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

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. 1. Introduction The investigation of boundary value problems (denoted as BVPs for short) of ordinary differential equations is of great significance. On one hand, it makes a great impact on the studies of partial differential equations [1]. On the other hand, BVPs of ordinary differential equations can be used to describe a large number of mechanical, physical, biological, and chemical phenomena; see [2–5] for example. So far a lot of work has been carried out, including second-order, third-order, and higher-order BVPs with various boundary conditions. As far as we know, for a long term most of works focused on existence and uniqueness of solutions. The works relating to approximation of solutions are relatively rare. In recent years, some approximate methods, such as the shooting method [6], monotone iterative technique [7], homotopy analysis method [8], and general quasilinearization method have been applied to BVPs for obtaining approximations of solutions. Among these methods, the general quasilinearization method becomes more and more popular. The quasilinearization method was originally proposed by Bellman and Kalaba [9]. It is a very powerful approximation technique and unlike perturbation methods, is not dependent on the existence of a small or large parameter. The method, whose sequence of solutions of linear problems convergences to the solution of the original nonlinear problem, is quadratic and monotone, which is one of the reasons for the popularity of this technique. This method was generalized by Lakshmikantham and Vatsala [10] in which the convexity or concavity assumption on the nonlinear functions involved in the problems is relaxed. So far, the general quasilinearization method, coupled with the method of upper and lower solutions, has been applied to obtain approximation of solutions for a large number of nonlinear problems, for example, BVPs of ordinary differential equations, such as first-order BVP with nonlinear boundary condition [11] and second-order BVPs with Dirichlet boundary condition [12], periodic boundary