A Second-Order Boundary Value Problem with Nonlinear and Mixed Boundary Conditions: Existence, Uniqueness, and Approximation. (20th July 2010) - Zhou, Zheyan; Shen, Jianhe
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 whichL is a formally self-adjoint second-order differential operator. Under appropriate assumptions onL, f, andg, 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.