Author | Saperstone, Stephen H. author |
---|---|
Title | Semidynamical Systems in Infinite Dimensional Spaces [electronic resource] / by Stephen H. Saperstone |
Imprint | New York, NY : Springer New York, 1981 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-5977-0 |
Descript | 492 p. online resource |
I. Basic Definitions and Properties -- 1. Introduction -- 2. Semidynamical Systems: Definitions and -- Conventions -- 3. A Glimpse of Things to Come; An Example from a Function Space -- 4. Solutions -- 5. Critical and Periodic Points -- 6. Classification of Positive Orbits -- 7. Discrete Semidynamical Systems -- 8. Local Semidynamical Systems; Reparametrization -- 9. Exercises -- 10. Notes and Comments -- II. Invariance, Limit Sets, and Stability -- 1. Introduction -- 2. Invariance -- 3. Limit Sets: The Generalized Invariance Principle -- 4. Minimality -- 5. Prolongations and Stability of Compact Sets -- 6. Attraction: Asymptotic Stability of Compact Sets -- 7. Continuity of the Hull and Limit Set Maps in Metric Spaces -- 8. Lyapunov Functions: The Invariance Principle -- 9. From Stability to Chaos: A Simple Example -- 10. Exercises -- 11. Notes and Comments -- III. Motions in Metric Space -- 1. Introduction -- 2. Lyapunov Stable Motions -- 3. Recurrent Motions -- 4. Almost Periodic Motions -- 5. Asymptotically Stable Motions -- 6. Periodic Solutions of an Ordinary Differential Equation -- 7. Exercises -- 8. Notes and Comments -- IV. Nonautonomous Ordinary Differential Equations -- 1. Introduction -- 2. Construction of the Skew Product Semidynamical System -- 3. Compactness of the Space ? -- 4. The Invariance Principle for Ordinary Differential Equations -- 5. Limiting Equations and Stability -- 6. Differential Equations without Uniqueness -- 7. Volterra Integral Equations -- 8. Exercises -- 9. Notes and Comments -- V. Semidynamical Systems in Banach Space -- 1. Introduction -- 2. Nonlinear Semigroups and Their Generators -- 3. The Generalized Domain for Accretive Operators -- 4. Precompactness of Positive Orbits -- 5. Solution of the Cauchy Problem -- 6. Structure of Positive Limit Sets for Contraction Semigroups -- 7. Exercises -- 8. Appendix: Proofs of Theorems 2.4 and 2.16 -- 9. Notes and Comments -- VI. Functional Differential Equations -- 1. Why Hereditary Dependence, Some Examples from Biology, Mechanics, and Electronics -- 2. Definitions and Notation: Functional Differential Equations with Finite or Infinite Delay. The Initial Function Space -- 3. Existence of Solutions of Retarded Functional Equations -- 4. Some Remarks on the Semidynamical System Defined by the Solution to an Autonomous Retarded Functional Differential Equation: The Invariance Principle and Stability -- 5. Some Examples of Stability of RFDEโs -- 6. Remarks on the Asymptotic Behavior of Nonautonomous Retarded Functional Differential Equations -- 7. Critical Points and Periodic Solutions of Autonomous Retarded Functional Differential Equations -- 8. Neutral Functional Differential Equations -- 9. A Flip-Flop Circuit Characterized by a NFDE โ The Stability of Solutions -- 10. Exercises -- 11. Notes and Comments -- VII. Stochastic Dynamical Systems -- 1. Introduction -- 2. The Space of Probability Measures -- 3. Markov Transition Operators and the Semidynamical System -- 4. Properties of Positive Limit Sets -- 5. Critical Points for Markov Processes -- 6. Stochastic Differential Equations -- 7. The Invariance Principle for Markov Processes -- 8. Exercises -- 9. Notes and Comments -- VIII. Weak Semidynamical Systems and Processes -- 1. Introduction -- 2. Weak Semidynamical Systems -- 3. Compact Processes -- 4. Uniform Processes -- 5. Solutions of Nonautonomous Ordinary Differential Equations Revisited โ A Compact Process -- 6. Solutions of a Wave Equation โ A Uniform Process -- 7. Exercises -- 8. Notes and Comments -- Appendix A -- 0. Preliminaries -- 1. Commonly Used Symbols -- 2. Nets -- 3. Uniform Topologies -- 4. Compactness -- 5. Linear Spaces -- 6. Duality -- 7. Hilbert Spaces -- 8. Vector Valued Integration -- 9. Sobolev Spaces -- 10. Convexity -- 11. Fixed Point Theorems -- 12. Almost Periodicity -- 13. Differential Inequalities -- Appendix B -- 1. Probability Spaces and Random Variables -- 2. Expectation -- 3. Convergence of Random Variables -- 4. Stochastic Processes; Martingales and Markov Processes -- 5. The Ito Stochastic Integral -- References -- Index of Terms -- Index of Symbols