A Second-Order Boundary Value Problem with Nonlinear and Mixed Boundary Conditions: Existence, Uniqueness, and Approximation. (20th July 2010)

• Record Type: Journal Article
• Authors: Zhou, Zheyan; Shen, Jianhe
• Other Names: Eloe Paul Academic Editor.
• Is Part Of: Abstract and applied analysis. Volume 2010(2010)
• Publication Date: 2010-07-20
• Subjects: Mathematical analysis -- Periodicals; Mathematical analysis; Applied Mathematics; Mathematical Analysis; Periodicals; 515.05
• DOI: 10.1155/2010/287473
• Languages: English
• ISSNs: 1085-3375

Abstract

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.