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 ′ ( t ) = A u ( t ) + f ( t ) , t ≥ 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.