Stability Study For Impulsive Integro-differential Systems With Variable Times

In this paper,we study impulsive integro-differential system with variable times and one class of delay impulsive integro-differential system in system(â… ), Every solution of it meet each hypersurface in turn exactly once. In system (â…¡), we mainly study their stability.Impulsive integro-differential system has widespread application background in the natural sciences, as the systems extensively occur in the mathematical modeling of phys-ical problems,the governing systems in the problems of biological sciences such as spreading of disease by the dispersal of infections individuals, the reaction-diffusion models in ecology to estimate the speed of invasion are some examples of impulsive integro-differential systems.The research of it has aroused expert's interest and attention, and has made important progress in recent years,especially for impulsive integro-differential system with invariable times. But impulsive integro-differential system with variable times as an extension of impulsive integro-differential system with invariable times has more important application background. However,there is a number of diffi-culties related to the phenomena of beating and bifurcation,the research is slow. In the findings appeared up to now,article [5] gave one existence result of solutions, and article [6] gave several criterions for asymptotic stability of this system by using the Lyapunov functional method. On the whole, it is still at the start stage for the stability study of this system and there are also multiply works to do.In the nature, many thing's variance not only relied on conditions then,but also re-lied on conditions for its past time,namely there is a lag between reaction and the reason which caused it,and often with impulsive phenomenon followed.All these mathematical model can attach to the delay impulsive differential systems,which is more richer and realistic than the impulsive differential systems.There were many achievement for its study in the recent ten years,and the 3/2 stability method is very useful. The research of impulsive integro-differential system has aroused expert's interest and attention,and has made important progress in recent years, but the research of delay impulsive integro-differential systems is quite scarce,and there are also multiply works to do.In chapter one we do some related preparation first,after that establish a new com-parison principle by comparing with a scalar integro-differential system without impulse, then study the stability of system (â… ) in terms of two measures and obtain a stability criterion,finally an example is given to show the effectiveness of the theorems.In chapter two we investigate stability of system (â… ) 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 several criterion for stability and asymptotic stability of system (â… ),in these stability criterion we weakens the condition of Lyapunov function at impulsive point,and the Lyapunov function does not require to have a negative definite first derivative along trajectories of the system, further we may not require its derivative to restrict it increasing growth, and we employ a combined estimate of continuous and discrete portions instead of giving conditions on them respectively. An example is given to show the effectiveness of the theorems.In chapter three we give some related preparation knowledge of the delay impulsive integro-differential system(â…¡) first,then investigate its stability and get some criterions for uniformly stability and asymptotic stability.