Semidynamical Systems in Infinite Dimensional Spaces

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 RFDE’s.- 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 — 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 — A Compact Process.- 6. Solutions of a Wave Equation — 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. LinearSpaces.- 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.