Solution Sets for Differential Equations and Inclusions

ISBN-13: 9783110293449 / Angielski / Twarda / 2012 / 472 str.

Sma L. Djebali; Lech G. Rniewicz; Abdelghani Ouahab

Opis

The series is devoted to the publication of high-level monographs which cover the whole spectrum of current nonlinear analysis and applications in various fields, such as optimization, control theory, systems theory, mechanics, engineering, and other sciences. One of its main objectives is to make available to the professional community expositions of results and foundations of methods that play an important role in both the theory and applications of nonlinear analysis. Contributions which are on the borderline of nonlinear analysis and related fields and which stimulate further research at the crossroads of these areas are particularly welcome. Please submit book proposals to Jurgen Appell. Titles in planning include Said Abbas, Mouffak Benchohra, John R. Graef, and Johnny Henderson, Implicit Fractional Differential and Integral Equations: Existence and Stability (2018)
Lucio Damascelli and Filomena Pacella, Morse Index of Solutions of Nonlinear Elliptic Equations (2019)
Rafael Ortega, Periodic Differential Equations in the Plane: A Topological Perspective (2019)
Cyril Tintarev, Profile Decompositions and Cocompactness: Functional-Analytic Theory of Concentration Compactness (2020)
Takashi Suzuki, Semilinear Elliptic Equations: Classical and Modern Theories (2021)

Recenzje

"This interesting and self-contained book offers both classical and recent results on the existence of solutions for differential equations and inclusions, and on the topological structure of solution sets." Mathematical Reviews

"In this excellent book, a comprehensive description of methods concerning the topological structure of fixed point sets and solution sets for differential equations and inclusions is presented." Zentralblatt für Mathematik

Spis treści

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 L1 . . . . . . . . . . . . . . . . . . . . . . 63

1.5.2 Decomposability in L1(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 C0¡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 ½(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 ½(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 . . . . . . . . . . . 266

4.4.3 Using solution sets to prove existence results . . . . . . . . . . . 282

I SUPPLEMENTS 287

5 PRELIMINARY NOTIONS OF TOPOLOGY 289

5.1 Extension and embedding properties . . . . . . . . . . . . . . . . . . . 289

5.2 Homotopical properties of spaces . . . . . . . . . . . . . . . . . . . . . 296

5.3 ·Cech homology (cohomology) functor . . . . . . . . . . . . . . . . . . . 303

5.4 Maps of spaces of ¯nite type . . . . . . . . . . . . . . . . . . . . . . . . 304

5.5 ·Cech homology functor with compact carriers . . . . . . . . . . . . . . 311

5.6 Acyclic sets and Vietoris maps . . . . . . . . . . . . . . . . . . . . . . . 313

5.7 Homology of open subsets of Euclidean spaces . . . . . . . . . . . . . . 317

5.8 Lefschetz number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

5.9 Coincidence problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

6 BACKGROUND IN MULTI-VALUED ANALYSIS 335

6.1 Continuity of multivalued mappings . . . . . . . . . . . . . . . . . . . . 337

6.1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

6.1.2 Upper semi-continuity . . . . . . . . . . . . . . . . . . . . . . . 339

6.1.3 Lower semi-continuity . . . . . . . . . . . . . . . . . . . . . . . 344

6.1.4 Hausdor® continuity . . . . . . . . . . . . . . . . . . . . . . . . 347

6.2 Selection theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

6.2.1 Partitions of unity . . . . . . . . . . . . . . . . . . . . . . . . . 349

6.2.2 Michael's selection theorem . . . . . . . . . . . . . . . . . . . . 350

6.2.3 ¾¡selectionable mappings . . . . . . . . . . . . . . . . . . . . . 353

6.2.4 The Kuratowski-Ryll-Nardzewski selection theorem . . . . . . . 356

6.2.5 Hausdor®-measurable multivalued maps . . . . . . . . . . . . . 371

6.2.6 The Scorza-Dragoni property . . . . . . . . . . . . . . . . . . . 373

6.2.7 The Bressan-Colombo-Fryszkowski selection theorem . . . . . . 379

6.3 The Bochner integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

6.3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

6.3.2 Nemytski·i operators . . . . . . . . . . . . . . . . . . . . . . . . 383

6.3.3 Integration of multivalued maps . . . . . . . . . . . . . . . . . . 386

6.4 Compactness in C([a; b];E) and PC([a; b];E) . . . . . . . . . . . . . . . 388

6.5 Further auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . 391

O autorze

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