Massera problem for non-autonomous retarded differential equations

In this paper, under Acquistapace-Terreni conditions, we study the existence of periodic solutions for some non-autonomous semilinear partial functional differential equation with delay. We assume that the linear part is non-densely defined. The delayed part is assumed to be ω-periodic with respect to the first argument. Using a fixed point theorem for multivalued mapping, some sufficient conditions are given to prove the existence of periodic solutions. An example is shown to illustrate our results.