Abstract and Applied Analysis
Volume 2014 (2014), Article ID 142391, 14 pages
http://dx.doi.org/10.1155/2014/142391

Research Article

Positive Solutions for -Type Singular Fractional Differential System with Coupled Integral Boundary Conditions

Ying Wang,1,2 Lishan Liu,1,3 Xinguang Zhang,3,4 and Yonghong Wu³

¹School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China
²School of Science, Linyi University, Linyi, Shandong 276000, China
³Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia
⁴Department of Mathematics and Informational Science, Yantai University, Yantai, Shandong 264005, China

Received 19 April 2014; Revised 27 August 2014; Accepted 15 September 2014; Published 3 November 2014

Academic Editor: Marco Donatelli

Copyright © 2014 Ying Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study the positive solutions of the -type fractional differential system with coupled integral boundary conditions. The conditions for the existence of positive solutions to the system are established. In addition, we derive explicit formulae for the estimation of the positive solutions and obtain the unique positive solution when certain additional conditions hold. An example is then given to demonstrate the validity of our main results.

1. Introduction

This paper is motivated by the boundary value problem

which arises in a variety of disciplinary areas such as mechanics, chemical physics, mathematical biology, flows, fluid electrical networks, and viscoelasticity (see [1–6] and the references cited therein). In problem (1), the nonlinearity may be singular at and ; may be singular at and .

Research on fractional order integrodifferential operators dates back to the end of the 19th century, when Riemann and Liouville introduced the first definition of the fractional derivative. However, this field of study started to become attractive to engineers only in the late 1960s, when fractional derivative description of some real systems was observed. It was found that fractional operators are nonlocal and are more suitable for constructing models possessing memory effect in a long time period, and hence fractional differential equations possess many advantages.

In this paper, we consider the existence of positive solutions for a nonlinear singular fractional differential system with coupled boundary conditions:

where , , and is the standard Riemann-Liouville derivative. , is right continuous on , left continuous at , and nondecreasing on , , and denotes the Riemann-Stieltjes integrals of with respect to . and are two continuous functions, and may be singular at and , while may be singular at and .

Coupled boundary value problem arises naturally in the research of Sturm-Liouville problems, reaction-diffusion equations, mathematical biology, and so on. In recent years, there has been a significant development in ordinary and partial differential equations involving fractional derivatives with coupled boundary conditions, as shown by [7–16] and the references therein. By using the nonlinear alternative of the Leray and Schauder theorem and the Krasnoselskii fixed point theorem in a cone, Bai and Fang in [17] obtained some results of existence of positive solutions by considering the singular coupled system of nonlinear fractional differential equations:

where , , , are two standard Riemann-Liouville fractional derivative, and are two given continuous functions and are singular at .

Wang et al. [18] study the following system of nonlinear fractional differential equations:

where , , , , , are continuous functions, and , are two standard Riemann-Liouville fractional derivatives. By using the Banach fixed point theorem and the nonlinear alternative of Leray-Schauder type, the existence and uniqueness of a positive solution are obtained.

In this paper, we consider the existence and uniqueness of positive solutions for the singular system (2). The work presented in this paper has the following new features. Firstly, until now, coupled integral boundary value problems for fractional differential system as system (2) have seldom been considered when may be singular at and , and may be singular at and . Also denotes the Riemann-Stieltjes integral, and thus system (2) includes the multipoint problems and integral problems as special cases. Secondly, by using the well-known fixed point theorem due to Guo-Krasnoselskii, we not only obtain the existence of positive solutions for system (2), but also obtain the uniqueness of system (2).

A vector is said to be a positive solution of system (2) if and only if satisfies (2) and , for any .

2. Preliminaries and Lemmas

In what follows, we present some necessary definitions about fractional calculus theory.

Definition 1 (see [2, 19]). Let and let be piecewise continuous on and integrable on any finite subinterval of . Then, for , we call

the Riemann-Liouville fractional integral of of order .

Definition 2 (see [2, 19]). The Riemann-Liouville fractional derivative of order , , , is defined as

where denotes the natural number set and the function is times continuously differentiable on .

Lemma 3 (see [2]). Let ; then,

where and is the smallest integer greater than or equal to .

Lemma 4. Let , and the following condition holds:

Then the system subjected to the coupled boundary conditions

has an integral representation

where

Proof. By Lemma 3, the system (9) is equivalent to the following integral equations system:

Integrating (13) and (14) with respect to and , respectively, we have

which yield

It follows from

that the system of (16) has a unique solution, which can be represented as

Substituting (18) into (13) and (14), we have

So (10) holds. This completes the proof of the lemma.

Lemma 5. For , the functions and    defined by (11) possess the following properties:

where

Proof. By [20], for any , we have

It follows from (11) and (25) that

As for the proof of (26), we have

that is, (20) holds.
By [20], for any , we have

So, by (11) and (28), we have

Proceeding as for the proof of (29), we have

thus (21) holds.
On the other hand, it follows from (11) and (25) that

which implies that (22) holds. Similarly, we also have

This completes the proof of the lemma.

From Lemma 5, we have the following conclusion.

Remark 6. For , we have

where , .

