The Massera Theorem for almost periodic solutions of linear periodic ordinary differential equations of the form , where is almost periodic, is stated and proved. Furthermore, it is extended to abstract functional differential equations , where is the generator of a compact semigroup, is periodic and is almost periodic. The main techniques used in the proofs involve a new variation of constants formula in the phase space and a decomposition theorem for almost periodic solutions.
"Massera's theorem for almost periodic solutions of functional differential equations." J. Math. Soc. Japan 56 (1) 247 - 268, January, 2004. https://doi.org/10.2969/jmsj/1191418705