Massera's theorem for almost periodic solutions of functional differential equations

Abstract

The Massera Theorem for almost periodic solutions of linear periodic ordinary differential equations of the form $(*)x^{\prime }=A\left(t\right)x+f\left(t\right)$, where $f$ is almost periodic, is stated and proved. Furthermore, it is extended to abstract functional differential equations $(**)x^{\prime }=Ax+F\left(t\right)x_t+f\left(t\right)$, where $A$ is the generator of a compact semigroup, $F$ is periodic and $f$ 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.