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

Abstract

In this paper, a class of mixed nonlinear impulsive differential equations is studied. When the delay σ (t) is variable, each given interval is divided into two parts on which the quotients of x(t-σ (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 σ (t) = 0 and by Guo et al. (Abstr. Appl. Anal. 2012:351709, 2012) for σ (t) = σ 0 (σ 0 ≥ 0) are developed. Moreover, an example illustrating the effectiveness and non-emptiness of our results is also given.