T-periodic solutions and a priori bounds

Using Schaeferʹs fixed-point theorem, enabling us to show that if there is an a priori 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.