempty compact-valued multi-valued map such that: (a) .t; x; u/ 7! F .t; x; u/ is L ˝ B measurable; (b) u 7! F .t; x; u/ is lower semi-continuous for a.e. .t; x/ 2 Ja  Jb . Then the hyperbolic initial value problem (3.37) has at least one solution. Proof. 3.47 (2) and 3.49 (1) imply that F is of lower semi-continuous type. Then, from Theorem 6.138, there exists a continuous function f W C.Ja  Jb ; Rn / ! L1 .Ja  Jb ; Rn / such that f .u/ 2 F.u/ for all u 2 C.Ja  Jb ; Rn /. Consider the problem 8 2 < @ u.t; x/ D f .u.t; x//; a.e. .t; x/ 2 Ja  Jb ; (3.40) @t @x : u.0; x/ D .x/; x 2 Jb ; u.t; 0/ D '.t /; t 2 Ja and the operator G W C.Ja  Jb ; Rn / ! C.Ja  Jb ; Rn / defined by Z tZ x f .u/.s; y/dsdy; a.e. .t; x/ 2 Ja  Jb : G.y/.t / D .x/ C '.t /  .0/ C 0

0

Clearly, the fixed points of the operator G are solutions of problem (3.40) on Ja  Jb . As in [247], we can prove that operator G has at least one fixed point which is solution of problem (3.37).

160

Chapter 3 Solution sets for differential equations and inclusions

Also, we can prove a second existence result when F is Hd -Lipschitz. The proof is omitted. Theorem 3.50. Suppose that (1) F W Ja  Jb  Rn ! Pcp .Rn /; .t; x; u/ 7! F .t; x; u/ is measurable for each u 2 Rn . (2) There exists a function l 2 L1 .Ja  Jb ; RC / such that Hd .F .t; x; u/; F .t; x; u//  l.t; s/ku  uk; for a.e. .t; s/ 2 Ja  Jb and all u 2 Rn ; and Hd .0; F .t; x; 0//  l.t; s/;

for a.e. .t; x/ 2 Ja  Jb :

Then problem (3.37) has at least one solution.

3.6.2 Solution sets First, consider the hyperbolic single-valued problem: 8 2 < @ u.t; x/ D f .t; x; u.t; x//; a.e. t 2 Jb  Kd WD Œa; b  Œc; d ; @t @x : u.a; x/ D .x/; x 2 Jd ; u.t; c/ D $.t / t 2 Jb ; (3.41) where f W Ja  Jd  Rn ! Rn , 2 C.Jb ; Rn /, and ' 2 C.Jd ; Rn / are given functions with .c/ D '.a/. Denote by S.f; ; '/ the set of all solutions of problem (3.41). We are in a position to state and prove an Aronsajn type result for this problem. Our first result in this section is: Theorem 3.51. Assume that (1) f W Jb  Jd  Rn ! Rn is a Carathéodory function, (2) there exist a function p 2 C.Jb  Jd ; RC / and a continuous nondecreasing function  W Œ0; 1/ ! Œ0; 1/ such that jf .t; x; y/j  p.t; x/.kxk/; with

Z

b

Z

for a.e. .t; x/ 2 Jb  Jd and each y 2 Rn Z

d

1

p.s; t /dsdt < a

Then the set S.f; ; '/ is Rı .

c

k k1

du  .u/

161

Section 3.6 Hyperbolic differential inclusions

Proof. Let F W C.Jb  Jd ; Rn / ! C.Ja  Jb ; Rn / be defined by Z tZ

x

F .u/.t; x/ D .x/ C '.t /  '.a/ C

f .s; y; u.s; y//dsdy; a

.t; x/ 2 Jb  Jd :

c

Thus, Fix F D S.f; ; '/. By Theorem 3.47, we know that S.f; ; '/ 6D ; and there exists M > 0 such that kuk1  M ; Define

for every y 2 S.f; /:

8 ˆ f .t; x; u/; ˆ < ! e .t; x; u/ D f M u ˆ ˆ : f t; x; kuk ;

if kuk  M if kuk  M :

e is Carathéodory too and is integrably Since f is L1 -Carathéodory, the function f 1 bounded by 3.51 (2). So there exists h 2 L .Jb  Jd ; RC / such that e.t; s; u/k  h.t /; kf

a.e. .t; s/ 2 Jb  Jd and all u 2 Rn :

(3.42)

Consider the modified problem 8 2 < @ u.t; x/ e.t; x; u.t; x//; Df @t @x : u.a; x/ D .x/; x 2 Jd u.t; c/ D '.t /;

a.e. .t; x/ 2 Jb  Jd ; t 2 Jb :

e; / D FixF e ; where F e W C.Jb  Jd ; Rn / ! C.Jb  Therefore, S.f; / D S.f Jd ; Rn / is as defined by e .u/.t; x/ D .x/ C '.t /  '.a/ C F

Z tZ a

x

e .s; x; u.s; x//dsdx; f

.t; x/ 2 Jb  Jd :

c

By the inequality (3.42), we deduce that e.u/k1  k kD C khkL1 .J / WD R: kF b e is uniformly bounded. As in Theorem 3.47, we can prove that F e W C.Jb  Then F n n Jd ; R / ! C.Jb  Jd ; R / is compact which allows us to define the compact perture e .u/ which is a proper map. From the compactness bation of the identity G.u/ D uF e of F and the Lasota–Yorke approximation theorem, we can easily prove that all cone; a; c/ D G e1 .0/ is ditions of Theorem 1.32 are met. Therefore, the solution set S.f an Rı -set, hence an acyclic space. Now, let S. ; '/ denote the set of all solutions of problem (3.37). We have:

162

Chapter 3 Solution sets for differential equations and inclusions

Theorem 3.52. Let F W Ja Jb Rn ! Pcp;cv .Rn / be a Carathéodory and an mLLselectionable multi-valued map which satisfies the conditions of Theorem 3.47. Then, for every 2 C.Jb ; Rn /; ' 2 C.Ja ; Rn /, the set S. ; '/ is contractible. Proof. Let f  F be a measurable, locally Lipschitz selection and consider the single-valued problem 8 2 < @ u.t; x/ D f .t; x; u/; a.e. .t; x/ 2 Jb  Jd ; (3.43) @t @x : u.0; x/ D .x/; x 2 Jd u.t; 0/ D '; t 2 Jb : Using the Banach fixed point theorem, we can prove that problem (3.43) has exactly one solution for every 2 C.Jd ; Rn /; ' 2 C.Jb ; Rn /. Define the homotopy h W S. ; '/  Œ0; 1 ! S. ; '/ by ( u.t; x/; for .t; x/ 2 Œa; ˛a C .1  ˛/b  Jd ; h.u; ˛/.t; x/ D u.t; x/; for .t; x/ 2 Œ˛a C .1  ˛/b; b  Jd ; where u D S.f; ; '/ is the unique solution of problem (3.43). By the same argument used in Theorem 3.10, we can prove that h is a continuous homotopy, proving our claim.

Chapter 4

Impulsive differential inclusions: existence and solution sets 4.1 Motivation Historically, differential equations with impulses were considered for the first time by Milman and Myshkis [343] and then followed by a period of active research which culminated with the monograph by Halanay and Wexler [240]. Many phenomena and evolution processes in physics, chemical technology, population dynamics, and natural sciences may change state abruptly or be subject to short-term perturbations (see for instance [5, 303, 304] and the references therein) and these perturbations may be treated as impulses. Impulsive problems arise also in various applications in communications, mechanics (jump discontinuities in velocity), electrical engineering, medicine, and biology. A comprehensive introduction to the basic theory is well developed in the monographs by Bainov and Simeonov [48], Laskshmikantham et al. [315], Benchohra et al. [63], Samoilenko and Perestyuk [418] or the survey paper by Rogovchenko [410] or the more recent books [394, 444]. For instance, in the periodic treatment of some diseases, impulses correspond to the administration of a drug treatment. In environmental sciences, impulses correspond to seasonal changes of the water level of artificial reservoirs. Their models are described by impulsive differential equations and inclusions.

4.1.1 Ecological model with impulsive control strategy For the impulsive model with distributed time delay, papers [273, 342, 433] have investigated some ecological models with distributed time delay and impulsive control

164

Chapter 4 Impulsive differential inclusions: existence and solution sets

strategy. The model can be described by the following differential equations: 8 x.t / ˇa1 x.t /z.t / ˆ ˆ /  a2 x.t /y.t /  ; x 0 .t / D r x.t /.1  ˆ ˆ k b1 C x.t / C c1 z.t / ˆ ˆ ˆ ˆ t 6D nT; t 6D .n C l  1/T ˆ ˆ ˆ ˆ ˆ ˆ Rt ˆ ˆ y 0 .t / D dy.t / 1 F .t  s/x.s/ds  m1 y.t /; ˆ ˆ ˆ ˆ ˆ ˆ t 6D nT; t 6D .n C l  1/T ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ e1 ˇa1 x.t /z.t / ˆ ˆ  m2 z.t /; t 6D nT; < z 0 .t / D b1 C x.t / C c1 z.t / ˆ ˆ t 6D .n C l  1/T ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ x.t / D ı1 x.t /; t D .n C l  1/T ˆ ˆ ˆ ˆ ˆ y.t / D ı2 y.t /; t D .n C l  1/T; ˆ ˆ ˆ ˆ ˆ z.t / D ı3 z.t /; t D .n C l  1/T ˆ ˆ ˆ ˆ ˆ x.t /D0 ˆ ˆ ˆ ˆ ˆ y.t / D 0 ˆ ˆ : z.t / D p;

(4.1)

where x.t /; y.t /; z.t / are the densities of one prey and two predators at time t; respectively, x.t / D x.t C /  x.t /; y.t / D y.t C/  y.t /; z.t / D z.t C /  z.t / 

< l < 1 is used to describe the intervals of time between the pulsed use of controls.



r is the intrinsic growth rate, ai .i D 1; 2/ are the cropping rates,



e1 denotes the efficiency with which resources are converted to new consumers,



k is the carrying capacity of the prey,



b1 is a saturation constant, c1 scales the impact of predator interference,



mi .i D 1; 2/ are the mortality rates for each predator,



d denotes the product of the per-capita rate of predation and the rate of converting prey into predator,



ˇ is the relative superiority of predator .z/.

4.1.2 Leslie predator-prey system In 1948, Leslie [321] introduced the famous Leslie predator-prey system 8 0 < x .t / D x.t /.a  bx.t //  p.x/y.t /; y.t / : y 0 .t / D y.t /.e  f /; x.t /

(4.2)

165

Section 4.1 Motivation

where x.t /; y.t / stand for the population (the density) of the prey and the predator at time t , respectively, and p.x/ is the so-called predator functional response to prey. In biomathematics, when p.x/ D cx; the functional response p.x/ is called type 1; when 2 , the functional response p.x/ is called type 2; when p.x/ D dcx ; p.x/ D dcx Cx Cx 2 the functional response p.x/ is called type 3. In [480], the authors consider a ratiodependent Leslie predator-prey model with impulses 8  c.t /x1 .t /x2 .t /  ˆ 0 ˆ .t / D x .t / b.t /  a.t /x .t /  x 1 1 ˆ 1 ˆ ˆ h2 x22 .t / C x12 .t / ˆ <  x2 .t /  (4.3) 0 x ; t 6D tk ; .t / D x .t / e.t /  f .t / ˆ 2 2 ˆ x .t / ˆ 1 ˆ ˆ ˆ : xi .tkC / D .1 C hik /xi .tk /; xi .0/ > 0; i D 1; 2; where xi .t /; i D 1; 2 denote the density of prey and predator at time t , respectively. b; a; c; d; e; f; p 2 C.R; RC /; i D 1; 2 are all !-periodic functions of t I h2 is a positive constant, denoting the constant of capturing half-saturation.

4.1.3 Pulse vaccination model The pulse vaccination proposes to vaccinate a fraction p of the entire susceptible population in a single pulse, applied every  years, the following standard SIR model and constant vaccination: 8 0 2 ˆ < S .t / D ˇIS C   S; I 0 .t / D ˇIS 2  . C /S; (4.4) ˆ : 0 R .t / D I  R; evolves from its initial state without being further affected by the vaccination schemes until the next pulse is applied, when the pulse vaccination is incorporated into the system (4.4), the system may be rewritten as 8 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ :

S 0 .t / D ˇIS 2 C   S; I 0 .t / D ˇIS 2  . C /S; R0 .t / D I  R; S.n C / D .1  p/S.n  /; I.n C / D I.n  /; R.n C / D R.n  / C pS.n  /;

t t t t t t

6D n 6D n 6D n D n D n D n;

(4.5)

where the fractions of the population that are susceptible, infectious, and recovered with immunity are denoted by S; I and R, respectively. For more information about this model, we refer the reader to [264].

166

Chapter 4 Impulsive differential inclusions: existence and solution sets

4.2 Semi-linear impulsive differential inclusions In this section, we consider the following impulsive problem for first-order semi-linear differential inclusions: 8 0 ˆ < .y  Ay/.t / 2 F .t; y.t //; a.e. t 2 J; yt Dtk D Ik .y.tk //; k D 1; : : : ; m; (4.6) ˆ : y.0/ D a; where 0 D t0 < t1 <    < tm < tmC1 D b; J D Œ0; b. F W J  E ! P.E/ is a multifunction, and a 2 E. The operator A is the infinitesimal generator of a C0 -semigroup fT .t /gt 0 on a separable Banach space .E; j  j/, Ik 2 C.E; E/ .k D 1; : : : ; m/; and yjt Dtk D y.tkC /  y.tk /. The notations y.tkC / D lim y.tk C h/ and y.tk / D h!0C

lim y.tk  h/ stand for the right and the left limits of the function y at t D tk ,

h!0C

respectively. We shall be concerned with some existence results and structure of solution set for problem .4.6/. First, we discuss several results about the existence of solutions for problem .4.6/ as well as some properties of operator solutions. Finally, we prove some geometric properties of solution sets such that acyclicity, AR; Rı , and Rı -contractibility. Let Jk D .tk ; tkC1 ; k D 0; : : : ; m, and let yk be the restriction of a function y to Jk . In order to define mild solutions for problem .4.6/, consider the space PC D fyW Œ0; b ! E; yk 2 C.Jk ; E/; k D 0; : : : ; m; such that y.tk / and y.tkC / exist and satisfy y.tk / D y.tk / for k D 1; : : : ; mg: Endowed with the norm kykP C D maxfkyk k1 ;

k D 0; : : : ; mg; yk D yjJk ;

PC is a Banach space. Definition 4.1. A function y 2 PC is said to be a mild solution of problem (4.6) if there exists v 2 L1 .J; E/ such that v.t / 2 F .t; y.t //; a.e. on J and Z

t

y.t / D T .t /a C

T .t  s/v.s/ds C 0

X

T .t  tk /Ik .y.tk //:

0 0 and continuous functions k W RC ! RC such that jIk .x/j  ck k .jxj/;

for each x 2 E; k D 1; : : : ; m:

(A3 ) E is a reflexive Banach space, and the semigroup fT .t /gt >0 is compact in E. (A4 ) There exists p 2 L1 .Œ0; b; RC / such that for every bounded subset D in E .F .t; D//  p.t /.D/; and there exist Lk > 0; k D 0; : : : ; m such that Z tkC1 !tkC1 sup e Lk .t s/ p.s/ds < 1; qk WD 2Me

k D 0; : : : ; m;

t 2Œtk ;tkC1  tk

where  is the Hausdorff MNC. Remark 4.2. Note that if fT .t /g is compact for t > 0, then it is so for any t 0 > t since T .t 0 / D T .t 0  t / ı T .t / and T .t / is bounded. Theorem 4.3. Assume that F satisfies (A1 ), (A2 ) and (A3 ). Then the set of solutions for problem (4.6) is nonempty and compact. Moreover, the operator solution S W a ! P.S.a// is u.s.c., where S.a/ D fy 2 PCj y is a mild solution of .4.6/g: Proof. Step 1. S.a/ 6D ;. Consider the operator N W PC ! P.PC/ defined for y 2 PC by N.y/ D fh 2 PCg with Z t X h.t / D T .t /a C T .t  s/v.s/ds C T .t  tk /Ik .y.tk //; t 2 J; 0

0 0 be such that 2 CJC V .K.M /; X/. Then (a) either there exist  2 .0; 1/ and x 2 S.M / such that x 2 ˆ.x/. (b) or Fix .ˆ/ 6D ; and hence 0 2 . Proof. Since ' 2 CJC V .B.r; 0/; X/; then ˆ 2 CJC .B.r; 0/; X/. Consider the O defined by multi-valued map ˆ f

F O D f ı F ı r W X r!K.M / !P.X/!X; ˆ

with the radial retraction 8 jxj  M ˆ < x; r .x/ D Mx ˆ : ; jxj > M : jxj We know that X 2 AR and F ı r is u.s.c. with Rı -values. Furthermore, X 2 ANR, f is continuous function, and O ˆ.X/ D .f ı F ı r /.X/ D .f ı F /.r .X// D .f ı F /.K.M // D ˆ.K.M //: O is compact. From Proposition 4.25, there exists x 2 X such that This implies that ˆ   O x 2 ˆ.x/. Assume that jxj  M ; then x 2 .f ı F / M x which implies that jxj

Mx M M x  2 ˆ ; jxj jxj jxj leading to a contradiction with Fix .ˆ \ S.r // D ;: Then Fix .ˆ/ 6D ; and 0 2 '.x/.

