In this article we obtain two sequence of infinitely many periodic solutions for discrete second order Hamiltonian systems with an oscillating potential. One sequence of solutions are local minimizers of the functional corresponding to the system, the other sequence are minimax type critical points of the functional.

Two theorems about the existence of periodic solutions with prescribed energy for second order Hamiltonian systems are obtained. One gives existence for almost all energies under very natural conditions. The other yields existence for all energies under a further condition.

In this paper, we study that the local minimizers of a class of functionals (not necessarily differentiable) in the C 1 -topology are still their local minimizers in