The qualitative theory of dynamic equations is a rapidly developing area of research. In the last 50 years, the Oscillation Theory of ordinary, functional, neutral, partial and impulsive differential equations, and their discrete versions, has inspired many scholars. Hundreds of research papers have been published in every major mathematical journal. Many books deal exclusively with the oscillation of solutions of differential equations, but most of these books appeal only to researchers who already know the subject. In an effort to bring Oscillation Theory to a new and broader audience, the authors present a compact, but thorough, understanding of Oscillation Theory for second order differential equations. They include several examples throughout the text not only to illustrate the theory, but also to provide new direction.

Preface

vii

Preliminaries

Introduction

1

(1)

Initial Value Problem, Oscillation and Nonoscillation

2

(1)

Continuability and Boundedness

3

(9)

Some Basic Results for Second Order Linear Ordinary Differential Equations

12

(15)

Some Useful Criteria for First Order Functional Differential Equations

27

(2)

Some Useful Results from Analysis and Fixed Point Theorems

29

(3)

Notes and General Discussions

32

(1)

References

33

(2)

Oscillation of Differential Equations with Deviating Arguments

Introduction

35

(1)

Oscillation Theorems (I)

36

(15)

Oscillation Theorems (II)

51

(13)

Comparison Theorems for Second Order Functional Differential Equations

64

(16)

Oscillation of Functional Equations with a Damping Term

80

(27)

Oscillation of Second Order Linear Delay Differential Equations

107

(9)

Oscillation of Forced Functional Differential Equations

116

(13)

Oscillation of Functional Equations with Damping and Forcing Terms

129

(7)

Necessary and Sufficient Conditions for the Oscillation of Forced Equations

136

(13)

Oscillation for Perturbed Differential Equations

149

(11)

Asymptotic Behavior of Oscillatory Solutions of Functional Equations

160

(11)

Notes and General Discussions

171

(5)

References

176

(9)

Oscillation of Neutral Functional Differential Equations

Introduction

185

(1)

Oscillation of Nonlinear Neutral Equations

186

(16)

Oscillation of Neutral Equations with Damping

202

(7)

Oscillation of Forced Neutral Equations

209

(11)

Oscillation of Neutral Equations of Mixed Type

220

(15)

Necessary and Sufficient Conditions for Oscillations of Neutral Equations with Deviating Arguments

235

(24)

Comparison and Linearized Oscillation Theorems for Neutral Equations

259

(7)

Existence of Nonoscillatory Solutions of Neutral Delay Differential Equations

266

(30)

Asymptotic Behavior of Nonoscillatory Solutions of Neutral Nonlinear Delay Differential Equations

296

(16)

Notes and General Discussions

312

(2)

References

314

(4)

Conjugacy and Nonoscillation for Second Order Differential Equations

Introduction

318

(1)

Conjugacy of Linear Second Order Ordinary Differential Equations

318

(15)

Nonoscillation Theorems

333

(8)

Integral Conditions and Nonoscillation

341

(7)

Notes and General Discussions

348

(4)

References

352

(3)

Oscillation of Impulsive Differential Equations

Introduction

355

(1)

Oscillation Criteria for Impulsive Delay Differential Equations

355

(19)

Oscillation of Second Order Linear Differential Equations with Impulses

374

(12)

Oscillation of Second Order Nonlinear Differential Equations with Impulses

386

(15)

Notes and General Discussions

401

(1)

References

401

(2)

Subject Index

403

What is included with this book?

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.