New Results for Second Order Discrete Hamiltonian Systems

Abstract

In this paper, we deal with the second order discrete Hamiltonian system $\Delta[p(n) \Delta u(n-1)] - L(n) u(n) + \nabla W(n,u(n)) = 0$, where $L\colon \mathbb{Z} \to \mathbb{R}^{N \times N}$ is positive definite for sufficiently large $|n| \in \mathbb{Z}$ and $W(n,x)$ is indefinite sign. By using critical point theory, we establish some new criteria to guarantee that the above system has infinitely many nontrivial homoclinic solutions under the assumption that $W(n,x)$ is asymptotically quadratic and supquadratic, respectively. Our results generalize and improve some existing results in the literature.