Massera problem for some nonautonomous functional differential equations of neutral type with finite delay

The paper considers the existence of periodic solutions for some nonautonomous nonlinear partial functional differential equations of neutral type with finite delay. We suppose that the linear part is non-densely defined and satisfies the Acquistapace-Terreni conditions. The delayed part is assumed to be $\omega$-periodic with respect to the first argument. The existence of periodic solutions will be studied in the linear case by using the existence of bounded solutions. In the nonlinear case, a fixed point theorem for multivalued mapping and some sufficient conditions are given to prove the existence of periodic solutions. An example is given to illustrate the theoretical results.