## Boundary Value Problems

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

Boundary Value Problems20122012:73

DOI: 10.1186/1687-2770-2012-73

Accepted: 18 May 2012

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

where is a real number, is the Riemann-Liouville fractional derivative and is continuous, (f is singular at ). 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 [111].

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 [2, 4, 5, 11].

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 [12]. In [13], 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 [1419].

Very recently, Shurong Sun et al. [20] discussed the existence and uniqueness of a positive solution to the singular nonlinear fractional differential equation boundary value problem

where is a real number, is the Riemann-Liouville fractional derivative and is continuous, (f is singular at ), is nondecreasing for all .

Motivated by the above mentioned work, in this paper we investigate the existence and uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem
(1)
(2)

where is a real number, is the Riemann-Liouville fractional derivative and is continuous, (f is singular at ), for all is nondecreasing with respect to the first component, and it is decreasing with respect to the second component. Our approach is based on a recent coupled fixed point theorem on ordered metric spaces established by Harjani et al. [17]. We end the paper with an example that illustrates our main result.

## 2 Preliminaries

In this section, we recall some basic definitions and properties from fractional calculus theory. For more details about fractional calculus, we refer the readers to [1, 3, 10].

Definition 2.1 The Riemann-Liouville fractional derivative of order of a continuous function is given by

where , denotes the integer part of number α, provided that the right side is pointwise defined on .

Definition 2.2 The Riemann-Liouville fractional integral of order of a function is given by

provided that the right side is pointwise defined on .

From the definition of the Riemann-Liouville derivative, we can obtain the following statement.

Lemma 2.1 (see [10])

Let. If we assume, then the fractional differential equation

has, , as unique solutions, where N is the smallest integer greater than or equal to α.

Lemma 2.2 (see [10])

Assume thatwith a fractional derivative of orderthat belongs to. Then

for some, , where N is the smallest integer greater than or equal to α.

The Green function of fractional differential equation boundary value problem is given by

Lemma 2.3 (see [10])

Letand. The unique solution to
(3)
(4)
is
where

Hereis called the Green function of boundary value problem (3)-(4).

The following properties of the Green function will be used later.

Lemma 2.4 (see [10])

The following properties hold:
1. (i)

for;

2. (ii)

, ;

3. (iii)

, for;

4. (iv)

, , where.

Let be a partially ordered set endowed with a metric d such that is complete metric space. Let be a given mapping.

Definition 2.3 We say that is directed if for every there exists such that and .

Definition 2.4 We say that is regular if the following conditions hold: () = if is a nondecreasing sequence in X such that , then for all n;; () = if is a decreasing sequence in X such that , then for all n..

Example 2.1 Let , , be the set of real continuous functions on . We endow X with the standard metric d given by
We define the partial order on X by

Let . For , that is, for all , we have and . This implies that is directed. Now, let be a nondecreasing sequence in X such that as , for some . Then, for all , is a nondecreasing sequence of real numbers converging to . Thus we have for all n, that is, for all n. Similarly, if is a decreasing sequence in X such that as , for some , we get that for all n. Then we proved that is regular.

Definition 2.5 (see [15])

An element is called a coupled fixed point of F if and .

Definition 2.6 (see [15])

We say that F has the mixed monotone property if for all , , we have

Denote by Φ the set of functions satisfying: () = φ is continuous;; () = φ is nondecreasing;; () = ..

The following two lemmas are fundamental in the proofs of our main results.

Lemma 2.5 (see [17])

Letbe a partially ordered set and suppose that there exists a metric d on X such thatis a complete metric space. Letbe a mapping having the mixed monotone property on X such that
(5)
for allwithand, where. Suppose also thatis regular and there existsuch that

Then F has a coupled fixed point. Moreover, ifandare the sequences in X defined by

then
(6)

Lemma 2.6 (see [17])

Adding to the hypotheses of Lemma 2.5 the conditionis regular, we obtain the uniqueness of the coupled fixed point. Moreover, we have the equality.

## 3 Main result

Let Banach space be endowed with the norm . We define the partial order on E by
(7)
In Example 2.1, we proved that with the classic metric given by

satisfies the following properties: is directed and is regular.

Define the closed cone by

where 0 denotes the zero function.

Definition 3.1 (see [15])

We say that is a coupled lower and upper solution to (1)-(2) if
and

Our main result is the following.

Theorem 3.1 Let, , is continuous, andis continuous on. Assume that there existssuch that forwith, and,
(8)
where, . Suppose also that (1)-(2) has a coupled lower and upper solution. Then the boundary value problem (1)-(2) has a unique positive solution. The sequencesanddefined by

converge uniformly to.

