Eberlein-weakly almost periodic (in Stepanov-like sense) functions and application

Abstract

In this article, we prove a number of properties concerning a (new) class of (Stepanov-like) Eberlein-weakly almost periodic (S P -E.w. a. p.) functions with values in a Banach space. We use the results obtained to study the asymptotic behaviour of solutions to the evolution equation: where A is the generator of an integral resolvent family in a Banach space 𝕏, a ∈ L 1(ℝ), and f is a given 𝕏-valued function on ℝ. The objective is to deduce Eberlein-weakly almost periodicity (in Stepanov-like sense) of the solution u from corresponding properties on the part f.