Solution Sets for Differential Equations and Inclusions (eBook)

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

Preface 7
Notations 11
1 Topological structure of fixed point sets 21
1.1 Case of single-valued mappings 21
1.1.1 Fundamental fixed point theorems 21
1.1.1.1 Banach’s fixed point theorem 21
1.1.1.2 Brouwer’s fixed point theorem 25
1.1.1.3 Schauder’s fixed point theorem 27
1.1.2 Approximation theorems 31
1.1.3 Browder-Gupta theorems 33
1.1.4 Acyclicity of the solution sets of operator equations 40
1.1.5 Nonexpansive maps 43
1.1.5.1 Existence theory 43
1.1.5.2 Solution sets 46
1.2 The case of multi-valued mappings 47
1.2.1 Approximation of multi-valued maps 47
1.2.2 Fixed point theorems 50
1.2.3 Multi-valued contractions 53
1.2.4 Fixed point sets of multi-valued contractions 55
1.2.5 Fixed point sets of multi-valued nonexpansive maps 58
1.2.6 Fixed point sets of multi-valued condensing maps 59
1.2.6.1 Measure of noncompactness 59
1.2.6.2 Condensing maps 63
1.3 Admissible maps 64
1.3.1 Generalities 64
1.3.2 Fixed point theorems for admissible multi-valued maps 73
1.3.3 The general Brouwer fixed point theorem 78
1.3.4 Browder-Gupta type results for admissible mappings 80
1.3.5 Topological dimensions of solution sets 82
1.4 Topological structure of fixed point sets of inverse limit maps 85
1.4.1 Definition 85
1.4.2 Basic properties 86
1.4.3 Multi-maps of inverse systems 87
2 Existence theory for differential equations and inclusions 92
2.1 Fundamental theorems 92
2.1.1 Existence and uniqueness results 92
2.1.2 Picard-Lindelöf theorem 93
2.1.2.1 Maximal solutions 95
2.1.3 Peano and Carathéodory theorems 97
2.1.3.1 Peano theorem 97
2.2 The extendability problem 99
2.2.1 Global existence theorems 99
2.2.2 Existence results on noncompact intervals 102
2.2.2.1 The Lipschitz case 102
2.2.2.2 The Lipschitz-Nagumo case 103
2.2.2.3 The Nagumo case 106
2.2.3 A boundary value problem on the half-line 108
2.3 The case of differential inclusions 114
2.3.1 Initial value problems 114
2.3.1.1 A Nagumo type nonlinearity 114
2.3.1.2 A Lipschitz nonconvex nonlinearity 117
2.3.2 Boundary value problems 119
2.3.2.1 The convex case 120
2.3.2.2 The nonconvex case 123
3 Solution sets for differential equations and inclusions 125
3.1 General results 125
3.1.1 Kneser-Hukuhara theorem 125
3.1.2 Problems on bounded intervals 128
3.1.3 Problems on unbounded intervals 129
3.1.4 Second-order differential equations 131
3.1.5 Abstract Volterra equations 133
3.1.6 Aronszajn type results for differential inclusions 134
3.2 Second-order differential inclusions 142
3.2.1 The convex case 142
3.2.2 The nonconvex case 147
3.2.3 Solution sets 150
3.3 Higher-order differential inclusions 154
3.4 Neutral differential inclusions 155
3.4.1 The convex case 156
3.4.2 The nonconvex case 162
3.4.3 Solutions sets 166
3.5 Nonlocal problems 166
3.5.1 Main results 167
3.5.2 A viability problem 169
3.6 Hyperbolic differential inclusions 174
3.6.1 Existence results 175
3.6.1.1 The convex case 175
3.6.1.2 The nonconvex case 179
3.6.2 Solution sets 180
4 Impulsive differential inclusions: existence and solution sets 183
4.1 Motivation 183
4.1.1 Ecological model with impulsive control strategy 183
4.1.2 Leslie predator-prey system 184
4.1.3 Pulse vaccination model 185
4.2 Semi-linear impulsive differential inclusions 186
4.2.1 Existence results 186
4.2.1.1 The convex case 187
4.2.1.2 The nonconvex case 201
4.2.2 Structure of solution sets 206
4.3 A periodic problem 217
4.3.1 Existence results: 1 . .(T(b)) 218
4.3.2 The convex case: a direct approach 219
4.3.3 The convex case: an MNC approach 227
4.3.4 The nonconvex case 232
4.3.5 The parameter-dependant case 235
4.3.5.1 The convex case 235
4.3.5.2 The nonconvex case 237
4.3.6 Filippov’s Theorem 240
4.3.7 Existence of solutions: 1 . .(T(b)) 250
4.3.7.1 A nonlinear alternative 250
4.3.7.2 A Poincare translation operator 253
4.3.7.3 The MNC approach 253
4.4 Impulsive functional differential inclusions 256
4.4.1 Introduction 256
4.4.2 Existence results 257
4.4.3 Structure of the solution set 266
4.5 Impulsive differential inclusions on the half-line 270
4.5.1 Existence results and compactness of solution sets 270
4.5.1.1 The convex u.s.c. case 271
4.5.1.2 The nonconvex Lipschitz case 278
4.5.1.3 The nonconvex l.s.c. case 282
4.5.2 Topological structure via the projective limit 285
4.5.2.1 The nonconvex case 286
4.5.2.2 The convex case 291
4.5.2.3 The terminal problem 294
4.5.3 Using solution sets to prove existence results 303
5 Preliminary notions of topology and homology 308
5.1 Retracts, extension and embedding properties 308
5.2 Absolute retracts 314
5.3 Homotopical properties of spaces 316
5.4 Cech homology (cohomology) functor 324
5.5 Maps of spaces of finite type 326
5.6 Cech homology functor with compact carriers 333
5.7 Acyclic sets and Vietoris maps 335
5.8 Homology of open subsets of Euclidean spaces 339
5.9 Lefschetz number 343
5.10 The coincidence problem 350
6 Background in multi-valued analysis 357
6.1 Continuity of multi-valued mappings 359
6.1.1 Basic notions 359
6.1.2 Upper semi-continuity 361
6.1.2.1 Generalities 361
6.1.2.2 . — d u.s.c. mappings 364
6.1.2.3 U.s.c. maps and closed graphs 365
6.1.3 Lower semi-continuity 366
6.1.3.1 Generalities 366
6.1.3.2 . — d l.s.c. mappings 369
6.1.4 Hausdorff continuity 370
6.2 The selection problem 374
6.2.1 Michael’s selection theorem 375
6.2.2 Michael’s family of subsets 378
6.2.3 s—selectionable mappings 382
6.2.4 The Kuratowski-Ryll-Nardzewski selection theorem 386
6.2.5 Aumann and Filippov theorems 398
6.2.6 Hausdorff measurable multi-valued maps 402
6.2.7 Product-measurability and the Scorza-Dragoni property 403
6.3 Decomposable sets 410
6.3.1 The Bressan-Colombo-Fryszkowski selection theorem 410
6.3.2 Decomposability in L1(T,E) 410
6.3.3 Integration of multi-valued maps 412
6.3.4 Nemytski operators 413
Appendix 419
A.1 Axioms of the Cech homology theory 419
A.2 The Bochner integral 420
A.3 Absolutely continuous functions 423
A.4 Compactness criteria in C([a,b], E), Cb([0,8), E), and PC ([a, b], E) 425
A.5 Weak-compactness in L1 428
A.6 Proper maps and vector fields 430
A.7 Fundamental theorems in functional analysis 431
A.8 C0-Semigroups 432
References 435
Index 471

lt;P>"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