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

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On positive solutions for a class of singular nonlinear fractional differential equations Mohamed Jleli and Bessem Samet* *

Correspondence: [email protected] Department of Mathematics, King Saud University, Riyadh, Saudi Arabia

Abstract We study the existence and uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem Dα0+ u(t) = f (t, u(t), u(t)),

0 < t < 1,

u(0) = u(1) = u (0) = u (1) = 0, where 3 < α ≤ 4 is a real number, Dα0+ is the Riemann-Liouville fractional derivative and f : (0, 1] × [0, +∞) × [0, +∞) → [0, +∞) is continuous, limt→0+ f (t, ·, ·) = +∞ (f is singular at t = 0). 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

1 Introduction Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, fluid flows, electrical networks, viscoelasticity, aerodynamics, and many other branches of science. For details, see [–]. In the last few decades, fractional-order models are found to be more adequate than integer-order models for some real world problems. Recently, there have been some papers dealing with the existence and multiplicity of solutions (or positive solutions) of nonlinear initial fractional differential equations by the use of techniques of nonlinear analysis (fixed point theorems, Leray-Schauder theory, etc.); see [, , , ]. Recently, there have been many exciting developments in the field of fixed point theory on partially ordered metric spaces. The first result in this direction was given by Turinici []. In [], Ran and Reurings extended the Banach contraction principle in partially ordered sets with some applications to matrix equations. The obtained result by Ran and Reurings was further extended and refined by many authors; see [–]. Very recently, Shurong Sun et al. [] discussed the existence and uniqueness of a positive solution to the singular nonlinear fractional differential equation boundary value © 2012 Jleli and Samet; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Jleli and Samet Boundary Value Problems 2012, 2012:73 http://www.boundaryvalueproblems.com/content/2012/1/73

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problem   Dα+ u(t) = f t, u(t) ,

 < t < ,

u() = u() = u () = u () = , where  < α ≤  is a real number, Dα+ is the Riemann-Liouville fractional derivative and f : (, ] × [, +∞) → [, +∞) is continuous, limt→+ f (t, ·) = +∞ (f is singular at t = ), f (t, ·) is nondecreasing for all t ∈ (, ]