Abstract and Applied Analysis

Volume 2014, Article ID 142391, 14 pages

http://dx.doi.org/10.1155/2014/142391

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

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

*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