Set-Valued Analysis

Title page......Page 1
Date-line......Page 2
Dedication......Page 3
P. de Fermat......Page 4
Epigraph......Page 6
Acknowledgments......Page 9
Contents......Page 10
Introduction......Page 17
1 Continuity of Set-Valued Maps......Page 31
1.1.1 Definitions......Page 32
1.1.2 The Compactness Theorem......Page 39
1.1.3 The Duality Theorem......Page 40
1.1.4 Convex Hull of Limits......Page 42
1.2 Calculus of Limits......Page 43
1.2.1 Direct Images......Page 44
1.2.2 Inverse Images......Page 46
1.3 Set-Valued Maps......Page 49
1.4.1 Definitions......Page 54
1.4.2 Generic Continuity......Page 60
1.4.3 Example: Parametrized Set-Valued Maps......Page 62
1.4.4 Marginal Maps......Page 64
1.5 Lower Semi-Continuity Criteria......Page 65
2 Closed Convex Processes......Page 71
2.1 Definitions......Page 72
2.2 Open Mapping and Closed Graph Theorems......Page 73
2.3 Uniform Boundedness Theorem......Page 77
2.4 The Bipolar Theorem......Page 78
2.5 Transposition of Closed Convex Process......Page 83
2.6 Upper Hemicontinuous Maps......Page 90
3 Existence and Stability of an Equilibrium......Page 93
3.1 Ky Fan\'s Inequality......Page 96
3.2.1 The Equilibrium Theorem......Page 99
3.2.2 Fixed Point Theorems......Page 102
3.2.3 The Leray-Schauder Theorem......Page 105
3.3 Ekeland\'s Variational Principle......Page 107
3.4.1 Derivatives of Single-Valued Maps......Page 109
3.4.2 Constrained Inverse Function Theorems......Page 110
3.4.3 Pointwise Stability Conditions......Page 117
3.4.4 Local Uniqueness......Page 119
3.5.1 Monotone Maps......Page 120
3.5.2 Maximal Monotone Maps......Page 122
3.5.3 Yosida Approximations......Page 127
3.6 Eigenvectors of Closed Convex Processes......Page 130
4 Tangent Cones......Page 133
4.1.1 Contingent Cones......Page 137
4.1.2 Elementary Properties of Contingent Cones......Page 141
4.1.3 Adjacent and Clarke Tangent Cones......Page 142
4.1.5 Limits of Contingent Cones; Finite Dimensional Case......Page 146
4.1.6 Limits of Contingent Cones; Infinite Dimensional Case......Page 148
4.2 Tangent Cones to Convex Sets......Page 154
4.3.1 Intersection and Inverse Image......Page 162
4.3.2 Example: Tangent cones to subsets denned by equality and inequality constraints......Page 166
4.3.3 Direct Image......Page 169
4.4 Normal Cones......Page 172
4.5.1 Convex Kernel of a Cone......Page 175
4.5.2 Paratingent Cones......Page 176
4.5.3 Hypertangent Cones......Page 180
4.5.4 A Menagerie of Tangent Cones......Page 181
4.6 Tangent Cones to Sequences of Sets......Page 182
4.7 Higher Order Tangent Sets......Page 187
5 Derivatives of Set-Valued Maps......Page 195
5.1 Contingent Derivatives......Page 197
5.2.1 Definitions and Elementary Properties......Page 205
5.2.2 Limits of Differential Quotients......Page 207
5.2.3 Derivatives of monotone operators......Page 210
5.3 Chain Rules......Page 212
5.4.1 Stability and Approximation of Inclusions......Page 219
5.4.2 Localization of Inverse Images......Page 222
5.4.3 The Equilibrium Map......Page 223
5.4.4 Local Injectivity......Page 225
5.5 Qualitative Solutions......Page 226
5.6 Higher Order Derivatives......Page 231
6 Epiderivatives of Extended Functions......Page 235
6.1.1 Extended Functions and their Epigraphs......Page 238
6.1.2 Contingent Epiderivatives......Page 240
6.1.3 Fermat and Ekeland Rules......Page 248
6.1.4 Elementary Properties......Page 250
6.2.1 Adjacent and Circatangent Epiderivatives......Page 252
6.2.2 Other Convex Epiderivatives......Page 256
6.3 Epidifferential Calculus......Page 258
6.4.1 Subdifferentials and Generalized Gradients......Page 264
6.4.2 Limits of Subdifferentials and Gradients......Page 267
6.4.3 Local Subdifferentials and Superdifferentials......Page 268
6.4.4 Remarks......Page 270
6.5 Convex Functions......Page 271
6.6 Higher Order Epiderivatives......Page 275
6.6.1 Second Order Epiderivatives of Moreau-Yosida Approximations......Page 279
7 Graphical & Epigraphical Convergence......Page 281
7.1.1 Definitions......Page 283
7.1.2 Graphical Convergence of Closed Convex Processes......Page 285
7.2 Convergence Theorems......Page 286
7.3.1 Definitions and Elementary Properties......Page 290
7.3.2 Convergence of Infima and Minimizers......Page 297
7.3.3 Variational Systems......Page 300
7.4 Epilimits of Sums and Composition Products......Page 302
7.5 Conjugate Functions of Epilimits......Page 305
7.6 Graphical Convergence of Gradients......Page 310
7.6.1 Convergence of Gradients of Smooth Functions......Page 311
7.6.2 Convergence of Subdifferentials of Convex Functions......Page 313
7.7 Asymptotic Epiderivatives......Page 316
8 Measurability and Integration of Set-Valued Maps......Page 319
8.1 Measurable Set-Valued Maps......Page 322
8.2 Calculus of Measurable Maps......Page 326
8.3 Proof of the Characterization Theorem......Page 335
8.4 Limits of Measurable Maps and Selections......Page 338
8.5 Tangent Cones in Lebesgue Spaces......Page 340
8.6 Integral of Set-Valued Maps......Page 342
8.7.1 Finite dimensional case......Page 349
8.7.2 Infinite Dimensional Case......Page 356
8.8 The Bang-Bang Principle......Page 359
8.9.1 Linear Extension of Set-Valued Maps......Page 362
8.9.2 Invariant Measures......Page 366
9 Selections and Parametrization......Page 369
9.1 Case of lower semicontinuous maps......Page 371
9.2 Case of upper semicontinuous maps......Page 374
9.3 Minimal Selection......Page 376
9.4 The Steiner Selection......Page 380
9.4.1 Steiner Points of Convex Compact Sets......Page 381
9.4.2 The Intersection Lemma......Page 385
9.4.3 Lipschitz Selections of Lipschitz Maps......Page 388
9.5 Selections of Caratheodory maps......Page 389
9.6 Caratheodory Parametrization......Page 392
9.7 Measurable/Lipschitz Parametrization......Page 395
10 Differential Inclusions......Page 399
10.1 The Viability Theorem......Page 403
10.1.1 Solutions to Differential Inclusions......Page 404
10.1.2 Statements of the Viability Theorems......Page 405
10.1.3 Viability Kernels......Page 408
10.1.4 Viability and Equilibria......Page 409
10.2 Applications of the Viability Theorem......Page 410
10.2.2 Lyapunov Functions......Page 411
10.2.3 Tracking a Differential Inclusion......Page 414
10.3 Nonlinear Semi-Groups......Page 415
10.4 Filippov\'s Theorem......Page 416
10.5 Derivatives of the Solution Map......Page 419
Bibliographical Comments......Page 427
Bibliography......Page 437
Index......Page 473