Asian Resonance Uniform Asymptotic Stability of Functional Differential Equations with Infinite Delay

S Srivastava, Neha Wadhwa, Neeti Bhandari
unpublished
Introduction Impulsive differential equations arise naturally from a wide variety of applications such as aircraft control, inspection process in operations research and threshold theory in biology. Significant progress has been made in the theory of differential equations in recent years. But still there are number of difficulties one may face in developing the corresponding theory of impulsive delay differential equations. For example in the classical theory of delay differential equations,
more » ... e fact that the continuity of a function in implies the continuity of the functional in , plays a key role in establishing the existence of solution of delay differential equations [1]. However if the function is piecewise continuous, then the functional need not be piecewise continuous. In fact, it can be discontinuous everywhere. Existence and uniqueness results for impulsive delay differential equations have been presented in [2]. In [6,7], by using Lyapunov functions and Razumikhin techniques, some Razumikhin type theorems on stability are obtained for a class of impulsive functional differential equations with finite delay. However as pointed out in [8-10] even though for functional differential equations without impulses, stability results established for equations with finite delay are not obviously true in general for infinite delays. The common and main difficulty is that the interval is not compact and the images of a solution map of closed and bounded sets in , space may not be compact. Same situation arises in , space for impulsive differential equation with infinite delay. Recall that the stability theory of impulsive differential equations with infinite delays had received much attention in the literature [11-16].Here we extend the result develop in [3] to study infinite delay differential equations. Aim of the Study The purpose of present paper is to establish some criteria on uniform asymptotic stability for impulsive differential equations with infinite delay using Lyapunov functions and Razumikhin techniques. Review of Literature In past years there have been intensive studies on the stability of Impulsive Differential equations.In 1991, M. ramamohana Rao, investigates sufficient condition for uniform stability and uniform asymptotic stability of impulsive integro differential equations by constructing suitable piecewise continuous Lyapunov-like functionals without the decresent property. M.U.Akhmet, investigate the sufficient criteria for stability, asymptotic stability and instability for non-trivial solutions of the impulsive systems by Lyapunov's second method.Jianhua Shen and Jianli Li, investigates the sufficient criteria on asymptotic stability for system of volterra functional differential equations with nonlinear impulsive perturbations using Lyapunov like functions with Razumikhin technique or Lyapunov like functional. Abstract In this paper criteria on uniform asymptotic stability is established for impulsive functional differential equations with infinite delay. It is shown that certain impulsive perturbations may make unstable systems uniformly stable, even uniformly asymptotically stable.
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