Studies On Almost Periodicity And Almost Automorphicity Of Nonlinear Equations

In this thesis, we discuss mainly the existence of almost periodic, (weighted) pseudo almost periodic and (pseudo) almost automorphic solutions for some nonlinear equations. This thesis is divided into six chapters.In chapter 1, we introduce the research backgrounds and the main results of this thesis. However, it contains also some recent contributions to the theory of almost periodic as well as almost automorphic functions in abstract spaces.Chapter 2 is preliminaries, we wish to recall briefly some notations, definitions, meth-ods and lemmas that will be used throughout the thesis. This chapter mainly includes some definitions and basic properties about almost periodic functions, (weighted) pseudo almost periodic functions, Stepanov-like weighted pseudo almost periodic functions, al-most automorphic functions and pseudo almost automorphic functions. Moreover, we introduce briefly the definitions, basic facts and related notations of Co-semigroups as well as integral resolvent families.In chapter 3, we consider the existence and uniqueness of almost periodic and pseudo almost periodic solutions for the following neutral differential equations in the form (du(t))/t=Au(t)+d/(dt)F1(t,u(h1(t))+F2(t,u(h2(t))), t∈R. Upon making some suitable assumptions, we establish some new theorems about the existence and uniqueness of (pseudo) almost periodic mild solutions to the above problems.In chapter 4, we are mainly concerned with the existence of almost automorphic and pseudo almost automorphic mild solutions to the following abstract differential equations in the form (du(t))/dt=Au(t)+d/(dt)F1(t,u(h1(t)))+F2(t,u(h2(t))), t∈R. Under some appropriate assumptions on A, Fi, and hi, i=1,2, the existence of an almost automorphic and pseudo almost automorphic mild solutions to the above equations is obtained by using both composition theorems and the Banach's fixed point principle.Chapter 5 is devoted to the existence of pseudo almost automorphic solutions to the class of semi-linear integral equations in the abstract form x(t)=∫-∞tα(t-s)[Ax(s)+f(s, x(s))]ds, t∈R Under some suitable hypotheses, the first task there consisted of establishing some suffi-cient conditions which ensure the existence and uniqueness of pseudo almost automorphic mild solutions with diverse Lipschitz conditions on the nonlinear term f. Next, we ob- tain an existence theorem of pseudo almost automorphic mild solutions with no Lipschitz conditions on the nonlinear term f. And finally, as a corollary, we present some existence theorems of pseudo almost automorphic mild solutions to the above integral equations in the scalar case.In the last chapter, we investigate the existence of weighted pseudo almost periodic mild solutions to the following semi-linear differential equations x'(t)= Ax(t)+/(t, Bx(t)), t∈R. To do so, the first purpose there consisted of establishing a new composition theorem of Stepanov-like weighted pseudo almost periodic functions under conditions which are dif-ferent from Lipschitz conditions in the literatures. Next, we apply this new composition theorem along with the Schauder's fixed point theorem to obtain new existence theorems for weighted pseudo almost periodic mild solutions to a semi-linear differential equation in Banach spaces, which extend some known ones in the sense of weighted pseudo almost periodicity.