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 (u) over dot (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.