Two Positive Solutions of Third-Order BVP with Integral Boundary Condition and Sign-Changing Green's Function

Abstract

We are concerned with the following third-order boundary value problem with integral boundary condition:      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. An example is also included to illustrate the main results obtained.

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 for some third-order three-point boundary value problems (BVPs for short) and for some third-order BVPs with integral boundary conditions has received much attention from many authors; see [2–15] and the references therein. It is necessary to point out that all the abovementioned papers are achieved when the corresponding Green’s functions are nonnegative, which is a very important condition.

In 2008, Palamides and Smyrlis [16] 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.

Inspired greatly by [16], Sun and Zhao discussed the third-order three-point BVP with sign-changing Green’s function: They proved the existence of at least one or three positive solutions when by using the Guo-Krasnoselskii and Leggett-Williams fixed point theorems in [17, 18] and obtained the existence of a positive solution when by using iterative technique in [19]. For similar results, one can refer to [20, 21].

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 [22–24].

In this paper, we consider the following third-order BVP involving integral boundary condition and sign-changing Green’s function: Throughout this paper, we always assume that , and . By imposing suitable conditions on , we obtain the existence of at least two positive and decreasing solutions for the BVP (3). The main tool used 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 [25].

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 , one defines 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 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. Main Results

For any , we consider the BVP

After a direct computation, one may obtain the expression of Green’s function of the BVP (8) as follows: where

Remark 4. Consider the following: .

Remark 5. Consider the following: for .

Remark 6. has the following properties: Moreover, for , and, for ,

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 real Banach space be equipped with the norm and where . Then is a cone in . Now, we define an operator as follows: Obviously, if is a fixed point of in , then is a nonnegative and decreasing solution of the BVP (3).

Lemma 7. is completely continuous.

Proof. Let . Then, for , we have In view of , , Remark 5, and and , we get For , we have In view of , , Remark 5, and and , we get So, it follows from for that is decreasing on , which together with implies that , . At the same time, since for , we know that is concave on . Therefore, Consequently, This shows that . Furthermore, although is not continuous, it follows from known textbook results, for example, [26], that is completely continuous.

Lemma 8. For any and ,

Proof. Let . Then it follows from , , Remarks 5 and 6, and and that, for any ,

In what follows, we impose conditions on such that the BVP (3) has at least two positive and decreasing solutions. For convenience, we denote

Theorem 9. Suppose that there exist numbers with such that Then the BVP (3) 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, we assert that for all .
To prove this, let ; that is, . Then In view of Lemma 8, (25), (31), and and , we have Then, we assert that for all .
To see this, suppose ; that is, . Since , we have In view of Remark 6, (26), (33), and and , we get Finally, we assert that , and for all .
In fact, the constant function . Moreover, for , that is, . Then In view of Lemma 8, (27), (35), and and , we get To sum up, all the hypotheses of Theorem 3 are satisfied. Hence has at least two fixed points; that is, the BVP (3) has at least two positive and decreasing solutions and such that

3. An Example

Example 1. Consider the BVP where Since and , if we choose , then a simple calculation shows that Thus, if we let , and , then it is easy to verify that all the conditions of Theorem 9 are satisfied. So, it follows from Theorem 9 that the BVP (38) has at least two positive and decreasing solutions.

Conflict of Interests

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