Oscillation Criteria for Second‐Order Nonlinear Dynamic Equations on Time Scales

Shao-Yan Zhang,Qi-Ru Wang

doi:10.1155/2012/743469

Abstract

This paper is concerned with oscillation of second‐order nonlinear dynamic equations of the form on time scales. By using a generalized Riccati technique and integral averaging techniques, we establish new oscillation criteria which handle some cases not covered by known criteria.

Highlights

The theory of time scales was introduced by Stefan Hilger in his Ph.D. thesis in 1988 in order to unify continuous and discrete analysis
Let f : T → R be a function, f is called right-dense continuous rd-continuous if it is continuous at right-dense points in T and its left-sided limits exist finite at left-dense points in T
A function F : T → R is called an antiderivative of f provided FΔ t f t holds for all t ∈ Tk

Summary

Introduction

The theory of time scales was introduced by Stefan Hilger in his Ph.D. thesis in 1988 in order to unify continuous and discrete analysis. Can this theory of the so-called “dynamic equations” unify theories of differential equations and difference equations and extend these classical cases to cases “in between”, for example, to the so-called q-difference equations. In 2010, Sun et al 5 considered the second-order quasiliner neutral delay dynamic equation rtytptyτtΔγΔ q1xα τ1 t q2xβ τ2 t. 0, 1.3 on a time scale T, where p ∈ Crd T, 0, 1 , fi ∈ C T × R, R , i 1, 2, γ > 0 is a quotient of odd positive integers.

Preliminaries and Lemmas

Main Results

Examples