Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations

by Agarwal, Ravi P.; Grace, Said R.; O'Regan, Donal

• ISBN13: 9781402008023
• ISBN10: 1402008023
• Format: Hardcover
• Copyright: 2002-08-01
• Publisher: Kluwer Academic Pub

Summary

In this monograph, the authors present a compact, thorough, systematic, and self-contained oscillation theory for linear, half-linear, superlinear, and sublinear second-order ordinary differential equations. An important feature of this monograph is the illustration of several results with examples of current interest. This book will stimulate further research into oscillation theory.This book is written at a graduate level, and is intended for university libraries, graduate students, and researchers working in the field of ordinary differential equations.

Table of Contents

Preface	xi
Preliminaries
Introduction	1	(1)
Continuability, Boundedness and Convergence to Zero	2	(8)
Some Useful Results from Analysis and Fixed Point Theorems	10	(2)
Notes and General Discussions	12	(1)
References	12	(1)
Oscillation and Nonoscillation of Linear Ordinary Differential Equations
Introduction	13	(1)
Sturm Comparison Theorem	13	(2)
Nonoscillatory Characterizations and Comparison Theorems	15	(10)
Nonoscillation Criteria	25	(10)
Oscillation Criteria	35	(5)
Oscillation Criteria-Integral Averaging	40	(30)
Oscillation Criteria-Integrable Coefficients	70	(12)
Forced Oscillations	82	(8)
Notes and General Discussions	90	(1)
References	91	(2)
Oscillation and Nonoscillation of Half-Linear Differential Equations
Introduction	93	(1)
Preliminaries	94	(1)
Sturm-Type Comparison Theorems	94	(6)
Levin-Type Comparison Theorems	100	(5)
Liapunov's Inequality	105	(2)
An Oscillation Criterion for Almost-Periodic Sturm-Liouville Equations	107	(6)
On Zeros of Solutions of a Singular Half-linear Differential Equation	113	(27)
Nonoscillation Criteria	140	(23)
Nonoscillatory Characterization	140	(7)
Comparison Theorems	147	(1)
Sufficient Conditions for Nonoscillation of Equation (3.1.1)	148	(6)
Further Results on Nonoscillation of Equation (3.1.1)	154	(9)
Oscillation Criteria	163	(5)
Oscillation Criteria-Integral Averaging	168	(28)
Oscillation Theorems for Equation (3.1.1)	168	(11)
Interval Criteria for the Oscillation of Equation (3.1.1)	179	(9)
The Weighted-Average Oscillation Criteria	188	(8)
Oscillation Criteria-Integrable Coefficients	196	(9)
Oscillation of Damped and Forced Equations	205	(5)
Oscillation of Equation (3.11.1)	206	(1)
Oscillation of Equation (3.11.2)	207	(3)
Distance of Zeros of Oscillatory Solutions	210	(8)
Oscillation and Nonoscillation of Half-linear Equations with Deviating Arguments	218	(49)
Some Useful Lemmas	219	(4)
Oscillation of Equation (3.13.1)	223	(9)
Some Extensions and More Oscillation Criteria	232	(5)
Classification of Nonoscillatory Solutions	237	(14)
A Comparison Theorem	251	(2)
Oscillation of Equation (3.13.2)	253	(6)
Nonoscillation of Solutions of Equation (3.13.127)	259	(8)
Notes and General Discussions	267	(1)
References	268	(5)
Oscillation Theory for Superlinear Differential Equations
Introduction	273	(1)
Superlinear Oscillation Criteria	273	(23)
Further Results on Superlinear Oscillations	296	(32)
Existence of Nonoscillatory Solutions and Comparison Results	328	(13)
A Necessary Condition for Oscillation	328	(4)
Comparison of Nonlinear Equations of the Same Form	332	(7)
Linearization of Oscillation Theorems	339	(2)
Oscillation Criteria for Certain Nonlinear Differential Equations	341	(17)
Superlinear Forced Oscillations	358	(4)
Oscillation of Nonlinear Differential Equations with Nonlinear Damping Term	362	(9)
Damping with Nonlinear Growth-Oscillation Criteria	362	(5)
Damping with Nonlinear Growth-A Nonoscillation Theorem	367	(4)
Notes and General Discussions	371	(1)
References	372	(4)
Oscillation Theory for Sublinear Differential Equations
Introduction	376	(1)
Sublinear Oscillation Criteria	376	(11)
Further Sublinear Oscillation Criteria	387	(42)
Linearization of Sublinear Oscillation Theorems	429	(4)
Nonoscillation Theorems for Sublinear Differential Equations	433	(11)
Oscillation Criteria for Certain Nonlinear Differential Equations	444	(9)
Notes and General Discussions	453	(1)
References	453	(4)
Further Results on the Oscillation of Differential Equations
Introduction	457	(1)
Oscillation Criteria of Olech-Opial-Wazewski Type	457	(9)
Some Definitions and Notation	458	(1)
Oscillation Criteria	459	(4)
Further Results	463	(3)
Oscillation Criteria for Half-linear Differential Equations	466	(10)
Preliminary Results	467	(2)
Oscillation Criteria	469	(7)
Oscillation Criteria via Liapunov's Second Method	476	(13)
Preliminaries	477	(2)
Oscillation Criteria	479	(10)
Notes and General Discussions	489	(1)
References	490	(1)
Oscillation Results for Differential Systems
Introduction	491	(1)
Oscillation of Nonlinear Two-dimensional Differential Systems	491	(12)
Superlinear Case	493	(5)
Sublinear Case	498	(2)
Further Oscillation Criteria	500	(3)
Oscillation Theorems for Linear Differential Systems	503	(30)
Oscillation Theorems for Differential Systems with Functionally Commutative Matrix Coefficients	533	(7)
Comparison Theorems for Operator-Valued Linear Differential Equations	540	(9)
Oscillation Results for Differential Systems with Forcing Terms	549	(3)
Notes and General Discussions	552	(2)
References	554	(4)
Asymptotic Behavior of Solutions of Certain Differential Equations
Introduction	558	(1)
Asymptotic Behavior of Solutions of Nonlinear Differential Equations	558	(26)
Asymptotic Behavior of Nonoscillatory Solutions	559	(3)
Bounded Asymptotically Linear Solutions	562	(5)
Unbounded Asymptotically Linear Solutions	567	(7)
Further Extensions and Improvements	574	(10)
Asymptotic Behavior of Positive Solutions of Singular Emden-Fowler-Type Equations	584	(38)
Positive Solutions of Equation (8.2.1)	585	(10)
Positive Decaying Solutions of Equation (8.2.2)	595	(19)
Positive Solutions of Singular Emden-Fowler-Type Systems	614	(8)
Notes and General Discussions	622	(2)
References	624	(1)
Miscellaneous Topics
Introduction	625	(1)
Comparison and Nonoscillation Results	625	(7)
Nonoscillatory Solutions of Forced Differential Equations	632	(6)
Limit Circle Criteria and Related Properties for Nonlinear Equations	638	(12)
Properties of Certain Differential Equations	650	(4)
Notes and General Discussions	654	(1)
References	654	(2)
Nonoscillation Theory for Multivalued Differential Equations
Introduction	656	(1)
Preliminaries	656	(1)
Differential Inclusions	657	(7)
Neutral Inclusions	664	(4)
Notes and General Discussions	668	(1)
References	668	(2)
Subject Index	670