Asymptotically Almost Automorphic Mild Solutions For Two Classes Of Differential Equations

The research of almost automorphic functions is aroused more and more attention of mathematical workers at home and abroad in recent years, and its research value has been widely approved. Asymptotically almost automorphic functions is the generalization of almost automorphic functions, asymptotically almost automorphic functions has more general significance than almost automorphic functions to solve some problems.The main research contents in this paper are that asymptotically almost automorphic functions are applied to two classes of the differential equations, on the basis, the existence and uniqueness of asymptotically almost automorphic mild solutions to two classes of differential equations are discussed. The research contents of this paper are divided into three parts and its specific contents are as follows:First, some basic knowledge concerning about essay itself is introduced, which can lay a foundation for the proof of the second part and the third part. The basic theories of almost automorphic functions, asymptotically almost automorphic functions, c0, semigroup and Lebesgue’s dominated convergent theorem are specific introduced.Second, the existence and uniqueness of asymptotically almost automorphic mild solutions to a class of semilinear differential equations are discussed by using fixed point theorem and based on the results which have been investigated.Finally, the main properties of exponentiauy asymptotically almost automorphic functions and their applications tc differential equations are investigated.On the basis of some literature for asymptotically almost periodic mild solutions for some differential equations, the existence and uniqueness of asymptotically almost automorphic mild solutions to semilinear differential equations are discussed. Meanwhile, the applications to differential equations of exponentiauy asymptotically almost automorphic functions are discussed.This will make the conclusions be used more widely.