Positive Solutions for a Nonlinear Higher Order Differential System with Coupled Integral Boundary Conditions

Abstract

We investigate the existence of positive solutions for a nonlinear higher order differential system, where the differential system is coupled not only in the differential system but also through the boundary conditions. By constructing a special cone and using the fixed point theorem of cone expansion and compression of norm type, the existence of single and multiple positive solutions is established. As an application, we give some examples to demonstrate our results.

1. Introduction

In this paper, we consider the following nonlinear higher order differential system with coupled integral boundary conditions: where , , , , , are bounded linear functions on given by involving Stieltjes integrals. In particular, are functionals of  bounded variation with positive measures.

In recent years, there were many works to be done for a variety of  nonlinear higher order ordinary differential system. However, most papers only focus on paying attention to the differential system with uncoupled boundary conditions (see [1–5] and the reference therein). Coupled boundary conditions arise in the study of reaction-diffusion equations and Sturm-Liouville problems (see [6]) and have wide applications in various fields of sciences and engineering, for example, the heat equation [7, 8].

In a recent article [9], by applying a nonlinear alternative of  Leray-Schauder type and Guo-Krasnoselskii’s fixed point theorem on cone, the authors established the existence of multiple positive solutions of the following system with four-point coupled boundary conditions: where is the Riemann-Liouville’s fractional derivative.

In [10], by using fixed point index theory, Yang studied the following system with uncoupled boundary conditions: where are linear functionals defined by Stieltjes integrals.

The work of above-mentioned papers and wide applications of coupled boundary value conditions motivate us to study the system (1). Further, the system is coupled not only in the differential system but also through the boundary conditions. By constructing a special cone and using the fixed point theorem on cone expansion and compression, the existence of single and multiple positive solutions is established.

2. Preliminaries

Let ; we write . Clearly, is a Banach space. For each , we write . Define where is some subset of ; consider where is defined by the following Lemma 2. Clearly, is a Banach space and is a cone of .

Lemma 1. Let . Then differential system has the following integral representation where

Proof. By Taylor’s formula, we have So, we reduce the equation of problems (7) to the following equivalent integral equation: Let ; we have By substituting and into (12), we have that is, By applying and to (15), combined with the conditions , , respectively, we obtain Therefore and so By substituting (18) into (15), we obtain Therefore which is equivalent to system (8).

Lemma 2 (see [11]). The continuous function has the following properties: (i), for all , where ; (ii), for all , where .

Remark 3. By combining (i) and (ii), we can easily see

Remark 4. From Remark 3, and (9), for , we have
Define the operator by where operators are defined by Moreover, by Lemma 1, if is a fixed point of the operator , then is a solution of the system (1).

Lemma 5. The operator is completely continuous.

Proof. By Remark 4, for , we obtain Therefore, By Lemma 2 and Remark 3, for , we have Moreover, we have In the same way, we can prove that Thus Then operator is continuous since , , , , are continuous. Standard applications of Arzelà-Ascoli theorem; it is easy to prove that operator is completely continuous.

Lemma 6 (see [12]). Suppose is a real Banach space and is cone in , and let , be bounded open sets in such that , . Let operator be completely continuous. Suppose that one of the following two conditions holds: (i), for all ; , for all , (ii), for all ; , for all ,then operator has at least one fixed point in .

3. Main Results

In this section, we show the existence of positive solutions to the system (1). For convenience, we first introduce the following notations: where or . Let , , where

Theorem 7. If and , then system (1) has at least one positive solution.

Proof. At first, it follows from the assumption that there exists and a sufficiently small such that where satisfies and .
Set . For any , by (24), (25), and (33), we have Therefore Further, it follows from the the assumption that there exists and a sufficiently small such that where satisfies and . Let ; set . Then implies that . So, by (24), (25), and (36), we have Therefore, By applying Lemmas 5 and 6 to (35) and (38), it follows that operator has at least one fixed point in . This means that system (1) has at least one positive solution .

Using similar arguments as those used in the proof of Theorem 7, we can also obtain the following result.

Theorem 8. If and , then system (1) has at least one positive solution.

Next we discuss the multiplicity of positive solutions for system (1).

Theorem 9. If , and there exist an such that , for for all , where , then system (1) has at least two positive solutions.

Proof. At first, it follows from the assumption that there exists an and a sufficiently small such that where satisfies .
Set and . By (24), we have Further, by using , there exists and a sufficiently small such that where satisfies . Set , where . For all , by (24), we have By assumption, for for all , we have From (40)–(43), it is easy to know that two conditions of Lemma 6 are both satisfied. By applying Lemmas 5 and 6 to (40)–(43), it follows that operator has at least a fixed point and a fixed point . Both are positive solutions of system (1) and satisfy . This means system (1) has at least two positive solutions.

Similarly, we have the following results.

Theorem 10. If , and there exist an such that , for for all , where , then system (1) has at least two positive solutions.

Theorem 11. If , and there exist an such that , for all , where . Then system (1) has at least two positive solutions.

Theorem 12. If , and there exist an such that , for all , where . Then system (1) has at least two positive solutions.

4. Some Examples

In order to illustrate our result, we consider some examples.

Example 1. Consider the following system where , , , , , , , , . By direct calculation, we can obtain that , , . Choose ; then conditions of  Theorem 7 are satisfied. This means that system (44) has at least one position solution.

Example 2. Consider the following system where , , , , , ,  ,  , . By direct calculation, we can obtain that , , . Choose ; then conditions of Theorem 9 are satisfied. This means that system (45) has at least two position solutions.

Conflict of Interests

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

Authors’ Contribution

The work presented here was carried out in collaboration between all authors. All authors read and approved the final paper.

Acknowledgments

The authors are grateful to the referees for their careful reading. This research is supported by the Nature Science Foundation of Anhui Provincial Education Department (Grant nos. KJ2014A252 and KJ2013A248) and Professors (Doctors) Scientific Research Foundation of Suzhou University (Grant no. 2013jb04).