Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 126192, 8 pages
doi:10.1155/2010/126192
Research Article

Existence of Solution and Positive Solution for a Nonlinear Third-Order 𝑚 -Point BVP

Jian-Ping Sun and Fan-Xia Jin

Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

Received 5 November 2010; Accepted 14 December 2010

Academic Editor: Tomonari Suzuki

Copyright © 2010 Jian-Ping Sun and Fan-Xia Jin. 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

In this paper, we are concerned with the following nonlinear third-order 𝑚 -point boundary value problem: 𝑢 ( 𝑡 ) + 𝑓 ( 𝑡 , 𝑢 ( 𝑡 ) , 𝑢 ( 𝑡 ) ) = 0 , 𝑡 ∈ [ 0 , 1 ] , 𝑢 ( 0 ) = 𝐴 , 𝑢 ∑ ( 1 ) − 𝑚 − 2 𝑖 = 1 𝑎 𝑖 𝑢 ( 𝜉 𝑖 ) = 𝐵 , 𝑢 ( 0 ) = 𝐶 . Some existence criteria of solution and positive solution are established by using the Schauder fixed point theorem. An example is also included to illustrate the importance of the results obtained.