Oscillation of Second‐Order Nonlinear Delay Dynamic Equations on Time Scales

H A Agwa,A M M Khodier,Heba A Hassan

doi:10.1155/2011/863801

Abstract

In this work, we use the generalized Riccati transformation and the inequality technique to establish some new oscillation criteria for the second‐order nonlinear delay dynamic equation , on a time scale 𝕋, where γ is the quotient of odd positive integers and p(t) and q(t) are positive right‐dense continuous (rd‐continuous) functions on 𝕋. Our results improve and extend some results established by Sun et al. 2009. Also our results unify the oscillation of the second‐order nonlinear delay differential equation and the second‐order nonlinear delay difference equation. Finally, we give some examples to illustrate our main results.

Highlights

The theory of time scales was introduced by Hilger 1 in order to unify, extend, and generalize ideas from discrete calculus, quantum calculus, and continuous calculus to arbitrary time scale calculus
Many authors have expounded on various aspects of this new theory, see 2–4
We deal with the oscillation behavior of all solutions of the second-order nonlinear delay dynamic equation p t xΔ tγΔqtfxτt

Summary

Introduction

The theory of time scales was introduced by Hilger 1 in order to unify, extend, and generalize ideas from discrete calculus, quantum calculus, and continuous calculus to arbitrary time scale calculus. The new theory of the so-called “dynamic equation” unify the theories of differential equations and difference equations, and extends these classical cases to the so-called q-difference equations when T qN0 : {qt : t ∈ N0 for q > 1} or T qZ qZ ∪ {0} which have important applications in quantum theory see 5 It can be applied on different types of time scales like T hZ, T N20, and the space of the harmonic numbers T Tn. In the last two decades, there has been increasing interest in obtaining sufficient conditions for oscillation nonoscillation of the solutions of International Journal of Differential Equations different classes of dynamic equations on time scales, see 6–9. In 2007, Erbe et al considered the nonlinear delay dynamic equation 1.4 and obtained some new oscillation criteria which improve the results of Sahiner.

Some Preliminaries on Time Scales

Several Lemmas

Main Results

Examples