Generalized Ordinary Differential Equations in Abstract Spaces and Applications

1 Cover

2 Title Page Generalized Ordinary Differential Equations in Abstract Spaces and Applications Edited by Everaldo M. Bonotto Universidade de São Paulo São Carlos, SP, Brazil Márcia Federson Universidade de São Paulo São Carlos, SP, Brazil Jaqueline G. Mesquita Universidade de Brasília Brasília, DF, Brazil

3 Copyright

4 Dedication

5 List of Contributors

6 Foreword

7 Preface

8 1 Preliminaries 1.1 Regulated Functions 1.2 Functions of Bounded ‐Variation 1.3 Kurzweil and Henstock Vector Integrals Appendix 1.A: The McShane Integral

9 2 The Kurzweil Integral 2.1 The Main Background 2.2 Basic Properties 2.3 Notes on Kapitza Pendulum

10 3 Measure Functional Differential Equations 3.1 Measure FDEs 3.2 Impulsive Measure FDEs 3.3 Functional Dynamic Equations on Time Scales 3.4 Averaging Methods 3.5 Continuous Dependence on Time Scales

11 4 Generalized Ordinary Differential Equations 4.1 Fundamental Properties 4.2 Relations with Measure Differential Equations 4.3 Relations with Measure FDEs

12 5 Basic Properties of Solutions 5.1 Local Existence and Uniqueness of Solutions 5.2 Prolongation and Maximal Solutions

13 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

14 7 Continuous Dependence on Parameters 7.1 Basic Theory for Generalized ODEs 7.2 Applications to Measure FDEs

15 8 Stability Theory 8.1 Variational Stability for Generalized ODEs 8.2 Lyapunov Stability for Generalized ODEs 8.3 Lyapunov Stability for MDEs 8.4 Lyapunov Stability for Dynamic Equations on Time Scales 8.5 Regular Stability for Generalized ODEs

16 9 Periodicity 9.1 Periodic Solutions and Floquet's Theorem 9.2 ( θ, T )-Periodic Solutions

17 10 Averaging Principles 10.1 Periodic Averaging Principles 10.2 Nonperiodic Averaging Principles

18 11 Boundedness of Solutions 11.1 Bounded Solutions and Lyapunov Functionals 11.2 An Application to MDEs

19 12 Control Theory 12.1 Controllability and Observability 12.2 Applications to ODEs

20 13 Dichotomies 13.1 Basic Theory for Generalized ODEs 13.2 Boundedness and Dichotomies 13.3 Applications to MDEs 13.4 Applications to IDEs

21 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

22 15 Applications to Functional Differential Equations of Neutral Type 15.1 Drops of History 15.2 FDEs of Neutral Type with Finite Delay

23 References

24 List of Symbols

25 Index

26 End User License Agreement

1 Chapter 9Figure 9.1 Trajectory of .Figure 9.2 Trajectory of .

1 Cover

2 Table of Contents

3 Title Page Generalized Ordinary Differential Equations in Abstract Spaces and Applications Edited by Everaldo M. Bonotto Universidade de São Paulo São Carlos, SP, Brazil Márcia Federson Universidade de São Paulo São Carlos, SP, Brazil Jaqueline G. Mesquita Universidade de Brasília Brasília, DF, Brazil

4 Copyrigt

5 Dedication

6 List of Contributors

7 Foreword

8 Preface

9 Begin Reading

10 References

11 List of Symbols

12 Index

13 End User License Agreement

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