Throughout this paper, we will work in the space , which is a Banach space if it is endowed with the norm

Let

then is a cone in . For , denote

In what follows, we list some conditions to be used later:

is continuous, is nondecreasing in and nonincreasing in , and there exist such that

  is continuous, is nonincreasing in and nondecreasing in , and there exist such that

Remark 7. By , we have

This together with yields

From the above assumptions , for any , we define an integral operator by

where

Now we claim that is well defined for . In fact, for any , we have

Let be a positive number such that , . From and (44), we have

Hence, for any , by Lemma 5 and (45), we have

Similarly, for any , we have

Together with the continuity of and   , it is easy to see that , for . Therefore is well defined.

Obviously, is a positive solution of system (2) if and only if is a fixed point of in .

Lemma 8. Assume that hold. Then is a completely continuous operator.

Proof. First, we show .
For any , , , by Remark 6, we have

Then, we have

that is,

In the same way as (48), we can prove that

Therefore, we have .
Next, we show is continuous.
Let , such that . Obviously, , for all ; choose such that , , . So, by Lemma 5, , and (43), we have

By (52), for any , we can find a sufficiently large natural number , for all , such that

where . On the other hand, for each and , we have

where . Since and are uniformly continuous in , we have

hold uniformly on . Then the Lebesgue dominated convergence theorem yields that

So, for the above , there exists a sufficiently large natural number such that when , we have

It follows from (53) and (57) that

This implies that the operator is continuous. Similarly, we can prove is continuous. So is continuous.
Finally, we show is compact.
Let be any bounded set; then, for any , we have . Choose , such that , . By (45), for any , , we have

So is bounded in .
In what follows, we show that is equicontinuous. In fact, by (59), for any , there exists a sufficiently large natural number , for all , such that

Let

where .
By the uniform continuity of , on , for the above , there exists such that, for any , , , we have

Thus, when , , , for any , we get

This means that is equicontinuous. By the Arzela-Ascoli theorem, is a relatively compact set. In the same way, we can show that is a relatively compact set. So is compact.
From the above discussion, together with the fact that is continuous, we get that is completely continuous. This completes the proof of the lemma.

In order to obtain the existence of positive solutions of system (2), we will use the following cone compression and expansion fixed point theorem.

Lemma 9 (see [21]). Let be a positive cone in a Banach space , and are bounded open sets in , , , and is a completely continuous operator. If the following conditions are satisfied:

or

then has at least one fixed point in .

3. Main Results

Theorem 10. Assume that hold; then system (2) has at least one positive solution , and there exists a real number satisfying

where , .

Proof. First, we show that system (2) has at least one positive solution.
Choose , such that

For any , we have

By Lemma 5, Remark 7, and , for any , we get

This guarantees

On the other hand, for any , we have

By Lemma 5, , and , for any , , we get

In the same way as (72), we have

So,

It follows from (70), (74), and Lemmas 8 and 9 that has a fixed point with . It is easy to see that is a positive solution of system (2).
Next, we show there exists a real number such that the positive solution in system (2) satisfies

where , .
From Lemma 8, we know . So, we have

Choose , such that , ; by Lemma 5 and , for , we have

where

In the same way as (77), we also have , . Choose

then we have

This completes the proof of Theorem 10.

Theorem 11. Assume that hold. If and , then system (2) has a unique positive solution for .

Proof. Assume that system (2) has two different positive solutions and . Denote ; by Theorem 10, there exists , , such that

Thus, we have

Obviously, . Let 

It is easy to see that , and

By , we have

where . Consider the following:

From (85), for , we have

Similarly, from (86), we also have

Therefore, we obtain

Notice that , ; we have , so it is a contradiction with the maximality of . Therefore, system (2) has a unique positive solution . This completes the proof of Theorem 11.

Remark 12. By the proof of Theorems 10 and 11, we obtained the positive solutions of system (2) suppose hold; and the uniqueness of the solution to the system is established provided that system (2) satisfies the additional conditions ( and ) in theorem.

4. Example

Now we consider the existence and uniqueness of positive solutions for the fractional differential system (1). Obviously,

We also have

So, the condition holds. For

it is easy to see that is continuous, is nondecreasing in and nonincreasing in , is continuous, and is nonincreasing in and nondecreasing in . Take

Then, we know that the condition holds. As

the condition also holds.

Therefore, by Theorem 10, we get that system (1) has at least one positive solution . For

by Theorem 11, we get that is the unique positive solution of system (1).

Remark 13. The example not only implies that can be singular at , , and can be singular at , , but also indicates that there is a large number of functions that satisfy the conditions of the theorems which we discuss in this paper. Also, the conditions in our theorems are easy to check.

5. Conclusions

In this paper, the -type singular fractional differential system with coupled boundary conditions has been investigated. Based on the well-known Guo-Krasnoselskii fixed point theorem, the existence of solutions for the -type fractional differential system with coupled integral boundary conditions is presented; also the uniqueness of positive solution is established when certain additional conditions are satisfied. The example given demonstrates the effectiveness and feasibility of our results.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The first, second, and third authors were supported financially by the National Natural Science Foundation of China (11371221), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001), and the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province. The fourth author was supported financially by the Australia Research Council through an ARC Discovery Project Grant.

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