Journal of Applied Mathematics

Volume 2014 (2014), Article ID 901094, 7 pages

http://dx.doi.org/10.1155/2014/901094

Research Article

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

Yaohong Li1,2 and Haiyan Zhang¹

¹School of Mathematics and Statistics, Suzhou University, Anhui 234000, China

²School of Mathematics, University of Science and Technology of China, Anhui 230022, China

Received 7 April 2014; Accepted 13 August 2014; Published 24 August 2014

Academic Editor: Yong Shi

Copyright © 2014 Yaohong Li and Haiyan Zhang. 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 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).

References

J. Henderson and R. Luca, “Positive solutions for a system of higher-order multi-point boundary value problems,” Computers & Mathematics with Applications, vol. 62, no. 10, pp. 3920–3932, 2011.

S. Xie and J. Zhu, “Positive solutions of the system for th-order singular nonlocal boundary value problems,” Journal of Applied Mathematics and Computing, vol. 37, no. 1-2, pp. 119–132, 2011.

J. Henderson and R. Luca, “On a system of higher-order multi-point boundary value problems,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 2012, no. 49, pp. 1–14, 2012.

J. Xu and Z. Yang, “Positive solutions for a system of nth order nonlinear boundary value problems,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 2012, no. 4, pp. 1–16, 2012.

X. Zhang, C. Zhu, and Z. Wu, “Solvability for a coupled system of fractional differential equations with impulses at resonance,” Boundary Value Problems, vol. 2013, article 80, 2013.

H. Amann, “Parabolic evolution equations and nonlinear boundary conditions,” Journal of Differential Equations, vol. 72, no. 2, pp. 201–269, 1988.

Z. Lin and C. Xie, “The blow-up rate for a system of heat equations with nonlinear boundary conditions,” Nonlinear Analysis, vol. 34, no. 5, pp. 767–778, 1998.

M. Pedersen and Z. Lin, “Blow-up analysis for a system of heat equations coupled through a nonlinear boundary condition,” Applied Mathematics Letters, vol. 14, no. 2, pp. 171–176, 2001.

C. Yuan, D. Jiang, D. O’Regan, and R. P. Agarwal, “Multiple positive solutions to systems of nonlinear semipositone fractional differential equations with coupled boundary conditions,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 2012, no. 13, pp. 1–17, 2012.

Z. Yang, “Positive solutions to a system of second-order nonlocal boundary value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 62, no. 7, pp. 1251–1265, 2005.

D. Xie, C. Bai, Y. Liu, and C. Wang, “Positive solutions for nonlinear semipositone nth-order boundary value problem,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 2008, no. 7, pp. 1–12, 2008.

D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5, Academic Press, San Diego, Calif, USA, 1988.