Journal of Function Spaces
Volume 2016 (2016), Article ID 6874643, 6 pages
http://dx.doi.org/10.1155/2016/6874643

Research Article

Existence of Positive Solution to System of Nonlinear Third-Order Three-Point BVPs

Jian-Ping Sun, Xue-Mei Yang, and Ya-Hong Zhao

Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, China

Received 13 April 2016; Accepted 16 June 2016

Academic Editor: Enrique Llorens-Fuster

Copyright © 2016 Jian-Ping Sun et al. 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 are concerned with the following system of third-order three-point boundary value problems: , , , , , , , and , where and . By imposing some suitable conditions on and , we obtain the existence of at least one positive solution to the above system. The main tool used is the theory of the fixed-point index.

1. Introduction

Third-order differential equations arise from a variety of different areas of applied mathematics and physics, for example, in the deflection of a curved beam having a constant or varying cross section, a three-layer beam, electromagnetic waves, or gravity driven flows, and so on [1].

Recently, there are a lot of papers concerning the existence of positive solutions to third-order three-point boundary value problems (BVPs for short); see [2–13] and the references therein. Yet, only in a few papers has the problem of existence of positive solutions to systems of third-order three-point BVPs been considered.

It is worth mentioning that there are some excellent works on systems of second-order or higher-order multipoint BVPs; see [14–18].

In this paper, we study the existence of positive solution for the following system of nonlinear third-order three-point BVPs:

Throughout this paper, we always assume that the following conditions are fulfilled:

and . and , , .

If satisfies the differential equations and boundary conditions in system (1), then is said to be a solution of system (1). If is a solution of system (1) and , , , then is said to be a positive solution of system (1).

To end this section, we state the following results on the theory of the fixed-point index [19].

Let be a real Banach space, a cone, and the zero element in . For , we denote

Theorem 1. Let be a completely continuous operator. If there exists such that

then .

Theorem 2. Let be a completely continuous operator which has no fixed point on . If for all , then .

2. Preliminaries

Let be equipped with the maximum norm. Then is a Banach space.

Lemma 3 (see [7]). For any , the BVP

has a unique solution

where

For convenience, we denote

Obviously, and .

Lemma 4 (see [7]). For any , .

Lemma 5. For any , .

Proof. Since , we only need to consider .
If , then

If , then

If , then

Therefore, for any , .

Lemma 6. Let and , . Then the unique solution of BVP (4) satisfies

Proof. Since , , it follows from Lemma 4 that for . In view of , , we have , . This shows that , , which indicates that

On the other hand, it follows from the fact for that is concave down on , which together with implies that . In view of (12), we get that .

Lemma 7. There exists such that

Proof. Let

Then it is obvious that

This indicates that there exists such that , which together with the fact that

implies that there exists such that ; that is, (13) is satisfied.

In the remainder of this paper, we always assume that is defined as in Lemma 7 and , .

Corollary 8. is an eigenvalue of the eigenvalue problem

and is an eigenfunction corresponding to the eigenvalue .

Proof. By Lemma 7, it is obvious.

Let

Then and are cones in . Now, we define a linear operator as follows:

Lemma 9. Consider .

Proof. In view of Lemmas 3, 4, and 6, it is not difficult to verify that .

Lemma 10. Consider and

Proof. Obviously, is continuous and , , which indicates that . And it follows from Lemma 3 and Corollary 8 that (20) is satisfied.

3. Main Results

Obviously, is a solution of system (1) if and only if is a solution of the following system:

Moreover, system (21) can be written as the integral equation:

If we define an operator on by

then it is easy to verify that is completely continuous. Moreover, if is a fixed point of and , , then is a solution of system (1).

Theorem 11. Assume that the following conditions are fulfilled:

There exists a constant such that

uniformly on .There exists a constant such that

uniformly on .

Then system (1) has at least one positive solution.

Proof. First, let

Then we may assert that is a bounded subset of .
In fact, if , then there exists such that , which together with Lemma 10 implies that

where is defined as follows:

In view of (27) and Lemma 9, we have . This implies that

On the other hand, from (1) of , we know that there exist positive constants and such that

which together with Jensen inequality and Lemma 4 implies that

where .
From (2) of , we know that there exists such that

where .
So, it follows from (31), (32), and Lemmas 4 and 5 that

where .
Since , we have

In view of (33) and (34), we get

So,

and so,

which together with (29) indicates that

This shows that is a bounded subset of . Therefore, there exists a sufficiently larger such that

So, it follows from Theorem 1 that

Next, from (1) of , we have

which together with , , implies that

Let

where . Then from (2) of and , , we know that there exists such that

By (44), for any , we have

which together with (42) shows that

This indicates that , . So, it follows from Theorem 2 that

In view of (40) and (47), we get

Therefore, has at least one fixed point and . Let

Then is a solution of system (1).
Thirdly, we prove for .
Suppose on the contrary that there exists such that . Since , , we know that , . So, is concave down on , which together with implies that . At the same time, it follows from , , that , . So, , ; that is, , , which contradicts with the fact that . Therefore, , .
Finally, we verify for .
First, we may assert that . Indeed, if , , in view of , , then we get

which is a contradiction. This shows that . So, by using the similar method to prove for , we may obtain , .
To sum it up, is a positive solution of system (1).

Competing Interests

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

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