Abstract and Applied Analysis

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

Zheyan Zhou and Jianhe Shen

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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 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.

Article information

Source
Abstr. Appl. Anal., Volume 2010 (2010), Article ID 287473, 20 pages.

Dates
First available in Project Euclid: 1 November 2010

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1288620761

Digital Object Identifier
doi:10.1155/2010/287473

Mathematical Reviews number (MathSciNet)
MR2680412

Zentralblatt MATH identifier
1204.34025

Citation

Zhou, Zheyan; Shen, Jianhe. A Second-Order Boundary Value Problem with Nonlinear and Mixed Boundary Conditions: Existence, Uniqueness, and Approximation. Abstr. Appl. Anal. 2010 (2010), Article ID 287473, 20 pages. doi:10.1155/2010/287473. https://projecteuclid.org/euclid.aaa/1288620761


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