233

Section 4.3 A periodic problem

4.3.7.2 A Poincaré translation operator By a Poincaré operator for a differential system, we mean the translation operator (or the Poincaré–Andronov, or Levinson operator, or simply the T -operator [291]) along the trajectories of the associated differential system, and the first return map defined on the cross section of the torus by means of the flow generated by the vector field. Both of these operators are single-valued when the uniqueness of solutions of initial value problems is assumed. In the absence of uniqueness, it is often possible to approximate the right-hand side of the given system by locally Lipschitzian ones (hence implying uniqueness), and then apply a standard limiting argument which may be rather complicated for discontinuous right-hand sides. However, set-valued analysis allows us to handle such problems effectively. For further details, we refer to the paper [17] and the monographs [22, 215]. Consider the following impulsive problem 8 0 a.e. t 2 J nft1; : : : ; tm g; ˆ < .y  Ay/ 2 G.t; y.t //; (4.52) y.tkC/  y.tk / D Ik .y.tk //; k D 1; : : : ; m; ˆ : y.0/ D x; where G W J  X ! P.X/ is a u-Carathéodory multifunction map. Now, define a multi-valued map SG W X ! P.PC/ by SG .x/ D fy j y is a solution of problem .4.52/g; where x 2 X. For some positive real number b; consider the operator Pb defined by Pb D ‰b ı SG called the Poincaré translation map associated with the impulsive Cauchy problem (4.52) where SG

‰b

X !P.PC/!P.X/ and ‰b .y/ D y.0/  y.b/: The following lemma is easily proved. Lemma 4.27. Let G W J  X ! Pcv;cp .X/ be a u-Carathéodory multi-valued map. Then the periodic problem (4.52) has a solution if and only if for some x 2 X; 0 2 Pb .x/; where Pb is the Poincaré map associated with (4.52). 4.3.7.3 The MNC approach Let X be ANR; we are in a position to state our main existence result. Throughout this subsection, fT .t /gt 0 is assumed to be uniformly continuous.

234

Chapter 4 Impulsive differential inclusions: existence and solution sets

Theorem 4.28. Let G W J  X ! Pcp;cv .X/ be a u-Carathéodory multi-valued map with the upper -Scorza–Dragoni property. Assume that (R1 ) There exist a function p 2 L1 .J; RC / and a continuous nondecreasing function W Œ0; 1/ ! Œ0; 1/ such that kG.t; y/kP  p .t /.jyj/; with

Z

for each .t; y/ 2 J  X; Z

b

1

p .s/ds < 0

1

du  .u/

(R2 / There exist constants c k > 0 and continuous functions k W RC ! RC such that jIk .x/j  c k k .jxj/; for each x 2 X; k D 1; : : : ; m: (R3 ) There exists p 2 L1 .Œ0; b; RC / such that for every bounded subset D in X .G.t; D//  p.t /.D/ and there exist Lk > 0; k D 0; : : : ; m such that Z qk W D 2Me

!tkC1

tkC1

sup

e Lk .t s/ p.s/ds < 1; k D 0; : : : ; m:

t 2Œtk ;tkC1  tk

Here  is the Hausdorff MNC. Then problem (4.51) has at least one solution. Proof. (a) From Theorem 4.3 we know that problem (4.52) has at least one solution, and the solution set SG .x/ is nonempty and compact for each x 2 X. From [Theorem 6.125 Chapter 6], G is -Ca-selectionable and so, for every x 2 X; SG .x/ is an Rı -set. In addition, the mapping ‰ W PC ! X defined by y 7! ‰.y/ D y.0/  y.:/ is continuous. Indeed, let fyn g be a sequence such that yn ! y in PC. Then, j‰.yn /.t /  ‰.y/.t /j  2kyn  ykP C ! 0; as n ! 1: (b) Using the conditions .R1 /–.R2 /; we can prove, as mentioned at the end of the proof of Theorem 4.3, that there exists M > 0 independent of x such that, for every y solution of problem (4.52), we have kykP C  M . Let K D fx 2 X j jxj  2M C 1g:

235

Section 4.3 A periodic problem

We have to show that Pb 2 CJC V .K; X/. Let x 2 Pt .x/ D .‰t ı SG /.x/ for some  2 .0; 1/. Then, there exists y 2 PC such that y 2 SG .x/. This yields y.0/ D x and x D .x  y.t //; x 2 S.2M C 1/. For t 2 J , we have the estimates jxj  jy.0/j C jy.t /j  2kykP C  2M which is a contradiction to jxj D 2M C 1. (c) Making use of [Lemma 6.24, Chapter 6], we will show that SG is u.s.c. by proving that the graph of SG

G D f.x; y/ j y 2 SG .x/g is closed. Let .xn ; yn / 2 G , i.e., yn 2 SG .xn / and let .xn ; yn / ! .x; y/; as n ! 1. Since yn 2 SG .xn /, there exists vn 2 SG;yn such that Z

t

yn .t / D T .t /xn C

T .t s/vn .s/ds C 0

X

T .t tk /Ik .yn .tk //; t 2 J: (4.53)

0 0; there corresponds T ."/ > 0 such that jx.t /  x.1/j < " for any t  T ."/ and x 2 M . Proof of Theorem 4.38. Step 1. Existence of solutions. Consider the operator N1 W PC ! P.PC/ defined for y 2 PC by N1 .y/ D fh 2 PCb g with Z

t

h.t / D a C

v.s/ds C 0

X

Ik .y.tk //;

a.e. t 2 J;

(4.68)

0 0; 9 n0 2 N; 8 n  n0 ; .An  A" /

n!1

H) lim sup An  A lim sup An  A ”

n!1 lim Hd .A; An / n!1

n!1 Hd.lim sup n!1

An ; A/ D 0

D 0 H) A  lim inf An D 0: n!1

Before ending with limit sets, we prove a useful result for the sequel. Lemma 6.51 (see, e.g., [38], Lemma 1.1.9). Let .Kn /n2N  K  X be a subset of sequences where K is compact in the separable Banach space X. Then \ [ co .lim sup Kn / D co . Kn /: n!1

N >0

nN

Proof. (a) We have [ nN

Kn  co.

[

nN

Kn / )

\ [ N >0 nN

Kn 

\ N >0

co.

[

nN

Kn /:

354

Chapter 6 Background in multi-valued analysis

Hence, lim sup Kn  n1

co.

N >0

and so co.lim sup Kn /  co. n1

\

\

co.

N >0

[

[

Kn /;

nN

Kn// D

nN

\ N >0

co.

[

Kn /:

nN

p (b)T Conversely, S assume X is a finite-dimensional space, say X D R and let A D N >0 co. nN Kn/. For any x 2 A, there exists a sequence .vN /N such that Pp Pp vN D j D1 ajN xNj and x D lim vN with ajN  0, j D1 ajN D 1, xNj 2 KNj N !1

and Nj  N; j D 1; : : : ; p. The sequence defined by aN D .a1N ; : : : ; apN / is bounded in Rp hence admits, up to a subsequence, a limit a D .a1 ; : : : ; ap / 2 Rp S P with p Nj N KNj  K, the sequence .xNj / admits, j D1 aj D 1. Since .xNj /  up to a subsequence, a limit xj 2 lim supn!1 Kn. We have x D lim vN D lim N !1

N !1

p X j D1

ajN xNj D

p X j D1

aj xj

\ [

Kn :

N >0 nN

Therefore, x is a convex combination of xj0 s 2 lim supn!1 Kn, and then for all N , we have x 2 co.lim supn1 Kn /. The reverse inclusion is then obtained, ending the proof of the lemma.

6.2 The selection problem First, we start with some auxiliary notions and results needed in this section. Partitions of unity play an important role in the existence of continuous selections of some classes of lower semi-continuous and in approximation of upper multi-valued maps. Definition 6.52. Let fU j  2 ƒg and fVˇ j ˇ 2 ƒ0 g be two coverings of a space. fU j  2 ƒg is said to refine (or be a refinement of) fVˇ j ˇ 2 ƒ0 g if for each U , there is some Vˇ with U  Vˇ . Definition 6.53. Let fU j  2 ƒg be a covering of X. If ƒ0 is contained in ƒ and fU j  2 ƒ0 g is again a covering, then ƒ0 called a subcovering. Definition 6.54. A covering fU j  2 ƒg of a topological space X is called locally finite if for every x 2 X; there exists a neighbourhood V of x such that U \ V is nonempty only for a finite number of indexes. Definition 6.55. A Hausdorff (T2 separated space) space is called paracompact if each open covering has a locally finite open refinement.

355

Section 6.2 The selection problem

Lemma 6.56. A closed subset of a paracompact space is paracompact. Lemma 6.57 (Stone’s theorem). (see, e.g., [80,128,170]) Every metric space is paracompact. Definition 6.58. Let X be a Hausdorff space. A family f  j  2 ƒg of continuous maps  W X ! Œ0; 1 is called a partition of unity on X if (1) the support (i.e., supp.  / D fx 2 X j  .x/ 6D 0g) of  is closed locally finite, X  .x/ D 1. (2) for each x 2 X; 2ƒ

If fU j  2 ƒg is a given open covering of X, we say that a partition f  j  2 ƒg of unity is subordinated to fU j  2 ƒg if for every  2 ƒ; Supp.  /  U . Theorem 6.59 ([37]). Let X be a metric space. To any locally finite open covering fU j  2 ƒg of X; we can associate a locally Lipschitzian partition of unity subordinated to it.

6.2.1 Michael’s selection theorem The most famous continuous selection theorem is the following result proved by Michael in 1956 (see [215]). Theorem 6.60. Let X be a metric space, E a Banach space and 'W X ! Pcl;cv .E/ a l.s.c. map. Then there exists f W X ! E, a continuous selection of ' .f  '/, i.e., f .x/ 2 '.x/ for every x 2 X. Proof. Step 1. Let us begin by proving the following claim: given any convex (not necessarily closed) valued l.s.c. map ˆW X ! P.E/ and every " > 0, there exists a continuous gW X ! E such that dist.g.x/; ˆ.x//  ", i.e., g.x/ 2 O" .ˆ.x//, for every x 2 X. In fact, for every x 2 X, let yx 2 ˆ.x/ and let ıx > 0 be such that B.yx ; "/ \ ˆ.x 0 / 6D ;, for x 0 in B.x; ıx /. Since X is metric, it is paracompact by Stone’s theorem. Hence, there exists a locally finite refinement fUx gx2X of fB.x; ıx /gx2X . Let fLx gx2X be a partition of unity subordinated to it. The mapping gW X ! E defined by X Lx .u/  yx g.u/ D x2X

is continuous since it is locally a finite sum of continuous functions. Now fix n 2 X. Whenever Lx .u/ > 0, n 2 B.x; ıx /, hence yx 2 O".ˆ.u//. Since this latter set is convex, any convex combination of such y’s belongs to it.

356

Chapter 6 Background in multi-valued analysis

Step 2. We claim that we can define a sequence ffn g of continuous mappings from X to E with the following properties 1 ; n D 1; 2; : : : ; u 2 X; 2n 1 .2/ kfn .u/  fn1 .u/k  n2 ; n D 2; 3; : : : ; u 2 X: 2 dist.fn .u/; '.u// 

.1/

(6.21)

For n D 1 it is enough to take in Step 1, ˆ D ' and " D 1=2. Assume we have defined mappings fn satisfying 6.21 (1) up to n D k. We shall define fkC1 satisfying 6.21 (1) and 6.21 (2) as follows. Consider the set ˆ.u/ D B.fk .u/; 1=2k / \ '.u/. By 6.21 (1), it is a nonempty convex set. By Proposition 6.34, the map ˆ is l.s.c.; so by the claim in Step 1 there exists a continuous g such that dist.g.x/; ˆ.x//
0, there corresponds ı > 0 such that for any A 2 M.X/, any k 2 N, and any x1 ; : : : ; xk 2 Oı .A/, one has A.x1 ; : : : ; xk /  O" .A/. (4) A \ B.x; r / 2 M.X/ for all A 2 M.X/, x 2 X, and r > 0. Then we say that M.X/ is a Michael family of subsets of X. It is an easy remark that in the Michael selection theorem, the notion of convexity can be replaced by a Michael family. Namely, we obtain: Proposition 6.68. Let X; Y 2 M and let ˆW X ! P.Y D [f;g be a lower semicontinuous multifunction. If Y is complete and there exists a Michael family M.Y / of subsets of Y such that ˆ.x/ 2 M.Y / for each x 2 X then, for any nonempty closed set X0  X, every continuous selection f0 from ˆjX0 admits a continuous extension f over X such that f .x/ 2 ˆ.x/ for all x 2 X. The preceding result gains in interest if we realise that significant classes of sets are examples of Michael families. Example 6.69. (1) Let X be a convex subset of a normed space and let M.X/ be the class of all sets A  X such that A D ; or A is closed and convex in X. Then M.X/ is a Michael family of subsets of X. (2) Let X 2 M and let M.X/ be the family of all simplicially convex closed subsets of X (in the sense of [70]) or closed convex sets with respect to an abstract convex structure (see [492]). Then M.X/ is a Michael family of subsets of X. Example 6.70. Let A1 ; : : : ; An be compact, nonempty subsets of RN . Let f1 ; : : : ; fn W Œ0; b  RN ! RN be single-valued maps satisfying the Carathéodory conditions. We define a multi-valued map F W Œ0; b  RN ! P.RN / by putting F .t; x/ D f1 .t; x/A1 C : : : f2 .t; x/An :

360

Chapter 6 Background in multi-valued analysis

Let xi be a point in Ai ; i D 1; : : : ; n. Define a map f W Œ0; b  RN ! RN as follows: f .t; x/ D f1 .t; x/x1 C : : : f2 .t; x/xn : Then f is a C -map and evidently f  F . Observe that if ffi gniD1 are maps satisfying the Lipschitz condition with respect to the second variable and are Lebesgue measurable with respect to the first variable, then f is an L-map. So we are able to construct C -selection (L-selection) multi-valued maps. In 6.69 (2), we only mentioned some non-typical examples of Michael families. The following definition is crucial in what follows Definition 6.71. Let X 2 M, let ˆW X ! P.X/ [ f;g be a lower semi-continuous multifunction, and let D  M. We say that ˆ has the selection property with respect to D, when for any Y 2 D, any pair of continuous functions f W Y ! X and hW Y ! 0; C1Œ such that ‰.y/ D ˆ.f .y// \ B.f .y/; h.y// 6D ;;

y 2 Y;

and for any nonempty closed set Y0  Y , every continuous selection g0 from ‰jY0 admits a continuous extension g over Y fulfilling g.y/ 2 ‰.y/ for all y 2 Y . If D D M, then we say that ˆ has the selection property (in short, ˆ 2 SP.X/). The above notion has some meaningful features, as is pointed out below. Example 6.72. Let X 2 M and let ˆW X ! 2X be a l.s.c. mapping. If X is complete and there exists a Michael family M.X/ of subsets such that ˆ.x/ 2 M.X/ for all x 2 X, then ˆ 2 SP.X/ (see Proposition 6.68). Now, we establish the following result: Theorem 6.73. Let X be a nonempty closed subset of L1 .T; E/ and let 'W X ! P.X/ be a lower semi-continuous map, with decomposable values. Then ' has the selection property with respect to the family D of all separable metric spaces. Proof. Throughout this proof, we write 0 to denote the zero vector of L1 .T; E/ with k  kL1 .T;E/ . Pick Y 2 D and a pair of continuous functions f W Y ! X and hW Y ! .0; C1/ such that .y/ D cl .'.f .y// \ B.f .y/; h.y/// 6D ; for all y 2 Y . If Y0 is a nonempty closed subset of Y and g0 denotes a continuous selection from jY0 , then the function k0 W Y0 ! L1 .T; E/ defined by k0 .y/ D h.y/1 Œg0 .y/  f .y/;

for y 2 Y0 ;

is a continuous selection of jY0 , where

.y/ D cl .h.y/1 Œ'.f .y//  f .y/ \ B.0; 1//;

for y 2 Y:

361

Section 6.2 The selection problem

Evidently, the proof will be completed as soon as we show that k0 admits a continuous extension k over Y with the property k.y/ 2 .y/ for every y 2 Y . We first define ( fk0 .y/g; if y 2 Y0 ; .y/ D 1 h.y/ Œ'.f .y//  f .y/; if y 2 Y n Y0 : It is a simple matter to see that the multi-valued map W Y ! L1 .T; E/ is lower semicontinuous and has decomposable values. Hence, by Theorem 3 of [81], for any y 2 Y and any u 2 .y/ \ B.0; 1/, there exists a continuous selection ky;u W Y ! L1 .T; E/ of such that ky;u .y/ D u. Let Vy;u D fz 2 Y j kky;u .z/k1 < 21 .1 C kuk1 /g: The family of sets fVy;u j y 2 Y , u 2 .y/ \ B.0; 1/g is an open covering of the separable metric space Y , so it has a countable neighbourhood finite refinement fVn j n 2 Ng. For each n 2 N, choose yn 2 Y and un 2 .yn / \ B.0; 1/ such that Vn  Vyn ;un , and define kn D kyn ;un . Let fpn gn2N be a continuous partition of unity subordinated to the covering fVn g and let fhn gn2N be a sequence of continuous functions from Y into Œ0; 1, fulfilling the conditions hn .y/ D 1 on supp pn , supp hn  Vn, n 2 N. For any y 2 Y , let 'n .y/.t / D kkn.y/.t /k; for t 2 Œ0; a and n 2 N; 1 1 i1 X X 1h 1 C kun k1 l.y/ D 1  pn .y/ hn .y/: 2 2 nD0

nD1

Since un 2 B.0; 1/ and the above summations are locally finite, the function l is well defined, positive, and continuous. Therefore, there exists a continuous function r W Y ! 0; C1Œ and a family fAr; j r > 0,  2 Œ0; 1g of measurable subsets of T satisfying (see Lemma 2 in [81]): (a) Ar;1  Ar;2 ; if 1  2 , (b) .Ar1 ;1 Ar2 ;2 /  j1  2 j C 2jr1  r2 j and .Ar; / D .T /, (c) for each y 2 Y ,  2 Œ0; 1, and n 2 N, if hn .y/ D 1 then Z ˇZ ˇ ˇ ˇ 'n .y/.t / d   'n .y/.t / dˇ < ˇ Ar.y/;

