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Uniqueness and existence of positive solutions for a multi-point boundary value problem of singular fractional differential equations
Advances in Difference Equations volume 2013, Article number: 114 (2013)
In this paper, we study the uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem , , , , where is a real number, is the standard Riemann-Liouville differentiation and , with . Our analysis relies on a fixed-point theorem in partially ordered set. As an application, an example is presented to illustrate the main result.
MSC:26A33, 34B15, 34K37.
Fractional differential equations have gained considerable importance due to their application in various sciences, such as physics, mechanics, chemistry, engineering, etc. In recent years, there has been a significant development in ordinary and partial differential equations involving fractional derivatives; see the monographs of Kilbas et al. , Miller and Ross , Oldham and Spanier , Podlubny , Samko , and the papers [6–16] and the references therein.
However, there are few papers, which have considered the singular boundary value problems of fractional differential equations; see [17–23]. In particular, Delbosco and Rodino  considered the existence of a solution for the nonlinear fractional differential equation , where and , is a given continuous function in . They obtained some results for solutions by using the Schauder fixed-point theorem and the Banach contraction principle.
Qiu and Bai  considered the existence of a positive solution to boundary value problems of the nonlinear fractional differential equation
where is the Caputo fractional derivative, and , with (i.e., f is singular at ). They obtained the existence of positive solutions by means of the Guo-Krasnosel’skii fixed-point theorem and nonlinear alternative of Leray-Schauder type in a cone. In , the uniqueness of the solution is not treated.
From the above works, we can see a fact, although the fractional boundary value problems have been investigated by some authors, to the best of our knowledge, there have been few papers which deal with the problem (1.1)-(1.2) for nonlinear singular fractional differential equation. Motivated by all the works above, this paper is mainly concerned with the uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem
where is a real number, , satisfy , and is the standard Riemann-Liouville differentiation, and , with . In this article, by using a fixed- point theorem in partially ordered set, existence and uniqueness results of a positive solution for the problem (1.1)-(1.2) are given.
The paper is organized as follows. In Section 2, we give some preliminary results that will be used in the proof of the main results. In Section 3, we establish the uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem (1.1)-(1.2). In the end, we illustrate a simple use of the main result.
2 Preliminaries and lemmas
The Riemann-Liouville fractional integral of order of a function is given by
provided that the right side is pointwise defined on , where Γ is the gamma function.
The Riemann-Liouville fractional derivative of order of a continuous function is given by
provided that the right side is pointwise defined on . Here, and denotes the integer part of α.
Lemma 2.1 
Let . If we assume , then the fractional differential equation
as unique solutions, where N is the smallest integer greater than or equal to α.
Lemma 2.2 
Assume that with a fractional derivative of order that belongs to . Then
for some , , where N is the smallest integer greater than or equal to α.
Lemma 2.3 
Let and , , satisfy that , then the unique solution of
is given by
Lemma 2.4 
Let and , , satisfy that , then the unique solution of the problem (2.1)-(2.2)
is nonnegative on , where
The following two lemmas are fundamental in the proofs of our main result.
Lemma 2.5 
Let be a partially ordered set and suppose that there exists a metric d in E such that is a complete metric space. Assume that E satisfies:
Let be a nondecreasing mapping such that
where is continuous and nondecreasing function such that φ is positive in , and . If there exists with , then f has a fixed point.
If we consider that satisfies the following condition:
then we have the following result.
Lemma 2.6 
Adding condition (2.7) to the hypotheses of Lemma 2.5, we obtain uniqueness of the fixed point of f.
3 Main results
Theorem 3.1 Let , , satisfy that , is continuous, and . Suppose that there exists a constant σ: such that is a continuous function on . Then the unique solution of the problem (2.1)-(2.2) is given by
and is continuous on .
Proof By the continuity of , it is easy to check that . The proof is divided into three cases.
Case 1. , .
Since is continuous in , there exists a constant , such that , . Hence,
where denotes the beta function.
Case 2. , .
Case 3. , . The proof is similar to that of Case 2, so we omit it. □
Let Banach space be endowed with the norm . Note that this space can be equipped with a partial order given by
It is easy to check that with the classic metric given by
satisfies condition (2.6) of Lemma 2.5. Moreover, for , as the function is continuous in , satisfies condition (2.7).
Theorem 3.2 Let , , , satisfy that , is continuous, with , and is continuous function on . Assume that there exists λ satisfying
such that for with and ,
where is continuous and nondecreasing, satisfies
φ is positive in .
Then the problem (1.1)-(1.2) has an unique positive solution.
Proof Define the cone by
Note that, as is a closed subset of E, is a complete metric space.
Suppose that u is a solution of boundary value problem (1.1) and (1.2). Then
Define an operator as follows:
By Theorem 3.1, . Moreover, in view of Lemma 2.4 and for , by hypothesis, we get
In what follows, we check that hypotheses in Lemmas 2.5 and 2.6 are satisfied. Firstly, the operator is nondecreasing. By hypothesis, for , we get
Besides, for , by (3.4), we get
As the function is nondecreasing, for , we get
By the last inequality, we get
Put . Obviously, is continuous, nondecreasing, positive in , .
Thus, for , we get
Finally, take into account that for the zero function, , by Lemma 2.5, our problem (1.1)-(1.2) has at least one nonnegative solution. Moreover, this solution is unique, since satisfies condition (2.7) and Lemma 2.6. This completes the proof. □
In the sequel, we present an example which illustrates Theorem 3.2.
4 An example
Consider the following fractional boundary value problem:
where , . In this case, for , . Note that f is continuous in and . Moreover, for and , we have
Because is nondecreasing on , and
With the aid of a computer, we obtain that
So, by Theorem 3.2, the problem (4.1)-(4.2) has an unique positive solution.
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Dedicated to Professor Hari M Srivastava.
The authors are thankful to the referees for their careful reading of the manuscript and insightful comments. The research is supported by the National Natural Science Foundation of China (11161027, 11262009).
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript. All authors read and approved the final manuscript.
About this article
- boundary value problem
- singular fractional differential equations
- Riemann-Liouville fractional derivative
- partially ordered set