Aubin J.-P. Set-valued analysis / H.Frankowska, J.-P.Aubin. - Boston: Birkhäuser Boston, 1990 (2009). - xi, 461 p. - (Modern Birkhäuser Classics). - Originally published in the series: Systems & control: foundations & applications. - Bibliogr.: p.421-455. - Ind.: p.461. - ISBN 978-0-8176-4847-3  Место хранения:   03 | Центральная библиотека Иркутского научного центра СО РАН | Иркутск

Оглавление / Contents

Introduction .................................................... 1 1 Continuity of Set-Valued Maps ............................... 15 1.1 Limits of Sets ......................................... 16 1.1.1 Definitions ..................................... 16 1.1.2 The Compactness Theorem ......................... 23 1.1.3 The Duality Theorem ............................. 24 1.1.4 Convex Hull of Limits ........................... 26 1.2 Calculus of Limits ..................................... 27 1.2.1 Direct Images ................................... 28 1.2.2 Inverse Images .................................. 30 1.3 Set-Valued Maps ........................................ 33 1.4 Continuity of Set-Valued maps .......................... 38 1.4.1 Definitions ..................................... 38 1.4.2 Generic Continuity .............................. 44 1.4.3 Example: Parametrized Set-Valued Maps ........... 46 1.4.4 Marginal Maps ................................... 48 1.5 Lower Semi-Continuity Criteria ......................... 49 2 Closed Convex Processes ..................................... 55 2.1 Definitions ............................................ 56 2.2 Open Mapping and Closed Graph Theorems ................. 57 2.3 Uniform Boundedness Theorem ............................ 61 2.4 The Bipolar Theorem .................................... 62 2.5 Transposition of Closed Convex Process ................. 67 2.6 Upper Hemicontinuous Maps .............................. 74 3 Existence and Stability of an Equilibrium ................... 77 3.1 Ky Fan's Inequality .................................... 80 3.2 Equilibrium and Fixed Point Theorems ................... 83 3.2.1 The Equilibrium Theorem ......................... 83 3.2.2 Fixed Point Theorems ............................ 86 3.2.3 The Leray-Schauder Theorem ...................... 89 3.3 Ekeland's Variational Principle ........................ 91 3.4 Constrained Inverse Function Theorem ................... 93 3.4.1 Derivatives of Single-Valued Maps ............... 93 3.4.2 Constrained Inverse Function Theorems ........... 94 3.4.3 Pointwise Stability Conditions ................. 101 3.4.4 Local Uniqueness ............................... 103 3.5 Monotone and Maximal Monotone Maps .................... 104 3.5.1 Monotone Maps .................................. 104 3.5.2 Maximal Monotone Maps .......................... 106 3.5.3 Yosida Approximations .......................... 111 3.6 Eigenvectors of Closed Convex Processes ............... 114 4 Tangent Cones .............................................. 117 4.1 Tangent Cones to a Subset ............................. 121 4.1.1 Contingent Cones ............................... 121 4.1.2 Elementary Properties of Contingent Cones ...... 125 4.1.3 Adjacent and Clarke Tangent Cones .............. 126 4.1.4 Sleek Subsets .................................. 130 4.1.5 Limits of Contingent Cones; Finite Dimensional Case ............................... 130 4.1.6 Limits of Contingent Cones; Infinite Dimensional Case ............................... 132 4.2 Tangent Cones to Convex Sets .......................... 138 4.3 Calculus of Tangent Cones ............................. 146 4.3.1 Intersection and Inverse Image ................. 146 4.3.2 Example: Tangent cones to subsets defined by equality and inequality constraints ............ 150 4.3.3 Direct Image ................................... 153 4.4 Normal Cones .......................................... 156 4.5 Other Tangent Cones ................................... 159 4.5.1 Convex Kernel of a Cone ........................ 159 4.5.2 Paratingent Cones .............................. 160 4.5.3 Hypertangent Cones ............................. 164 4.5.4 A Menagerie of Tangent Cones ................... 165 4.6 Tangent Cones to Sequences of Sets .................... 166 4.7 Higher Order Tangent Sets ............................. 171 5 Derivatives of Set-Valued Maps ............................. 179 5.1 Contingent Derivatives ................................ 181 5.2 Adjacent and Circatangent Derivatives ................. 189 5.2.1 Definitions and Elementary Properties .......... 189 5.2.2 Limits of Differential Quotients ............... 191 5.2.3 Derivatives of monotone operators .............. 194 5.3 Chain Rules ........................................... 196 5.4 Inverse Set-Valued Map Theorem ........................ 203 5.4.1 Stability and Approximation of Inclusions ...... 203 5.4.2 Localization of Inverse Images ................. 206 5.4.3 The Equilibrium Map ............................ 