Of concern is the nonlinear second order difference equation
where and for some positive integer and for all . One supposes there exists a function such that the gradient of in coincides with . Let be a given positive integer. In this paper, the existence of -periodic solutions of the above difference equation is studied, under different hypotheses on and . The method used here is from the critical point theory. These results are the discrete analogues of some theorems obtained in the continuous case for the second order differential equation , .