Viability, Invariance and Applications
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Table of contents
- Cover
- Viability, Invariance and Applicationsiii
- Copyright Pageiv
- Table of Contentsv
- Prefaceix
- Chapter 1. Generalities1
- 1.1 Basic facts on Banach spaces1
- 1.2 The Bochner integral and Lp spaces3
- 1.3 Compactness theorems7
- 1.4 C0-semigroups10
- 1.5 Mild solutions12
- 1.6 Evolutions governed by m-dissipative operators15
- 1.7 Examples of m-dissipative operators19
- 1.8 Differential and integral inequalities24
- Chapter 2. Specific preliminary results29
- 2.1 Brezis-Browder Ordering Principle29
- 2.2 Projections31
- 2.3 Tangent sets32
- 2.4 Bouligand–Severi tangent vectors35
- 2.5 Other types of tangent vectors41
- 2.6 Multi-functions46
- 2.7 Measures of noncompactness48
- 2.8 Scorza Dragoni type theorems53
- Part 1. Ordinary differential equations and inclusions59
- Chapter 3. Nagumo type viability theorems61
- 3.1 Necessary conditions for viability61
- 3.2 Sufficient conditions for viability64
- 3.3 Existence of ε-approximate solutions66
- 3.4 Convergence of ε-approximate solutions70
- 3.5 Extension to the nonautonomous case72
- 3.6 Global solutions77
- Chapter 4. Problems of invariance81
- 4.1 Preliminary facts81
- 4.2 Sufficient conditions for local invariance86
- 4.3 Viability and comparison imply invariance87
- 4.4 When tangency does imply exterior tangency?90
- 4.5 Local invariance and monotonicity91
- Chapter 5. Viability under Carathéodory conditions93
- 5.1 Necessary conditions for Carathéodory viability93
- 5.2 Sufficient conditions for Carathéodory viability95
- 5.3 Existence of (E,L)-approximate Carathéodory solutions97
- 5.4 Convergence of (E,L)-approximate Carathéodory solutions102
- 5.5 Noncontinuable Carathéodory solutions105
- Chapter 6. Viability for differential inclusions107
- 6.1 Necessary conditions for exact viability107
- 6.2 Sufficient conditions for exact viability113
- 6.3 Existence of ε-approximate exact solutions115
- 6.4 Convergence of ε-approximate exact solutions119
- 6.5 The nonautonomous u.s.c. case124
- 6.6 Global (almost) exact solutions126
- 6.7 Sufficient conditions for invariance127
- Chapter 7. Applications131
- 7.1 Viability of the relative closure131
- 7.2 Viability of the epigraph133
- 7.3 Monotone solutions135
- 7.4 A Banach-type fixed point theorem137
- 7.5 Positive solutions to pseudoparabolic PDEs138
- 7.6 Hukuhara Theorem140
- 7.7 Kneser Theorem143
- 7.8 Lyapunov functions144
- 7.9 The characteristics method for a first order PDE147
- Part 2. Evolution equations and inclusions153
- Chapter 8. Viability for single-valued semilinear evolutions155
- 8.1 Necessary conditions for mild viability155
- 8.2 Sufficient conditions for mild viability158
- 8.3 Existence of ε-approximate mild solutions160
- 8.4 Proof of the sufficiency of Theorem 8.2.1165
- 8.5 The quasi-autonomous case167
- 8.6 A class of reaction-diffusion systems170
- 8.7 Convergence in the case of Theorem 8.6.2174
- 8.8 Global mild solutions177
- Chapter 9. Viability for multi-valued semilinear evolutions179
- 9.1 Necessary conditions for mild viability179
- 9.2 Sufficient conditions for mild viability184
- 9.3 Existence of ε-approximate mild solutions185
- 9.4 Proof of Theorem 9.2.1191
- 9.5 Proof of Theorem 9.2.3193
- 9.6 Proof of Theorem 9.2.4193
- 9.7 The quasi-autonomous case194
- 9.8 Global mild solutions197
- Chapter 10. Viability for single-valued fully nonlinear evolutions199
- 10.1 Continuous perturbations of m-dissipative operators199
- 10.2 Existence of ε-approximate C0-solutions202
- 10.3 Convergence in the case of Theorems 10.1.1 and 10.1.3206
- 10.4 Convergence in the case of Theorem 10.1.2208
- 10.5 The quasi-autonomous noncylindrical case209
- 10.6 Noncontinuable C0-solutions211
- 10.7 A class of fully nonlinear reaction-diffusion systems212
- 10.8 Convergence in the case of Theorem 10.7.2216
- 10.9 Convergence in the case of Theorem 10.7.4218
- 10.10 Sufficient conditions for C0-local invariance220
