Open Access Research

On positive solutions for a class of singular nonlinear fractional differential equations

Mohamed Jleli and Bessem Samet*

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Department of Mathematics, King Saud University, Riyadh, Saudi Arabia

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Citation and License

Boundary Value Problems 2012, 2012:73  doi:10.1186/1687-2770-2012-73

Published: 12 July 2012

Abstract

We study the existence and uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M2">View MathML</a> is a real number, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M3">View MathML</a> is the Riemann-Liouville fractional derivative and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M4">View MathML</a> is continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M5">View MathML</a> (f is singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/73/mathml/M6">View MathML</a>). Our approach is based on a coupled fixed point theorem on ordered metric spaces. An example is given to illustrate our main result.

MSC: 34A08, 34B16, 47H10.

Keywords:
singular fractional differential equation; positive solution; coupled fixed point; coupled lower and upper solution; ordered metric space