Homoclinic orbits for discrete Hamiltonian systems with local super-quadratic conditions

Abstract

Consider the second order self-adjoint discrete Hamiltonian system$\triangle [p(n)\triangle u(n-1)]-L(n)u(n)+\nabla W(n, u(n)) = 0, $where $p(n), L(n)$ and $W(n, x)$ are $N$-periodic on $n$, and 0 lies in a gap of the spectrum $σ(\mathcal{A})$of the operator $\mathcal{A}$, which is bounded self-adjoint in $l^2(\mathbb{Z}, \mathbb{R}^{\mathcal{N}})$ defined by $(\mathcal{A}u)(n) = \triangle [p(n)\triangle u(n-1)]-L(n)u(n)$. We obtain a sufficient condition on the existence of nontrivial homoclinic orbits for the above system under a much weaker condition than $\lim_{|x|\to ∞}\frac{W(n, x)}{|x|^2} = ∞$ uniformly in $ n∈ \mathbb{Z}$, which has been a common condition used in the existing literature. We also give three examples to illustrate our result.