Massera Type Theorems 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 u̇(t) = Au(t)+[Bu](t)+f(t), where A is a closed operator in a Banach space X, 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 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 A + 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.