T

Finally, let us define, for y 2 Y and n 2 N, 0 .y/ D 0; n .y/ D

X

pm .y/;

mn

y;n D Ar.y/;n .y/ nAr.y/;

n1 .y/

;

1 : 4l.y/

362

Chapter 6 Background in multi-valued analysis

and k.y/ D

1 X

y;n  kn .y/:

nD1

It is easy to see that the function kW Y ! L1 .T; E/ is continuous. Furthermore, for any y 2 Y one has k.y/ 2 .y/, because .y/ is decomposable. Thus, to complete the proof, we only need to show that kk.y/k1 < 1 at all points ofP Y . Fix.y/ 2 Y and observe that if I.y/ D fn 2 N j pn .y/ > 0g then 1  ]I.y/  1 nD1 hn .y/. From (a)–(c), we deduce the estimates: Z kk.y/.t /kd T X Z 'n .y/.t / d  n2I.y/

Ar.y/;n .y/nAr .y/;n1 .y/

X hZ

D

n2I.y/

Z

Z

'n .y/.t / d  n .y/ Ar.y/;n .y/

'n .y/.t / d T

Z Z i 'n .y/.t / d C n1 .y/ 'n .y/.t / d C pn .y/ 'n .y/.t / d

 Ar.y/;n1 .y/

T

T

1 1 1 X 1 X 1 C kun k1 ]I.y/ X 1 C kun k1 C pn .y/  pn .y/: hn .y/ C < 2l.y/ 2 2l.y/ 2 nD1

nD1

nD1

Hence, by the definition of l, kk.y/k1 < 1 as required.

6.2.3  selectionable mappings The Michael selection theorem is not true for u.s.c. mappings; but under some natural assumptions u.s.c. mappings are -selectionable. Selectionable mappings play an important role in the theory of topological structure of solutions set for differential inclusions; for more information about this subject, we refer the reader to [37, 38, 146, 215, 258, 259, 276, 468]. Definition 6.74. We say that a map 'W X ! P.Y / is -selectionable, if there exists a decreasing sequence of compact-valued u.s.c. maps 'n W X ! P.Y / satisfying: (1) 'n has a continuous selection, for all n  0, T (2) '.x/ D n 'n .x/, for all x 2 X. Definition 6.75. Assume that F W X ! P.Y / is a multi-valued map and Fn W X ! P.Y /; n D 1; 2; : : : is a sequence of multi-valued mappings such that: (1) FnC1 .x/  Fn .x/

363

Section 6.2 The selection problem

(2) F .x/ D \n0 Fn .x/, for every x 2 X and n D 1; 2; : : :. We say that (3) F is   L-selectionable, provided Fn is L-selectionable for every n (i.e., for every n there exists a Lipschitz continuous map such that fn  Fn ) (4) F is   LL-selectionable, provided Fn is LL-selectionable for every n (i.e., for every n, there exists a locally Lipschitz continuous map such that fn  Fn ) (5) F is   C a-selectionable, provided Fn is C a-selectionable for every n (i.e., for every n, there exists a Carathéodory map such that fn  Fn ) (6) F is   m-selectionable, provided Fn is m-selectionable for every n (i.e., for every n, there exists a measurable map such that fn  Fn ) (7) F is  c-selectionable, provided Fn is c-selectionable for every n (i.e., for every n, there exists a continuous map such that fn  Fn ). (8) F is   mLL-selectionable provided Fn is mLL-selectionable for every n (i.e., for every n D 0; 1; 2; : : : ; there exists a measurable-locally Lipschitz map fn W Œa; b  X ! Y such that fn  Fn ). Let .X; d / and .Y; d 0 / be two metric spaces. Recall that a single-valued map f W Œa; b  X ! Y is said to be measurable-locally-Lipschitz (mLL) if f .; x/ is measurable for every x 2 X and for every x 2 X; there exists a neighbourhood Vx of x 2 X and an integrable function Lx W Œa; b ! Œ0; 1/ such that d 0 .f .t; x1 /; f .t; x2 //  Lx .t /d.x1 ; x2 / for every t 2 Œa; b and x1 ; x2 2 Vx : We are going to prove: Theorem 6.76. Let 'W X ! Pcp;cv .E/ be an u.s.c. multi-valued map from a metric space X to a Banach space E. If '.X/ is a compact set, then ' is -selectionable. Actually, there exists a sequence of u.s.c. mappings 'n from X to co .'.X// which approximate ' in the sense that, for all x 2 X, we have: 8 for all n  0; ˆ 0, there exists D ."; x/ such that cl .y; x/  implies '.y/  O".'.x//. Then there obviously exists n0 D n0 ."; x/ such that for n  n0 we have %n  =3. .n/ x D fi 2 I .n/ j x 2 B.xi ; %n /g. For the same reasons Let us define as before I.n/ as for '0 and '1 we can write: X .n/ .n/ Li .x/  Ci ; 'n .x/ D x i 2I.n/

x where Ci.n/ D conv '.B.xi.n/ ; 2%/  K. Then for all y 2 B.xi.n/ ; 2%/ with i 2 I.n/ we have: .n/

.n/

d.y; x/  d.y; xi / C d.xi ; x/  2%n C %n D 3%n < ;

if we take n > n0 :

366

Chapter 6 Background in multi-valued analysis .n/

x Thus, for all n  n0 we have '.y/  O" .'.x// for all y 2 B.xi ; 2%n / with i 2 I.n/ . .n/

 O" .'.x// and by But since O" .'.x// is closed and convex, we obtain: Ci convexity we infer 'n .x/  O" .'.x// for all n  n0 . Therefore, the proof of Theorem 6.22 is completed. Remark 6.78. (a) If X is compact, then F is compact. (b) If E D Rn , then any bounded u.s.c. map with convex, compact values satisfies the assumptions of Theorem 6.22.

6.2.4 The Kuratowski–Ryll-Nardzewski selection theorem Apart from semi-continuous multi-valued mappings, multi-valued measurable mappings will be of great importance in the sequel. Throughout this section, assume that X is a separable metric space and .; U; / is a complete -finite measurable space, i.e., a set  equipped with -algebra U of subsets and a countably additive measure  on U. A typical example is when  is a bounded domain in the Euclidean space Rk equipped with the Lebesgue measure. Definition 6.79. A multi-valued map F W  ! P.X/ is said (a) measurable, if for every closed subset C  X, we have FC1 .C / D f! 2  j F .!/ \ C 6D ;g 2

X ;

(b) weakly measurable, if for every open subset U  X, we have FC1 .U / D f! 2  j F .!/ \ U 6D ;g 2 U; (c) F ./ is said to be K-measurable if for every compact subset K  X, we have FC1 .K/ D f! 2  j F .!/ \ K 6D ;g 2 U; (d) graph measurable, if Gr .F / D f.!; x/ 2   X j x 2 F .!/g 2 U ˝ B.X/; where B.X/ is the -algebra generated by the family of open all sets from X.

367

Section 6.2 The selection problem

Another way of defining measurability is by requiring the measurability of the graph

' of ' in the product Y , equipped with the minimal -algebra U ˝B.Y / generated by the sets A  B with A 2 U and B 2 B.Y /, where B.Y / denotes the family of all Borel subsets of Y . For further reference, we collect some relations between these definitions in the following Proposition 6.80. Assume that '; W  ! P.Y / are two multi-valued mappings. Then the following holds true: 1 (1) ' is measurable if and only if 'C .A/ 2 U, for each closed A  Y , 1 .V / 2 U, for each open V  Y , (2) ' is weakly measurable if and only if 'C

(3) if ' is measurable, then ' is also weakly measurable, (4) if ' has compact values, measurability and weak measurability of ' are equivalent, (5) ' is weakly measurable if and only if the distance function fy W  ! R, fy .x/ D dist.y; '.x// is measurable for all y 2 Y , (6) if ' is weakly measurable, then the graph ' of ' is product measurable, (7) if ' and

are measurable, then so is ' [

,

(8) if ' and

are measurable, then so is ' \

,

(9) if ' and

are measurable, then so is ' 

.

The proof of Proposition 6.80 is straightforward and therefore is left to the reader. Of course, the composition of two measurable multi-valued mappings need not be measurable. Example 6.81. Let  D Œ0; 1 be equipped with the Lebesgue measure and let f W  ! R be a strictly increasing Cantor function which of course is measurable. It is well known that one may find a measurable set D  R such that f 1 .D/ is not measurable. If we define 'W  ! P.R/ and W R ! P.R/ by ( f1g if u 2 D; '.t / D ff .t /g for t 2 ; .u/ D f0g if u 62 D; then both ' and

are measurable, but

ı ' is not.

For further reference, we collect the results and counter-examples given so far on the conservation of semi-continuity or measurability properties in the following table where * holds when ' and have compact values.

368

Chapter 6 Background in multi-valued analysis

'; '[ '\ ' 'ı

u.s.c. yes yes yes yes

l.s.c. yes no yes yes

measurable yes yes yes no

In what follows, we present the Kuratowski–Ryll-Nardzewski selection theorem (see [37, 38, 286, 436]). Theorem 6.82. Let Y be a separable complete space. Then every measurable 'W  ! P.Y / has a selection. Proof. Without loss of generality we can change the metric of Y into an equivalent metric, preserving completeness and separability, so that Y becomes a bounded (say, with diameter M ) complete metric space. Now, let us divide the proof into two steps. Step 1. Let C be a countable dense subset of Y . Set "0 D M , "i D M=2i . We claim that we can define a sequence of mappings sm W  ! C such that: (1) sm is measurable, (2) sm .x/ 2 O"m .'.x//, (3) sm .x/ 2 B.sm1 .x/; "m1 /, m > 0. In fact, arrange the points of C into a sequence fcj gj D0;1;::: and define s0 by putting: s0 .x/ D c0 ;

for every x 2 :

Then 6.82 (1) and 6.82 (2) are clearly satisfied. Assume we have defined functions sm satisfying 6.82 (1) and 6.82 (2) up to m D p  1, and define sp satisfying 6.82 (1)–6.82 (3) as follows. Set 1 1 Aj D 'C .B.cj ; "p // \ sp1 .B.cj ; "p1 //;

E0 D A0 ;

Ej D Aj n .E0 [ : : : [ Ej 1 /:

We claim that D

1 [

Ej :

j D0

Of course Ej , j D 0; 1; : : : is measurable (see Proposition 6.80). In fact, let x 2  and consider sp1 .x/ and '.x/. By 6.82 (2) sp1 .x/ 2 O"p1 .'.x//; by the density of C there is a cj such that at once sp1 .x/ 2 B.cj ; "p1 / and '.x/ \ B.cj ; "p1 / 6D ;, i.e., x 2 A . Finally, either x 2 E, or it is in some Ei , i < j . In either case, we have Sj1 that x 2 j D0 Ej . Now define sp W  ! C by putting: sp .x/ D cj

whenever x 2 Ej :

Then sp satisfies 6.82 (1)–6.82 (3). Condition 6.82 (3) implies that fsm .x/g is a Cauchy sequence for every x 2 .

369

Section 6.2 The selection problem

We let sW  ! Y as follows: s.x/ D lim sm .x/; m!1

x 2 :

Since ' has closed values by (ii) we deduce that s.x/ 2 '.x/ for every x 2 . Step 2. It remains to show that s is measurable. This is equivalent to proving that counter images of closed sets are measurable. Let K be a closed subset of Y . Then 1 each set sm .O"m .K// is measurable. We shall complete the proof by showing that T 1 1 .O"m .K//. In fact on the one hand, when x 2 s 1 .K/, s.x/ 2 K and s .K/ D sm since d.sm .x/; s.x// < "m , we have sm 2 O"m .K/, for every m. On the other hand, 1 .O"m .K// for all m; sm .x/ 2 O"m .K/ and since fsm .x/g converges to when x 2 sm s.x/ and K is closed we get s.x/ 2 K. The proof of Theorem 6.82 is completed. The Kuratowski–Ryll-Nardzewski selection theorem was first published in 1965. In 1966, Castaing observed that it is possible to represent measurable multi-valued maps by the union of single-valued measurable maps. Recall Definition 6.83. A multi-valued map G W  ! PP .X/ has a Castaing representation if there exists a family measurable single-valued maps gn W  ! X such that G.!/ D fgn .!/ j n 2 Ng: The following result is due to Castaing (see [107]). Theorem 6.84. Let X be a separable metric space. Then the multi-valued map G W  ! P.X/ is measurable if and only if G has a Castaing representation. Proof. Let D D fxn j n 2 Ng be such that D D X. Assume that G is measurable. For each n; k 2 N, define the following family of multi-valued map Gn;k W  ! P.X/ by 8  1 < 1 G.!/ \ B xn ; k ; ! 2 GC .!/; Gn;k .!/ D 2 : G.!/ 1 .!/: ! 62 GC 1 .B.xn ; 21k // 2 U. Let V  X be an open set; then Since G is measurable, then GC 1 1 1 1 GC .V / D GC .B.xn ; 2k // [ nGC .B.xn ; 2k // \ GC .V / 2 U:

Hence, ! ! Gn;k .!/ is a measurable multifunction and from Proposition 6.80, the multi-function ! ! Gn;k .!/

370

Chapter 6 Background in multi-valued analysis

is measurable. Then from Theorem 6.82, there exists a family of single-valued measurable maps gn;k defined from  to X such that gn;k .!/ 2 Gn;k .!/;

for all ! 2 :

Now, we shall prove that G.!/ D ffn;k .!/ j n; k  1g: Let x 2 G.!/. For every " > 0, there exist n; k  1 such that B.xn ; 2k /. Hence,

1 2k1

< " and x 2

! 2 G 1 .B.xn ; 2k / and gn;k .!/ 2 B.xn ; 2k /: Then d.gn;k .!/; x/  d.gn;k .!/; xn / C d.xn; x/ 

1 2k1

 ":

This implies that for every " > 0, we have fgn;k .!/ W n; k  1g \ B.x; "/ 6D ;: Hence, G.!/ D fgn;k .!/ j n; k  1g: Conversely, let V  X be an open set; then G 1 .V / D f! 2  j G.!/ \ V 6D ;g D [n1 f! 2  j gn .!/ 2 V g [ gn1 .V / 2 U: D n1

Thus, G is measurable. A famous relation between measurability and continuity of single-valued functions is established by Lusin’s theorem, which states, roughly speaking, that f W  ! Y is measurable if and only if f is continuous up on to subsets of  of arbitrarily small measure. It is not surprising that this result has an analogue for multi-valued mappings (for details, see [31, 176]) which we shall sketch below. Definition 6.85. We will say that a multi-valued map 'W  ! Pcl .Y / has the Lusin property if, given ı > 0, one may find a closed subset ı   such that .nı /  ı and the restriction 'jı of ' to ı is continuous (of course we have assumed that  is a metric space).

Section 6.2 The selection problem

371

Lemma 6.86 ([270]). Let F W  ! Pcl .X/ be a measurable multifunction. Then for every " > 0, there exists a compact set "   with .n" / < " such that F restricted to " has a closed graph. Proof. Fix " > 0 and let D D fxngn2N be a dense subset in X. Then each mapping dn .!/ D d.xn ; F .!// is measurable and by Lusin’s theorem for single-valued functions, there exists a compact set n  ; with .nn /  2"n such that dn restricted to n is continuous. Take " D \1 nD1 n . It is clear that " is compact, .n" /  " and each dn restricted to " is continuous. We claim that F restricted to " has a closed graph. Let .!m ; ym / 2 Gr .F /\" X be a sequence which converges to .!; y/. For every fixed ı > 0, there exists xn 2 X such that d.xn ; y/ < ı. Then for sufficiently large m  m.n/; one has d.xn; F .!m //  d.xn ; ym / < ı. Thus, by the continuity of dn , we have dn .zn ; F .!//  ı. Therefore, d.y; F .!//  2ı. Since ı is arbitrary, we have d.y; F .!// D 0, which implies that .!; y/ 2 Gr .F /. A similar fact holds for the lower semi-continuity. Lemma 6.87. Let F W  ! Pcl .X/ be a measurable multi-valued map. Then for every " > 0, there exists a compact map "   with .n" /  " such that F restricted to " is l.s.c. Proof. Since F is measurable and X is a separable space, there exists a sequence fn j  ! X of measurable single-valued maps such that F .!/ D ffn.!/ j n 2 Ng: By Lusin’s theorem for single-valued functions, for every " > 0, there exists a compact set "   with .n" /  " such that each fn restricted to " is continuous. Now we prove that F is continuous on " . Let C  X be a closed subset; then 1 FC1 .C / D f! 2 " j F .!/ \ C 6D ;g D \1 nD1 fn .C \ " /:

Using the fact that fn are continuous functions, we deduce that FC1 .C / is a closed set. Now, we are in position to prove Theorem 6.88 ([270, 276]). A multifunction F W  ! Pcl .X/ is measurable if and only if it has the Lusin property. Proof. From Lemmas 6.86 and 6.87, we have that if F is measurable, then F has the Lusin property. Conversely, let C  X be a closed set in X. For arbitrary " > 0, we have a closed set "   such that .n" /  " and the restriction of F on " is continuous. Then FC1 .C / consists of a closed set FC1 .C / \ " / and a set FC1 .C / \ n" whose outer measure is less or equal " and therefore FC1 .C / is measurable.

