Oscillation criteria for certain third-order delay dynamic equations

Jiashan Yang

doi:10.1186/1687-1847-2013-178

Abstract

This paper is concerned with the oscillatory behavior of a certain class of third-order nonlinear variable delay neutral functional dynamic equations,

Highlights

Consider a third-order nonlinear variable delay dynamic equation r(t)φ a(t)y (t)+ P(t)F φ x δ(t) =, t ∈ T, t ≥ t, ( . )where y(t) = x(t) + B(t)g(x(τ (t))), φ(u) = |u|λ– u, λ ≥
The purpose of this article is to obtain new oscillation criteria for the oscillation of ( . ), these criteria can improve the restriction of the conditions for the equation, which promote some existing results
We should note that many of our results of this article are new for the corresponding third-order nonlinear differential and difference equations

Summary

Introduction

Consider a third-order nonlinear variable delay dynamic equation r(t)φ a(t)y (t). where y(t) = x(t) + B(t)g(x(τ (t))), φ(u) = |u|λ– u, λ ≥. Obtained the result that every solution of equation ), these criteria can improve the restriction of the conditions for the equation, which promote some existing results. We should note that many of our results of this article are new for the corresponding third-order nonlinear differential and difference equations. Let x(t) be an eventually position solution of equation Every solution x(t) of equation is either oscillatory or limt→+∞ x(t) =. We can obtain different conditions for oscillation of all solutions of Kamenev-type oscillation criteria for second-order linear differential equation was extended to third-order nonlinear variable delay dynamic equations on time scales.

Now we claim that

This implies