The boundary value problem of fourth-order ordinary differential equations (BVP for short) has attracted much attention due to its amazing application in engineering, physics, material mechanics, fluid mechanics and so on. Many authors use Banach contraction to study the existence of single or multiple positive solutions for certain third-order BVP-Guo (Orem), Guo-Krasnoselsky (Krasnoselsky) Fixed point theorem, Leray-Schauder nonlinear substitution, fixed point index theory of viewing cone, monotonic iterative technique, upper and lower solution method, degree theory, the Critical point theorem in a conical shell, etc. see  - .
However, it is necessary to point out that, in most of the existing literature, the Greens functions involved are nonnegative, which is an important condition in the study on BVP Positive Solution.
Recently, when the corresponding Green’s function is changing signs, some work has been done on the positive solution of the second or third order BVP. For example, Zhong and An  studied the existence of at least one positive solution of the following second-order periodic BVP with positive and negative transformation Green’s function
where . The main tool used is the fixed point index theory of cone
mapping 2008, for a singular third-order three-point BVP of Green’s function with infinite signature
where . Palamide and Smirlis  discussed the existence of at least
one positive solution. Their technique is a combination of Guo-Krasnosel’sski fixed point theory and the corresponding vector field characteristics. In 2012, Sun and Zhao   obtained single or multiple positive solutions with three-point positive and negative BVP by applying the fixed point theory of Guo-Krasnosel’skii and Leggett-Williams.
where . For relevant results, one can refer to  - . It is worth
mentioning that there are other types of achievements on either sign-changing or vanishing Green’s functions which prove the existence of sign-changing solutions, positive in some cases, see     .
Inspired and inspired by the above works, this article focuses on the following fourth-order three-point BVP with the iconic Green’s function.
Throughout this paper, we always assume that and .
Obviously, the BVP (2.1) is a special case of the BVP (2.2). However, it is necessary to point out that this paper is not a simple extension of , which is different from the restriction in . On the other hand, compared with , we can only prove that the obtained solution is concave on .
Our main tool is the following well-known Guo-Krasnoselskii fixed point theorem  :
Let K be a cone in a real Banach space E.
Definition 1.1. A functional is said to be increasing on K provided for all with , where if and only if .
Definition 1.2. Let be continuous. For each , one defines the set
Theorem 1.1. Let and be increasing, nonnegative, and continuous functionals on K, and let be a nonnegative continuous functional on K with such that, for some and ,
for all . Suppose there exist a completely continuous operator and such that
(H1) for all ;
(H1) for all ;
(H3) and for all .
Then T has at least two fixed points and in such that
The remainder of this paper, we assume that Banach space is equipped with the norm .
For the following BVP:
then we have the following lemma.
Lemma 2.1. The BVP (2.1) has only trivial solution.
Proof. Easy to check.
Now, for any , we consider the boundary value problems
After a direct computation, one may obtain the expression of Green’s function of the BVP (2) as following:
Lemma 2.2. It is not difficult to verify that has the following characteristics:
1) If , then is nonincreasing with respect to .
2) changes its sign on . In details, if , then . If , then .
3) If , then such that
for and for .
Moreover, if , then
if , then
Now, let is nonnegative and decreasing on .
Then is a cone in C [0, 1].
Lemma 2.3. Let and . Then u is the unique solution of the BVP (1.2) and . Moreover, is concave on
Proof. For , we have
since we get
At the same time, shows that
For , we have
In view of and , we get
Obviously, for , , . This shows that u is a solution of the BVP (2.2). The uniqueness follows immediately from Lemma 2.1. Since for and , we have for . So, . In view of for , we know that is concave on .
Lemma 2.4. Assume then the unique solution of the BVP (2.2) satisfies
where and .
Proof. From Lemma 2.2, we know that is concave on , thus for ,
In view of , we know that , which together with (2.3) implies that
according to that
3. Main Results
In this section, we are concerned with the existence of at least one positive solution of the problem (2.1). Assume that
(C1) For each , the mapping is decreasing;
(C2) For each , the mapping is increasing.
Then it is easy to see that K is a cone in .
Now, we define an operator by
distinctly, if u is a fixed point of A in K, then u is a positive and nondecreasing solution of the BVP (2.2), by lemma 2.3 and lemma 2.4 we know, although is not continuous, it follows from known textbook results, for example, see , that , is completely continuous. Set
Lemma 3.1. Suppose that (C1) and (C2) hold. Moreover, If there exist three constants a, b and c with such that
then boundary value problem (1.1) has at least two positive solutions
Proof. First, we define the increasing, nonnegative, and continuous functionals , and on K as follows:
Obviously, for any , . At the same time, for each , in view of , we have
Furthermore, we also note that for , .
Next, for any , we claim that
In fact, it follows from (C1), (C2), and
Now, we assert that for all . To prove this, let ; that is, and . Then
Since is decreasing on , it follows from (3.1), (3.2), (C2), (C1) and (F1) that
Then, we assert that for all . To see this, suppose that
; that is, and . Since , we have
In view of the properties of , (F2), (3.3), (C1) and (C2), we get
Finally, we assert that and for all .
In fact, the constant function . Moreover, for , that is and . Then
Since is decreasing on , it follows from (F3), (3.1), (3.4), (C1) and (C2) that
To sum up, all the hypotheses of Theorem 1.1 are satisfied. Consequently A has at least two fixed points; that is, the BVP (1.1) has at least two positive solutions and such that
The author expresses gratitude to the referees for their valuable comments and suggestions.
The Term of the Foundation
This work is partly supported by the National Natural Science Foundation of China (11561064) and partly supported by NWNU-LKQN-14-6.
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