Abstract and Applied Analysis
Volume 2010 (2010), Article ID 287473, 20 pages
Research Article

A Second-Order Boundary Value Problem with Nonlinear and Mixed Boundary Conditions: Existence, Uniqueness, and Approximation

School of Mathematics and Computer Science, Fujian Normal University, 350007 Fuzhou, China

Received 31 December 2009; Revised 14 April 2010; Accepted 3 July 2010

Academic Editor: Paul Eloe

Copyright © 2010 Zheyan Zhou and Jianhe Shen. 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.


A second-order boundary value problem with nonlinear and mixed two-point boundary conditions is considered, Lx=f(t,x,x), t(a,b), g(x(a),x(b),x(a),x(b))=0, x(b)=x(a) in which L is a formally self-adjoint second-order differential operator. Under appropriate assumptions on L, f, and g, existence and uniqueness of solutions is established by the method of upper and lower solutions and Leray-Schauder degree theory. The general quasilinearization method is then applied to this problem. Two monotone sequences converging quadratically to the unique solution are constructed.