On SS-asymptotically ωω-periodic functions and applications

Let (X,‖⋅‖)(X,‖⋅‖) be a Banach space and ω∈Rω∈R. A bounded function u∈C([0,∞);X)u∈C([0,∞);X) is called SS-asymptotically ωω-periodic if limt→∞[u(t+ω)−u(t)]=0limt→∞[u(t+ω)−u(t)]=0. In this paper, we establish conditions under which an SS-asymptotically ωω-periodic function is asymptotically ωω-periodic and we discuss the existence of SS-asymptotically ωω-periodic and asymptotically ωω-periodic solutions for an abstract integral equation. Some applications to partial differential equations and partial integro-differential equations are considered.