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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 237036, 12 pages
doi:10.1155/2012/237036
Research Article

Bounded Positive Solutions for a Third Order Discrete Equation

Zeqing Liu,1 Ming Jia,2 Shin Min Kang,3 and Young Chel Kwun4

1Liaoning Normal University, Department of Mathematics, Dalian, Liaoning 116029, China
2Dalian Hongwen Middle School, Dalian, Liaoning 116029, China
3Gyeongsang National University, Department of Mathematics and RINS, Jinju 660-701, Republic of Korea
4Dong-A University, Department of Mathematics, Pusan 614-714, Republic of Korea

Received 21 February 2012; Accepted 12 April 2012

Academic Editor: Josef Diblík

Copyright © 2012 Zeqing Liu et al. 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

This paper studies the following third order neutral delay discrete equation Δ ( 𝑎 𝑛 Δ 2 ( 𝑥 𝑛 + 𝑝 𝑛 𝑥 𝑛 − 𝜏 ) ) + 𝑓 ( 𝑛 , 𝑥 𝑛 − 𝑑 1 𝑛 , … , 𝑥 𝑛 − 𝑑 𝑙 𝑛 ) = 𝑔 𝑛 , 𝑛 ≥ 𝑛 0 , where 𝜏 , 𝑙 ∈ ℕ , 𝑛 0 ∈ ℕ ∪ { 0 } , { 𝑎 𝑛 } 𝑛 ∈ ℕ 𝑛 0 , { 𝑝 𝑛 } 𝑛 ∈ ℕ 𝑛 0 , { 𝑔 𝑛 } 𝑛 ∈ ℕ 𝑛 0 are real sequences with 𝑎 𝑛 ≠ 0 for 𝑛 ≥ 𝑛 0 , { 𝑑 𝑖 𝑛 } 𝑛 ∈ ℕ 𝑛 0 ⊆ ℤ with l i m 𝑛 → ∞ ( 𝑛 − 𝑑 𝑖 𝑛 ) = + ∞ for 𝑖 ∈ { 1 , 2 , … , 𝑙 } and 𝑓 ∈ 𝐶 ( ℕ 𝑛 0 × ℝ 𝑙 , ℝ ) . By using a nonlinear alternative theorem of Leray-Schauder type, we get sufficient conditions which ensure the existence of bounded positive solutions for the equation. Three examples are given to illustrate the results obtained in this paper.