Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part

Qinqin Zhang

doi:10.3934/cpaa.2015.14.1929

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2015, Volume 14, Issue 5: 1929-1940. Doi: 10.3934/cpaa.2015.14.1929

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Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part

Qinqin Zhang^1,

1.
College of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006

Received: November 2014

Revised: January 2015

Published: September 2015

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Abstract

Based on a generalized linking theorem for the strongly indefinite functionals, we study the existence of homoclinic orbits of the second order self-adjoint discrete Hamiltonian system \begin{eqnarray} \triangle [p(n)\triangle u(n-1)]-L(n)u(n)+\nabla W(n, u(n))=0, \end{eqnarray} where $p(n), L(n)$ and $W(n, x)$ are $N$-periodic on $n$, and $0$ lies in a gap of the spectrum $\sigma(\mathcal{A})$ of the operator $\mathcal{A}$, which is bounded self-adjoint in $l^2(\mathbb{Z}, \mathbb{R}^{\mathcal{N}})$ defined by $(\mathcal{A}u)(n)=\triangle [p(n)\triangle u(n-1)]-L(n)u(n)$. Under weak superquadratic conditions, we establish the existence of homoclinic orbits.

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Mathematics Subject Classification: Primary: 39A11; Secondary: 58E05, 70H05.

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