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Semidynamical Systems in Infinite Dimensional Spaces

Where do solutions go, and how do they behave en route? These are two of the major questions addressed by the qualita­ tive theory of differential equations. The purpose of this book is to answer these questions for certain classes of equa­ tions by recourse to the framework of semidynamical systems...

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Bibliographic Details
Main Author: Saperstone, Stephen H.
Format: eBook
Language:English
Published: New York, NY Springer New York 1981, 1981
Edition:1st ed. 1981
Series:Applied Mathematical Sciences
Subjects:
Mathematical Physics
Theoretical, Mathematical And Computational Physics
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 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
  • 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
  • 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
  • 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

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