372

Chapter 6 Background in multi-valued analysis

Remark 6.89. The notion of measurable multi-valued maps used in this book is called strong measurable multifunction (see, for example, [147, 276]). Lemma 6.90 ([276]). Let E be a Banach space, J  R be an interval, and F W J ! Pcp .E/ be a measurable multi-valued map. S Then F is almost separable (i.e., there is a subset I  J with .I / D 0 such that fF .t / j t 2 J nI g is separable). Lemma 6.91 (see [505], Lemma 3.2). Let E be a separable Banach space, G W Œa; b ! Pcl .E/ a measurable multifunction, and u W Œa; b ! E a measurable function. Then for any measurable v W Œa; b ! RC , there exists a measurable selection g of G such that for a.e. t 2 Œa; b; ku.t /  g.t /k  d.u.t /; G.t // C v.t /: Proof. By Theorem 6.84, there is a sequence of measurable selections fgn j n 2 Ng of G such that G.t / D fgn .t / j n 2 Ng; for all t 2 Œa; b: Let Tn D ft 2 Œa; bj kgn .t /  u.t /k  d.u.t /; G.t // C r .t /g and consider the single-valued map ‰n W Œa; b ! RC defined by ‰n .t / D kgn .t /  u.t /k  d.u.t /; G.t // C r .t /; t 2 Œa; b; It clear that ‰n is a measurable map; then ‰n1 ..1; 0/ D ft 2 Œa; b j kgn .t /  u.t /k  d.u.t /; G.t // C r .t /g D Tn : The maps Tn ; n 2 N are then measurable and we can easily show that Œa; b D n1 [1 nD1 Tn up to a negligible set. Let E1 D T1 ; E2 D T2 nE1 ; : : : ; En D Tn n [i D1 1 1 Ei ,. . . . Then Œa; b D [i D1 Ei up to a negligible set and fEi gi D1 is a disjoint sequence of measurable sets. Let 1 X En .t /gn .t /; g.t / D nD1

where En represents the characteristic function of the set En . Then g is a measurable selection of G satisfying the requirement of the lemma. Corollary 6.92. Let G W Œ0; b ! Pcp .E/ be a measurable multifunction and g W Œ0; b ! E be a measurable function. Then there exists a measurable selection u of G such that ku.t /  g.t /k  d.g.t /; G.t //:

373

Section 6.2 The selection problem

Proof. Let v" W Œ0; b ! RC be defined by v" .t / D " > 0. From Lemma 6.91, there exists a measurable selection u" of G such that ku".t /  g.t /k  d.g.t /; G.t // C ": Take " D 1=n; n 2 f1; 2; : : :g; hence for every n 2 f1; 2; : : :g; we have kun .t /  g.t /k  d.g.t /; G.t // C 1=n: Using the fact that G has compact values, we may pass to a subsequence if necessary to get that un .:/ converges to some measurable function u in E. Then ku.t /  g.t /k  d.g.t /; G.t //: Corollary 6.93. Let E be a reflexive Banach space, G W Œ0; b ! Pcl;cv .E/ be a measurable multifunction and there exists k 2 L1 .Œ0; b; E/ such that G.t /  k.t /B.0; 1/; t 2 Œ0; b; where B.0; 1/ denotes the closed ball in E and g W Œ0; b ! E a measurable function. Then there exists a measurable selection u of G such that ku.t /  g.t /k  d.g.t /; G.t //: Lemma 6.94. Let X be a separable metric space with fxk g a countable dense subset of X and Y be a Banach space. Let F W X ! Pcp;cv .Y / be a u.s.c. mapping; then the mapping G W X ! P.Y / defined by G.x/ D \1 nD1 co .[fF .xk / j d.x; xk / < 1=ng/;

x2X

satisfies the conditions: (i) For any x 2 X; we have G.x/ is nonempty and G.x/  F .x/. (ii) G is u.s.c. Proof. Consider the family of multi-valued maps Gn W X ! Pcl;cv .Y /; n 2 f1; 2; : : :g defined by Gn .x/ D co f[F .xk / j d.xk ; x/ < 1=ng ; x 2 X; and let

G.x/ D \1 nD1 Gn .x/:

Firstly, we show that G.x/ is nonempty for every x 2 X. For any n; we can take kn 22 f1; 2; : : :g such that d.xkn ; x/ < 1=n. Then co .[i n F .xki //  Gn .x/:

374

Chapter 6 Background in multi-valued analysis

Since fxkn j n 2 Ng [ fxg is compact and F is u.s.c., then [1 nD1 F .xkn / [ F .x/ is compact; hence co .[i n F .xki / is compact. Thus, 1 G.x/ D \1 i D1 Gn .x/ \nD1 co .[i n F .xki // 6D ;:

Using the fact F is u.s.c., we deduce that G.x/  F .x/. Indeed for every " > 0 and for n sufficiently large, we have that d.xk ; x/ < 1=n implies F .xk /  .F .x//" . Since .F .x//" is convex, then G.x/  Gn .x/  .F .x//" ; which implies that G.x/  F .x/. Now we prove that G is u.s.c. Let C be any closed 1 subset of Y and fzq g be a sequence of GC .C / converging to some limit z 2 X. For each n; choose zq such that d.zq ; z/ < 1=2n. If d.xk ; zq / < 1=2n; then d.xk ; z/ < 1=n; hence G2n .zq /  Gn .z/ and ; 6D G2n .zq / \ C  Gn .z/ \ C . Since F is u.s.c. there exists jn > n such that d.xk ; z/ < j1n implies F .xk /  .F .z//1=n . Thus, ; 6D Gjn .z/ \ C  .F .z//1=n . Then there exists yn 2 Gjn .z/ \ C such that d.yn ; F .z//  1=n. Since F .z/ is compact, then some subsequence .ym / of .yn / converges to an element y of C . If jm > n; then Gjm .z/ \ C  Gn .z/ \ C: Hence, y 2 Gn .z/ \ C; This implies that

for all n:

y 2 \1 nD1 Gn .z/ \ C D G.z/ \ C

1 1 and z 2 GC .C /. Hence GC .C / is closed and G is u.s.c.

Now, we shall be concerned with multi-valued mappings which are defined on the topological product of some measurable set with the Euclidean space Rn . We are particularly interested in Carathéodory multi-valued mappings and Scorza–Dragoni multi-valued mappings. Apart from their fundamental importance in all fields of multivalued analysis, such multi-valued mappings are useful in differential inclusions. Let  D Œ0; a be equipped with the Lebesgue measure and Y D Rn . Definition 6.95. A map 'W Œ0; a  Rn ! Pcp .Rn / is called u-Carathéodory (resp. l-Carathéodory; resp. Carathéodory) if it satisfies: (1) t ! '.t; x/ is measurable for every x 2 Rn , (2) x ! '.t; x/ is u.s.c. (resp. l.s.c.; resp. continuous) for almost all t 2 Œ0; a, As before, by U ˝ B.Rn /, we denote the minimal -algebra generated by the Lebesgue measurable sets A 2 U and the Borel subsets of Rn , and then the term “product-measurable” means measurability with respect to U ˝ B.Rn /. More precisely, let a topological space E and a family A of subsets of E. We have

375

Section 6.2 The selection problem

Definition 6.96. A is called a -algebra if it verifies the following properties: (a) ; 2 A. (b) O 2 A ) E n O 2 A. (c) On 2 A; n D 1; 2;    )

S

n1 On

2 A.

Let E be a Banach space and A a subset of J  E. Definition 6.97. A is called L˝B measurable if A belongs to the -algebra generated by all sets of the form I  D where I is Lebesgue measurable in J and D is Borel measurable in E. Now, we prove Proposition 6.98. Let 'W Œ0; aRn ! P.Rm / be a Carathéodory multi-valued map. Then ' is product-measurable. Proof. Consider the countable dense subset Qn  Rn of rationals. For closed A  Rn , a 2 Qn and k, the set Gk .A; a/ D ft 2 Œ0; a j '.t; a/ \ O1=k .A/ 6D ;g  B.a; 1=k/ belongs to U ˝ B.Rn /. Since ' is l.s.c. in the second variable, we have: 1 'C .A/ 

1 [ \

Gk .A; a/;

kD1 a2Qn

while the u.s.c. of ' implies the reverse inclusion. The proof is completed. The following example shows that an l-Carathéodory multi-valued map needs not to be product-measurable. Example 6.99. Let 'W Œ0; 1  R ! P.R/ be defined by ( f0g; if u D 0; '.t; u/ D Œ0; 1; otherwise: Then ' is l-Carathéodory but not u-Carathéodory. An analogous example can be constructed for u-Carathéodory mappings. Let 'W Œ0; a  Rn ! P.Rn / be a fixed multi-valued map. We are interested in the existence of Carathéodory selections, i.e., Carathéodory functions f W Œ0; a  Rn ! Rn such that f .t; u/ 2 '.t; u/, for almost all t 2 Œ0; a and all n 2 Rn . It is evident that, in the case when ' is u-Carathéodory, there does not exist a selection in general

376

Chapter 6 Background in multi-valued analysis

(the reason is exactly the same as in Michael’s selection theorem). For l-Carathéodory multi-valued maps ', however, this is an interesting problem. In order to study this problem, we shall use the following notation: C.Rn ; Rn / D ff W Rn ! Rn j f is continuousg: We shall understand that C.Rn ; Rn / is equipped with the topology on uniform convergence on compact subsets of Rn . In fact this topology is metrisable. Moreover, as usual by L1 .Œ0; a; Rn /, we shall denote the Banach space of Lebesgue integrable functions. There are two ways, essentially, to deal with the above selection problem. Let 'W Œ0; a  Rn ! P.Rn / be an l-Carathéodory mapping. On the one hand, we may show that the multi-valued map ˆW Œ0; a ! P.C.Rn ; Rn //; ˆ.t / D fuW Rn ! Rn j u.x/ 2 '.t; u.x// and u is continuousg is measurable. Then, if we assume that ' has convex values, in view of the Michael selection theorem, we obtain that ˆ.t / 6D ; for every t . Moreover, let us observe that every measurable selection of ˆ will give rise to a Carathéodory selection of '. On the other hand, we may show that the multi-valued map: ‰W Rn ! P.L1 .Œa; 1; Rn //; ‰.x/ D fuW Œa; 1 ! Rn j u.t / 2 '.t; u.t //; for almost all t 2 Œ0; ag is a l.s.c. mapping. Consequently, continuous selections of ‰ will give rise to Carathéodory selections of '. Hence, our problem can be solved by using Michael and Kuratowski–Ryll-Nardzewski selection theorems. Let us formulate, only for informative purposes, the following result owed to A. Cellina. Theorem 6.100. Let 'W Œ0; a  Rn ! Pcp;cv .Rn / be a multi-valued map. If '.  ; x/ is u.s.c. for all x 2 Rn and '.t;  / is l.s.c. for all t 2 Œ0; a, then ' has a Carathéodory selection. Proposition 6.101. Let X be a separable metric space with fxk g a countable dense subset of X and Y a separable Banach space. Let F W   X ! Pcp;cv .Y / be an upper Carathéodory multifunction. Then the mapping G W   X ! Pcl;cv .Y / defined by G.!; x/ D \1 nD1 co fF .!; xk / j d.xk ; x/ < 1=ng satisfies the following conditions: (1) For each ! 2  and x 2 X; ; 6D G.!; x/  F .!; x/.

377

Section 6.2 The selection problem

(2) For each ! 2 ; G.!; :/ is u.s.c. (3) G is L ˝ B-measurable. Proof. Let Gn W   X ! P.Y / be a sequence of multi-valued maps defined by Gn .!; x/ D [fF .!; xk / j d.xk ; x/ < 1=ng;

.!; x/ 2   X:

Together with 6.101 (1) and 6.101 (2), we apply Lemma 6.94; then we have F .t; x/ Hn .!; x/ 6D ;;

for every .!; x/ 2   X:

and Hn.!; :/ is u.s.c. Now, we prove that Hn.:; :/ is L ˝ B is measurable. Indeed, for any open subset V of Y , we have Hn1 C .V / D f.!; x/ 2   X j Hn .!; x/ \ V 6D ;g D [1 nD1 f! 2  j F .!; x/ \ V 6D ;g  fx 2 X j d.x; xk / < 1=ng 2 L ˝ B: Then the multi-map Gn W   X ! P.Y / defined by Gn .!; x/ D co .Hn .!; x// is measurable (see [38]). If we show that 1 1 .C / D \1 GC nD1 .Gn /C .Cn1 /;

where

C D \1 nD1 Cn1

and Cn1 D fy 2 Y j d.y; C / < 1=ng ; then we can conclude that G is L ˝ B measurable. It is obvious that 1 1 .C /  \1 GC nD1 .Gn /C .Cn1 /:

Conversely, let

1 .!; x/ 2 \1 nD1 .Gn1 /C .cn1 /I

then Gn .!; x/ \ Cn1 6D ; for all n 2 N. Since F .!; :/ is u.s.c. by the same way as in the proof of Lemma 6.94, we have ; 6D \1 nD1 Gn .!; x/ \ Cn1 D G.!; x/ \ C: Hence,

1 1 \1 nD1 .Gn /C .Cn1 /  GC .C /:

378

Chapter 6 Background in multi-valued analysis

6.2.5 Aumann and Filippov theorems The following important result is due to J. R. Aumann (see [127, 254, 286, 367]). Theorem 6.102 (Aumann). If G W  ! Pcp .X/ is a multi-valued map such that the graph Gr .G/ of G is measurable, then G possesses a measurable selector. For the proof of Aumann’s theorem, we need a very useful projection property enjoyed by complete measurable spaces. Lemma 6.103. Let X be a complete separable metric space and G 2 U ˝ B.X/. Then its projection is measurable: … .G/ WD f! 2 j 9 x 2 X; .!; x/ 2 Gg 2 U: Proof of Theorem 6.102. Let C be a closed set in X; then 1 GC .C / D … .Gr .G/ \   C /

is measurable by Lemma 6.103. Hence, G is a measurable multifunction. From the Kuratowski–Ryll-Nardzewski selection theorem, G has a measurable selection. Proposition 6.104 ([107, 505]). Let X be a complete separable Banach space, G W Œt0 ; b ! Pcl .X/ be a Lebesgue measurable multi-valued map (i.e., for every open 1 subset V  X, the set GC .V / is Lebesgue measurable), and f W Œt0 ; b ! X; C g W Œt0 ; b ! R be measurable single-valued maps. Then the maps t ! co G.t /; and

t ! B.f .t /; g.t //;

t ! d.f .t /; G.t //

e G.t / .g.t / D fx 2 G.t /j dX .x; f .t // D dY .f .t /; G.t //g …

are measurable. Consequently, if fv 2 G.t /j kv  g.t /k  k.t /g 6D ;;

a.e. in Œt0 ; b;

then there exists a measurable selection u.t / 2 G.t / such that for a.e. t 2 Œt0 ; b, we have ku.t /  f .t /k  k.t /: Proof. Since G is a measurable multifunction and X is a separable Banach space, then from Theorem 6.84, there is a sequence of measurable selections fgn .:/j n  1g such that G.t / D fgn .t /j n 2 Ng:

379

Section 6.2 The selection problem

Let fng be a sequence of Pnon-negative rational numbers such that there are only finitely many n 6D 0 and 1 nD1 n D 1. The set 1 nX

gn .:/j .n /n1 2 QC

o

nD1

is a countable family of measurable functions. Using the fact that fgn .t /j n 2 Ng  G.t /; we have

1 nX

o gn .:/ j .n /n1 2 QC  co G.t /

nD1

and co G.t / 

1 nX

o gn .:/j .n /n1 2 QC :

nD1

Hence, co G.t / D

1 nX

o gn .:/j .n /n1 2 QC I

nD1

then we conclude that co G is measurable. Now, we show that t ( B.f .t /; g.t // is a measurable multifunction. We can easily verify that B.f .t /; g.t // D f .t / C g.t /B.0; 1/; t 2 Œt0 ; b: Since X is a separable space, then there exists fxn j n 2 Ng a countable subset in B.0; 1/ such that fxn j n  1g D B.0; 1/: Set ff .t / C g.t /xn j n  1g  B.f .t /; g.t //: Hence, B.f .t /; g.t // D ff .t / C g.t /xn j;

n  1g:

