Stability Analysis For Impulsive Integro-differential Systems With Variable Times

In this paper, we consider the stability properties about the following impulsive integro-differential equations with variable times where f(t,x, Tx)=F(t,x)+R(t,x,Tx), Tx=(?) L(t,s,x(s))ds,L∈C(R+2×Rn,Rn). Every solu-tion of it meet each hypersurface in turn exactly once.Impulsive integro-differential system have extensive practical background and application in the natural sciences,as the systems extensively occur in the mathematical modeling of cir-cuit simulation in physics and neuronal networks in biology.The research of impulsive integro-differential system has aroused expert's interest and attention,and various interesting results have been obtained in the past years, especially for impulsive integro-differential system with fixed times.For example,article[1,3,4]studied boundedness of solutions of this system and got some direct results.However, impulsive integro-differential equations with variable times as an extension of systems with fixed times have more application.But the theory of impulsive integro-differential systems with variable times is relatively less developed due to the difficulties created by the phenomena of beating and bifurcation etc.Up to now, the results about impul-sive systems with variable times are really few. In the findings appeared,article [10] gave one existence result of solutions,and article [11] gave several criteria for asymptotic stability of this system from which the impulsive effects on the stability are displayed in the result ob-tained. However, the study of stability of this system is in underway phase,and there are many problems which are not solved.Therefore,we have a large number of work to do.In this paper we study the properties of stability of impulsive integro-differential system with variable times,and we get some new results.In chapter one,using the method of cone-valued Lyapunov functions and comparison prin- ciple,we discuss the property of impulsive integro-differential system with variable times.we firstly give the definition of the cone,and furthermore we introduce the conception of cone-valued Lyapunov functions and its derivative along the solution of system (Ⅰ).In section 3 of this chap-ter, we established a new comparison principle by comparing with a scalar differential system on cone.Then base on the comparison theorem,we get several stability criteria in terms of two measures for system (Ⅰ) in section 4.In chapter two,by employing the method of cone-valued Lyapunov functions,variational Lyapunov functions and comparison principle,several stability criteria for the impulsive per-turbed integro-differential systems (Ⅰ) are established through the corresponding unperturbed differential systems and the comparison systems.In section 2 of this chapter,we introduce the definition of (h0,h)-stability.In section 3,the cone-valued variational comparison principle of system (Ⅰ) is obtained by cone-valued variational Lyapunov functions.In corollaries,by taking the relevant function of comparison theorem,the solution of the perturbed system (Ⅰ) and unper-turbed one are connected through the maximal solutions of the comparison system.Then on the base of comparison theorem,we get the comparison criteria of stability in terms of two measures of system (Ⅰ).At the same time,an example is given to show the effective of the theorems.In chapter three,by employing the variational Lyapunov method and Razumikhin technique which are used in the study of impulsive functional differential systems,a Razumikhin type of the variational Lyapunov method is obtained.By using above method,several stability criteria in terms of two measures of system (Ⅰ) are obtained,in these stability and instability criterion we weakens the condition of Lyapunov functions at impulsive point.One example is given to illus-trate the advantages of them as well.