The Kamenev type interval oscillation criteria of mixed nonlinear impulsive differential equations under variable delay effects

Xiaoliang Zhou,Ruyun Chen,Changdong Liu

doi:10.1186/s13662-018-1931-1

Xiaoliang Zhou, Ruyun Chen + Show 1 more

Open Access

https://doi.org/10.1186/s13662-018-1931-1

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Abstract

In this paper, a class of mixed nonlinear impulsive differential equations is studied. When the delay sigma(t) is variable, each given interval is divided into two parts on which the quotients of x(t-sigma(t)) and x(t) are estimated. Then, by introducing binary auxiliary functions and using the Riccati transformation, several Kamenev type interval oscillation criteria are established. The well-known results obtained by Liu and Xu (Appl. Math. Comput. 215:283–291, 2009) for sigma(t)=0 and by Guo et al. (Abstr. Appl. Anal. 2012:351709, 2012) for sigma(t)=sigma_{0} (sigma_{0}geq0) are developed. Moreover, an example illustrating the effectiveness and non-emptiness of our results is also given.

Highlights

Unlike the methods of [22, 25], we introduce a binary auxiliary function, divide each given interval into two parts and estimate the quotients of x(t – σ (t)) and x(t)
2 Main results First, we define a functional space C–(I, R) as follows: C–(I, R) := y : I → R | I is a real interval, y is continuous on I \ {ti} and y ti– = y(ti), i ∈ N
For any given intervals [cj, dj] (j = 1, 2), we suppose that k(cj) < k(dj) (j = 1, 2), there exist impulse moments τk(cj)+1, . . . , τk(dj) in [cj, dj] (j = 1, 2) and we have the following cases to consider

Summary

Introduction

They used Riccati transformation and univariate ω functions to obtain some generalized interval oscillation results. 2 Main results First, we define a functional space C–(I, R) as follows: C–(I, R) := y : I → R | I is a real interval, y is continuous on I \ {ti} and y ti– = y(ti), i ∈ N . For any given intervals [cj, dj] (j = 1, 2), we suppose that k(cj) < k(dj) (j = 1, 2), there exist impulse moments τk(cj)+1, .

Results

Conclusion