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 Tx= ∫t0t P(t, s,x(s))ds,P:R+2×Rnâ†'Rn, Z+ is the positive integer.Every solu-tion of it meet each hypersurface in turn finite times.Impulsive Integro-differential systems have extensive practical background and ap-plication in the natural sciences.For example, in biology, the control of pest invasion speed; in medicine,The diseases spread through the source and so on. Therefore, this type of systems has important application value.the research of impulsive Integro-differential systems has around expert’s interest and attention and various interesting results have been obtained in the past years[1-4,11-21],especially for impulsive Integro-differential sys-tems with fixed time [1-4].For example article[1,3]studied boundedness of solution of this system and got some direct results.Howevr,impulsive Integro-differential equation 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 devel-oped due to the difficulties created by the phenomena of beating and bifurcation etc.Up to now,the results about impulsive system with variable times are really few[10-12].In the finding appeared,article[10]gave one existence result of solution,and article[11]gave sev-eral criteria for asymptotic stability of this system from which the impulsive effects on the stability are displayed in the result obtained.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 focus research work on the boundedness and stability criteria for impulsive Integro-differential equation with variable times using several Lyapunov functions including partial components coupled with the Razurnikhin technique,and we get some new results.In chapter one,using the method of several Lyapunov functions including partial components coupled with the Razumikhin technique,we discuss the stability,uniform sta-bility and uniform asymptotical stability of impulsive Integro-differential equation with variable times.And a example is given to verify the validity of the results.In chapter two we obtain the sufficient condition of boundedness for the (I). Finally, two examples are given to verify the validity of the results.In this paper, the method adopted is different from [11,12], but it have taken the thought of [8,9],avoid using the auxiliary function P meanwhile using the method of Razumikhin. So there have more effective when we determine the stability of the sys-tems (I).In the theorems 1.3.1,1.3.2 of this paper,we use the method of several Lyapunov functions including partial components coupled with the Razumikhin technique,and the conditions of criteria of the Razumikhin-type theorems is weakened by avoiding using the auxiliary function P under less restrictive condition.However, to our knowledge, there has been very little existing work on the bounded-ness and stability criteria for impulsive Integro-differential equation with variable times using several Lyapunov functions including partial components coupled with the Razu-mikhin technique.