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

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.