Homoclinic orbits of a class of second-order difference equations

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:396

Article

Zhang, Xu¹  Shi, Yuming¹ 

[1] Shandong Univ, Dept Math, Jinan 250100, Shandong, Peoples R China

关键词: Hamiltonian system;    Homoclinic orbit;    Spectral theory;    Variational method;   

DOI  :  10.1016/j.jmaa.2012.07.016

来源: Elsevier

【 摘 要 】

In this paper, we apply the variational method and the spectral theory of difference operators to investigate the existence of homoclinic orbits of the second-order difference equation Delta(2)x(t - 1) - L(t)x(t) + V-x' (t, x(t)) = 0 in the two cases that V(t, .) is superquadratic and subquadratic. Under the assumptions that L(t) is positive definite for sufficiently large vertical bar t vertical bar is an element of Z, we show that there exists at least one non-trivial homoclinic orbit of the difference equation. Further, if V(t, x) is superquadratic and even with respect to x, then it has infinitely many different non-trivial homoclinic orbits. At the end, two illustrative examples are provided. (C) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

Free