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

Abstract

Abstract In this paper, a class of mixed nonlinear impulsive differential equations is studied. When the delay σ(t) $\sigma(t)$ is variable, each given interval is divided into two parts on which the quotients of x(t−σ(t)) $x(t-\sigma(t))$ and x(t) $x(t)$ 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 $\sigma(t)=0$ and by Guo et al. (Abstr. Appl. Anal. 2012:351709, 2012) for σ(t)=σ0 $\sigma(t)=\sigma_{0}$ ( σ0≥0 $\sigma_{0}\geq0$) are developed. Moreover, an example illustrating the effectiveness and non-emptiness of our results is also given.

Keywords

• Interval oscillation
• Impulsive differential equation
• Variable delay
• Interval delay function