The Kamenev type oscillation criteria of nonlinear differential equations under impulse effects

Xiaoliang Zhou,Wu-Sheng Wang

doi:10.1186/s13662-016-0911-6

Abstract

In this paper, a general class of second-order nonlinear damped differential equations under impulse effects is studied. By introducing an auxiliary function of two variables and using Riccati transformation, several Kamenev type interval oscillation criteria are established, which generalize and extend some known ones, such as those of Huang and Feng. Moreover, examples are given to illustrate the effectiveness and non-emptiness of our results.

Highlights

Introduction and preliminariesIn, Rogovchenko and Rogovchenko [ ] first studied the oscillation of the nonlinear second-order differential equation with nonlinear damping of the form r(t)k x(t), x (t) + p(t)k x(t), x (t) x (t) + q(t)h x(t) = . ( . )Later, Tiryaki and Zafer [ ], Zhao and Feng [ ], Zhao et al [ ] and Huang and Meng [ ] obtained several oscillation criteria for solutions of ( . ), which extended and improved the results in [ ]
Our methods are different from those of Özbekler and Zafer [ ] and Li et al [ ] because we use an auxiliary function of two variables and divide the considered interval into two parts to study oscillation of ( . )
In many published papers on the subject of interval oscillation criteria, the authors have insisted on the use of the function ρ(t) as a multiplier in the Riccati transformation ( . )

Summary

Introduction

). Later, some interval oscillation criteria were given by Shi [ ] and Huang and Meng [ ] for a forced equation of the form r(t)k x(t), x (t) + p(t)k x(t), x (t) x (t) + q(t)h x(t) = f (t). Interval oscillation of impulsive differential equations aroused the interest of many researchers; see, for example, [ – ].

Results

Conclusion