Stepanov-like Almost Automorphic Functions With Their Applications

In this thesis, we discuss mainly some fundamental properties of Stepanov-likeweighted pseudo almost automorphic functions with their applications to some nonlin-ear evolution equations. This thesis is divided into seven chapters.In chapter1, we introduce the research backgrounds and the main results of this thesis.However, it contains also some recent contributions to the theory of almost automorphicfunctions in abstract spaces.Chapter2is preliminaries, we wish to recall briefly some notations, definitions, meth-ods and lemmas that will be used throughout the thesis. This chapter mainly includes somedefinitions and basic properties about almost automorphic functions, weighed pseudo al-most automorphic functions, Stepanov-like weighed pseudo almost automorphic functions,Stepanov-like weighed pseudo almost automorphic functions of class h and Stepanov-likeweighed pseudo almost automorphic functions of class∞. Moreover, we introduce brieflythe definitions, basic facts and related notations of hyperbolic evolution families, integralresolvent families as well as infinite delay spaces.In chapter3, we consider the existence and uniqueness of weighted pseudo almostautomorphic solutions for the following nonautonomous evolution equations in the formu (t)=A(t)u(t)+f (t, u(t)), t∈R.We establish two new composition theorems for Stepanov-like weighed pseudo almost au-tomorphic functions. We then apply the obtained results to investigate the existence anduniqueness of weighted pseudo almost automorphic mild solutions to the class of nonau-tonomous evolution equations with Sp-weighted pseudo almost automorphic coefcients.In chapter4, we are mainly concerned with the existence of weighted pseudo almostautomorphic mild solutions to the class of semi-linear fractional diferential equations inthe abstract formThe main results are established by sectorial operator theory, suitable composition theo-rems combined with fixed point technique.In chapter5, we are mainly concerned with the existence of weighted pseudo almostautomorphic mild solutions to the class of semi-linear integral equations in the abstractform The main results are proved by using integral resolvent families, suitable compositiontheorems combined with the theory of fixed points.Chapter6is devoted to the existence of weighted pseudo almost automorphic solutionsto the nonautonomous semi-linear evolution equations with a constant delay in the abstractformu (t)=A(t)u(t)+f(t, u(t h)), t∈R.We obtained our main results using the Leray-Shauder Alternative theorem.In the last chapter, we investigate the existence of weighted pseudo almost automor-phic solutions to the following nonautonomous functional diferential equationsTo do so, the first purpose there consisted of introducing the concept of Sp-weightedpseudo almost automorphic functions of class h, Sp-weighted pseudo almost automorphicfunctions of class∞, then we establish some basic results not only on the completenessof the space but also on the compositions of such functions. And finally, we establishthe extistence and uniqueness of weighted pseudo almost automorphic mild solutions withSp-weighted pseudo almost automorphic coefcients for nonautonomous partial neutralfunctional diferential equation with infinite delay.