This implies that B.f .:/; g.:// is measurable. The map t ! d.f .t /; G.t // is measurable. Let r > 0 and ft 2 Œt0 ; bj d.f .t /; G.t // < r g D D

1 [ nD1 1 [ nD1

ft 2 Œt0 ; bj kf .t /  fn .t /k < r g n.r; 1/;

380

Chapter 6 Background in multi-valued analysis

where n.tS / D kfn .t /  f .t /k. We can easily prove that n are measurable functions, then 1 nD1 n ..r; 1// is a measurable set; we conclude that d.f .:/; G.:// is a measurable single-valued function and e G.t /.g.t // D G.t / \ fv 2 Y j dX .v; f .t // D dY .f .t /; G.t //g: … Theorem 6.105 ([38]). Let .; A; / be a complete -finite measurable space, X a complete separable metric space, and F W  ! P.X/ be a measurable multi-valued map with closed images. Consider a Carathéodory multi-valued map G from   X to a complete separable metric space Y . Then the map  3 ! ! G.!; F .!// 2 P.Y / is measurable. While this result characterises the measurability, the following lemma is a measurable selection result. It is crucial in the proof that the control system coincides with the differential inclusion problem. It is known as Filippov’s theorem. Lemma 6.106 (see [38], Theorem 8.2.10). Consider a complete -finite measurable space .; A; / (A is a -algebra and  is a positive measure). Let X; Y be two complete separable metric spaces. Let F W X ! P.Y / be a measurable multi-valued map with closed nonempty values and g W   X ! Y a Carathéodory map. Then for every measurable map h W  ! Y satisfying h.!/ 2 g.!; F .!//;

for almost all ! 2 ;

there exists a measurable selection f .!/ 2 F .!/ such that h.!/ D g.!; f .!//;

for almost all ! 2 :

Proof. Define the multi-valued map H W  ! P.X/ by letting H.!/ D F .!/ \

1 \ ˚

 x 2 Xj dy .g.!; x/; h.!// < 1=n :

nD1

Let W   X !   Y be a measurable function defined by .!; x/ D .!; g.!; x//

.!; x/ 2   X

and the multifunction G W  ! P.X/ defined by ˚  G.!/ D x 2 Xj dy .g.!; x/; h.!// < 1=n :

381

Section 6.2 The selection problem

Observe that

e Gr .G/ D 1 .Gr .G//;

e is a multi-valued map defined by where G e G.!/ D B .h.!/; 1=n/ : Now, we show that g.:; :/ is measurable function. Since X is separable Banach space, then there exists a set D D fxn j n 2 Ng  X such that D D X and C be a closed subset in Y; then g 1 .C / D

1 [ \

f! 2 j g.!; v/ 2 Cng

nD1 v2D

 fx 2 Xj dX .x; v/ < 1=ng 2 A ˝ B.X/; where Cn D fy 2 Y j dY .y; C / < 1=ng : Then g is a measurable single-valued map which implies that is measurable. Hence Gr .G/ 2 A ˝ B.X/. From Proposition 6.80, G is measurable; also H is measurable multifunction. Therefore, by Theorem 6.82, H has a measurable selection f . Then for every n 2 N, we have dY .g.!; f .!//; h.!//  1=n: Hence, h.!/ D g.!; h.!//;

for almost every ! 2 :

Lemma 6.107 ([468]). Let X; Y be complete separable metric spaces and F W Œ0; b X ! Pcl .Y / be a L ˝ B.X/ measurable multifunction. Then for any continuous function x W Œ0; b ! Y , the multifunction t ! F .t; x.t // is measurable and has a strongly measurable selector. Proof. Let B  Y be a closed set and put C D ft 2 Œ0; b j F .t; x.t // \ B 6D ;g: Let V D f.t; u/ 2 Œ0; b  X j F .t; u/ \ B 6D ;g: Then the set V is L ˝ B.X/-measurable, hence W D f.t; u/ 2 V j u D x.t /g measurable. From Theorem 3.5 and Proposition 2.2 in [254], if follows that the set C D ft 2 Œ0; b j .t; x.t // 2 W g is measurable. This implies that t ! F .t; x.t // is measurable. By Theorem 5.6 in [254], we obtain that there exists a strong measurable selector.

382

Chapter 6 Background in multi-valued analysis

Lemma 6.108 ([258]). Let .; †/ be a measurable space, X; Y are separable metric spaces and F W   X ! Pcl .Y / be a multifunction. Assume that for every x 2 X t ! F .t; x/ is measurable and for every t 2 , we have x ! F .t; x/

is continuous or Hd  continous:

Then .t; x/ ! F .t; x/

is  ˝ B.X/  measurable:

6.2.6 Hausdorff measurable multi-valued maps In this subsection, we study of some Hausdorff measurability properties of multivalued maps. Let .; †/ be a measurable space and let  be a non-negative measure on . We say that the measurable space .; †/ is complete if the -algebra † coincides with  the Lebesgue completion of  with respect to  and a metric space X. Definition 6.109. A multifunction F W  ! P.X/ is said to be: (1) d -measurable if for every x 2 X, the function ! ! d.x; F .!// is measurable on I (2) Hd -measurable if for every C 2 P.X/, the functions ! ! Hu.F .!/; C / and ! ! Hl .C; F .!// are measurable on I e max -measurable if for every C 2 P.X/ the function ! ! Hd .F .!/; C / is (3) H measurable on . Proposition 6.110. Let F W  ! P.X/ be a multifunction. We have: (1) F is Hd -measurable if and only if F is h-measurable, e max -measurable) if and only if for every C 2 P.X/, (2) F is Hd -measurable (resp. H ! ! Hu.F .!/; C / and ! ! Hl .C; F .!// are measurable on  (resp. ! ! Hd .F .!/; C / is measurable on . ); e max -measurable; (3) F is Hd -measurable implies that F is H (4) F is Hd -measurable implies that F is d -measurable. Definition 6.111. A multi-valued map F W  ! P.X/ is called simple if there exists an admissible partition fk g (i.e., a countable family fk g of nonempty measurable pairwise disjoint subsets k of ) whose union is  and such that F restricted to each k is constant. An analogous notion applies to single-valued maps.

Section 6.2 The selection problem

383

Remark 6.112. Each simple multi-valued map F W  ! P.X/ is weakly measurable and Hd -measurable. The following proposition is a variant of the theorem of Kuratowski and RyllNardzewski. e max-measurable multifunction, Proposition 6.113. Let F W  ! Pcl;b .X/ be an H whose range F ./ is a separable subset of Pcl;b .X/. Then we have: (1) there is a sequence fFn j n 2 Ng of simple multi-valued maps Fn W  ! Pcl;b .X/. converging to F uniformly on I (2) if X is complete, then F has a measurable selector. e max-measurable multifunction, Proposition 6.114. Let F W  ! Pcl;b .X/ be an H whose range F ./ is a separable subset of Pcl;b .X/. Then we have: (1) if X is separable, then F is Hd -measurable , F is weakly measurable , F is d -measurableI (2) if F ./ is a separable subset of Pcl;b .X/, then e max -measurable , F is weakly measurable: F is H Corollary 6.115. Let .; L/ be a Borel space, where  is a metric space. Then each Hd -u.s.c. or Hd -l.s.c. multi-valued map F W  ! Pcl;b .X/ is Hd -measurable. Proposition 6.116. Let  be a complete separable metric space. Let  beX a nonnegative finite measure defined on the completion L of the Borel  -algebra . Let X be a metric space. For a multi-valued F W  ! Pcl;b .X/, the following statements are equivalent: (1) F is Lusin measurable, (2) F is Hd -measurable, and there exists a set 0 L with .0 / D 0 such that F .n0 / is a separable subset of Pcl;b .X/. More details on the above results may be found in [141].

6.2.7 Product-measurability and the Scorza–Dragoni property We shall end this section by introducing mappings having the Scorza–Dragoni property. First, we recall some definitions. Let A be a subset of J  B.E/.

384

Chapter 6 Background in multi-valued analysis

Definition 6.117. A is L ˝ B.E/ measurable if A belongs to the -algebra generated by all sets of the form N  D, where N is Lebesgue measurable in J and D is Borel measurable in B.E/. Definition 6.118. (a) A map 'W Œa; b  Rn ! P.Rn / is said to be integrably bounded if there exists an integrable function  2 L1 .Œa; b/ such that kyk  .t / for every x 2 Rn , t 2 Œa; b and y 2 '.t; x/. (b) ' has linear growth if there exists an integrable function  2 L1 .Œa; b/ such that kyk  .t /.1 C kxk/ for every x 2 Rn , t 2 Œa; b and y 2 '.t; x/. In fact, the class of nonlinearities ' satisfying the Nagumo–Bernstein condition is more general than that of ' with linear growth. Definition 6.119. We say that a multi-valued map 'W Œ0; a  Rn ! Pcl .Rn / has the u-Scorza–Dragoni property (resp. l-Scorza–Dragoni property; resp. Scorza–Dragoni property) if, given ı > 0, one may find a closed subset Aı  Œ0; a such that the measure .Œ0; a n Aı /  ı and the restriction e ' of ' to Aı  Rn is u.s.c. (resp. l.s.c.; resp. continuous). Let us observe that the Scorza–Dragoni property plays the same role for multivalued mappings of two variables as the Lusin property for multi-valued mappings of one variable. In addition, there is a close connection between Carathéodory multivalued mappings and multi-valued mappings having the Scorza–Dragoni property. Proposition 6.120. Let 'W Œ0; a  Rm ! Pcp .Rn / be a multi-valued map. Then we have: (1) ' is Carathéodory if and only if ' has the Scorza–Dragoni property, (2) if ' has the u-Scorza–Dragoni property, then ' is u-Carathéodory, (3) if ' has the l-Scorza–Dragoni property, then ' is l-Carathéodory, (4) if ' is a product-measurable, l-Carathéodory multi-valued map, then ' has the l-Scorza–Dragoni property. (5) Assume that ' satisfies the Filippov condition, i.e., for every open set U , V  Rn , the set ft 2 Œ0; a j '.t; U /  V g is Lebesgue measurable; then ' is uCarathéodory multi-valued map if and only if ' has the u-Scorza–Dragoni property.

Section 6.2 The selection problem

385

Proposition 6.120 is taken from [31]. All proofs are rather technical and need sometimes long calculations. Therefore, we shall present below only two examples showing that l-Carathéodory (u-Carathéodory) maps need not have the l-Scorza–Dragoni (uScorza–Dragoni) property. Example 6.121. Let 'W Œ0; 1  R ! P.R/ be the map defined by 8 if u D t and t 2 Œ0; 1 n A; ˆ < f0g; '.t; u/ D f1g; if u D t and t 2 A; ˆ : Œ0; 1; otherwise; where A is a nonmeasurable subset of Œ0; 1. Then obviously ' is l-Carathéodory but does not have the l-Scorza–Dragoni property. Moreover, ' is not product measurable. Example 6.122. Let 'W Œ0; 1  R ! P.R/ be defined by ( Œ0; 1; if t D u and t 2 A; '.t; u/ D otherwise; f0g; where A is a nonmeasurable subset of Œ0; 1. It is not hard to see that ' is u-Carathéodory but does not have the u-Scorza–Dragoni property. Until the end of this section, X is a metric separable space and  a complete measure space. We also assume that 'W X ! Pcp .X/ is a product-measurable multi-valued mapping. First, we shall prove: Proposition 6.123. If 'W   X ! P.X/ is product measurable, then the function f W   X ! Œ0; C1/ defined by the formula: f .!; x/ D dist.x; '.!; x// is also product measurable. Proof. We have: f.!; x/ 2   X j f .!; x/ < r g D f.!; x/ 2   X j '.!; x/ \ Or .fxg/ 6D ;g: Therefore, our assertion follows from the assumption that ' is measurable. The following Scorza–Dragoni type result describes possible regularisation of Carathéodory maps. For the proof, we refer to [138, 271, 414]. Theorem 6.124. Let X be a compact subset of Rn and 'W Œ0; aX ! Pcp;cv .Rn / be an upper-Carathéodory map. Then there exists a u-Scorza–Dragoni W Œ0; a  X ! Pcp;cv .Rn / such that:

386 (1)

Chapter 6 Background in multi-valued analysis

.t; x/  '.t; x/ for every .t; x/ 2 Œ0; a  X,

(2) if   Œ0; a is measurable, uW  ! Rn and vW  ! X are measurable maps and u.t / 2 '.t; v.t // for almost all t 2 , then u.t / 2 .t; v.t // for almost all t 2 . Now, we prove: Theorem 6.125. Let E; E1 be two separable Banach spaces and 'W Œa; b  E ! Pcp;cv .E1 / be a u-Scorza–Dragoni map; then ' is -Ca-selectionable: '.t; z/ D T 1 kD1 'k .t; z/. For k 2 f1; 2; : : :g, the maps 'k W Œa; b  E ! P.E1 / are u-Scorza– Dragoni and we have  [  '.t; x/ : 'k .t; x/  x2E

Moreover, if ' is integrably bounded, then ' is -mLL-selectionable. Proof. Consider the family fB.y; rk /gy2E , where rk D .1=3/k , k D 1; 2; : : : Using Stone’s theorem for every k D 1; 2; : : :, we get a locally finite subcovering fUik gi 2Ik of fB.y; rk /gy2E . For every i 2 Ik , k D 1; 2; : : :, we fix the centre yik 2 E such that Uik  B.yik ; rk /. Now, let ki W E ! Œ0; 1 be a locally Lipschitz partition of unity subordinated to fUik gi 2I k . Define ik W Œ0; a ! P.E/ and fik W Œ0; a ! E as follows: k i .t /

 D conv

[

 '.t; y/ ;

y2B.yik ;2rk /

and let fik be a measurable selection of ik which exists in view of the Kuratowski– Ryll-Nardzewski theorem. Finally, define 'k W Œa; b  E ! P.E1 / and fk W Œa; b  E ! E1 as follows: X X

ki .z/  ik .t /; and fk .t; z/ D

ki .z/  fik .t /: 'k .t; z/ D i 2Ik

i 2Ik

T Then fk  'k . Fix t 2 Œa; b. If '.t;  / is u.s.c., then '.t; z/ D 1 kD1 'k .t; z/ and 'kC1 .t; z/  'k .t; z/, for every z 2 E. By the assumptions on ', the map '.t;  / is u.s.c. for almost all t 2 Œ0; a, and the first part of Theorem 6.125 is proved. The second claim is an immediate consequence of the first one. The following -selectionability lemma can be proved as in [Proposition 4.1 of De Belasi [136]] or [Lemma of Papageorgiou [386]]. Lemma 6.126. Assume that X is a Polish space (a complete separable metrisable space), Y is a separable Banach space, and F W Œ0; b  X ! Pcl;cv .Y / satisfies

387

Section 6.2 The selection problem

(1) t ! F .t; x/ is measurable; (2) x ! F .t; x/ is Hd -u.s.c. (i.e., "  ı-u.s.c.); (3) kF .t; x/kP  l.t / a.e. on Œ0; b with l 2 Lp .Œ0; b; RC /; 1  p < 1. Then, there exists a sequence of Fn W Œ0; b  X ! Pcl;cv .Y / such that for every n 2 N and x 2 X, there exists ln .x/ > 0 and "n > 0 such that if x1 ; x2 2 B.x; "n /; then Hd .Fn .t; x1 /; Fn .t; x2 //  ln .x/p.t /jx1  x2 j;

a.e. on Œ0; b;

F .t; x/  : : : Fn .t; x/  Fn1 .t; x/ : : : ; kFn .t; x/kP  p.t /; Hd .Fn .t; x/; F .t; x// ! 0; as n ! 1

a.e. on Œ0; b;

for all .t; x/ 2 Œ0; b  X;

and there exists un W Œ0; b  X ! Y measurable in t 2 Œ0; b, locally Lipschitz in x 2 X and un .t; x/ 2 Fn .t; x/ for .t; x/ 2 Œ0; b  X and n 2 N. Remark 6.127. The above lemma tells us that if F W Œ0; b  X ! Pcp;cv .Y / is an integrably bounded u-Carathéodory multi-map, then F is -mLL-selectionable. This result is often used in applications when dealing with the topological structure of solution sets (see Chapters 3 and 4). It corresponds to the u-Carathéodory version of Theorem 6.125. For u.s.c. multi-valued maps, there exists a useful approximation result similar to Lasota–Yoke lemma [318]. Lemma 6.128 ( [146, Lemma 2.2]). Let  be a subset of a Banach space, E be a n Banach space and F W  ! Pcl;cv .E/ S be a multi-valued map. Let rn D 3 , fU gƒ be a locally finite refinement of  D !2 B.!; rn /; f  g2ƒ be a locally Lipschitz partition of unity subordinate to fU g2ƒ ; pick ! 2 U  B.! ; rn / \  and let X Fn .!/ D  .!/C ; 2ƒ

where C D co F .B.! ; 2rn / \ . Then (1) F .!/  FnC1 .!/  Fn .!/  co F .B.!; 3rn / \ / on . (2) If F is uniformly locally bounded, then Fn is locally Lipschitz for large n. (3) If F is "  ı  u.s.c., then Hd .Fn .!/; F .!// ! 0 on , as n ! 1. The following result is in part the multi-valued version of a well-know theorem of Scorza–Dragoni and Lusin’s theorem which states, roughly speaking, that every measurable function is continuous almost everywhere.

