Oscillation of higher order nonlinear dynamic equations on time scales

Said R Grace,Ağacık Zafer,Ravi P Agarwal

doi:10.1186/1687-1847-2012-67

Abstract

Some new criteria for the oscillation of n th order nonlinear dynamic equations of the form

Highlights

Consider the nth order nonlinear delay dynamic equation x n (t) + q (t) xσ (ξ (t)) λ = 0 (1:1)on an arbitrary time-scale T ⊆ R with sup T = ∞ and 0 ∈ T, where n ≥ 2 is a positive integer, l is the ratio of positive odd integers, q : T → R+ = (0, ∞) and ξ : T → T are real-valued rd-continuous functions, ξ(t) ≤ t, ξΔ(t) ≥ 0, and limt®∞ξ(t) = ∞
On an arbitrary time-scale T ⊆ R with sup T = ∞ and 0 ∈ T, where n ≥ 2 is a positive integer, l is the ratio of positive odd integers, q : T → R+ = (0, ∞) and ξ : T → T are real-valued rd-continuous functions, ξ(t) ≤ t, ξΔ(t) ≥ 0, and limt®∞ξ(t) = ∞
There are very few results regarding the oscillation of higher order equations

Summary

Introduction

There has been an increasing interest in studying the oscillatory behavior of first-and second-order dynamic equations on time-scales, see [1,2,3,4,5,6,7]. The purpose of this article is to obtain new criteria for the oscillation of Equation (1.1). This topic is fairly new for dynamic equations on time scales. The article is organized as follows: In Section 2, some preliminary lemmas and notations are given, while Section 3 is devoted to the study of Equation (1.1) via comparison with a set of second-order dynamic equations whose oscillatory character is. Let x(t) be an eventually positive solution of inequality (2.3). ≥ h −2 (t, σ (τ )) x −1 (τ ) τ for > 1

We claim that x

Define w η