Asymptotically Almost Periodic And Asymptotically Almost Automorphic Mild Solutions For Two Classes Of Nonlinear Differential Equations

From the perspective of real life,periodic movement is perfect with great qualities whose overall characteristics can be inferred according to its local characteristics.But periodic movement is a idealized movement which is very hard to describe various phenomenon of real life.When it comes to theoretical research,the sum of two periodic functions will not be periodic,so the class of periodic functions does not form a linear space.Here it is become,unable to constitute a Banach space with any norms.Therefore,in order to overcome this problem,the theories of almost periodic functions and almost automorphic functions are proposed,And these theories are applied to many research fields.After years of research by mathematical workers,asymptotically almost periodic function and asymptotically almost automorphic function have been derived basing on these two types of functions,and applied to other branches of mathematics,especially to discuss the existence and uniqueness of almost periodic type solutions and almost automorphic type solutions of various types of differential equations and integral equations corresponding to practical problems.Thus,the existence and uniqueness of asymptotically almost periodic mild solution and asymptotically almost automorphic mild solution for two classes of nonlinear differential equations are discussed in this paper.The main research contents of this paper are divided into two parts:In the first part,the existence and uniqueness of asymptotically almost periodic mild solutions for a class of nonlinear differential equations are discussed by using Banach compressed image principle,the exponential dichotomy of the convolution,the Lipschitz condition and some theories on asymptotically almost periodic functions.In the second part,the existence and uniqueness of asymptotically almost automorphic mild solutions for another nonlinear differential equations are discussed by using Banach compressed image principle,some theories onC₀-semigroup,the exponential dichotomy of the convolution,Lipschitz condition and some theories on asymptotically almost automorphic functions.