Solution Sets for Differential Equations and Inclusions

This monograph gives a systematic presentation of classical and recent results obtained in the last couple of years. It comprehensively describes the methods concerning the topological structure of fixed point sets and solution sets for differential equations and inclusions. Many of the basic techniques and results recently developed about this theory are presented, as well as 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. Several examples of applications relating to initial and boundary value problems are discussed in detail.

The book is intended to advanced graduate researchers and instructors active in research areas with interests in topological properties of fixed point mappings and applications; it also aims to provide students with the necessary understanding of the subject with no deep background material needed. This monograph fills the vacuum in the literature regarding the topological structure of fixed point sets and its applications.

Auteur

Smäil Djebali, Ecole Normale Supérieure, Algiers, Algeria; Lech Górniewicz, Nicolaus Copernicus University, Torun, Poland; Abdelghani Ouahab, Sidi-Bel-Abbès University, Algeria.

Contenu

1 TOPOLOGICAL STRUCTURE OF FIXED POINT SETS 11

1.1 Case of single-valued mappings . . . . . . . . . . . . . . . . . . . . . . 11

1.1.1 Fundamental ¯xed point theorems . . . . . . . . . . . . . . . . . 11

1.1.2 Approximation theorems . . . . . . . . . . . . . . . . . . . . . . 14

1.1.3 Browder{Gupta Theorems . . . . . . . . . . . . . . . . . . . . . 16

1.1.4 Acyclicity of the solution sets of operator equation . . . . . . . 21

1.1.5 Solution sets for nonexpansive maps . . . . . . . . . . . . . . . . 24

1.2 Case of multi-valued mappings . . . . . . . . . . . . . . . . . . . . . . . 25

1.2.1 Fixed point theorems . . . . . . . . . . . . . . . . . . . . . . . . 25

1.2.2 Multivalued contractions . . . . . . . . . . . . . . . . . . . . . . 27

1.2.3 Fixed point sets of multi-valued contractions . . . . . . . . . . . 29

1.2.4 Fixed point sets of multivalued condensing maps . . . . . . . . . 32

1.2.5 Approximation of multi-valued maps . . . . . . . . . . . . . . . 37

1.3 Admissible maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

1.3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

1.3.2 Fixed point theorems for admissible multivalued maps . . . . . 48

1.3.3 Browder{Gupta type results for admissible mappings . . . . . . 54

1.4 Topological structure of ¯xed point sets of inverse limit maps . . . . . . 58

1.4.1 De¯nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

1.4.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

1.4.3 Multi-maps of inverse systems . . . . . . . . . . . . . . . . . . . 60

1.5 Further results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

1.5.1 Semi-compactness in *L*1 . . . . . . . . . . . . . . . . . . . . . . 63

1.5.2 Decomposability in L*1(*T;E) . . . . . . . . . . . . . . . . . . . . 64

1.5.3 Michael family of subsets . . . . . . . . . . . . . . . . . . . . . . 66

2 EXISTENCE THEORY FOR DIFFERENTIAL EQUATIONS AND

INCLUSIONS 71

2.1 Case of di®erential equations . . . . . . . . . . . . . . . . . . . . . . . . 71

2.1.1 Existence and uniqueness results . . . . . . . . . . . . . . . . . 71

2.1.2 Picard-LindelÄof Theorem . . . . . . . . . . . . . . . . . . . . . . 72

2.1.3 Peano and Carath¶eodory theorems . . . . . . . . . . . . . . . . 77

2.1.4 Global existence theorems . . . . . . . . . . . . . . . . . . . . . 79

2.1.5 Existence results on non-compact intervals . . . . . . . . . . . . 82

2.1.6 A boundary value problem on the half-line . . . . . . . . . . . . 89

2.2 Case of di®erential inclusions . . . . . . . . . . . . . . . . . . . . . . . 94

2.2.1 Initial value problem . . . . . . . . . . . . . . . . . . . . . . . . 94

2.2.2 A boundary value problem . . . . . . . . . . . . . . . . . . . . . 99

3 SOLUTIONS SETS FOR DIFFERENTIAL EQUATIONS AND IN-

CLUSIONS 105

3.1 Solutions sets for di®erential equations . . . . . . . . . . . . . . . . . . 105

3.1.1 Problems on bounded intervals . . . . . . . . . . . . . . . . . . 105

3.1.2 Problems on unbounded intervals . . . . . . . . . . . . . . . . . 107

3.1.3 Kneser-Hukuhara Theorem . . . . . . . . . . . . . . . . . . . . . 109

3.2 Aronszajn-type results for di®erential inclusions . . . . . . . . . . . . . 111

3.3 Application to neutral di®erential inclusions . . . . . . . . . . . . . . . 118

3.3.1 The convex case . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

3.3.2 The nonconvex case . . . . . . . . . . . . . . . . . . . . . . . . . 125

3.3.3 Solutions sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

3.4 Application to second order di®erential inclusions . . . . . . . . . . . . 136

3.4.1 The convex case . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

3.4.2 The nonconvex case . . . . . . . . . . . . . . . . . . . . . . . . . 141

3.4.3 Solution sets to second-order di®erential equations . . . . . . . . 144

3.4.4 Solution sets to second-order di®erential inclusions . . . . . . . 146

3.5 Application to a nonlocal problem . . . . . . . . . . . . . . . . . . . . . 150

3.5.1 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . 150

3.5.2 Solutions set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

3.6 Application to a nonlocal viability problem . . . . . . . . . . . . . . . . 152

3.6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

3.6.2 Viable solutions on proximate retracts . . . . . . . . . . . . . . 154

3.7 Application to hyperbolic di®erential inclusions . . . . . . . . . . . . . 158

3.7.1 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . 158

3.7.2 Solution sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

3.8 Application to abstract Volterra operators . . . . . . . . . . . . . . . . 166

4 IMPULSIVE DIFFERENTIAL INCLUSIONS: EXISTENCE AND

SOLUTION SETS 169

4.1 Impulsive di®erential inclusions . . . . . . . . . . . . . . . . . . . . . . 169

4.1.1 C*0¡*Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

4.1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

4.1.3 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . 174

4.1.4 Structure of solution sets . . . . . . . . . . . . . . . . . . . . . . 190

4.2 A periodic problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

4.2.1 Existence results: 1 2 (T(b)) . . . . . . . . . . . . . . . . . . . 203

4.2.2 The convex case: direct approach . . . . . . . . . . . . . . . . . 204

4.2.3 The convex case: MNC approach . . . . . . . . . . . . . . . . . 211

4.2.4 The nonconvex case . . . . . . . . . . . . . . . . . . . . . . . . . 216

4.2.5 The parameter-dependant case . . . . . . . . . . . . . . . . . . 219

4.2.6 Filippov's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 224

4.2.7 Existence of solutions: 1 62 (T(b)) . . . . . . . . . . . . . . . . 232

4.3 Impulsive Functional Di®erential Inclusions . . . . . . . . . . . . . . . . 238

4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

4.3.2 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . 239

4.3.3 Structure of the solution set . . . . . . . . . . . . . . . . . . . . 247

4.4 Impulsive di®erential inclusions on the half-line . . . . . . . . . . . . . 251

4.4.1 Existence results and compactness of solution sets . . . . . . . . 252

4.4.2 Topological structure via the projective limit . . …