Existence of bounded asymptotic solutions of autonomous differential equations

We study the existence of bounded asymptotic mild solutions to evolution equations of the form $u^{\mathrm{\prime }}(t)=Au(t)+f(t),t\ge 0$ in a Banach space $\text{X}$, 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.