Abstract

In this paper, we first show that if x is a bounded solution of the difference equation x(n+1) = Ax(n)+f(n), and the sequence is totally ergodic, σΓ(A): = σ(A)∩Γ is countable and the sequence is asymptotically almost periodic, then the sequence is asymptotically almost periodic. As an application, we consider the asymptotical periodicity of mild solutions to periodic evolution equations.