A second-order boundary value problem with nonlinear and mixed boundary conditions: Existence, uniqueness, and approximation

Abstract

A second-order boundary value problem with nonlinear and mixed two-point boundary conditions is considered, L x = 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. © 2010 Zheyan Zhou and Jianhe Shen.