Interval oscillation criteria for nonlinear impulsive differential equations with variable delay

Abstract

In this paper, the interval qualitative properties of a class of second order nonlinear differential equations are studied. For the hypothesis of delay being variable $\tau(t)$, an "interval delay function" is introduced to estimate the ratio of functions $x(t-\tau(t))$ and $x(t)$ on each considered interval, then Riccati transformation and $H$ functions are applied to obtain interval oscillation criteria. The known results gained by Huang and Feng [Comput. Math. Appl. 59(2010), 18–30] under the assumption of constant delay $\tau$ are developed. Moreover, examples are also given to illustrate the effectiveness and non-emptiness of our results.

Keywords

• interval oscillation
• impulsive
• variable delay
• interval delay function