Periodic Solutions of Nondensely Nonautonomous Differential Equations with Delay
Periodic Solutions of Nondensely Nonautonomous Differential Equations with Delay
Abstract
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.