The Stability Of Impulsive Functional Differential Systems With Infinite Delays

In this paper, we mainly study the asymptotical stability and the strict stabilityof impulsive functional differential systems with infinite delayswhere f∈C(R₊×PC, Rⁿ),I_k∈C(Rⁿ,Rⁿ),k∈N^*,0＜t₁＜t₂＜...＜t_k....with t_k→+∞as k→+∞, x'(t)denote the right-hand derivative of x at t, andx_t(s)∈PC denote x_t(s)=x(t+s). s∈(-∞. 0].Impulsive functional differential function with infinite delays describe a kind ofsystem present in the real world, such as a predator-prey system, so its study isof great value. On the other hand, applications of impulsive fuuctional differentialsystems are more wide and a large number of mathematicians are interested in it. butthese are limited to impulsive functional differential systems with finite delays.The basic theory of impulsive functional differential systems with infinite delaysis established just now and the properties of the its solutions are seldom studied.Therefore, we have still much to do. It is well known that Lyapunov functions andthe Razumikhin technique has been very effective in the study of impulsive functionaldifferential equations, because of the impulse, there is further study. In addition,documentation[14] develop a new technique iu studying stability of FDE. in which thecomponents:x₁, x₂,..., x_n of x are divided into some groups. Correspondingly, some Lyapunov functions are adopted, then theorems of stability are established. Whereevery Lyapunov function satisfies weaker conditions and is easier to be construct.Based on the ideas above, this paper is divided into two chapter.In chapter one, we study the uniformly asymptotical stability of the system (â… ).In the second section, we use the method of Lyapunov function and Razumikhintechnique. To the best of the authors' knowledge, results for system (â… ) are rare. Inthe new theorem, by finding a Razumikhin condition we overcome the the difficultywith which infinite delays bring the properties of uniformly attractive. In addition,the Lyapunov function has proper increase in impulsive point instead of diminishingalong the solution curve of the system (â… ). Its application is flexible. In the end ofthis section, an example is given to illustrate the advantage of the obtained result. Inthe third section, different from earlier result, we generalizes the method of severalLyapunov functions to impulsive functional differential systems with infinite delaysand establish some theorems. These theorems generalize some of earlier findings andare applied to vector equations, so the application is more wide. At last. an exampleis given to illustrate the advantage of the obtained result.In chapter two, we study the strict stability of system (â… ). At some time, weneed learn some information about the rate of decay of the solutions, so the strictstability of the system (â… ) must be considered. In the second sectiou, we construct twoLyapunov functions and find suitable conditions, then obtain the stability of system(â… ) by combining Razumikhin technique.