Asymptotically Almost Automorphic Solutions And Square-mean Asymptotically Almost Automorphic Solutions For Two Classes Of Differential Equations

In the early 1960 s,Bochner S proposed the theory of almost automorphic functions.Furthermore,it has been used in a variety range of differential equations fields,such as biomathematics,communication theory,so on and so forth.The existence and uniqueness of almost automorphic solutions of differential equations have always been of great research value and application significance,and this has become one of the research contents that scholars are most interested in.Stochastic differential equations are used among many fields,including group dynamics,neural networks,and control theory,and so forth.The exploration of the square-mean almost automorphic type solutions has been a popular issue because stochastic differential equations may better incorporate problems with uncertain factors into mathematical descriptions.The existence and uniqueness of solutions for two types of differential equations are studied in the paper.Details are as follows:First and foremost,in this paper,the asymptotically almost automorphic mild solutions for a class of abstract functional differential equations are studied.The problem is to promote it on the basis of existing conclusions.The asymptotically almost automorphic of the composite function and the integral function are explored under the appropriate assumptions with the help of the definitions and correlation theories of the asymptotically almost automorphic functions,and combined with the theory of operator semigroup,the definition and property of evolution family.At the same time,the asymptotically almost automorphic of mild solutions are also discussed.The Banach fixed point theorem is used to investigate the existence and uniqueness of asymptotically almost automorphic mild solutions for this problem.Secondly,in the real separable Hilbert space,the square-mean asymptotically almost automorphic mild solutions for a class of non-autonomous stochastic differential equations are studied.Assumptions about equations are made first.For the operational derivation,the It? isometric integral and the Cauchy-Schwarz inequality are utilized.After that,using the definitions of square-mean almost automorphic and square-mean asymptotically almost automorphic functions,combing with Banach fixed point theorem,the exponential dichotomy of evolution family,as well as the Acquistapace-Terreni conditions(ATCs),the square-mean asymptotically almost automorphism of mild solutions for this kind of equations are discussed.At last,according to some stochastic analytics techniques,the existence of square-mean asymptotically almost automorphic mild solutions for such equations are studied,and the solutions are studied to be unique.