Abstract and Applied Analysis
Volume 2011 (2011), Article ID 693890, 23 pages
http://dx.doi.org/10.1155/2011/693890
Research Article

Existence of Nonoscillatory Solutions for a Third-Order Nonlinear Neutral Delay Differential Equation

Zeqing Liu,1 Lin Chen,1 Shin Min Kang,2 and Sun Young Cho3

1Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China
2Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea
3Department of Mathematics, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 4 April 2011; Accepted 28 June 2011

Academic Editor: Elena Braverman

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

The aim of this paper is to study the solvability of a third-order nonlinear neutral delay differential equation of the form { 𝛼 ( 𝑡 ) [ 𝛽 ( 𝑡 ) ( 𝑥 ( 𝑡 ) + 𝑝 ( 𝑡 ) 𝑥 ( 𝑡 − 𝜏 ) ) ] } + 𝑓 ( 𝑡 , 𝑥 ( 𝜎 1 ( 𝑡 ) ) , 𝑥 ( 𝜎 2 ( 𝑡 ) ) , … , 𝑥 ( 𝜎 𝑛 ( 𝑡 ) ) ) = 0 , 𝑡 ≥ 𝑡 0 . By using the Krasnoselskii's fixed point theorem and the Schauder's fixed point theorem, we demonstrate the existence of uncountably many bounded nonoscillatory solutions for the above differential equation. Several nontrivial examples are given to illustrate our results.