P-th Mean Almost Automorphic Type Mild Solutions For Two Classes Of Stochastic Differential Equations

Stochastic differential equations are a class of mathematical models,which are established to solve the problem of stochastic phenomena.It is an extension of differrential equations,and it has obtained a wide range of applications in many fields outside of Mathematics.The existence and uniqueness of p-th mean almost automorphic type mild solutions for two classes of stochastic differential equations are discussed in this paper.1.Firstly,based on the definition of square-mean asymptotically almost automorphic stochastic process,properties of range to square-mean almost automorphic stochastic process and Cauchy-Schwarz inequality,a property that closure of the range to square-mean asymptotically almost automorphic stochastic process contains range to corresponding square-mean asymptotically almost automorphic stochastic process is given.Then,compositeness of square-mean asymptotically almost automorphic stochastic process is studied by the property of range to square-mean asymptotically almost automorphic stochastic process combining Lipschitz assumption.Finally,by using the contraction mapping principle,?Ito isometry,the exponential stability of convolution family,the existence and uniqueness of square-mean asymptotically almost automorphic mild solutions of stochastic differential equations with delay are studied.2.The existence and uniqueness of p-th mean pseudo almost automorphic mild solutions of a class stochastic integrodifferential equations are discussed.Firstly,under the exponentially stable of resolvent opera-tor family and Lipschitz condition and other appropriate assumptions,by using defi-nition and some basic properties of p-th mean pseudo almost automorphic stocha-stic processes,and Lebesgue dominated convergence theorem,contracti-on mapping principle,?Ito stochastic integral property,the existence and uniqueness of p-th mean pseudo almost automorphic mild solutions for the class stochastic functionalintegrodifferential equations are discussed.Then,the existence of p-th mean pseudo almost automorphic mild solutions is discussed under the assumption which is weaker than the Lipschitz condition,by using the Schauder fixed point,?Ito stochastic integral property,Arzela-Ascoli lemma,H?lder inequality.