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

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.