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, 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.