Oscillation of second-order damped differential equations

Xiaoling Fu,Tongxing Li,Chenghui Zhang

doi:10.1186/1687-1847-2013-326

Abstract

AbstractWe study oscillatory behavior of a class of second-order differential equations with damping under the assumptions that allow applications to retarded and advanced differential equations. New theorems extend and improve the results in the literature. Illustrative examples are given.MSC:34C10, 34K11.

Highlights

1 Introduction This paper is concerned with oscillation of solutions to a second-order differential equation with damping r(t)x (t) + p(t)x (t) + q(t)f x τ (t) =, ( . )
The questions regarding the study of oscillatory properties of differential equations with damping or distributed deviating arguments have become an important area of research due to the fact that such equations arise in many real life problems; see the research papers [ – ] and the references cited therein
Secondorder damped differential equations are used in the study of NVH of vehicles

Summary

Introduction

This paper is concerned with oscillation of solutions to a second-order differential equation with damping r(t)x (t) + p(t)x (t) + q(t)f x τ (t) = ,. Yan [ ] established an important extension of the celebrated Kamenev oscillation criterion [ ] for a second-order damped equation r(t)x (t) + p(t)x (t) + q(t)x(t) =. Rogovchenko [ ] and Rogovchenko and Tuncay [ ] studied a nonlinear damped equation r(t)x (t) + p(t)x (t) + q(t)f x(t) =. Rogovchenko and Tuncay [ ] extended the results of [ ] to a general nonlinear damped equation r(t)ψ x(t) x (t) + p(t)x (t) + q(t)f x(t) =. Proof Let x be a nonoscillatory solution of ). Without loss of generality, we may assume that there exists T ≥ t such that x(t) > and x(τ (t)) > for all t ≥ T.

Applying the inequality

Assume now that

Now define