388

Chapter 6 Background in multi-valued analysis

Theorem 6.129 ([412]). Let X; Y be a sparable Banach spaces and F W J  X ! Pcl .Y / with J a measurable subset of R. Assume that Gr .F .t; :// is closed in X  Y b W J  X ! P.Y / such that for almost t 2 J . Then there exists F b .t; x/  F .t; x/ for all x 2 XI (1) for almost all t 2 J; F (2) if   J is measurable and u W  ! X; v W  ! Y are measurable functions b .t; u.t // a.e. in I with v.t / 2 F .t; u.t // a.e in ; then v.t / 2 F (3) for every " > 0, there exists a closed J"  J with .J nJ" / < " and the graph of b jJ" X is closed in J  X  Y . F If E is separable Banach space, we present the following result of Scorza–Dragoni type (essentially due to Rzézuchowski [412]): Theorem 6.130 ([412]). Let F W Œ0; b  D ! Pcl .E/ with D a closed convex subset of E. Assume that F satisfies the following conditions: (1) F is an upper-Carathédory map (2) F maps compact subsets of J  D into compact ones. Then there is a map F0 W Œ0; b  D ! P.E/ [ f;g such that (a) for all t 2 Œ0; b and x 2 D, F0 .t; x/  F .t; x/I (b) if   Œ0; b is measurable, u; v W  ! D; then v.t / 2 F0 .t; u.t // a.e. in ; (c) for any " > 0; there is a closed "  Œ0; b such that F0 restricted to "  D has nonempty values and is (jointly) upper semi-continuous. Finally, we presents some nice approximations of upper semi-continuous multivalued maps. Theorem 6.131 ( [141]). Let  be a complete separable metric space. Let  be a non-negative finite measure on the completion L of the Borel -algebra †. Let X be a complete separable metric space with Borel -algebra B.X/, and let E be a separable Banach space. Suppose that F W   X ! Pcl;cv;b .E/ is a bounded multi-valued map such that (1) F is L ˝ B.X/ weakly measurable; (2) for each ! 2 ; x ! F .!; x/ is Hd -u.s.c.. Then there exists a sequence fFn j n 2 Ng of multi-valued maps Fn W   X ! Pcv;cl;b .E/ and a sequence ffn j n 2 Ng of single functions fn W   X ! E satisfying, for every n 2 N, the following conditions:

389

Section 6.2 The selection problem

.a1 / for each x 2 X; ! ! Fn .!; x/ is L-weakly measurable .a2 / for each ! 2 ; x ! Fn .!; x/ is locally Lipschitz; .a3 / for each .!; x/ 2   X; F .t; x/  FnC1 .t; x/  Fn .t; x/  co

 [

 .F .t; x/ ;

for all n 2 NI

x2X

.a4 / for each .t; x/ 2   X; lim Hd .Fn .t; x/; F .t; x// D 0I n!1

.a5 / fn is a Carathéodory–Lipschitz selector of Fn . Another version is given by (see [141]) Theorem 6.132. Let  be a complete separable metric space. Let  be a non-negative finite measure on the completion L of the Borel -algebra †. Let X be a complete separable metric space with Borel -algebra B.X/, and let E be a separable Banach space. Suppose that F W   X ! Pcl;cv;b .E/ is a bounded multi-valued map such that (1) F is L ˝ B.X/ Hd -measurable; (2) for each ! 2 ; x ! F .!; x/ is Hd -u.s.c..(resp. Hd l.s.c.) (3) F .  X/ is a separable subset of Pcl;b .X/. Then for every " > 0, there exists a compact set K"  , with .nK" / < " such that F restricted to K"  X is Hd -u.s.c..(resp. Hd -l.s.c.) P Corollary 6.133 ( [141]). Let ; X; L; ;  be as in Theorem 6.132, and Y be a separable metric space. Let F W X ! Pcl;b;cp .Y / be a multi-valued map such that the conditions 6.132 (1) and 6.132 (2) of Theorem 6.132 hold. Then for every " > 0, there exists a compact set K"  , with .nK" / < " such that F restricted to K"  X is Hd -u.s.c..(resp. Hd -l.s.c.). P Theorem 6.134 ([141]). Let ; X; L; ; and  be as in Theorem 6.132, let Y be a metric space, and let F W   X ! Pcl;b;cp .Y / be a multi-valued map such that (1) for each x 2 X, t ! F .!; x/ is L-Hd -measurable and has F .fxg/ separable Pcl;b ; (2) for each t 2 ; x ! F .!; x/ is Hd -continuous. Then for every " > 0, there exists a compact set K"   with .nK" / < " such that F restricted to K"  X is Hd -continuous.

390

Chapter 6 Background in multi-valued analysis

6.3 Decomposable sets 6.3.1 The Bressan–Colombo–Fryszkowski selection theorem We end the selection problem with a theorem where convexity of values of multivalued functions is replaced by decomposability. Definition 6.135. A subset A of L1 .J; E/ is decomposable if for all functions u; v 2 A and measurable subset N  L1 .J; E/, the function uN C vJ N 2 A, where  stands for the characteristic function. The family of all nonempty closed and decomposable subsets of L1 .J; E/ is denoted by D. Definition 6.136. Let Y be a separable metric space and let N W Y ! P.L1 .J; E// be a multi-valued operator. We say that N has property (BC) if (1) N is lower semi-continuous (l.s.c.); (2) N has nonempty closed and decomposable values. Definition 6.137. Let F W J  E ! P.E/ be a multi-valued function with nonempty compact values. We say that F is of lower semi-continuous type (l.s.c. type) if its associated Niemytzki operator F is lower semi-continuous and has nonempty closed and decomposable values. Next, we state a selection theorem due to Bressan, Colombo, and Fryszkowski. Theorem 6.138 (see [81, 186, 187]). Let Y be separable metric space, E a Banach space, and let N W Y ! P.L1 .J; E// be a multi-valued operator which has property (BC). Then N has a continuous selection, i.e., there exists a continuous function (single-valued) f W Y ! L1 .J; E/ such that f .x/ 2 N.x/ for every x 2 Y .

6.3.2 Decomposability in L1.T; E / Below we establish a result which unifies and extends to a larger class of multi-valued contractions defined on arbitrary complete absolute retracts both Theorems 1.84 and 1.85. Let .T; F; / be a finite, positive, nonatomic measure space and let .E; k  k/ be a Banach space. As before, we denote by L1 .T; E/ the Banach space of all (equivalence classes) -measurable functions uW T ! E such that the function t ! ku.t /k is -integrable, equipped with the norm Z kukL1 .T;E/ D ku.t /k d: T

1

We always assume that the space L .T; E/ is separable. Now, we set

Section 6.3 Decomposable sets

391

Definition 6.139. A nonempty set K  L1 .T; E/ is said to be decomposable if for every u1 ; u2 2 K and every -measurable subset A of T , one has .A  u1 C .1  A /  u2 / 2 K; where A denotes the characteristic function of A  T . Some basic facts about decomposable sets in L1 .T; E/ are collected in the following: Remark 6.140. (1) It is easily seen that every decomposable subset of L1 .T; E/ is contractible and, consequently, infinitely connected. (2) Any closed decomposable subset of L1 .T; E/ is absolute retract. (3) A simple calculation shows that the open (or closed) ball unit ball of L1 .T; E/ is not decomposable. Let .; †; / be a -finite measure space, X a separable Banach space, F W  ! P.X/ a multi-valued map, and SF D ff W  ! X measurable with f 2 L1 .; X/g. The following important result characterises closed decomposable sets in the Lebesgue space L1 .; X/ to be nothing but selection sets of measurable multivalued maps; for the proof, we refer the reader to [387, Theorem 6.4.6]. Theorem 6.141. Let K  L1 .; X/ be a nonempty closed subset. Then K is decomposable if and only if K D SF for some measurable multi-valued map F W  ! Pcl .X/. Let X; Y be Banach spaces and F W J  X ! Pcl .Y / be a multi-map. Assign to F the multi-valued operator F W C.J; Y / ! P.L1 .J; Y // defined by F.x/ D SF ;x . The operator F is called the Nemyts’ki˘ı operator associated to F . Definition 6.142. Let F W J  X ! Pcp .Y / be a multi-map. We say that F is of lower semi-continuous type (l.s.c. type) if its associated Nemyts’ki˘ı operator F is lower semi-continuous and has nonempty closed and decomposable values. Along with this definition, the following lemma is very useful. Lemma 6.143 (see, e.g., [185]). Let F W J  X ! Pcp .Y / be an integrably bounded multi-map satisfying .Hlsc / F W J  X ! P.Y / is a nonempty compact-valued multi-map such that (a) the mapping .t; x/ 7! F .t; x/ is L ˝ B measurable; (b) the mapping x 7! F .t; x/ is l.s.c. for a.e. t 2 J . Then F is of lower semi-continuous type.

392

Chapter 6 Background in multi-valued analysis

A selection theorem for lower semi-continuous type multi-valued maps is given by [Theorem 6.138, Chapter 6]. For more details concerning the notion of decomposability, we recommend [186, 187].

6.3.3 Integration of multi-valued maps Let .; U; / be a complete -finite measure space and E be a separable Banach space. Definition 6.144. Let F W  ! P.E/ be a measurable multifunction. The integral of F is defined by Z Z F .!/d D f .!/dj f 2 F ; 



where SF D ff 2 L1 .; /j f .!/ 2 F .!/ a:e:g;

R and  f .!/d is Bochner-integrable. This definition was introduced by Aumann [40] as a natural generalisation of the integration of point-valued functions. Consider the following example [258]. Example 6.145. Let  D Œ0; 1 be equipped with the Lebesgue measure and let F be a multi-valued map F W Œ0; 1 ! P.R/ defined by ( !2A D1 ; F .!/ D ! 62 A; D2 ; where A is a nonmeasurable subset of Œ0; 1 and D1 , D2 are two infinity countable sets in Œ0; 1. It is clear that F is measurable but does not have a measurable selection. R1 Hence SF D ;, which implies that 0 F .!/d! is not well defined. Definition 6.146. F W J  E ! P.E/ is said to be integrably bounded if there exists p 2 L1 .J; RC / such that kF .t; y/kP  p.t / for a.e. t 2 J and each y 2 E: Let F W  ! Pcl .E/ be a multi-valued map and recall SF D fx 2 L1 .; E/ j x.!/ 2 F .!/g. Lemma 6.147. R Assume that F is integrably bounded and graph measurable. Then SF 6D ; and  F .!/d is bounded. Proof. From Theorems 6.82, 6.102 and using the fact that F is integrably bounded, we can easily prove that SF is nonempty and the Aumann integral of F is bounded.

Section 6.3 Decomposable sets

393

More precisely, we have (see, e.g., [478, Theorem 5.10]). Lemma 6.148. If F .:/ is graph-measurable, then SF is nonempty if and only if inf kxk 2 L1 .; RC /.

x2F .!/

6.3.4 Nemytskiˇı operators Definition 6.149. Let F W Œa; b  E ! P.E/ be a multi-valued map with nonempty compact values. Assign to F the multi-valued operator F W C.J; E/ ! P.L1 .J; E// by letting F.y/ D fv 2 L1 .J; E/ j v.t / 2 F .t; y.t //; for a.e. t 2 J g: The operator F is called the Nemyts’ki˘ı operator associated to F . Lemma 6.150. Let X and E be two separable Banach spaces and F W J  E ! Pcp;cv .E/ be an L1 u.s.c. (or L1 l.s.c.) multifunction. Then for every y 2 C.J; X/, we have F.y/ 6D ;. Proof. Let y 2 C.J; E/; then from Proposition 6.101, there exists G W J  X ! Pcp;cv .E/ a multi-valued map which is joint measurable in L ˝ B.E/ and such that G.t; x/  F .t; x/; for each t 2 J; and x 2 X: Hence G.t; y.t //  F .t; y.t //; t 2 J . Since X is a separable Banach space and G is a joint measurable multifunction, then by Theorem 6.102, there exists a measurable single-valued function f W J ! E such that f .t / 2 G.t; y.t // ) f .t / 2 F .t; y.t //: Using the fact that F is L1 -Carathéodory, we deduce that f 2 L1 ; hence F.y/ 6D ;. In the case where F is a lower Carathéodory multifunction, we have that F is joint measurable in L ˝ B.E/; then by the same method used for the u.s.c., we can prove that F is well defined. In this section, we study some important properties of the Nemyts’ki˘ı operator (also called superposition operator or composition operator) in some classical functional spaces. First, we give a definition. Definition 6.151. Let E be a separable Banach space and F W  ! P.E/ be a multivalued map. Assign to F , the multi-valued operator F W Lp .; E/ ! P.Lp .; E// defined by F.y/ D ff 2 Lp .; E/ j f .!/ 2 F .y.!//; a.eg;

394

Chapter 6 Background in multi-valued analysis

where 1  p  1 and Lp is the space of Bochner pth-integrable functions. The operator F is called the Nemyts’ki˘ı operator associated to F . Lemma 6.152 ( [252]). Let F W  ! Pcl .E/ be a measurable multifunction. If F 6D ;; then there exists a sequence of p-integrable functions .fn / contained in F such that F .!/ D ffn .!/ j n 2 Ng; for all ! 2 : Proof. Since F is a measurable multifunction, then from Theorem 6.84, there exists a sequence fgn g of measurable single-valued functions such that F .!/ D fgn .!/ j n 2 Ng;

for all ! 2 :

Taking a countable measurable fk g of  such that .k / < 1 and a function f 2 Lp .; E/ such that f .!/ 2 F .!/ for all ! 2 ; we define Bj mk D f! 2  j m  1  kgj .!/k < mg \ k fj mk D Bj mk gj C nBj mk f; j; m; k  1: It is clear that fj mk is a measurable function and fj mk .!/ 2 F .!/ for all ! 2 . Also we have Z p kf .!/kp .d.!//  m.k / C kf kLp < 1I 

thus, fj mk 2 F and F .!/ D ffj mk j j; m; k  1g for all ! 2 . Lemma 6.153. Let F W  ! Pcl .E/ be a measurable multifunction and .fn / be a p-integrable sequence in F such that F .!/ D ffn .!/ j n 2 Ng;

for all ! 2 :

Then for every f 2 F and " > 0; there exists a measurable partition f1 ; : : : ; k g of  such hat k X i fi kLp  ": kf  i D1

Proof. Let f .!/ 2 F .!/, for all ! 2 . Let  2 Lp .; RC / be such that Z "p : d < 3  Then there exists a countable measurable partition fBi g such that kf .!/  fi .!/kp < .!/;

! 2 Bi ; i  1:

395

Section 6.3 Decomposable sets

Choose an integer n such that 1 Z X

kf .!/kp d 

"p 2p 3

kf1 .!/kp d 

"p I 2p 3

i DnC1 Bi

and

1 Z X i DnC1 Bi

then define a finite measurable partition fA1 ; : : : ; Ak g of  as follows: A1 D 1 [

1 [

i and Ai D i 2  i  n:

i DnC1

Thus, kf 

n X i D1

nAi fi kp Lp



n Z X i D1 i 1 X

kf .!/  fi .!/kp d Z kf .!/  f1 .!/kp d

C

i DnC1 i

Z

.!/ C

 

Z

1 X

2p

i DnC1

Z

 kf1 .!/kp d < "p :

kf .!/kp d C i

i

Theorem 6.154. Let F1 ; F2 W  ! F.E/ be measurable multi-functions and F .!/ D F1 .!/ C F2 .!/, for ! 2 . (1) Then F is a measurable function. Moreover, if F1 and F2 are nonempty where 1  p < 1; then F D F1 C F2 . (2) co F1 is measurable and if F1 is nonempty where 1  p < 1; then co F1 D ff 2 Lp .; E/ j f .!/ 2 co F1 .!/; a:e:g: Proof. From Theorem 6.84, there exist measurable sequences .fn1 / and .fn2 / such that F1 .!/ D ffn1 .!/ j n 2 Ng and F2 .!/ D ffn2 .!/ j n 2 Ng:

396

Chapter 6 Background in multi-valued analysis

Hence, F1 .!/ C F2 .!/ D ffn1 .!/ C fn2 .!/ j n 2 Ng;

for all ! 2 :

This implies that F is measurable. If F1 6D ; and F2 6D ;. , from Lemma 6.152, we can take fn1 2 F1 and fn2 2 F2 for all n 2 N. It follows that F1 .!/ C F2 .!/ D ffn1 .!/ C fn2 .!/ j n 2 Ng;

for all ! 2 :

Now it is clear that fn1 C fn2 2 F, for every n 2 N; then ffn1 C fn2 j n 2 Ng  F ) ffn1 C fn2 j n 2 Ng  F: Conversely, let f 2 F and " > 0. By Lemma 6.153, we may choose a finite measurable partition 1 ; : : : ; k of  and integer i1 ; : : : ; in ; j1 ; : : : ; jn such that kf 

n X

i .fi1 C fi 2 /kLp  ":

i D1

Therefore, F  F1 C F2 . Then we conclude that F D F1 C F2 . Now we prove 6.154 (2). Let .fn / be a measurable selection of F1 such that F1 .!/ D ffn .!/ j n 2 Ng: Let U D fgj g D

m X

˛i fi ; ˛i ;

i D1

m X

˛i D 1; ˛i 2 Q; m 2 Ng:

i D1

It is clear that for every g 2 U , we have g.!/ 2 co F .!/ and U is a countable subset of the family of measurable co F . Let g 2 co F ; then there exist P selections of P m n i 2 Œ0; 1; i D 1; : : : ; m; g D m  f and i D1 i i i D1 i D 1. Let ˛i 2 Œ0; 1 \ QC n be such that lim ˛i D i ; hence, n!1

g D lim

n!1

m X

˛ingi ) g 2 U :

i D1

Thus, co F .!/ D fg.!/ j g 2 U g; and so co F is measurable for every g 2 G and " > 0, where G D ff 2 Lp .; E/ j f .!/ 2 co F1 .!/; a:e:g: By Lemma 6.153, we can choose a partition f1 ; : : : ; k g of  and functions g1 ; : : : ; gn 2 U such that kf 

m X i D1

i gi kLp  ":

397

Section 6.3 Decomposable sets

Since gi 2 U; then there exists m 2 N and ˛i k 2 QC with

m X

˛i k D 1 and fk 2 F1

kD1

such that gi D

m X

˛i k fk ; i D 1; : : : ; m:

kD1

Hence, n X i D1

i gi D

n X i D1

i

m X kD1

i fk D

X

.˛1i1 ; : : : ; ˛nin /

.i1 ;:::;im /

n X

k fik ;

kD1

where Pn .i1 ; : : : ; im / is taken for 1  ik  m and 1  k  n. This shows that kD1 k gk is a convex combination of F. Hence f 2 co F1 . We end this section with a useful result in applications: Lemma 6.155. [317]. Let X be a Banach space, F W Œa; b  X ! Pcp;c .X/ be an L1 -Carathéodory multi-valued map with SF ;y 6D ;, and be a linear continuous mapping from L1 .Œa; b; X/ into C.Œa; b; X/. Then the operator

ı SF W C.Œa; b; X/ ! Pcp;c .C.Œa; b; X//; y 7! . ı SF /.y/ WD .SF ;y / is a closed graph operator in C.Œa; b; X/  C.Œa; b; X/.