- Chapter 11. Viability for multi-valued fully nonlinear evolutions223
- 11.1 Necessary conditions for C0-viability223
- 11.2 Necessary and/or sufficient conditions for C0-viability226
- 11.3 Existence of ε-approximate C0-solutions227
- 11.4 Convergence in the case of Theorem 11.2.1232
- 11.5 Convergence in the case of Theorem 11.2.4233
- 11.6 The fully nonlinear multi-valued quasi-autonomous case234
- 11.7 Noncontinuable C0-solutions236
- Chapter 12. Carathéodory perturbations of m-dissipative operators239
- 12.1 Necessary and sufficient conditions for C0-viability239
- 12.2 Existence of ε-approximate C0-solutions242
- 12.3 Convergence in the case of Theorems 12.1.1 and 12.1.3248
- 12.4 Convergence in the case of Theorem 12.1.2250
- 12.5 Noncontinuable C0-solutions252
- Chapter 13. Applications255
- 13.1 Viability in the first approximation255
- 13.2 Lyapunov pairs260
- 13.3 A comparison result for a semilinear diffusion equation264
- 13.4 A comparison result for a predator–pray system267
- 13.5 A comparison result for a nonlinear diffusion inclusion271
- 13.6 Comparison for a fully nonlinear reaction-diffusion system277
- 13.7 A controllability problem281
- 13.8 Periodic solutions284
- Solutions to the proposed problems288
- Bibliographical notes and comments309
- Bibliography325
- Name Index335
- Subject Index339
- Notation Index343
The book is an almost self-contained presentation of the most important concepts and results in viability and invariance. The viability of a set K with respect to a given function (or multi-function) F, defined on it, describes the property that, for each initial data in K, the differential equation (or inclusion) driven by that function or multi-function) to have at least one solution. The invariance of a set K with respect to a function (or multi-function) F, defined on a larger set D, is that property which says that each solution of the differential equation (or inclusion) driven by F and issuing in K remains in K, at least for a short time.
The book includes the most important necessary and sufficient conditions for viability starting with Nagumo’s Viability Theorem for ordinary differential equations with continuous right-hand sides and continuing with the corresponding extensions either to differential inclusions or to semilinear or even fully nonlinear evolution equations, systems and inclusions. In the latter (i.e. multi-valued) cases, the results (based on two completely new tangency concepts), all due to the authors, are original and extend significantly, in several directions, their well-known classical counterparts.
- New concepts for multi-functions as the classical tangent vectors for functions
- Provides the very general and necessary conditions for viability in the case of differential inclusions, semilinear and fully nonlinear evolution inclusions
- Clarifying examples, illustrations and numerous problems, completely and carefully solved
- Illustrates the applications from theory into practice
- Very clear and elegant style
The book includes the most important necessary and sufficient conditions for viability starting with Nagumo’s Viability Theorem for ordinary differential equations with continuous right-hand sides and continuing with the corresponding extensions either to differential inclusions or to semilinear or even fully nonlinear evolution equations, systems and inclusions. In the latter (i.e. multi-valued) cases, the results (based on two completely new tangency concepts), all due to the authors, are original and extend significantly, in several directions, their well-known classical counterparts.
- New concepts for multi-functions as the classical tangent vectors for functions
- Provides the very general and necessary conditions for viability in the case of differential inclusions, semilinear and fully nonlinear evolution inclusions
- Clarifying examples, illustrations and numerous problems, completely and carefully solved
- Illustrates the applications from theory into practice
- Very clear and elegant style