This paper deals with a non-autonomous discrete second-order Hamiltonian system under asymptotically linear conditions. The existence of a periodic solution is obtained via the saddle point theorem.

We investigate the non-trivial solutions for a partial discrete Dirichlet nonlinear problem with $p$-Laplacian by applying Ricceri's variational principle and a two non-zero critical points theorem. In addition, we identify open intervals of the parameter $\lambda$ under appropriate constraints imposed on the nonlinear term. This allows us to ensure that the nonlinear problem has at least one or two non-trivial solutions.