This paper provides sufficient conditions for the existence and uniqueness of positive solutions to a singular differential system with integral boundary value. The emphasis here is that the boundary conditions are coupled and this is where the main novelty of this work lies. By mixed monotone method, the existence and uniqueness results of the problem are established. An example is given to demonstrate the main results.

1. Introduction

In recent years, differential system has been studied extensively in the literature (see, for instance, [117] and their references). Most of the results told us that the equations had at least single and multiple positive solutions. In papers [6], the authors obtained some of the newest results for differential system with four-point coupled boundary conditions. But there is no result on the uniqueness of solution in them.

In this paper, we discuss the existence and uniqueness of the positive solutions for a new class of boundary value problems of singular differential system. Precisely, we consider the following problem: where and are right continuous on , left continuous at , and nondecreasing on , with ; and denote the Riemann-Stieltjes integrals of with respect to and , respectively; , ; that is, may be singular at , , and and may be singular at , , and . By a positive solution of the system (1), we mean that , satisfies (1), and and on .

2. Preliminaries

For each , we write . Clearly, is a Banach space. Similarly, for each , we write . Clearly, is a Banach space.

Throughout this paper, we shall use the following notation: It is well known that is the Green function of the following second order boundary value problem: and is nonnegative continuous function. It is easy to verify that for ,

We first list the following assumptions for convenience.(), is nondecreasing in and nonincreasing in , and there exist ,   such that , is nonincreasing in and nondecreasing in , and there exist , such that (),  .(),  ,  , where

Remark 1. By () and (), we can get

Remark 2. (i) (5) and (6) imply that Conversely, (11) implies (5) and (12) implies (6).
(ii) (7) and (8) implies that Conversely, (13) implies (7) and (14) implies (8).

Lemma 3. Assume that () holds. Let ; then the system of BVPs has integral representation where

Proof. It is easy to see that (15) is equivalent to the system of integral equations Integrating (18) and (19) with respect to and , respectively, on gives Therefore, and so Substituting (22) into (18) and (19), we have which is equivalent to the system (16).

Remark 4. From (4) and (), for , we have where

Denote where . It can be easily seen that is a cone in . For any real constant , define .

Define an operator by where operators are defined by Now we claim that is well defined for . In fact, since , we can see that

Let be a positive number such that and . From () and Remark 2, we have Hence, for any , by Remark 4 and equation (30), we get Thus, is well defined on .

Lemma 5. Assume that (), (), and () hold. Then, for any , is a completely continuous operator.

Proof. Firstly, we show that . By Remark 4, for , we obtain Hence, for , , we have Then and ; that is, . In the same way, we can prove that . Therefore, .
Next, we prove that is a compact operator. That is, for any bounded subset , we show that is relatively compact in . Since is a bounded subset, there exists a constant such that for all . Notice that, for any , we have and from (), (), Remarks 2 and 4, (16), and (18), we obtain Therefore, is uniformly bounded.
In the following, we shall show that is equicontinuous on .
For , , using Lemma 3, we have Differentiating with respect to and combining () and (), we obtain Exchanging the integral order, we have From the absolute continuity of the integral, we know that is equicontinuous on . Thus, according to the Ascoli-Arzela theorem, is a relatively compact set. In the same way, we can prove that is relatively compact. Therefore, is relatively compact.
Finally, it remains to show that is continuous. We need to prove only that are continuous. Suppose that , and . Let . Then we may still choose positive constants such that and . From () and Remark 2, we get
For any , by (), there exists a positive number such that On the other hand, for and , we have Since and are uniformly continuous in , we have holds uniformly on . Then the Lebesgue dominated convergence theorem yields that Thus, for above , there exists a natural number , for ; we have It follows from (39)–(44) that when This implies that is continuous. Similarly, we can prove that is continuous. So, is continuous. Summing up, is completely continuous.

Our main tool of this paper is the following cone compression and expansion fixed point theorem.

Lemma 6 (see [18]). Let be a Banach space and a cone in . Suppose that and are two bounded open subsets of with , . If   is a completely continuous operator satisfying then has a fixed point in .

3. Main Results

In this section, we present our main results.

Theorem 7. Suppose that conditions (), (), and () hold. Then, if and , the differential system (1) has a unique positive solution .

Proof. We divide the rather long proof into three steps.(i) The differential system (1) has at least one positive solution .
Choose , such that
Clearly . By Lemma 5, is completely continuous.
Extend (denote yet) to which is completely continuous.
Then, for , we have By Remarks 1 and 2, (), and (), we get This guarantees that On the other hand, for , we have Therefore, This guarantees that By the complete continuity of , (50) and (53), and Lemma 6, we obtain that has a fixed point in . Consequently, (1) has a positive solution in .(ii) Suppose that is a positive solution of the differential system (1).
Then there exist real numbers such that From Lemma 5, we know that . So, we have Let be a constant such that and . By Lemma 3, we get In the same way, we can prove that , . Then we may pick out such that , which implies that (54) holds.(iii) The differential system (1) has a unique positive solution .
Assuming the contrary, we find that the differential system (1) has a positive solution different from . By (54), there exist , such that Hence, we have Clearly, . Put It is easy to see that , and So, by (), we have where such that . Therefore, we have Similarly, we can get Noticing that , , we get to a contradiction with the maximality of . Thus, the differential system (1) has a unique positive solution . This completes the proof of Theorem 7.

4. An Example

In this section, we give an example to illustrate the usefulness of our main results. Let us consider the singular differential system with couple boundary value problem

Let then So all conditions of Theorem 7 are satisfied for (64), and our conclusion follows from Theorem 7.


The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original paper. This project is supported by the National Natural Science Foundation of China (11371221, 11071141), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001), the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province, the Postdoctoral Science Foundation of Shandong Province, and Foundation of SDUST.