Asymptotically periodic solutions of abstract differential equations

In this paper, we introduce the concept of SS-asymptotically ωω-periodicity in the Stepanov sense. We study some properties of the space consisting of SS-asymptotically ωω-periodic functions, and we use this notion to establish the existence of SS-asymptotically ωω-periodic solutions to linear and semi-linear first-order abstract differential equations. In particular, we characterize the strongly continuous semigroups of bounded linear operators defined on reflexive spaces which are SS-asymptotically ωω-periodic as those semigroups which are the direct sum of an ωω-periodic and a strongly stable semigroup.