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.
Highlights
The investigation of boundary value problems denoted as BVPs for short of ordinary differential equations is of great significance
Two monotone sequences of iterations converging to the unique solution quadratically are obtained
Several definitions and lemmas needed to the main results are given first
Summary
The investigation of boundary value problems denoted as BVPs for short of ordinary differential equations is of great significance. 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 firstorder BVP with nonlinear boundary condition and second-order BVPs with Dirichlet boundary condition , periodic boundary condition , three-point boundary condition , four-point boundary condition , and m-point boundary condition ; BVPs of partial differential equations, such as parabolic initial-boundary value problem , elliptic problems with nonlinear boundary condition and p-Laplacian equations with nonlinear boundary condition , and so forth; BVPs of impulsive differential equations and impulse functional differential equations with anti-periodic boundary condition ; BVPs of some practically nonlinear problems, such as Duffing equation involving both integral and nonintegral forcing terms with Robin boundary condition and forced Duffing equation with discontinuous-type integral boundary condition ; as well as some abstract problems such as fixed point theorems in ordered Banach space. Two monotone sequences of iterations converging to the unique solution quadratically are obtained
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