Qualitative Analysis For Impulsive Integro-differential Systems With Variable Times

In this paper,we consider the stability and boundedness properties about the following impulsive Integro-differential equations with variable times (?)where(?)?R~n,Z_+is the positive integer.Every solu-tion of it meet each hypersurface in turn finite times and the initial time to doesn't touch any a hypersurface.As an important embranchment of nonlinear impulsive differential systems,impulsive Integro-differential systems have extensive practical applications in nature science. For example.in physics.the use of circuit simulation;in biology.neuronal networks and the control of pest invasion speed;In medicine,The diseases spread through the source and so on.These mathematic models can all remain with impulsive Integro-differential sys-tems to analyze and research.So it has important application value.Now various impor-tant results of stability and boundedness of the systems have been obtained[1-12,15-23].For example,article[1-5] studied boundedness of solution of the systems and got some direct results,and article[1] has established the comparison results of stability for the systems(I).Because impulsive Integro-differential systems with variable times depend on the state of system,their movement is very complicated.It is more hard to study it,so their research progress slowly.Up to now,the results about impulsive systems with vari-able times are really few[10-12].Among them,article[10] gave one existence result of solu-tion,and article[11]gave several criterias for asymptotic stability of this system from which the impulsive effects on the stability are displayed in the result obtained.However,the re-search of the systems(I) is still not perfect and there are many problems which are not solved,therefore,we have a large number of work to do.In this paper,we focus research work on the boundedness and stability for impulsive Integro-differential equation with variable times using several Lyapunov functions including partial components coupled with the Razumikhin technique,and we get some new results.This paper is divided into three parts.In chapter one,we discuss impulsive Integro-differential equation with variable times using the method of several Lyapunov functions including partial components coupled with the Razumikhin technique.The uniform boundedness theorem and uniform ulti-mately boundedness theorem haved beed get.And an example is given to verify the validity of the results.In previous research on the boundedness of the systems,people usually use Lyapunov functions and the Razumikhin technique.so we must choose to the right function P and put all the components of x in the same function V.Because it is hard to select the appropriate function P and the conditions of the Lyapunov functions are strong,these greatly reduce the superiority of Razumikhin-type theorems.In view of this,the method of I adopted is different from [11,12],but it have taken the thought of [13,14],avoiding using the auxiliary function P meanwhile can use several Lyapunov functions including partial components.In this paper,the results significantly improve the known results.On the one hand,it is more easy to use;On the other hand,there are fewer restrictions about the stability and boundedness.In chapter two,we also discuss impulsive Integro-differential equation with variable times using the method of several Lyapunov functions including partial components cou-pled with the Razumikhin technique.The uniform stability and asymptotical stability theorems haved beed get.And an example is given to verify the validity of the results. In the latest research results,a list of constants dk are used to limit the conditions of the function V in impulsive time. In this chapter,I use a list of functions ?k which belong to function ?1 to limit the conditions of function V in impulsive time.it is more general.In chapter three,we discuss the stability of the systems(I) in terms of two measures directly by the idea of using Lyapunov functions coupled with Razumikhin technique which are used in the study of impulsive functional differential systems and get sev-eral criterions of stability and asymptotical stability of the systems.In these results we weaken the condition of Lyapunov functions at impulsive point,and we don't require the Lyapunov functions to have a negative definite first derivative along trajectories of the system.Indeed we needn't require its derivative to restrict its increasing growth and we give a combined estimate of continuous and discrete portions instead of giving respective conditions on them.In the end,an example is given to verify the validity of the results.To our knowledge,there has been very little existing work on the boundedness and stability criteria for impulsive Integro-differential equation with variable times using sev-eral Lyapunov functions including partial components coupled with the Razumikhin tech-nique.