Oscillation of impulsive functional differential equations with oscillatory potentials and Riemann-Stieltjes integrals

Zhi Liu,Yuangong Sun

doi:10.1186/1687-1847-2012-175

Abstract

This paper addresses the oscillation problem of a class of impulsive differential equations with delays and Riemann-Stieltjes integrals that cover many equations in the literature. In the case of oscillatory potentials, both El-Sayed type and Kamenev type oscillation criteria are established by overcoming the difficulty caused by impulses and oscillatory potentials in the estimation of the delayed argument. The main results not only generalize some existing results but also drop a restrictive condition imposed on impulse constants. Finally, two examples are presented to illustrate the theoretical results. MSC:34K11.

Highlights

Recent years have witnessed a rapid progress in the theory of impulsive differential equations which provide a natural description of the motion of several real world processes subject to short time perturbations
We are here concerned with the oscillation problem of impulsive functional differential equations
We investigate the oscillation of the following impulsive differential equation with delay and Riemann-Stieltjes integral:

Summary

Introduction

Recent years have witnessed a rapid progress in the theory of impulsive differential equations which provide a natural description of the motion of several real world processes subject to short time perturbations. We investigate the oscillation of the following impulsive differential equation with delay and Riemann-Stieltjes integral:. Some oscillation results for the second-order forced mixed nonlinear impulsive differential equation were established in [ , ]. We will overcome difficulties caused by oscillatory potentials, delayed argument and impulses, and establish both El-Sayed type and Kamenev type interval oscillation criteria for Eq ( ). The redundant restriction on impulse constants ck and dk that dk ≥ ck is removed by introducing particular El-Sayed type functions in [ ] and using Kong’s technique in [ ] many times based on the number of impulse moments in a bounded interval Both impulse, delay and Riemann-Stieltjes integral are taken into consideration in this paper. Multiplying both sides of ( ) by H(t, a ), integrating it from a to tφ(a ), and using integration by parts, we obtain

Dividing and by

For any