Boundedness and Almost Periodicity of Solutions of Partial Functional Differential Equations

We study necessary and sufficient conditions for the abstract functional differential equation x=Ax+Fxt+f(t) to have almost periodic, quasi periodic solutions with the same structure of spectrum as f. The main conditions are stated in terms of the imaginary solutions of the associated characteristic equations and the spectrum of the forcing term f. The obtained results extend recent results to abstract functional differential equations.