Asymptotically Almost Periodic Solutions of Evolution Equations in Banach Spaces

Tile linear abstract evolution equation (∗) u′(t) = Au(t) + ƒ(t), t ∈ R, is considered, where A: D(A) ⊂ E → E is the generator of a strongly continuous semigroup of operators in the Banach space E. Starting from analogs of Kadets′ and Loomis′ Theorems for vector valued almost periodic Functions, we show that if σ(A) ∩ iR is countable and ƒ: R → E is [asymptotically] almost periodic, then every bounded and uniformly continuous solution u to (∗) is [asymptotically] almost periodic, provided e−λtu(t) has uniformly convergent means for all λ ∈ σ(A) ∩ iR. Related results on Eberlein-weakly asymptotically almost periodic, periodic, asymptotically periodic and C0-solutions of (∗), as well as on the discrete case of solutions of difference equations are included.