Stability Study For Impulsive Functional Differential Systems With P-delay

In this paper, we study stability and boundedness for impulsive functional differential systems with p-delay as followsImpulsive functional differential systems with p-delay are very important impulsive functional differential systems which contain many impulsive functional differential systems with finite delay as well as with unbounded delay and have extensive applications in nature science, therefore it is important to be studied. In the last two years, the basic theory on impulsive functional differential systems with p-delay is just established . but the properties of its solutions are seldom studied. Therefore, we still have much work to do. As we all know, it. is an effetive tool for Lyapunov functions coupled with Razu-mikhin technique to investigate the stability for impulsive functional differential systems. It can guarantee the stability under less restrictive conditions which can be applied more conveniently. In addition, a new approach is introduced in [22], that is, the stability for functional differential systems can be investigated by the method of several Lyapunov functions containing partial components of x. where every Lyapunov function satisfies weaker conditions and is easier to be constructed. Based on the ideas above, we study the stability and the boundedness for system (1). This paper is divided into two chapters.In chapter one, we first introduce the conception of the p-function. Then we give one comparison Lemma on Lyapunov function from which we get the comparison criteria on stability and practical stability in terms of two measures of system (1). And we also gain the direct, results of uniform stability, uniformly asymptotic stability and practical stability in terms of two measures by the method of Lyapunov functions and Razumikhin technique. At the same time, an example is given to show the effectiveness of the theorems. At last, we establish the criteria on (h₀, h)-uniformly strongly practical stabilityby two Lyapunov functions which satisfy less restrictive conditions. The results in this chapter improve and generalize some of the earlier foundings of IFDE with finite delay and infinite delay, therefore the applications are more extensive.In chapter two, we study the boundedness of system (1) mainly by the method of several Lyapunov functions containing partial components coupled with Razumikhin technique. The derivative of V function along trajectories of system (1) is no longer required to be nonpositive or negative definite and can be weakened to be positive which can be applied more conveniently. Different from earlier results, we generalize the method of several Lyapunov functions containing partial components to two measures and set up the [h0j. hj)— boundedness theorems on partial components which are seldem known at present. Finally we give an example to illustrate the applications of our results.