On the Oscillation for Second-Order Half-Linear Neutral Delay Dynamic Equations on Time Scales

Quanxin Zhang,Xia Song

doi:10.1155/2014/321764

Abstract

We discuss oscillation criteria for second-order half-linear neutral delay dynamic equations on time scales by using the generalized Riccati transformation and the inequality technique. Under certain conditions, we establish four new oscillation criteria. Our results in this paper are new even for the cases of𝕋=ℝand𝕋=ℤ.

Highlights

In recent years, the research results relevant to oscillation of second-order dynamic equations on time scales are emerging, such as [1,2,3,4,5,6,7]
The derivative is defined by fΔ
A function f : T → R is said to be rd-continuous if it is continuous at each right-dense point and if there exists a finite left limit at all left-dense points

Summary

Introduction

The research results relevant to oscillation of second-order dynamic equations on time scales are emerging, such as [1,2,3,4,5,6,7]. On the basis of the above work, we will study the oscillatory behavior of all solutions of second-order half-linear neutral delay dynamic equation in this paper, which is given as follows:. T0 a (t) the two famous results of Philos [24] about oscillation of second-order linear differential equations are extended to (1) in this paper. We establish four new oscillatory criteria when the condition (4) or (5) holds, respectively, for the solutions of (1)

Some Preliminaries

Several Lemmas

Main Results