T-Periodic solutions and a priori bounds

Using Schaefer's fixed-point theorem, enabling us to show that if there is an a prion: bound on all possible T-periodic solutions of a Volterra-type difference equation, then there is a T-periodic solution. The a priori bound is established by means of a Lyapunov functional on which no bound is required. Thus, in addition to the periodic solution, the equation may have both bounded and unbounded solutions. (C) 2000 Elsevier Science Ltd. All rights reserved.