Practical Stability of Differential Equations with State Dependent Delay and Non-instantaneous Impulses

Abstract

In this paper some practical stability results for nonlinear differential equations with non-instantaneous impulses and state dependent delays are presented. The impulses start abruptly at some points and their action continue on given finite intervals. The delay depends on both the time and the state variable which is a generalization of time variable delay. Some sufficient conditions for practical stability and strong practical stability are obtained by the help with the appropriate modification of Razumikhin method and an appropriate definition of the derivative of the Lyapunov function. Examples are given to illustrate the results.