Multiplicity Results for Positive Solutions to Differential Systems of Singular Coupled Integral Boundary Value Problems

Abstract

By constructing a special cone and using a fixed-point theorem in cone, this paper investigates the existence of multiple solutions of coupled integral boundary value problems for a nonlinear singular differential system.

1. Introduction

In recent years, singular uncoupled boundary value problems to differential systems have been studied widely and there are many excellent results (see [1–18] and references therein). Naturally we hope there are the same excellent results on singular uncoupled boundary value problems to differential systems with coupled boundary conditions. Many researchers put their efforts to study the existence of solutions for differential systems with coupled boundary conditions (see [19–30] and references therein).

In [19], Asif and Khan addressed the question of the existence of coupled four-point boundary value conditions where the parameters , , , and satisfy , . The main tool in [19] is the Guo-Krasnosel’skiĭ fixed-point theorem.

In [29], the authors studied the following nonlinear semipositone fractional differential equation with four-point coupled boundary value problem: where is a parameter, , , , and satisfy , , is a real number and , and is Riemann-Liouville’s fractional derivative. The existence of positive solutions is established by using a nonlinear alternative of Leray-Schauder type and Guo-Krasnosel’skiĭ fixed-point theorem in a cone.

In [20], Cui and Sun, using fixed-point index theory, studied the existence of positive solutions for superlinear differential system where and are bounded linear functionals on given by involving Stieltjes integrals; in particular, and are functions of bounded variation with positive measures.

We should note that the nonlinear terms in two equations for the above problems have the same features. For instance, the nonlinear terms in two equations are both superlinear [20, 29] or both sublinear [19]. However, to the best of our knowledge, only a few papers discuss differential system under the case that the nonlinear terms of the system have different behaviors. Motivated by [19, 20, 29], the purpose of this paper is to establish the existence of multiple positive solutions for differential system with coupled integral boundary value problems (3) when is superlinear in and and is sublinear in and . Also suppose that may be singular at , and may be singular at , , , and . Our main features are threefold. Firstly, our study is on singular nonlinear differential systems with general boundary value conditions. Secondly, is allowed to be not only singular at and but also singular at and . Finally, a special cone is constructed to overcome difficulties due to singularities of nonlinear term.

In the rest of this section, let us list the following assumptions:

, , and , where

   satisfy and there exist constants , (, , ) such that, for , ,

   and there exist constants ,   (,  ,  ) such that, for , , And they satisfy one of the following conditions:

  , , ,

  , , ,

  , , ,

  , ,   , where

Remark 1. Equations (7)-(8) imply Conversely, (11) implies (7)-(8).

Remark 2. Equation (7) implies

Remark 3. implies that is nondecreasing in and .
implies that is nonincreasing in and .
implies that is nondecreasing in and nonincreasing in .
implies that is nonincreasing in and nondecreasing in .

2. Main Results

For each , we write . Clearly, is a Banach space. Similarly, for each , we write . For any real constant , define . Define where is given by (9). Clearly, is a Banach space and is a cone of .

Remark 4. The cone defined by (13) is completely different from the cone used in the uncoupled boundary value problems. This means that the cone has the following property:which is crucial in the definition of and in the proof of Lemma 9.

Our main result is the following theorems.

Theorem 5. Assume that , , and are satisfied. Then differential system (3) has at least two positive solutions such that .

Note. We need only to prove this theorem under condition , since the proof is similar when or or is satisfied.

The proof of Theorem 5 is based on the following theorem in [31].

Lemma 6. Let be a Banach space and a cone in . Suppose that and are two bounded open subsets of with , . Let operator be completely continuous. Suppose that one of the two conditions is satisfied. Then has a fixed point in .

Lemma 7 (see [20]). Assume that holds. Let ; then the system of BVPs has integral representationwhere

Remark 8 (see [20]). From (13) and , for , we haveDefine an operator by where operators are defined by

Lemma 9. Assume that , , and hold. Then, for any , is a completely continuous operator.

Proof. For , let be a positive number such that and . From , , and Remark 2, we haveHence for any , by Remark 8, we getThus, is well defined on .
Next we show that . By Remark 8, for , we obtain Hence, for , , we haveThen and , that is, . In the same way, we can prove that . Therefore, .
Moreover, is a completely continuous operator. This is a standard textbook result using Ascoli-Arzela theorem (see, e.g., [31]) and is omitted.

Proof of Theorem 5. By and , we can get This implies that Choose constants and such thatIt follows from Lemma 9 that is a completely continuous operator. Moreover, by Lemma 7, if is a fixed point of , then is a solution of differential system (3).
For any , , it follows from the definition of cone that Thus for any , by and Remarks 2 and 8, we haveConsequently, Again, for any , , we have Thus for any , noting that , we have This guarantees On the other hand, for any , , it follows from the definition of cone that Thus for any , by and Remarks 2 and 8, we haveThat is, Therefore, from (31), (37), and Lemma 6, it follows that differential system (3) has one positive solution with . In the same way, from (34), (37), and Lemma 6, it follows that differential system (3) has one positive solution with .

3. 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 coupled boundary value problemwhere , , .

Let thenSo all conditions of Theorem 5 are satisfied for (38), and our conclusion follows from Theorem 5.

Conflicts of Interest

The author declares that they have no conflicts of interest.

Acknowledgments

This project is supported by NNSF of China (11371221, 11571207) and the Tai’shan Scholar Engineering Construction Fund of Shandong Province of China.