Proof Suppose that u is a solution of boundary value problem (1)-(2). Then
(9)
We define the operator by
• Step 1. We shall prove that
(10)
Let . Let us prove that . We have
By the continuity of in , it is easy to check that . Now, let . We have to prove that
We distinguish three cases:
Case 1. . Since is continuous on , there exists a constant such that for all . We have
Using Lemma 2.3, we have

where denotes the beta function.

Case 2. and . In this case,
Now, we have

Case 3. and . The proof is similar to that of Case 2, so we omit it.

Thus we proved that is continuous on for all . Moreover, taking into account Lemma 2.4 and as for all , , our claim (10) is proved. Now the mapping

is well defined.

• Step 2. We shall prove that F has the mixed monotone property with respect to the partial order given by (7).

Let such that and . From (8), we have
for all . This implies that
that is,
which gives us that
for all , and then we have

Then F has the mixed monotone property.

• Step 3. We shall prove that F satisfies the contractive condition (5) for some .

Let such that and . For all , using (8), we have
Thus we have
(11)
Now, let . We have
which implies that
Now, using the above inequality, (11) and the fact that , we get
Thus we proved that for all such that and , we have

where and .

• Step 4. Existence of such that and .

We take , the coupled lower and upper solution to (1)-(2).

Now, from Lemmas 2.5 and 2.6, there exists a unique such that , that is is the unique positive solution to (1)-(2). The convergence of the sequences and to follows immediately from (6). □

Now, we end this paper with the following example.

Example 3.1 Consider the boundary value problem
(12)
(13)
In this case, , for . Note that f is continuous on and . Let and . For all with , and , we have
On the other hand,

Consider now, the pair defined by and . Using Lemma 2.4(iv), one can show easily that is a coupled lower and upper solution to (12)-(13).

Finally, applying Theorem 3.1, we deduce that (12)-(13) has one and only one positive solution .

## Declarations

### Acknowledgement

This work was supported by the Research Center, College of Science, King Saud University.

## Authors’ Affiliations

(1)
Department of Mathematics, King Saud University

## References

1. Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.MATH
2. Babakhani A, Gejji VD: Existence of positive solutions of nonlinear fractional differential equations. J. Math. Anal. Appl. 2003, 278: 434-442. 10.1016/S0022-247X(02)00716-3
3. Bai Z, Lu H: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 2005, 311: 495-505. 10.1016/j.jmaa.2005.02.052
4. Delbosco D, Rodino L: Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 1996, 204: 609-625. 10.1006/jmaa.1996.0456
5. Gejji VD, Babakhani A: Analysis of a system of fractional differential equations. J. Math. Anal. Appl. 2004, 293: 511-522. 10.1016/j.jmaa.2004.01.013
6. Lakshmikantham V, Vatsala AS: Basic theory of fractional differential equations. Nonlinear Anal. 2008, 69(8):2677-2682. 10.1016/j.na.2007.08.042
7. Nonnenmacher TF, Metzler R: On the Riemann-Liouville fractional calculus and some recent applications. Fractals 1995, 3: 557-566. 10.1142/S0218348X95000497
8. Oldham KB, Spanier J: The Fractional Calculus. Academic Press, New York; 1974.MATH
9. Sabatier J, Agrawal OP, Machado JAT (Eds): Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht; 2007.
10. Xu X, Jiang D, Yuan C: Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation. Nonlinear Anal. 2009, 71: 4676-4688. 10.1016/j.na.2009.03.030
11. Zhang S: Existence of positive solution for some class of nonlinear fractional differential equations. J. Math. Anal. Appl. 2003, 278: 136-148. 10.1016/S0022-247X(02)00583-8
12. Turinici M: Abstract comparison principles and multivariable Gronwall-Bellman inequalities. J. Math. Anal. Appl. 1986, 117: 100-127. 10.1016/0022-247X(86)90251-9
13. Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004, 132: 1435-1443. 10.1090/S0002-9939-03-07220-4
14. Agarwal RP, El-Gebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008, 87: 109-116. 10.1080/00036810701556151
15. Gnana Bhaskar T, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 2006, 65: 1379-1393. 10.1016/j.na.2005.10.017
16. Ćirić L, Cakić N, Rajović M, Ume JS: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2008., 2008:
17. Harjani J, López B, Sadarangani K: Fixed point theorems for mixed monotone operators and applications to integral equations. Nonlinear Anal. 2011, 74: 1749-1760. 10.1016/j.na.2010.10.047
18. Nieto JJ, López RR: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. Engl. Ser. 2007, 23: 2205-2212. 10.1007/s10114-005-0769-0
19. Samet B: Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces. Nonlinear Anal. 2010, 72: 4508-4517. 10.1016/j.na.2010.02.026
20. Sun S, Zhao Y, Han Z, Xu M: Uniqueness of positive solutions for boundary value problems of singular fractional differential equations. Inverse Probl. Sci. Eng. 2011.