207 5.4.4 Local Injectivity .............................. 209 5.5 Qualitative Solutions ................................. 210 5.6 Higher Order Derivatives .............................. 215 6 Epiderivatives of Extended Functions ....................... 219 6.1 Contingent Epiderivatives ............................. 222 6.1.1 Extended Functions and their Epigraphs ......... 222 6.1.2 Contingent Epiderivatives ...................... 224 6.1.3 Fermat and Ekeland Rules ....................... 232 6.1.4 Elementary Properties .......................... 234 6.2 Other Epiderivatives .................................. 236 6.2.1 Adjacent and Circatangent Epiderivatives ....... 236 6.2.2 Other Convex Epiderivatives .................... 240 6.3 Epidifferential Calculus .............................. 242 6.4 Generalized Gradient .................................. 248 6.4.1 Subdifferentials and Generalized Gradients ..... 248 6.4.2 Limits of Subdifferentials and Gradients ....... 251 6.4.3 Local Subdifferentials and Superdifferentials ............................. 252 6.4.4 Remarks ........................................ 254 6.5 Convex Functions ...................................... 255 6.6 Higher Order Epiderivatives ........................... 259 6.6.1 Second Order Epiderivatives of Moreau-Yosida Approximations ................................. 263 7 Graphical & Epigraphical Convergence ....................... 265 7.1 Graphical Limits ...................................... 267 7.1.1 Definitions .................................... 267 7.1.2 Graphical Convergence of Closed Convex Processes ...................................... 269 7.1.3 Monotone and Maximal Monotone Maps ............. 270 7.2 Convergence Theorems .................................. 270 7.3 Epilimits ............................................. 274 7.3.1 Definitions and Elementary Properties .......... 274 7.3.2 Convergence of Infima and Minimizers ........... 281 7.3.3 Variational Systems ............................ 284 7.4 Epilimits of Sums and Composition Products ............ 286 7.5 Conjugate Functions of Epilimits ...................... 289 7.6 Graphical Convergence of Gradients .................... 294 7.6.1 Convergence of Gradients of Smooth Functions ... 295 7.6.2 Convergence of Subdifferentials of Convex Functions ...................................... 297 7.7 Asymptotic Epiderivatives ............................. 300 8 Measurability and Integration of Set-Valued Maps ........... 303 8.1 Measurable Set-Valued Maps ............................ 306 8.2 Calculus of Measurable Maps ........................... 310 8.3 Proof of the Characterization Theorem ................. 319 8.4 Limits of Measurable Maps and Selections .............. 322 8.5 Tangent Cones in Lebesgue Spaces ...................... 324 8.6 Integral of Set-Valued Maps ........................... 326 8.7 Proofs of the Convexity of the Integral ............... 333 8.7.1 Finite dimensional case ........................ 333 8.7.2 Infinite Dimensional Case ...................... 340 8.8 The Bang-Bang Principle ............................... 343 8.9 Invariant Measures & Poincare's Recurrence Theorem .... 346 8.9.1 Linear Extension of Set-Valued Maps ............ 346 8.9.2 Invariant Measures ............................. 350 9 Selections and Parametrization ............................. 353 9.1 Case of lower semicontinuous maps ..................... 355 9.2 Case of upper semicontinuous maps ..................... 358 9.3 Minimal Selection ..................................... 360 9.4 The Steiner Selection ................................. 364 9.4.1 Steiner Points of Convex Compact Sets .......... 365 9.4.2 The Intersection Lemma ......................... 369 9.4.3 Lipschitz Selections of Lipschitz Maps ......... 372 9.5 Selections of Caratheodory maps ....................... 373 9.6 Caratheodory Parametrization .......................... 376 9.7 Measurable/Lipschitz Parametrization .................. 379 10 Differential Inclusions .................................... 383 10.1 The Viability Theorem ................................. 387 10.1.1 Solutions to Differential Inclusions ........... 388 10.1.2 Statements of the Viability Theorems ........... 389 10.1.3 Viability Kernels .............................. 392 10.1.4 Viability and Equilibria ....................... 393 10.2 Applications of the Viability Theorem ................. 394 10.2.1 Linear Differential Inclusions ................. 395 10.2.2 Lyapunov Functions ............................. 395 10.2.3 Tracking a Differential Inclusion .............. 398 10.3 Nonlinear Semi-Groups ................................. 399 10.4 Filippov's Theorem .................................... 400 10.5 Derivatives of the Solution Map ....................... 403 Bibliographical Comments ...................................... 411 Bibliography .................................................. 421 Index ......................................................... 457