Generalized Ordinary Differential Equations in Abstract Spaces and Applications

Оглавление

Группа авторов. Generalized Ordinary Differential Equations in Abstract Spaces and Applications

Table of Contents

List of Illustrations

Guide

Pages

Generalized Ordinary Differential Equations in Abstract Spaces and Applications

List of Contributors

Foreword

Preface

1 Preliminaries

1.1 Regulated Functions

1.1.1 Basic Properties

1.1.2 Equiregulated Sets

1.1.3 Uniform Convergence

1.1.4 Relatively Compact Sets

1.2 Functions of Bounded ‐Variation

1.3 Kurzweil and Henstock Vector Integrals

1.3.1 Definitions

Definition 1.42 (Indefinite Vector Integrals):

1.3.2 Basic Properties

1.3.3 Integration by Parts and Substitution Formulas

1.3.4 The Fundamental Theorem of Calculus

1.3.5 A Convergence Theorem

Appendix 1.A: The McShane Integral

2. The Kurzweil Integral

2.1 The Main Background. 2.1.1 Definition and Compatibility

2.1.2 Special Integrals

2.2 Basic Properties

2.3 Notes on Kapitza Pendulum

3 Measure Functional Differential Equations

3.1 Measure FDEs

3.2 Impulsive Measure FDEs

3.3 Functional Dynamic Equations on Time Scales

3.3.1 Fundamentals of Time Scales

3.3.2 The Perron -integral

3.3.3 Perron -integrals and Perron–Stieltjes integrals

3.3.4 MDEs and Dynamic Equations on Time Scales

3.3.5 Relations with Measure FDEs

3.3.6 Impulsive Functional Dynamic Equations on Time Scales

3.4 Averaging Methods

3.4.1 Periodic Averaging

3.4.2 Nonperiodic Averaging

3.5 Continuous Dependence on Time Scales

4 Generalized Ordinary Differential Equations

4.1 Fundamental Properties

4.2 Relations with Measure Differential Equations

4.3 Relations with Measure FDEs

5 Basic Properties of Solutions

5.1 Local Existence and Uniqueness of Solutions

5.1.1 Applications to Other Equations

5.2 Prolongation and Maximal Solutions

5.2.1 Applications to MDEs

5.2.2 Applications to Dynamic Equations on Time Scales

6 Linear Generalized Ordinary Differential Equations

6.1 The Fundamental Operator

6.2 A Variation-of-Constants Formula

6.3 Linear Measure FDEs

6.4 A Nonlinear Variation-of-Constants Formula for Measure FDEs

7 Continuous Dependence on Parameters

7.1 Basic Theory for Generalized ODEs

7.2 Applications to Measure FDEs

8 Stability Theory

8.1 Variational Stability for Generalized ODEs

8.1.1 Direct Method of Lyapunov

8.1.2 Converse Lyapunov Theorems

8.2 Lyapunov Stability for Generalized ODEs

8.2.1 Direct Method of Lyapunov

8.3 Lyapunov Stability for MDEs

8.3.1 Direct Method of Lyapunov

8.4 Lyapunov Stability for Dynamic Equations on Time Scales

8.4.1 Direct Method of Lyapunov

8.5 Regular Stability for Generalized ODEs

8.5.1 Direct Method of Lyapunov

8.5.2 Converse Lyapunov Theorem

9. Periodicity

9.1 Periodic Solutions and Floquet's Theorem

9.1.1 Linear Differential Systems with Impulses

9.2 (θ, T)-Periodic Solutions

9.2.1 An Application to MDEs

10. Averaging Principles

10.1 Periodic Averaging Principles

10.1.1 An Application to IDEs

10.2 Nonperiodic Averaging Principles

11. Boundedness of Solutions

11.1 Bounded Solutions and Lyapunov Functionals

11.2 An Application to MDEs

11.2.1 An Example

12 Control Theory

12.1 Controllability and Observability

12.2 Applications to ODEs

13 Dichotomies

13.1 Basic Theory for Generalized ODEs

13.2 Boundedness and Dichotomies

13.3 Applications to MDEs

13.4 Applications to IDEs

14 Topological Dynamics

14.1 The Compactness of the Class

14.2 Existence of a Local Semidynamical System

14.3 Existence of an Impulsive Semidynamical System

14.4 LaSalle's Invariance Principle

14.5 Recursive Properties

15 Applications to Functional Differential Equations of Neutral Type

15.1 Drops of History

15.2 FDEs of Neutral Type with Finite Delay

References

List of Symbols

Index

WILEY END USER LICENSE AGREEMENT

Отрывок из книги

Edited by

Everaldo M. Bonotto

.....

Theorem 1.49: The following assertions hold:

.....