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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 543035, 13 pages
http://dx.doi.org/10.1155/2011/543035
Research Article

Uniqueness of Positive Solutions for a Class of Fourth-Order Boundary Value Problems

Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria, Campus de Tafira Baja, 35017 Las Palmas de Gran Canaria, Spain

Received 31 December 2010; Accepted 23 March 2011

Academic Editor: Yuri V. Rogovchenko

Copyright © 2011 J. Caballero et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The purpose of this paper is to investigate the existence and uniqueness of positive solutions for the following fourth-order boundary value problem: ๐‘ฆ ( 4 ) ( ๐‘ก ) = ๐‘“ ( ๐‘ก , ๐‘ฆ ( ๐‘ก ) ) , ๐‘ก โˆˆ [ 0 , 1 ] , ๐‘ฆ ( 0 ) = ๐‘ฆ ( 1 ) = ๐‘ฆ ๎…ž ( 0 ) = ๐‘ฆ ๎…ž ( 1 ) = 0 . Moreover, under certain assumptions, we will prove that the above boundary value problem has a unique symmetric positive solution. Finally, we present some examples and we compare our results with the ones obtained in recent papers. Our analysis relies on a fixed point theorem in partially ordered metric spaces.