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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 297618, 20 pages
doi:10.1155/2012/297618
Research Article

Infinitely Many Homoclinic Orbits for 2 𝑛 th-Order Nonlinear Functional Difference Equations Involving the 𝑝 -Laplacian

Xiaofei He1,2

1Department of Mathematics and Computer Science, Jishou University, Hunan, Jishou 416000, China
2Zhangjiajie College of Jishou University, Zhangjiajie 427000, China

Received 14 October 2011; Accepted 18 November 2011

Academic Editor: Donal O'Regan

Copyright © 2012 Xiaofei He. 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 establishing a new proper variational framework and using the critical point theory, we establish some new existence criteria to guarantee that the 2 𝑛 th-order nonlinear difference equation containing both advance and retardation with 𝑝 -Laplacian Ξ” 𝑛 ( π‘Ÿ ( 𝑑 βˆ’ 𝑛 ) πœ‘ 𝑝 ( Ξ” 𝑛 𝑒 ( 𝑑 βˆ’ 1 ) ) ) + π‘ž ( 𝑑 ) πœ‘ 𝑝 ( 𝑒 ( 𝑑 ) ) = 𝑓 ( 𝑑 , 𝑒 ( 𝑑 + 𝑛 ) , … , 𝑒 ( 𝑑 ) , … , 𝑒 ( 𝑑 βˆ’ 𝑛 ) ) , 𝑛 ∈ β„€ ( 3 ) , 𝑑 ∈ β„€ , has infinitely many homoclinic orbits, where πœ‘ 𝑝 ( 𝑠 ) is 𝑝 -Laplacian operator; πœ‘ 𝑝 ( 𝑠 ) = | 𝑠 | 𝑝 βˆ’ 2 𝑠 ( 1 < 𝑝 < ∞ ) π‘Ÿ , π‘ž , 𝑓 are nonperiodic in 𝑑 . Our conditions on the potential are rather relaxed, and some existing results in the literature are improved.