Asymptotically Periodic Solutions of Abstract Differential Equations

In this paper, we introduce the concept of S-asymptotically ?-periodicity in the Stepanov sense. We study some properties of the space consisting of S-asymptotically ?-periodic functions, and we use this notion to establish the existence of S-asymptotically ?-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 S-asymptotically ?-periodic as those semigroups which are the direct sum of an ?-periodic and a strongly stable semigroup. © 2012 Elsevier Ltd. All rights reserved.