Massera Type Theorem for Abstract Functional Differential Equations

The paper is concerned with conditions for the existence of almost periodic solutions of the following abstract functional differential equation $\dot u(t) = Au(t) + [{\cal B}u](t) +f(t),$ where $A$ is a closed operator in a Banach space $\X$, $\cal B$ is a general bounded linear operator in the function space of all $\X$-valued bounded and uniformly continuous functions that satisfies a so-called {\it autonomous} condition. We develop a general procedure to carry out the decomposition that does not need the well-posedness of the equations. The obtained conditions are of Massera type, which are stated in terms of spectral conditions of the operator ${\cal A}+{\cal B}$ and the spectrum of $f$. Moreover, we give conditions for the equation not to have quasi-periodic solutions with different structures of spectrum. The obtained results extend previous ones.