A Philos-type theorem for second-order neutral delay dynamic equations with damping

Jiashan Yang,Tongxing Li

doi:10.1186/s13662-016-0767-9

Abstract

We establish a Philos-type oscillation theorem for a class of nonlinear second-order neutral delay dynamic equations with damping on a time scale by using the Riccati transformation and integral averaging technique. An illustrative example is provided to show that our theorem has practicability and maneuverability.

Highlights

Oscillation, as a kind of physical phenomena, widely exists in the natural sciences and engineering
The assorted phenomena can be unified into oscillation theory of equations; see [ ]
On the basis of these background details, we are concerned with the oscillation of a nonlinear second-order damped delay dynamic equation of neutral type

Summary

Introduction

Oscillation, as a kind of physical phenomena, widely exists in the natural sciences and engineering. There has been a great deal of interest in studying oscillatory behavior of solutions to dynamic equations on various classes of time scales; see, for example, [ – ] and the references therein. ), whereas Bohner and Li [ ] and Erbe et al [ ] considered oscillation of solutions to a nonlinear second-order dynamic equation. Agarwal et al [ ] and Zhang et al [ ] established several oscillation results for a second-order linear neutral dynamic equation r(t) x(t) + p(t)x τ (t). It should be noted that the topic of this paper is new for dynamic equations on time scales due to the fact that the results reported in [ , , , , , – , , , ] cannot be applied to a general equation

Define the function W by

Define another function V by