Periodic Solutions of Nondensely Nonautonomous Differential Equations with Delay

In this paper we study the Massera problem for the existence of a periodic mild solution of a class of non-densely non-autonomous semilinear differential equations with delay. We assume that the linear part satisfies the conditions introduced by Tanaka. First, we prove that the existence of a periodic solution for non-autonomous inhomogeneous linear differential equations with delay is equivalent to the existence of a bounded solution on the right half real line. Next, we undertake the analysis of the existence of periodic solutions in the semilinear case. To this end, we use a fixed point Theorem concerning set-valued maps. To illustrate the obtained results, we consider a periodic reaction diffusion equation with delay.