Journal of Applied Mathematics
Volume 2012 (2012), Article ID 262139, 12 pages
http://dx.doi.org/10.1155/2012/262139
Research Article

Existence of Solutions of Nonlinear Mixed Two-Point Boundary Value Problems for Third-Order Nonlinear Differential Equation

Yongxin Gao and Fengqin Wang

College of Science, Civil Aviation University of China, Tianjin 300300, China

Received 17 February 2012; Revised 18 June 2012; Accepted 2 July 2012

Academic Editor: Mehmet Sezer

Copyright © 2012 Yongxin Gao and Fengqin Wang. 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 authors use the upper and lower solution method to study the existence of solutions of nonlinear mixed two-point boundary value problems for third-order nonlinear differential equation 𝑦 ′ ′ ′ = 𝑓 ( 𝑥 , 𝑦 , 𝑦 ′ , 𝑦 ′ ′ ) , 𝑦 ′ ( 𝑏 ) = ℎ ( 𝑦 ′ ( 𝑎 ) ) , 𝑝 ( 𝑦 ( 𝑎 ) , 𝑦 ( 𝑏 ) , 𝑦 ′ ( 𝑎 ) , 𝑦 ′ ( 𝑏 ) ) = 0 , 𝑔 ( 𝑦 ( 𝑎 ) , 𝑦 ( 𝑏 ) , 𝑦 ′ ( 𝑎 ) , 𝑦 ′ ( 𝑏 ) , 𝑦 ′ ′ ( 𝑎 ) , 𝑦 ′ ′ ( 𝑏 ) ) = 0 . Some new existence results are obtained by developing the upper and lower solution method. Some applications are also presented.