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

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 BVPhas a unique solution where

For convenience, we denoteObviously, 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 thatOn 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 thatwhere is defined as follows: In view of (27) and Lemma 9, we have . This implies thatOn 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 thatwhere .
From (2) of , we know that there exists such thatwhere .
So, it follows from (31), (32), and Lemmas 4 and 5 thatwhere .
Since , we haveIn view of (33) and (34), we get So, and so, which together with (29) indicates thatThis shows that is a bounded subset of . Therefore, there exists a sufficiently larger such that So, it follows from Theorem 1 thatNext, from (1) of , we have which together with , , implies thatLet where . Then from (2) of and , , we know that there exists such thatBy (44), for any , we have which together with (42) shows that This indicates that , . So, it follows from Theorem 2 thatIn 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.