Qualitative Analysis Of Impulsive Integro-Diferential Systems

In this paper,we mainly study the following impulsive integro-diferential systems.Where Tx=f_t0^tW （t, s, x（s））ds, W : R₊²Ã—Rⁿâ†'Rⁿ.Qua an important embranchment of nonlinear impulsive diferential systems[1], im-pulsive Integro-Diferential systems have extensive applications in nature-science. Forexample,mathematic models of circuit simulation in physics and neuronal networks inbiology remain with impulsive diferential systems to analyze and discuss. So it is valu-able to be studied and also attracts many experts’attention and interesting[2 7]. Inthe process of the study of this system, article got the comparison criteria of stabilityof the trivial solution of impulsive integro-diferential systems, and article[2-4,6] studiedboundedness of solutions of this system and got some direct results. However, the studyof stability of this system is in underway phase, and there are many problems which arenot solved, Therefore, we have a large number of work to do. In this paper we study theproperties of stability and boundedness of the systems, and we get some new results.As everyone knows,Lyapunov’s second method plays an important role while study-ing the stability of impulsive diferential systems,in past research,people usually makeuse of one order of derivatives of Lyapunov function to discuss various kinds of natureof the impulsive diferential systems,and always to setting up the independently con-dition on continuous portion and dispersed portion of the systems.But,the article [11]proposed one new method,the derivative of Lyapunov function alone the system railline no longer confine to negative or definitely negative,and allow the continuous partof Lyaunov function alone the systems rail line increase progressively,or after jumpingincreasing in pulse,but this thought,they defined the generalized second order derivativesof Lyapunov function.they meet under the ground prerequisite of certain terms the gen-eralized second order derivatives of mixing on continuous portion and dispersed portionand dispersed portion and dispersed portion of the systems to estimate synthetically.We call this kind of method the method of generalized second order derivatives of Lyapunovfunction briefly here.While using this method,we needn’t consider the symbol questionsof one order of derivatives.So,when the symbols of one order of derivative for Lyapunovfunction are uncertain,and the generalized second order of derivatives of system exist andthe symbol is confirmed,use this method study impulsive diferential system very muchefvetive.In recent years,the article of stability of impulsive diferential system employinggeneralized second order derivative method have been many,but the article of applierthis method research stability and boundedness of impulsive Integro-Diferential systemsis very rare,therefore there is a lot of work to be done . This method is applied to studythe stability and boundedness of impulsive Integro-Diferential systems in this article.This paper is divided into three parts. In chapter one,we mainly introduce the re-search background and practical example of the impulsive diferential systems, especiallyimpulsive integro-diferential system , and show the generalized second order derivativesthat the method, analyzes the general research situation of the generalized second or-der derivatives,and show the feasibility and superiority of the generalized second orderderivative method in impulsive integro-diferential system.In chapter two,we have studied the stability of the impulsive Integro-Diferentialsystems firstly.As everyone knows§the Lyapunov function method and unifies the Razu-mikhin skill is one kind of efective tool by studying the stability of the impulsive function-al diferential system.In this chapter,we mainly use the generalized second order derivativemethod to study the stability of the trivial solution of the system by the idea of usingLyapunov functions coupled with Razumkhin technique which are used in the study ofimpulsive functional diferential systems,and under the stability theory of the systems intwo measures foundation,we then utilizing method of generalized second order derivativeto study stability in terms of two measures of systems （1）.Introduce function in a certainblock or it’s disconnected there is concept that circles increase while studying the stabil-ity among them,it has limited the growth of Lyapunov function.In chapter three,we mainly study the boundedness of impulsive Integro-Diferentialsystems,which is still using the generalized second derivative method with Lyapunovfunction and Razumikhin skills,and gives the boundedness theorem and uniform bound-ness theorem of the trivial solution of impulsive Integro-Diferential systems and givesthe proof.And under the boundedness theory of impulsive Integro-Diferential system-s in two measures foundation,we get the boundedness theorem and uniform boundnesstheorem in terms of two measures of impulsive Integro-Diferential systems by using thethe generalized second derivative method,and finally illustrates the usefulness of theorem.