Oscillation of second-order nonlinear neutral dynamic equations with distributed deviating arguments on time scales

Shao-Yan Zhang,Qi-Ru Wang

doi:10.1186/s13662-014-0337-y

Abstract

This paper concerns second-order nonlinear neutral dynamic equations with distributed deviating arguments on time scales of the form $$\bigl(r(t) \bigl(\bigl(y(t)+p(t)y\bigl(\tau(t)\bigr)\bigr)^{\Delta}\bigr)^{\gamma}\bigr)^{\Delta}+\int_{a}^{b}f \bigl(t,y\bigl(\delta (t,\xi)\bigr)\bigr)\Delta\xi=0, $$ where $\gamma>0$ is a quotient of odd positive integers. By using the generalized Riccati technique and integral averaging techniques, we derive new oscillation criteria for the above equations, which generalize and improve some existing results in the literature.

Highlights

In this paper, we consider second-order nonlinear neutral dynamic equations with distributed deviating arguments of the following form:b r(t) y(t) + p(t)y τ (t) γ + f t, y δ(t, ξ ) ξ = ( . )a on a time scale T satisfying inf T = t and sup T = ∞
In Section, we present the main results
2 Some preliminaries we present several technical lemmas which will be used in the proofs of the main results

Summary

Introduction

We consider second-order nonlinear neutral dynamic equations with distributed deviating arguments of the following form:. Oscillation of some second-order nonlinear delay dynamic equations on time scales has been discussed; see [ – ] and the references therein. Proof Since y(t) is an eventually positive solution of Eq T ∈ T, one of the following conditions is satisfied:. We may assume that y(t) is eventually positive. Integrating the above inequality from T to t for t ≥ T, we get t T s ≤ w(T) – w(t) < w(T) Taking limsup on both sides as t → ∞, we obtain a contradiction to condition (b).

Then w

By integration by parts we obtain t

Ts ss