Appendix

In this appendix, we gather together some auxiliary notions from homology and functional analysis, and semigroup theory used throughout this book.

A.1

ˇ Axioms of the Cech homology theory

ˇ The construction of the Cech homology theory is outlined. Interested readers are invited to consult the fundamental reference [166] or [207, 263, 338]. Recall that by a pair it is meant .X; A/ such that A  X. A map of pairs is a function f W .X; A/ ! .Y; B/ such that f .A/  B; f is called a morphism. Continuity of f is assumed whenever X; Y are topological spaces (spaces for short). A category C is a collection of objects. With two objects A; B 2 C, we associate a set of morphisms HomC .A; B/ represented by arrows: A ! B. We say that the following diagram f

A ! B g

h

&

. C

commutes if the morphism h is a composition of g and f . A functor from a category C to a category C 0 is a rule that transforms objects from C to objects from C 0 and for any f W A ! B the functor produces an arrow FA;B .f / W F .A/ ! F .B/: A functor with maps between sets of morphisms is called a covariant functor. Let C be a category and .I; / an ordered set. Assume that for each i 2 I , we associate an object Xi 2 C such that for any pair of indices i  j we have fij 2 HomC .Xi ; Xj /. If the family fij satisfies fi i D IdXi and fij ı fjk D fi k for any triple i  j  k, then we say that .fXi i 2I ; fij g/ is a projective or inverse system in the category C. In order to set the axioms for homology (or the Eilenberg–Steenrod axioms for homology), consider a category C together with three functions. (1) The first function assigns to each pair .X; A/ and each integer k 2 Z an Abelian group Hk .X; A/, the k-th homology group of X modulo A.

400

Appendix

(2) The second one attaches to every map f W .X; A/ ! .Y; B/ and each k 2 Z, the group homomorphism induced by f : f W Hk .X; A/ ! Hk .Y; B/: (3) The third function associates to each pair .X; A/ and each integer k 2 Z, a homomorphism called the boundary operator @ W Hk .X; A/ ! Hk .Y; B/: In order to form a homology theory H , the three functions are assumed to satisfy the seven Eilenberg–Steenrod axioms: Axiom 1. If f is the identity, then so is f . Axiom 2. .gf / D g f . Axiom 3. @f D .f j A/ @. Axiom 4. (Exactness axiom). Given a pair .X; A/, the following sequence is exact @

i

j

@

i

: : : ! Hk .A/ ! Hk .X/ ! Hk .X; A/ ! Hk1 .A/ ! : : : where i and j are the homomorphisms induced by the inclusions i W A ! X and j W X ! .X; A/. Axiom 5. (Homotopy axiom) Two homotopic maps f; g W .X; A/ ! .Y; B/ have identical induced homomorphisms. Axiom 6. (Excision axiom) Let .X; A/ be a pair and U  X an open subset such that ı

U A and the inclusion map i W .X n U; A n U / ! .X; A/ is admissible. Then the induced homomorphism i is an isomorphism. Axiom 7. (Dimension axiom) For all k 6D 0, we have Hk .fx0 g/ D 0 where fx0 g is a singleton in the category.

A.2

The Bochner integral

In this section, we present some properties of the Bochner integral of vector-valued functions. We shall consider only those properties needed in this book, and we shall assume that the reader is familiar with the basic facts about measure and integration of scalar-valued functions. Recall that a measure space .; †; / is called -finite if there exists fn j n 2 Ng  † such that .n / < 1 and  D [n2N n ; it is called finite if ./ < 1. The measure space .; †; / is complete if every null -measure set is measurable. Let E be a complex Banach space and let J be an interval (bounded or unbounded) in R.

401

Section A.2 The Bochner integral

Definition A.1. A function f : J ! E is called simple if there is a finite sequence n  J; n D 1; : : : ; m of Lebesgue measurable sets such that p \ l D ;I and J D

lDm [

for p 6D l

l ;

lD1

where f .t / D

lDm X

al l .t /;

al 2 E

lD1

and Lebesgue measurable sets l  J with finite Lebesgue measure .l /; i.e., f is constant on the measurable set l ; for each l 2 f1; : : : ; mg. Definition A.2. A function f W J ! E is called measurable if there is a sequence of simple functions ffngn2N such that lim kgn .t /  f .t /k D 0

n!1

for almost all t 2 J . Proposition A.3. If f W J ! E is measurable, then the real function kf k W I ! RC is measurable. Proof. Since f is measurable, then there exists a sequence of simple functions ffn gn2N such that lim kfn .t /  f .t /k D 0;

n!1

a.e. t 2 J:

Then kfn k are simple real functions for all n 2 N and jkfn .t /k  kf .t /kj  kfn.t /  f .t /k;

a.e. t 2 J:

We conclude that limn!1 kfn .t /k D kf .t /k a.e. in J and therefore kf .:/k is measurable. Remark A.4. It has to be mentioned that (in the case E D R), a function f W J ! R is measurable if and only if for every finite a 2 R, the set ft 2 J j f .t / > ag; ft 2 J j f .t /  ag (or equivalently ft 2 J j f .t / < ag; ft 2 J j f .t /  ag) is measurable. Definition A.5. A function f W J ! E is called almost separably valued if there is a null set 0 in J such that f .J n0 / WD ff .t / j t 2 J n0 g is separable (equivalently, f .n0 / is contained in a separable closed subspace of E).

402

Appendix

Definition A.6. A function f W J ! E is called weakly measurable if for each x  2 E  the real function x  .f / W J ! R is measurable. The concepts of measurability and weak measurability are closely related. The relation is given by the well known theorem of Pettis which we present below. Theorem A.7 (Pettis [193]). A function f W J ! E is measurable if and only if it is weakly measurable and almost separably valued. We will use the following result (see [161] Theorem III.6.12 and its corollaries). Theorem A.8 (Egorof’s theorem). Let fn W J ! E .n 2 N/ be a sequence of measurable functions such that lim kfn.t /  f .t /k D 0

n!1

almost everywhere in J . Then for every > 0, there is a measurable set H  J such that .J nH / < and lim kfn.t /  f .t /k D 0 n!1

uniformly on H . Corollary A.9 ([193]). Let f W J ! E be a function. Then the following statements hold: (1) the function f is measurable if and only if it is the uniform limit almost everywhere of a sequence of measurable, countably valued functions. (2) If E is separable, then f is measurable if and only if it is weakly measurable. (3) If f is continuous, then it is measurable. (4) If fn W J ! E (n 2 N) are measurable functions and fn ! f point-wise a.e., then f is measurable. For a simple function f W J ! E given by f .t / D Z f .t /dt D J

m X

PlDm lD1

al l .t /, define

al .l /;

lD1

where .l / is the Lebesgue measure ofPJ . It is routine to verify that the definition is independent of the representation f D m lD1 al .l /, and the integral so defined is linear.

403

Section A.3 Absolutely continuous functions

Definition A.10. A function f W J ! E is Bochner integrable if there exists a sequence of simple functions fn W J ! E such that the following two conditions are satisfied lim fn .t / D f .t / almost everywhere n!1

Z

and lim

n!1 J

kfn .t /  f .t /kdt D 0:

If f is Bochner integrable, then for all x  2 E  , we have E Z DZ  f .t /dt; x D hf .t /dt; x  idt: J

J

Theorem A.11. A function f W J ! E is Bochner integrable if and only if f is measurable and kf k is integrable. If f is Bochner integrable, then Z  Z   kf .t /kdt:  f .t /dt   J

J

Now we recall the following analogue of the dominated convergence theorem: Theorem A.12 (Dominated convergence [193]). Let fn W J ! E .n 2 N/ be Bochner integrable functions. Assume that f .t / D limn!1 f .t / exists a.e. and there exists an integrable function g W J ! RC such that kfn.t /k  g.t /;

for a.e. t and all n 2 N:

Then f is Bochner integrable and Z Z f .t /dt D lim fn .t /dt: J

n!1 J

Z

Furthermore,

kfn .t /  f .t /kdt D 0; as n ! 1: J

A.3

Absolutely continuous functions

Definition A.13. Let E be a Banach space. A function f W Œ0; b ! E is called absolutely continuous (we write f 2 AC) if for each " > 0, there exists ı."/ > 0 such that for all pairwise disjoint intervals of Œ0; b, we have 1 X

.bn  an / < ı H)

nD1

1 X nD1

kf .bn /  f .an /kE < ":

404

Appendix

Remark A.14. If f 2 AC, then f is of bounded variation, i.e., for any partition 0 D x0 < x1 < : : : < xn D b of Œ0; b, we have V .f / D

m X

sup x0 ;:::;xn

kf .xk /  f .xk1 /kE < 1:

kD1

Also, we have (see, e.g., [193, Theorem 2.2.17]): Theorem A.15. If f 2 AC and E is reflexive, then f is differentiable almost everywhere and we have the formula Z t f .t / D f .0/ C f 0 .s/ds: 0

Now, we define spaces of higher-order absolutely continuous functions and compare with Sobolev spaces. Definition A.16. AC 1;p .Œ0; b; E/ D

n

o f 2 AC such that f is differentiable a.e. and : 0 p f 2 L .Œ0; b; E/: Z

W

1;p

p

b

p

.0; bŒ; E/ D ff 2 L .0; bŒ; E/ j 9 g 2 L .0; bŒ; E/ W

f '0

0

Z

b

D 0

g'; 8 ' 2 C01 .0; bŒ; E/g:

Then, we have (see, e.g., [193], Theorem 2.2.24): Theorem A.17. W 1;p .Œ0; b; E/ D AC 1;p .Œ0; b; E/ Remark A.18. If E is reflexive, then f 2 AC 1;p .Œ0; b; E/ if and only if there exists g 2 Lp .0; bŒ; E/ such that Z

t

f .t / D f .0/ C

g.s/ds: 0

Section A.4 Compactness criteria in C.Œa; b; E/, Cb .Œ0; 1/; E/, and P C.Œa; b; E/

A.4

405

Compactness criteria in C.Œa; b; E /, Cb .Œ0; 1/; E /, and PC.Œa; b; E /

Let E be a real Banach space and let Œa; b be an interval. Definition A.19. A family A in C.Œa; b; E/ is equicontinuous at t in Œa; b if for each " > 0 there exists ı."; t / > 0 such that, for each s 2 Œa; b with jt  sj < ı."; t /, we have kf .t /  f .s/k < "; uniformly with respect to f 2 A. The family A is equicontinuous on Œa; b if it is equicontinuous at each point t 2 Œa; b; in the sense mentioned above. The family A is uniformly equicontinuous on Œa; b if it is equicontinuous on Œa; b, and ı."; t / can be chosen independently of t 2 Œa; b. Remark A.20. We leave to the reader the proof of the fact that a family A in C.Œa; b; E/ is equicontinuous on Œa; b if and only if it is uniformly equicontinuous on Œa; b. Theorem A.21 (Arzela–Ascoli [498]). A bounded subset A in C.Œa; b; E/ is relatively compact if and only if (1) A is equicontinuous on Œa; bI there exists a dense subset D in Œa; b such that, for each t 2 D; A.t / D ff .t / j f 2 Ag is relatively compact in E. Corollary A.22 ([477]). If A  C.Œa; b; E/ is relatively compact, then the set A.Œa; b/ D ff .t / j f 2 A; t 2 Œa; bg is relatively compact in E. Corollary A.23. Let C be nonempty and closed in E, g W Œa; bC ! E a continuous function, C D fu 2 C.Œa; b/ j u.t / 2 C; t 2 Œa; bg; and let G W C ! C.Œa; b; E/ the superposition operator associated to the function g; i.e., G.x/.t / D g.t; x.t // for each x 2 C and t 2 Œa; b. Then G is continuous from C in C.Œa; b; E/; both the domain and range being endowed with the norm topology k : k1 .

406

Appendix

Proof. Let fxngn2N be a sequence in C which converges to x 2 C in the norm k : k1 . Since fxn j n 2 Ng [ fxg is compact in C.Œa; b; E/, the set fxn j n 2 Ng; is relatively compact in C.Œa; b; E/. From [Corollary A.22, Chapter 6], we deduce that the set K D fxn .t / j n 2 N; t 2 Œa; bg is compact in E; this implies that Œa; b  K is compact in R  E; hence gjŒ a;bK is uniformly continuous. Then for each " > 0, there exists ı."/ > 0 such that, for every .t; x/; .s; y/ 2 Œa; b  K with jt  sj C kx  yk < ı."/, we have kg.t; x.t //  g.s; y.s//k < ": Let " and let ı."/ > 0 be as above. It clear that fxn gn2N converges uniformly to x on Œa; b and there exists n."/ 2 N such that for each n 2 N; n  n."/, we have kxn.t /  x.t /k < ";

for all t 2 Œa; b:

Therefore, kg.t; xn .t //  g.t; x.t //k < "; for each n  n."/ and every t 2 Œa; b. Then kG.xn/  G.x/k < ";

for each n  n."/:

Hence, lim G.x/ D G.x/

n!1

in .C.Œa; b; E/; k : k/. Let Jk D Œtk ; tkC1 /; k D 0; : : : ; m; t0 D a < t1 < : : : ; tm < tmC1 WD b, and let yk be the restriction of a function y to Jk . Define PC D

o n yW Œa; b ! E; y 2 C.J ; E/; k D 0; : : : ; m; such that k k : C   y.tk / and y.tk / exist and satisfy, y.tk / D y.tk / for k D 1; : : : ; m

Then endowed with the norm kykP C D maxfkyk k1 ;

k D 0; : : : ; mg;

P C is a Banach space, where yk D yjJk . Theorem A.24. Let A be a bounded set in P C . Assume that (1) A is equicontinuous on Œa; b (i.e., A is equicontinuous on C.Jk ;E/; k D 1; : : : ; m)

Section A.4 Compactness criteria in C.Œa; b; E/, Cb .Œ0; 1/; E/, and P C.Œa; b; E/

407

(2) there exists a dense subset D in Œa; b such that, for each t 2 D; the set A.t / D ff .t / j f 2 A; t 2 Œa; bg is relatively compact in E. Then A is relatively compact in P C . Proof. Let fxngn2N  A; then fxn gn2N  C.J 0 ; E/; J 0 D Œ0; t1 . From Theorem A.21, the set K0 D fxn j n 2 Ng is relatively compact in C.J 0 ; E/; thus there exists a subsequence of fxngn2N which converges to x0 in .C.J 0 ; E/; k : k1 /. Let ( yn .t / D

xn .t1C /; xn .t /;

t D t1 ; t 2 .t1 ; t2 :

It clear that fyn j n 2 Ng  C.J 1 ; E/; J 1 D Œt1 ; t2 . By the Ascoli–Arzéla theorem, the set K1 D fyn j n 2 Ng is relatively compact in C.J 1 ; E/; thus there exists a subsequence of fxngn2N converging to some limit x1 in .C.J 1 ; E/; k : k1 /. We continue this process taking into account that ( C /; t D tm ; xn .tm yn .t / D t 2 .tm ; b: xn .t /; By [Theorem A.21, Chapter 6] the set Km D fyn j n 2 Ng is relatively compact in C.J m ; E/ where Jm D Œtm ; b; thus there exists a subsequence of fxn gn2N converging to xm in .C.J m ; E/; k : k1 /. Hence, fxn j n 2 Ng has a subsequence in PC which converges to the limit x defined by 8 x0 .t /; ˆ ˆ ˆ ˆ ˆ < x2 .t /; x.t / D :: ˆ : ˆ ˆ ˆ ˆ xm .t /; :

if t 2 Œ0; t1 ; if t 2 .t1 ; t2 ; :: : if t 2 .tm ; b:

The following compactness criterion for subsets of Cb is a consequence of the well-known Arzéla–Ascoli theorem (see Avramesu [43], Corduneanu [126], Przeradzki [402], Staikos [442])

408

Appendix

Theorem A.25. Let B  Cb .Œ0; 1/; Rn / be a subset, assume the following conditions are satisfied: for every t 2 RC ; the set fx.t / j x 2 Bg is relatively compact,

(i)

(ii) for every ˛ > 0; the set B is equicontinuous on the interval Œ0; ˛, (iii) for every " > 0 there exist T D T ."/ and ı D ı."/ > 0 such that if x; y 2 B with kx.T /  y.T /k  ı; then kx.t /  y.t /k  " for all t 2 ŒT; 1/. Then the set B is compact in Cb WD Cb .Œ0; 1/; Rn /. As a consequence, we have Corollary A.26. Let M  Cb be the space of functions which have limits at positive infinity. Then M is relatively compact in Cb if the following conditions hold: (a) M is uniformly bounded in Cb . (b) The functions belonging to M are almost equicontinuous on RC , i.e., equicontinuous on every compact interval of RC . (c) The functions from M are equiconvergent at 1, that is, given " > 0; there corresponds T D T ."/ > 0 such that kx.t /  x.1/k < " for any t  T ."/ and x 2 M.

