A second-order boundary value problem with nonlinear and mixed two-point boundary conditions is considered, , , , in which is a formally self-adjoint second-order differential operator. Under appropriate assumptions on , , and , 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.
Zheyan Zhou. Jianhe Shen. "A Second-Order Boundary Value Problem with Nonlinear and Mixed Boundary Conditions: Existence, Uniqueness, and Approximation." Abstr. Appl. Anal. 2010 1 - 20, 2010. https://doi.org/10.1155/2010/287473