# American Institute of Mathematical Sciences

January  2019, 18(1): 425-434. doi: 10.3934/cpaa.2019021

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

 Center for Applied Mathematics, Guangzhou University, Guangzhou, Guangdong, 510006, China

* Corresponding author

Received  February 2018 Revised  April 2018 Published  August 2018

Fund Project: This work was partially supported by National Natural Science Foundation of China grant 11471085 and the Program for Changjiang Scholars and Innovative Research Team in University grant IRT1226.

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
 zhongwenzy$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.
Citation: Qinqin Zhang. Homoclinic orbits for discrete Hamiltonian systems with local super-quadratic conditions. Communications on Pure & Applied Analysis, 2019, 18 (1) : 425-434. doi: 10.3934/cpaa.2019021
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