Oscillation and Nonoscillation of Half-Linear Differential Equations

Abstract

In this chapter we shall present oscillation and nonoscillation criteria for second order half-linear differential equations. In recent years these equations have attracted considerable attention. This is largely due to the fact that half-linear differential equations occur in a variety of real world problems; moreover, these are the natural generalizations of second order linear differential equations. In Section 3.1, we shall provide some preliminaries for the study of half-linear differential equations. In Sections 3.2 and 3.3, respectively, Sturm’s and Levin’s type comparison theorems are developed. In Section 3.4, we shall establish a Liapunov type inequality. Section 3.5 presents an oscillation criterion for almost periodic Sturm-Liouville equations. A systematic study on the zeros of solutions of singular half-linear equations is made in Section 3.6. Nonoscillation characterizations (necessary and sufficient conditions), comparison results as well as several sufficient criteria for the nonoscillation are presented in Section 3.7. Section 3.8 is devoted to the study of oscillation of half-linear equations. In Section 3.9, we shall establish oscillation criteria by employing integral and weighted averaging techniques. Here, interval criteria for the oscillation of half-linear equations are also provided. Section 3.10 deals with the oscillation of half-linear equations with integrable coefficients. Section 3.11 addresses the oscillation of damped and forced equations. In Section 3.12, we shall derive lower bounds for the distance between consecutive zeros of an oscillatory solution. Finally, in Section 3.13, we shall present a systematic study of the oscillation and nonoscillation of half-linear equations with a deviating argument. Here, classifications of the nonoscillatory solutions, and the existence results which guarantee that the solutions have prescribed asymptotic behavior are also presented.