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.