In this paper we consider periodic functional differential and integral equations. Massera proved that the existence of a bounded solution implies the existence of a periodic solution for linear (inhomogeneous) ordinary differential equations. Chow proved a similar result for linear functional differential equations with finite delay. We give a new proof for Chow's result that works for linear functional differential equations with infinite delay and also for integral equations. We also give examples that do not satisfy uniform boundedness or uniform ultimate boundedness, the conditions frequently used to prove that there is a periodic solution.
"Periodic solutions of linear differential and integral equations." Differential Integral Equations 8 (8) 2177 - 2187, 1995.