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.

In this paper, we consider the existence of infinitely many periodic solutions for some ordinary $p(t)$-Laplacian differential systems by minimax methods in critical point theory.