## Abstract and Applied Analysis

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

#### Abstract

A second-order boundary value problem with nonlinear and mixed two-point boundary conditions is considered, $Lx=f(t,x,{x}^{\prime })$, $t\in (a,b)$, $g(x(a),x(b),{x}^{\prime}(a),{x}^{\prime }(b))=0$, $x(b)=x(a)$ 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