#### 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.