A.5

Weak-compactness in L1

Let L1 .; ; E/ and .; †; / be a finite measure space (i.e., ./ < 1/. Definition A.27. Let E be a Banach space. A subset A in L1 .; ; E/ is called uniformly integrable if for each " > 0 there exists ı."/ > 0 such that, for each measurable subset C 2 † with measure .C / < ı."/; we have Z jf .s/jd.s/  ": C

Remark A.28. Let A  L1 .; ; E/. (i)

If .; †; / is of totally bounded type, i.e., for each " > 0, there exists a finite covering fk W k D 1; : : : ; n."/g  † of  with .k /  " for k D 1; : : : ; n."/ and A is uniformly integrable, then it is norm bounded in L1 .; ; E/,

(ii) if ./ < 1 and A is bounded in Lp .; †; / for some p > 1; then it is uniformly integrable;

Section A.5 Weak-compactness in L1

409

Definition A.29. A subset K  Lp .Œ0; b; E/ (p  1) is said to be p-equi-integrable if it is uniformly integrable and Z bh p kf .t C h/  f .t /kE dt D 0; uniformly for all f 2 K: lim h!0 0

We have the Kolmogorov criterion of compactness in Lp .Œ0; b; E/ (see [84, 165, 193]): Theorem A.30. A subset K  Lp .Œ0; b; E/ .p  1/ is relatively compact if and only if it is p-equi-integrable and o nZ t f ./d  j f 2 K is relatively compact in E: 8 0 < s < t < b; the set s

Definition A.31. A sequence fvn gn2N  L1 .Œ0; b; E/ is said to be integrably bounded if there exists q 2 L1 .Œ0; b; RC / such that jvn .t /jE  q.t /;

for a.e. t 2 Œ0; b and every n 2 NI

Remark A.32. Every integrably bounded sequence is uniformly integrable. This follows from the fact that, for a finite measure space .; †; /, K  L1 .; ; E/ is uniformly integrable (see [165]) if and only if K is (uniformly) bounded and for each " > 0, there exists ı > 0 such that Z sup jf .w/jd  " f 2K A

for all A 2 † with .A/  ı. Now, we present two weak compactness criteria that follow from the well known Dunford–Pettis theorem (see [165, 499]). Lemma A.33 ([477]). Let .; †; / be a -finite measure space, let fk j k 2 Ng be a subfamily of † such that 8 .k1 / < 1 for k 2 N ˆ ˆ ˆ ˆ <  for k 2 N k1  k 1 [ ˆ ˆ ˆ k D ; ˆ : kD0

and let E be a Banach space. Let A 2 L1 .; ; E/ be bounded and uniformly integrable in L1 .k ; ; E/; for k 2 f0g [ N and Z lim jf .s/jd.s/ D 0; k!1 nk

410

Appendix

uniformly for f 2 A. If for each  > 0 and each k 2 N; there exist a weakly compact subset C;k  E and measurable subset ;k with .n;k /   and f .;k /  C;k for all f 2 A; then A is weakly compact in L1 .; ; E/. Corollary A.34. Let .; †; / be a -finite measure space, E reflexive, and K  L1 .; E/ be a bounded subset. Then K is relatively weakly compact if and only if K is uniformly integrable. Definition A.35. Let E be a Banach space. A sequence fvngn2N  L1 .Œ0; b; E/ is said to be semi-compact if (a) it is integrably bounded. (b) the image sequence fvn .t /gn2N is relatively compact in E for a.e. t 2 J . Finally, the following results follow from the Dunford–Pettis theorem. Lemma A.36 (see [276, Proposition 4.2.1] or [401, Proposition 3.6]). in case dim E < 1;) Every semi-compact sequence L1 .Œ0; b; E/ is weakly compact in L1 .Œ0; b; E/. Lemma A.37 (see [387, Corollary 6.4.11]). Let A  L1 .; E/ be a bounded decomposable set with  finite-measurable and E reflexive. Then A is weakly relatively compact in L1 .; E/.

A.6

Proper maps and vector fields

Let X; Y be two metric spaces and f W X ! Y a continuous map. Definition A.38. We say that f is proper if f 1 .K/ is compact for every compact subset K  Y . Notice that for finite-dimensional spaces, f proper means that f 1 .B/ is bounded for every bounded subset B. Proposition A.39. If f W X ! Y is proper, then it is closed. Proof. Let A be closed in X and B D f .A/. Consider a sequence .yn /n2N  Y converging to some limit y and let .xn /n2N be such that f .xn / D yn for each n. Since f 1 .fyn j n 2 Ng/ is compact and f is proper, .xn /n2N admits some converging subsequence to a limit x. By continuity of f , we deduce that y D f .x/, proving our claim. Proposition A.40. Let C  X be a nonempty, bounded, closed subset of a Banach space X and f D I  K W C ! X be a vector field associated with a compact mapping K. Then f is proper.

Section A.7 Fundamental theorems in functional analysis

411

Proof. Let B  X be compact and A D f 1 .B/. Using the Kuratowski measure of noncompactness, we obtain that ˛.A/ D ˛.A  B C B/  ˛.A  B/ C ˛.B/ D ˛.K.A// C ˛.B/ D 0; and our claim follows. Another sufficient condition is given by Proposition A.41. Let C  X be a nonempty unbounded subset of a Banach space X and F D I  K W C ! X be a vector field associated with a compact mapping K and satisfying the coerciveness condition: lim kF .x/k D 1:

kxk!1

Then F is proper.

A.7

Fundamental theorems in functional analysis

For this section, we recommend [83, 84, 161, 165, 339, 346, 499]. We start with the Eberlein–Šmulian theorem. Theorem A.42 (see [165], Theorem 8.12.1 and Theorem 8.12.7). Let K be a weakly closed subset of a Banach space X. Then the following are equivalent: (i)

K is weakly compact.

(ii) K is weakly sequentially compact. The following result is known as Eberlein–Kakutani theorem: Theorem A.43. A normed space is reflexive if and only if every bounded sequence admits a convergent subsequence. The following results are due to Mazur, 1933. Theorem A.44 (Mazur–Smˇulian theorem). The closure and weak closure of a convex subset of a normed space are the same. As a consequence, a convex subset of a normed space is closed if and only if it is weakly closed. Theorem A.45 (Mazur’s compactness theorem (see [161], Theorem 6, p. 416)). The closed convex hull of a (weakly) compact subset of a Banach space is itself (weakly) compact.

412

Appendix

In a reflexive Banach space, the unit ball is weakly sequentially compact. Moreover, we have a useful result when dealing with weak convergence: Corollary A.46 ( [346, Theorem 21.4] or [84, Corollary 3.8]). Let E be a normed space and fxk gk2N  E be a sequence weakly converging Pm to a limit x 2 E; then there exists a sequence of convex combinations ym D kD1 ˛mk xk ; where ˛mk > 0 P for k D 1; 2; : : : ; m and m kD1 ˛mk D 1; which converges strongly to x.

A.8

C0 -Semigroups

In all this subsection, B.E/ refers to the Banach space of linear bounded operators from E into E with norm kN kB.E/ D supfjN.y/j j jyj D 1g: Definition A.47. A semigroup is a one parameter family fT .t /W t  0g  B.E/ satisfying the conditions: (a) T .t / ı T .s/ D T .t C s/, for t; s  0; (b) T .0/ D I . Here I denotes the identity operator in E. Definition A.48. A semigroup T .t / is uniformly continuous if lim kT .t /  I kB.E/ D 0;

t ! 0C

that is if lim

jt sj! 0

kT .t /  T .s/kB.E/ D 0:

Definition A.49. We say that the semigroup fT .t /t 0 g is strongly continuous (or a C0 -semigroup) if the map t ! T .t /.x/ is strongly continuous, for each x 2 E, i.e., lim T .t /x D T .0/x;

t ! 0C

8 x 2 E:

Definition A.50. Let T .t / be a C0 -semigroup defined on E. The infinitesimal generator A 2 B.E/ of T .t / is the linear operator defined by A.x/ D lim

t ! 0C

where D.A/ D fx 2 EW lim

t ! 0C

T .t /.x/  T .0/x ; t T .t /.x/x t

for x 2 D.A/;

exists in Eg.

The following properties are classical (see Engel and Nagel [169]), Pazy [391], or Hill and Philips [253].

413

Section A.8 C0 -Semigroups

Proposition A.51. A linear operator A W D.A/  E ! E is the infinitesimal generator of the uniformly continuous semigroup if and only if A is a bounded linear operator. In this case, the semigroup can be defined by T .t / D e At ; t  0. Proposition A.52. (a) If fT .t /gt 0 is a C0 -semigroup of bounded linear operators, then there exist constants !  0 and M  1 such that kT .t /kB.E/  M exp.!t /;

for t  0:

(b) If A is the infinitesimal generator of a C0 -semigroup fT .t /gt 0 , then D.A/, the domain of A, is dense in X and A is a closed linear operator. Proposition A.53. Let fT .t /gt 0 be a uniformly continuous semigroup of a bounded linear operator. Then there exists some constant !  0 such that kT .t /kB.E/  exp.!t /;

for t  0:

Proposition A.54. If fT .t /gt 0 is a compact C0 -semigroup for t > 0; then it is uniformly continuous, for t > 0. Let A W E ! E be a linear operator. Definition A.55. The resolvent set ƒ.A/ of A consists of all complex numbers  for which the linear operator I  A is invertible, i.e., .I  A/1 is a bounded linear operator in E. The family R.; A/ D .I  A/1 ;  2 ƒ.A/ is called the resolvent of A. All complex numbers  not in ƒ.A/ form a set called the spectrum of A.

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Index

Hd -measurable 382 J mapping 284 Rı set 302 Rı contractible 315 ˇcondensing 44 1-proximally connected 283 -selectionable 362   C a-selectionable 363   L-selectionable 363   LL-selectionable 363   c-selectionable 363   m-selectionable 363 "  ı l.s.c. 349 "  ı u.s.c. 344 e max -measurable 382 H k-set contraction 52 k-th star 308 k-triad 308 r-map 294 u.s.c. 342 absolute retract 294 absolutely continuous 403 acyclic map 315 acyclic set 315 acyclically contractible 315 admissible condensing mappings 53 admissible contraction 35 admissible map 44, 45, 54 almost equicontinuous 21 almost separable 372 almost separably 401 Aronsajn result 160 Aronszajn result 108, 110, 111 Arzela-Ascoli 405 asymptotically compact 47

Bielecki’s norm 130 Bochner integrable 403 Bochner integral 400 Bouligand tangent cone 149 bounded linear operator 413 Bressan-Colombo-Fryszkowski 390 Browder–Gupta 13 Carathéodory selectionable 117 carrier 311 Castaing representation 369 category 399 characteristic function 372 closed graph 345 closed map 315 cofinal set 68 cohomological acyclicity 314 cohomology functor 304 coincidence theorem 10 compact absorbing contraction 48 compact attraction 47 compact carrier 313 compact pair 304 composition 339 composition operator 393 condensing map 43 conjugate basis 324 continuous map 347 continuous selection 390 contractible space 298 contractive field 3 convex hull 14 covering dimension 62 decomposable 390 decomposable set 391 decomposable set characterization 391 duality theorem 322

452 Eilenberg–Steenrod axioms 305 Eilenberg-Steenrod axioms 399 equicontinuous 405 Es space 288 eventually compact 47 extension 297 extension property 288 fibre product 319 Filippov theorem 220 finite measure space 408 fixed point 340 fixed point theorem (graph approximation) 30 Fredholm alternative 9 functor 399 generalized Lefschetz number 326, 327 global attractor 56, 57 graph 339 graph approximation 28 graph measurable 366 Hausdorff continuity 350 Hausdorff distance 351 Hausdorff measurability properties 382 Hausdorff metric space 351 Hausdorff-measure of noncompactness 39 homology, cohomology 304 homotopic maps 304 homotopic to 296 homotopically equivalent 298 homotopy 297 image 339 index of coincidence 330 infinitesimal generator 412 integrably bounded 409 integrably bounded map 384 inverse system 65, 69 K-measurable 366 Kuratowski-measure of noncompactness 39 Kuratowski-Ryll-Nardzewski 369 l.s.c. 346 Lasota-Yorke Lemma 11

Index Lebesgue measurable multivalued 378 Lefschetz Fixed Point Theorem 10, 56 Lefschetz number 10, 324, 327 Lefschetz set 53 Leray endomorphism 10, 327 Leslie predator-prey system 164 limit map 67 linear growth 384 local contraction principle 3 locally compact 47, 346 locally contractible 301 locally finite 354 lower semi-continuous 346 lower semi-continuous type 390 Lusin property 370 Lusin’s theorem 370 map coincidence 330 Mazur’s Lemma 411 measurable 366 measurable space 366 measure of noncompactness 39 Michael family 358 Michael selection theorem 359 mild solution 166 morphism 399 multi-maps of inverse systems 67 multi-valued contraction 37 multivalued mapping 339 multivalued nonexpansive maps 38, 39 neighbourhood retract 288 nerve 308 Niemytzki operator 390 Niemytzkiˇı operator 393 noncontractibility of sphere 6 nonexpansive map 23 nonlinear alternative 6, 8–10, 31, 32 nonretraction of sphere 6 ordinary Lefschetz number 324 pair 399 pair of spaces 304 Palais-Smale 15 partition of unity 355 polyhedral 311 posses an extension property 288

453

Index projective limit 65 proper map 13, 315 property (BC) 390 proximate retract 149 pull-back 319 quasi-compact multimap 346 quasi-open 348 refine 354 refinement 354 retract 288 Schauder Approximation 12 Scorza–Dragoni property 383 Scorza-Dragony property 384 selection property 360 selectionable mappings 362 semi-compact sequence 43 semigroup 412 simply connected 300 star 308 strong measurable multifunction 372 strongly acyclic set 322 strongly continuous 412 subcovering 354 superposition operator 393, 405 support 355 support chain 308 support set 308 Theorem of Arens-Eells 293 Theorem of Aronszajn 18 Theorem of Aumann 378 Theorem of Banach 1 Theorem of Bohnenblust-Karlin 31 Theorem of Bressan-ColomboFryszkowski 390 Theorem of Brouwer 5 Theorem of Browder-Göhde-KirkGoebel 25 Theorem of Browder-Gupta 13, 60 Theorem of Castaing 369 Theorem of Cellina 29, 376 Theorem of Covitz and Nadler 34 Theorem of domain invariance 3

Theorem of Duality 322 Theorem of Dugundji 290, 357 Theorem of Eilenberg-Montgomery 10 Theorem of Filippov 380 Theorem of Frigon 35 Theorem of Granas 10, 336 Theorem of Granas-Schauder 54 Theorem of Hyman 283 Theorem of Kakutani 31 Theorem of Klee 293 Theorem of Kneser-Hukuhara 105 Theorem of Krasnosel’skiˇı-PerovRabinowitz 19 Theorem of Lefschetz 10, 53, 54, 57 Theorem of Mayer-Vietoris 308 Theorem of Michael 355 Theorem of Peano 77 Theorem of Picard-Lindelöf 74 Theorem of Ricceri 37, 62 Theorem of Rothe 9 Theorem of Saint-Raymond 63 Theorem of Schaefer 9 Theorem of Schauder 7 Theorem of Schauder-Tykhonov 10 Theorem of Stone 355 Theorem of Szufla 17 Theorem of Tietze 289 Theorem of Van Kampen 323 Theorem of Vietoris-Begle 317 topological dimension 63, 64 trace 323 triad 308 u.s.c. dependence 121 uniformly equicontinuous 405 uniformly integrable 408 upper semi-continuous 342 Urysohn function 151 viable solutions 151 Vietoris map 315, 317 weakly measurable 366, 402 weakly nilpotent 329 weakly relatively compact 410