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

Almost automorphic type functions and mean-square almost automorphic type functions are applied to all kinds of equations and the problems of the existence to almost automorphic type solutions and mean-square almost automorphic type solutions for these equations are mainly discussed.From an application point of view,stochastic differential equations are particularly important because they incorporate randomness into mathematical descriptions.The existence and uniqueness of mean-square almost automorphic type mild solutions of two classes of stochastic differential equations are discussed in this paper.The main research contents of this paper are as follows:1.The existence and uniqueness of mean-square asymptotically almost automorphic mild solutions for a class of nonautonomous stochastic differential equations in real separable Hilbert Spaces are discussed in the first part of this paper.Firstly,the related theories of mean-square asymptotically almost automorphic function are introduced,and then the definition of mean-square asymptotically almost automorphic function,the condition of "Acquistapace-Terreni",the equidistance property of Ito integral isometric properties,the correlation property of convolution family generated by linear closed operator,and Banach fixed point theorem are mainly used to explore the condition that such equations exists unique mean-square asymptotically almost automorphic mild solutions.2.Because the mean-square pseudo almost automorphic mild solutions for all kinds of stochastic differential equations are more general than mean-square almost automorphic mild solutions,the mean-square pseudo almost automorphic mild solutions for another category of stochastic differential equations in real separable Hilbert spaces are investigated in the second part of this paper.Using the analytical semigroup of exponential stability,Lipschitz conditions of functions,gamma function,as well as the related theories of mean-square of pseudo almostautomorphic functions,and Banach fixed point theorem,the operator of the fractional power techniques,Lebesgue control convergence theorem and Ito integral isometric properties,the existence and uniqueness of mean-square pseudo almost automorphic mild solutions to this kind of eqution are discussed.