Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 406815, 6 pages

http://dx.doi.org/10.1155/2014/406815

Research Article

Positive Solutions of a Third-Order Three-Point BVP with Sign-Changing Green’s Function

Li-Juan Gao and Jian-Ping Sun

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

Received 3 April 2014; Revised 3 September 2014; Accepted 3 September 2014; Published 30 September 2014

Academic Editor: Zhen-Lai Han

Copyright © 2014 Li-Juan Gao and Jian-Ping Sun. 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 third-order three-point boundary value problem: and where and . Although the corresponding Green’s function is sign-changing, we still obtain the existence of at least two positive and decreasing solutions under some suitable conditions on by using the two-fixed-point theorem due to Avery and Henderson.

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, the existence of single or multiple positive solutions to some third-order three-point boundary value problems (BVPs for short) has received much attention from many authors; see [2–11] and the references therein. However, all the above-mentioned papers are achieved when the corresponding Green’s functions are nonnegative, which is a very important condition.

In 2008, Palamides and Smyrlis [12] studied the existence of at least one positive solution to the singular third-order three-point BVP with an indefinitely signed Green’s function:

where . Their technique was a combination of the Guo-Krasnoselskii fixed-point theorem and properties of the corresponding vector field.

Very recently, for the third-order three-point BVP with sign-changing Green’s function

Sun and Zhao proved the existence of at least one or three positive solutions when by using the Guo-Krasnoselskii and Leggett-Williams fixed-point theorems in [13, 14] and obtained the existence of a positive solution when by using iterative technique in [15].

In 2013, Li et al. [16] discussed the existence of a positive solution to the third-order three-point BVP with sign-changing Green’s function:

where and . The main tool used was the Guo-Krasnoselskii fixed-point theorem.

It is worth mentioning that there are other types of works on sign-changing Green’s functions which prove the existence of sign-changing solutions, positive in some cases; see Infante and Webb’s papers [17–19].

Motivated greatly by the above-mentioned works, in this paper, we consider the following third-order three-point BVP:

Throughout this paper, we always assume that and . Although the corresponding Green’s function is sign-changing, we still obtain the existence of at least two positive and decreasing solutions under some suitable conditions on . Our main tool is the well-known Avery-Henderson two-fixed-point theorem.

To end this section, we state some fundamental definitions and the two-fixed-point theorem due to Avery and Henderson [20].

Let be a cone in a real Banach space .

Definition 1. A functional is said to be increasing on provided for all with , where if and only if .

Definition 2. Let be continuous. For each , we define the set

Theorem 3. Let and be increasing, nonnegative, and continuous functionals on , and let be a nonnegative continuous functional on with such that, for some and ,

for all . Suppose that there exist a completely continuous operator and such that

and

(1) for all ;(2) for all ;(3), and for all .

Then has at least two fixed points and in such that

2. Preliminaries

For the BVP

we have the following lemma.

Lemma 4. The BVP (9) has only trivial solution.

Proof. It is simple to check.

Now, for any , we consider the BVP

After a direct computation, one may obtain the expression of Green’s function of the BVP (10) as follows:

where

It is not difficult to verify that the has the following properties:

Moreover, for ,

and for ,

Let

Then is a cone in .

Lemma 5. Let and , . Then is the unique solution of the BVP (10) and . Moreover, is concave on .

Proof. For , we have

Since and implies that , we get

At the same time, and shows that

For , we have

In view of and , we get

Obviously, for , , and . This shows that is a solution of the BVP (10). The uniqueness follows immediately from Lemma 4. Since for and , we have for . So, . In view of for , we know that is concave on .

Throughout this paper, for any , we define .

Lemma 6. Let . Then the unique solution of the BVP (10) satisfies

where and .

Proof. By Lemma 5, we know that is concave on ; thus, for ,

At the same time, it follows from that , which together with (23) implies that

Consequently,

3. Main Results

In the remainder of this paper, we always assume that is continuous and satisfies the following two conditions:

for each , the mapping is decreasing;for each , the mapping is increasing.

Let

Then it is easy to see that is a cone in .

Now, we define an operator as follows:

First, it is obvious that if is a fixed point of in , then is a nonnegative and decreasing solution of the BVP (4). Next, by Lemmas 5 and 6, we know that . Furthermore, although is not continuous, it follows from known textbook results, for example, see [21], that is completely continuous.

For convenience, we denote

Theorem 7. Suppose that there exist numbers , , and with such that

Then the BVP (4) has at least two positive and decreasing solutions.

Proof. First, we define the increasing, nonnegative, and continuous functionals , , and on as follows:

Obviously, for any , . At the same time, for each , in view of , we have

In addition, we also note that

Next, for any , we claim that

In fact, it follows from , , and that

Now, we assert that for all .

To prove this, let ; that is, and . Then

Since is decreasing on , it follows from (29), (35), (37), , and that

Then, we assert that for all .

To see this, suppose that ; that is, and . Since , we have

In view of the properties of , (30), (39), , and , we get

Finally, we assert that and for all .

In fact, the constant function . Moreover, for , that is, and , we know that

Since is decreasing on , it follows from (31), (35), (41), , and that

To sum up, all the hypotheses of Theorem 3 are satisfied. Hence, has at least two fixed points and ; that is, the BVP (4) has at least two positive and decreasing solutions and satisfying

Conflict of Interests

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

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