Abstract and Applied Analysis
Volume 2011 (2011), Article ID 328956, 16 pages
http://dx.doi.org/10.1155/2011/328956
Research Article

Positive Solutions for a Higher-Order Nonlinear Neutral Delay Differential Equation

Zeqing Liu,1 Ming Jia,1 Jeong Sheok Ume,2 and Shin Min Kang3

1Department of Mathematics, Liaoning Normal University, Liaoning, Dalian 116029, China
2Department of Applied Mathematics, Changwon National University, Changwon 641-773, Republic of Korea
3Department of Mathematics and the Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 23 May 2011; Accepted 31 July 2011

Academic Editor: Sergey V. Zelik

Copyright © 2011 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 deals with the higher-order nonlinear neutral delay differential equation ( 𝑑 𝑛 / 𝑑 𝑡 𝑛 ) [ 𝑥 ( 𝑡 ) + ∑ 𝑚 𝑖 = 1 𝑝 𝑖 ( 𝑡 ) 𝑥 ( 𝑇 𝑖 ( 𝑡 ) ) ] + ( 𝑑 𝑛 − 1 / 𝑑 𝑡 𝑛 − 1 ) 𝑓 ( 𝑡 , 𝑥 ( 𝛼 1 ( 𝑡 ) ) , … , 𝑥 ( 𝛼 𝑘 ( 𝑡 ) ) ) + ℎ ( 𝑡 , 𝑥 ( 𝛽 1 ( 𝑡 ) ) , … , 𝑥 ( 𝛽 𝑘 ( 𝑡 ) ) ) = 𝑔 ( 𝑡 ) , 𝑡 ≥ 𝑡 𝑜 , where 𝑛 , 𝑚 , 𝑘 ∈ ℕ , 𝑝 𝑖 , 𝜏 𝑖 , 𝛽 𝑗 , 𝑔 ∈ 𝐶 ( [ 𝑡 𝑜 , + ∞ ) , ℝ ) , 𝛼 𝑗 ∈ 𝐶 𝑛 − 1 ( [ 𝑡 𝑜 , + ∞ ) , ℝ ) , 𝑓 ∈ 𝐶 𝑛 − 1 ( [ 𝑡 𝑜 , + ∞ ) × ℝ 𝑘 , ℝ ) , ℎ ∈ 𝐶 ( [ 𝑡 𝑜 , + ∞ ) × ℝ 𝑘 , ℝ ) , and l i m 𝑡 → + ∞ 𝜏 𝑖 ( 𝑡 ) = l i m 𝑡 → + ∞ 𝛼 𝑗 ( 𝑡 ) = l i m 𝑡 → + ∞ 𝛽 𝑗 ( 𝑡 ) = + ∞ , 𝑖 ∈ { 1 , 2 , … , 𝑚 } , 𝑗 ∈ { 1 , 2 , … , 𝑘 } . By making use of the Leray-Schauder nonlinear alterative theorem, we establish the existence of uncountably many bounded positive solutions for the above equation. Our results improve and generalize some corresponding results in the field. Three examples are given which illustrate the advantages of the results presented in this paper.