Solution Sets for Differential Equations and Inclusions

Cover......Page 1
Title......Page 4
Copyright......Page 5
Preface......Page 8
Notations......Page 12
Contents......Page 16
1.1.1.1 Banach’s fixed point theorem......Page 22
1.1.1.2 Brouwer’s fixed point theorem......Page 26
1.1.1.3 Schauder’s fixed point theorem......Page 28
1.1.2 Approximation theorems......Page 32
1.1.3 Browder-Gupta theorems......Page 34
1.1.4 Acyclicity of the solution sets of operator equations......Page 41
1.1.5.1 Existence theory......Page 44
1.1.5.2 Solution sets......Page 47
1.2.1 Approximation of multi-valued maps......Page 48
1.2.2 Fixed point theorems......Page 51
1.2.3 Multi-valued contractions......Page 54
1.2.4 Fixed point sets of multi-valued contractions......Page 56
1.2.5 Fixed point sets of multi-valued nonexpansive maps......Page 59
1.2.6.1 Measure of noncompactness......Page 60
1.2.6.2 Condensing maps......Page 64
1.3.1 Generalities......Page 65
1.3.2 Fixed point theorems for admissible multi-valued maps......Page 74
1.3.3 The general Brouwer fixed point theorem......Page 79
1.3.4 Browder-Gupta type results for admissible mappings......Page 81
1.3.5 Topological dimensions of solution sets......Page 83
1.4.1 Definition......Page 86
1.4.2 Basic properties......Page 87
1.4.3 Multi-maps of inverse systems......Page 88
2.1.1 Existence and uniqueness results......Page 93
2.1.2 Picard-Lindelöf theorem......Page 94
2.1.2.1 Maximal solutions......Page 96
2.1.3.1 Peano theorem......Page 98
2.2.1 Global existence theorems......Page 100
2.2.2.1 The Lipschitz case......Page 103
2.2.2.2 The Lipschitz-Nagumo case......Page 104
2.2.2.3 The Nagumo case......Page 107
2.2.3 A boundary value problem on the half-line......Page 109
2.3.1.1 A Nagumo type nonlinearity......Page 115
2.3.1.2 A Lipschitz nonconvex nonlinearity......Page 118
2.3.2 Boundary value problems......Page 120
2.3.2.1 The convex case......Page 121
2.3.2.2 The nonconvex case......Page 124
3.1.1 Kneser-Hukuhara theorem......Page 126
3.1.2 Problems on bounded intervals......Page 129
3.1.3 Problems on unbounded intervals......Page 130
3.1.4 Second-order differential equations......Page 132
3.1.5 Abstract Volterra equations......Page 134
3.1.6 Aronszajn type results for differential inclusions......Page 135
3.2.1 The convex case......Page 143
3.2.2 The nonconvex case......Page 148
3.2.3 Solution sets......Page 151
3.3 Higher-order differential inclusions......Page 155
3.4 Neutral differential inclusions......Page 156
3.4.1 The convex case......Page 157
3.4.2 The nonconvex case......Page 163
3.5 Nonlocal problems......Page 167
3.5.1 Main results......Page 168
3.5.2 A viability problem......Page 170
3.6 Hyperbolic differential inclusions......Page 175
3.6.1.1 The convex case......Page 176
3.6.1.2 The nonconvex case......Page 180
3.6.2 Solution sets......Page 181
4.1.1 Ecological model with impulsive control strategy......Page 184
4.1.2 Leslie predator-prey system......Page 185
4.1.3 Pulse vaccination model......Page 186
4.2.1 Existence results......Page 187
4.2.1.1 The convex case......Page 188
4.2.1.2 The nonconvex case......Page 202
4.2.2 Structure of solution sets......Page 207
4.3 A periodic problem......Page 218
4.3.1 Existence results: 1 in p(T(b))......Page 219
4.3.2 The convex case: a direct approach......Page 220
4.3.3 The convex case: an MNC approach......Page 228
4.3.4 The nonconvex case......Page 233
4.3.5.1 The convex case......Page 236
4.3.5.2 The nonconvex case......Page 238
4.3.6 Filippov’s Theorem......Page 241
4.3.7.1 A nonlinear alternative......Page 251
4.3.7.3 The MNC approach......Page 254
4.4.1 Introduction......Page 257
4.4.2 Existence results......Page 258
4.4.3 Structure of the solution set......Page 267
4.5.1 Existence results and compactness of solution sets......Page 271
4.5.1.1 The convex u.s.c. case......Page 272
4.5.1.2 The nonconvex Lipschitz case......Page 279
4.5.1.3 The nonconvex l.s.c. case......Page 283
4.5.2 Topological structure via the projective limit......Page 286
4.5.2.1 The nonconvex case......Page 287
4.5.2.2 The convex case......Page 292
4.5.2.3 The terminal problem......Page 295
4.5.3 Using solution sets to prove existence results......Page 304
5.1 Retracts, extension and embedding properties......Page 309
5.2 Absolute retracts......Page 315
5.3 Homotopical properties of spaces......Page 317
5.4 Cech homology (cohomology) functor......Page 325
5.5 Maps of spaces of finite type......Page 327
5.6 Cech homology functor with compact carriers......Page 334
5.7 Acyclic sets and Vietoris maps......Page 336
5.8 Homology of open subsets of Euclidean spaces......Page 340
5.9 Lefschetz number......Page 344
5.10 The coincidence problem......Page 351
6 Background in multi-valued analysis......Page 358
6.1.1 Basic notions......Page 360
6.1.2.1 Generalities......Page 362
6.1.2.2 Epsilon - delta u.s.c. mappings......Page 365
6.1.2.3 U.s.c. maps and closed graphs......Page 366
6.1.3.1 Generalities......Page 367
6.1.3.2 Epsilon - delta l.s.c. mappings......Page 370
6.1.4 Hausdorff continuity......Page 371
6.2 The selection problem......Page 375
6.2.1 Michael’s selection theorem......Page 376
6.2.2 Michael’s family of subsets......Page 379
6.2.3 Sigma-selectionable mappings......Page 383
6.2.4 The Kuratowski-Ryll-Nardzewski selection theorem......Page 387
6.2.5 Aumann and Filippov theorems......Page 399
6.2.6 Hausdorff measurable multi-valued maps......Page 403
6.2.7 Product-measurability and the Scorza-Dragoni property......Page 404
6.3.2 Decomposability in L1(T,E)......Page 411
6.3.3 Integration of multi-valued maps......Page 413
6.3.4 Nemytskii operators......Page 414
A.1 Axioms of the Cech homology theory......Page 420
A.2 The Bochner integral......Page 421
A.3 Absolutely continuous functions......Page 424
A.4 Compactness criteria in C([a, b], E), Cb([0, Infinity], E), and PC([a, b], E)......Page 426
A.5 Weak-compactness in L1......Page 429
A.6 Proper maps and vector fields......Page 431
A.7 Fundamental theorems in functional analysis......Page 432
A.8 C0-Semigroups......Page 433
References......Page 436
Index......Page 472