In this paper, in the view of control functions, the existence of periodic solutions of the following second-order discrete Hamiltonian system

$${△}^{2}u(n-1)=\mathrm{\nabla }F(n,u(n)),n\in \mathbb{Z}$$

with a generalized sublinear condition is further explored.

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.