Studies On Asymptotically Almost Automorphy Of Two Kinds Of Nonlinear Equations

In recent years, the theory of almost automorphic function has got extensive devel-opment and applications, the asymptotically almost automorphic function is a naturalgeneralization of the almost automorphic function, so in this paper, we mainly discuss theexistence and uniqueness of asymptotically almost automorphic solutions for two kinds ofnonlinear equations. It is divided into four chapter.Chapter1, we introduce the research backgrounds, latest development and the mainresults of asymptotically almost automorphic function, and also the motivation and struc-ture arrangement of this paper.Chapter2is preliminaries, we shall introduce some concepts, notations, definitionsand lemmas which will be used throughout this work. This chapter mainly contains almostautomorphic functions, asymptotically almost automorphic functions and their associatedsome main properties. Moreover, we introduce the concepts of monotone operator, secto-rial linear operator, solution operator and related with result.In chapter3, we investigate the existence and uniqueness of asymptotically almostautomorphic solution to the following nonlinear delay integral equations: satisfies Lipschitz condition, then we used the Banach’s fixed point principle to prove theexistence and uniqueness of asymptotically almost automorphic mild solution to the abovefractional diferential equations. Secondly, we anew give to some appropriate hypothesesand without a Lipschitz condition for f, we obtain an existence of asymptotically almostautomorphic mild solution by Leray-Schauder alternative theorem. Finally, as a corollary,we establish an existence of asymptotically almost automorphic mild solution with f sat-isfies H¨older type condition.