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 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. The obtained conditions are of Massera type theorem, 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. Although we use the idea of decomposition, we need to develop an abstract procedure to carry out the decomposition.