Existence of bounded asymptotic solutions of autonomous differential equations

We study the existence of bounded asymptotic mild solutions to evolution equations of the form $u^{\prime }\left(t\right)=Au\left(t\right)+f\left(t\right),t\ge 0$ in a Banach space $X$, where A generates an (analytic) $C_0$-semigroup and f is bounded. We find spectral conditions on A and f for the existence and uniqueness of asymptotic mild solutions with the same “profile” as that of f. In the resonance case, a sufficient condition of Massera type theorem is found for the existence of bounded solutions with the same profile as f. The obtained results are stated in terms of spectral properties of A and f, and they are analogs of classical results of Katznelson-Tzafriri and Massera for the evolution equations on the half line. Applications from PDE are given.