Abstract and Applied Analysis
Volume 2011 (2011), Article ID 351562, 17 pages
http://dx.doi.org/10.1155/2011/351562
Research Article

Existence of Homoclinic Orbits for a Class of Asymptotically ๐‘ -Linear Difference Systems with ๐‘ -Laplacian

School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083, China

Received 30 March 2011; Accepted 2 June 2011

Academic Editor: Yong Zhou

Copyright ยฉ 2011 Qiongfen Zhang and X. H. Tang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

By applying a variant version of Mountain Pass Theorem in critical point theory, we prove the existence of homoclinic solutions for the following asymptotically ๐‘ -linear difference system with ๐‘ -Laplacian ฮ” ( | ฮ” ๐‘ข ( ๐‘› โˆ’ 1 ) | ๐‘ โˆ’ 2 ฮ” ๐‘ข ( ๐‘› โˆ’ 1 ) ) + โˆ‡ [ โˆ’ ๐พ ( ๐‘› , ๐‘ข ( ๐‘› ) ) + ๐‘Š ( ๐‘› , ๐‘ข ( ๐‘› ) ) ] = 0 , where ๐‘ โˆˆ ( 1 , + โˆž ) , ๐‘› โˆˆ โ„ค , ๐‘ข โˆˆ โ„ ๐‘ , ๐พ , ๐‘Š โˆถ โ„ค ร— โ„ ๐‘ โ†’ โ„ are not periodic in ๐‘› , and W is asymptotically ๐‘ -linear at infinity.