The Mean Square Asymptotic Stability Of Nonlinear Stochastic Delay Dierential Equations

This paper investigates stochastic differential delay equations (SDDEs) dx(t)=[a(t)｜x(t-r1(t))｜p+f(x(t),x(t-r2(t)),t]dt+[b(t)｜x(t-r1(t))｜q+g(t),x;(t-r2(t)),t))]dW(t),t>0. and analyses the global existence of solution, the mean square boundedness and the mean square asymptotic stability properties of stochastic differential delay equations by means of fixed point theory. In Part I LetO<p<1,0<q<1.We prove the global existence of solution, the mean square boundedness using Krasnoselskii’s fixed point theorem, an asymptotic mean square stability theorem with a necessary and sufficient condition is proved. Only under the same sufficient conditions, this paper generalizes the previous results. Finally, one example is given to illustrate our results. In Part Ⅱ Letp=1,q=1. we obtain the global existence of solution, the mean square boundedness and the mean square asymptotic stability using Schauder’s fixed point theorem. Finally, one example are given to illustrate our results.