Periodic Solutions of Infinite Delay Evolution Equations

For A t. and f t, x, y.T-periodic in t, we consider the differential equation with infinite delay in a general Banach space X, u9 t. qA t.u t. sf t, u t., ut., t)0, u s. sf s., sF0, 0.1. where the resolvent of the unbounded operator A t. is compact and f is continu- ous in its variables, and ut s.su tqs., sF0. We first show that the Poincar´e operator given by P f.suT f. i.e., T units along the unique solution u f. determined by the initial function f. is a condensing operator with respect to Kuratowski’s measure of non-compactness in a phase space Cg , and then derive periodic solutions from bounded solutions by using Sadovskii’s fixed point theorem. This extends the study of deriving periodic solutions from bounded solutions to infinite delay differential equations in general Banach spaces