A Set of Almost Automorphic Functions and Applications

Abstract

For a set  of real numbers, we introduce the concept of  -almost automorphic functions valued in a Banach space. It generalizes in particular the space of  -almost automorphic functions. Considering the space of  -almost automorphic functions, we give sufficient conditions of the existence and uniqueness of almost automorphic solutions of a differential equation with a piecewise constant argument of generalized type. This is done using the Banach fixed point theorem.