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On existence and uniqueness of positive solutions to a class of fractional boundary value problems

J Caballero*, J Harjani and K Sadarangani

Author Affiliations

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

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Boundary Value Problems 2011, 2011:25  doi:10.1186/1687-2770-2011-25

Published: 18 September 2011


The purpose of this paper is to investigate the existence and uniqueness of positive solutions for the following fractional boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/25/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/25/mathml/M1">View MathML</a>

where 2 < α ≤ 3 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/25/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/25/mathml/M2">View MathML</a> is the Riemann-Liouville fractional derivative.

Our analysis relies on a fixed-point theorem in partially ordered metric spaces. The autonomous case of this problem was studied in the paper [Zhao et al., Abs. Appl. Anal., to appear], but in Zhao et al. (to appear), the question of uniqueness of the solution is not treated.

We also present some examples where we compare our results with the ones obtained in Zhao et al. (to appear).

2010 Mathematics Subject Classification: 34B15

fractional boundary value problem; fixed-point theorem; positive solution