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.

Abstract

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.