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

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.

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 [1-11].

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 [14-19].

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

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.

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, whereNis the smallest integer greater than or equal toα.

Lemma 2.2 (see [10])

Assume thatwith a fractional derivative of orderthat belongs to. Then

for some, , whereNis the smallest integer greater than or equal toα.

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

Lemma 2.3 (see [10])

Letand. The unique solution to

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:

(i) for;

(ii) , ;

(iii) , for;

(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 metricdonXsuch thatis a complete metric space. Letbe a mapping having the mixed monotone property onXsuch that

for allwithand, where. Suppose also thatis regular and there existsuch that

ThenFhas a coupled fixed point. Moreover, ifandare the sequences inXdefined by

then

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.

Let Banach space be endowed with the norm . We define the partial order ⪯ on E by

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 3.1Let, , is continuous, andis continuous on. Assume that there existssuch that forwith, and,

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

We define the operator by

• Step 1. We shall prove that

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

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

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 .

The authors declare that they have no competing interests.

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

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

1. Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006)

2. Babakhani, A, Gejji, VD: Existence of positive solutions of nonlinear fractional differential equations. J. Math. Anal. Appl.. 278, 434–442 (2003). Publisher Full Text

3. Bai, Z, Lu, H: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl.. 311, 495–505 (2005). Publisher Full Text

4. Delbosco, D, Rodino, L: Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl.. 204, 609–625 (1996). Publisher Full Text

5. Gejji, VD, Babakhani, A: Analysis of a system of fractional differential equations. J. Math. Anal. Appl.. 293, 511–522 (2004). Publisher Full Text

6. Lakshmikantham, V, Vatsala, AS: Basic theory of fractional differential equations. Nonlinear Anal.. 69(8), 2677–2682 (2008). Publisher Full Text

7. Nonnenmacher, TF, Metzler, R: On the Riemann-Liouville fractional calculus and some recent applications. Fractals. 3, 557–566 (1995). Publisher Full Text

8. Oldham, KB, Spanier, J: The Fractional Calculus, Academic Press, New York (1974)

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.. 71, 4676–4688 (2009). Publisher Full Text

11. Zhang, S: Existence of positive solution for some class of nonlinear fractional differential equations. J. Math. Anal. Appl.. 278, 136–148 (2003). Publisher Full Text

12. Turinici, M: Abstract comparison principles and multivariable Gronwall-Bellman inequalities. J. Math. Anal. Appl.. 117, 100–127 (1986). Publisher Full Text

13. Ran, ACM, Reurings, MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc.. 132, 1435–1443 (2004). Publisher Full Text

14. Agarwal, RP, El-Gebeily, MA, O’Regan, D: Generalized contractions in partially ordered metric spaces. Appl. Anal.. 87, 109–116 (2008). Publisher Full Text

15. Gnana Bhaskar, T, Lakshmikantham, V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal.. 65, 1379–1393 (2006). Publisher Full Text

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.. 74, 1749–1760 (2011). Publisher Full Text

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.. 23, 2205–2212 (2007). Publisher Full Text

19. Samet, B: Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces. Nonlinear Anal.. 72, 4508–4517 (2010). Publisher Full Text

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)