The qualitative theory of dynamic equations is a rapidly developing area of research. In the last 50 years, many scholars have studied the oscillation theory of ordinary, functional, neutral, partial, and impulsive differential equations. Many books deal with oscillation theory, but in a way that appeals only to researchers already familiar with the subject. In an effort to bring the topic to a new and broader audience, the authors clearly explain oscillation theory for second-order differential equations. They include several examples to illustrate the theory and to inspire new direction. This text is ideal for students and researchers in applied mathematics, engineering science, and numerical analysis.

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

Preliminaries. Introduction. Initial Value Problem, Oscillation and Nonoscillation. Continuability and Boundedness. Some Basic Results for Second Order Linear Ordinary Differential Equations. Some Useful Criteria for First Order. Some Useful Results from Analysis and Fixed Point Theorems. Notes and General Discussions. References. Oscillations of Differential Equations with Deviating Arguments. Oscillation Theorems (I). Oscillation Theorems (II). Comparison Theorems for Second Order Functional Differential Equations. Oscillation of Functional Equations with a Damping Term. Oscillation of Second Order Linear Delay Differential Equations. Oscillation of Forced Functional Differential Equations. Oscillation of Functional Equations with Damping and Forcing Terms. Necessary and Sufficient Conditions for the Oscillation of Forced Equations. Oscillation for Perturbed Differential Equations. Asymptotic Behavior of Oscillatory Solutions of Functional Equations. Notes and General Discussions. References. Oscillation of Neutral Functional Differential Equations. Oscillation of Nonlinear Neutral Equations. Oscillation of Neutral Equations with Damping. Oscillation of Forced Neutral Equations. Oscillation of Neutral Equations with Mixed Type. Necessary and Sufficient Conditions for Oscillations of Neutral Equations with Deviating Arguments. Comparison and Linearized Oscillation Theorems for Neutral Equations. Existence of Nonoscillatory Solutions of Neutral Delay Differential Equations. Asymptotic Behavior of Nonoscillatory Solutions of Neutral Nonlinear Delay Differential Equations. Notes and General Discussions. References. Conjugacy and Nonoscillation for Second Order Differential Equations. Conjugacy of Linear Second Order Ordinary Differential Equations. Nonoscillation Theorems. Integral Conditions and Nonoscillations. Notes and General Discussions. References. Oscillation of Impulsive Differential Equations. Oscillation Criteria for Impulsive Delay Differential Equations. Oscillation of Second Order Linear Differential Equations with Impulses. Notes and General Discussions. References. Subject Index with Deviating Arguments. Oscillation of Neutral Functional Differential Equations. Conjugacy and Nonoscillation for Second Order Differential Equations. Oscillation of Impulsive Differential Equations.