As we all know, the earliest boundary value problem studied is Dirichlet problem. We need to find the solution of Laplace equation. Boundary value problems are most common in physics, such as wave equation. With the development of boundary value problems, many scholars began to pay attention to the study of higher-order boundary value problems. The third-order three-point problems have a wide range of applications in the fields of mathematics and physics      . Many works on the third-order boundary value problems have been established. In      , the authors have studied the third-order three-point boundary value problem and proved that the model has at least one positive solution. Recently, there have been many papers dealing with the positive solutions of boundary value problems for nonlinear differential equations with various boundary conditions. For example, Anderson  obtained some existence results for positive solutions for the following system:
where is continuous, f is nonnegative for and .
Moreover, Yao  considered the following system:
where , is a Caratheodory function. The author proved that (1.2) has at least one positive solution by Krasnoselskii fixed point theorems.
With the development of third-order boundary value problems, Guo et al.  considered the existence of a positive solution to the third-order three-point boundary value problem as follows
where is continuous; is continuous and not identically zero on , and . By using the Guo-Krasnoselskii fixed point theorem, they proved that the system (1.3) has at least one positive solution.
To our best knowledge, few papers can be found in the literature for three positive solutions of third-order three-point boundary value problems. Motivated greatly by the above-mentioned excellent works, in this paper, we will consider the following model
where , , , is a continuous function; is continuous and not identically zero on .
Obviously, this model is new because the nonlinear f depends not only on the unknown function but also the derivative of unknown function. In particular, the system (1.2) is special case of system (1.4). By the properties of the Green’s function, existence results of at least three positive solution for the third-order three-point boundary value problem are established by a new method which is different from the method in  . The paper is organized as follows. In Section 2, we present some notation and lemmas. In Section 3, we give the main results. In Section 4, an example is given to illustrate the main results of this paper.
Definition 2.1. Let E be a real Banach space. is a nonempty closed convex set. If it satisfies the following two conditions:
1) implies ;
2) implies .
Then, K is called a cone of E.
Definition 2.2. Suppose K is a cone. The map is continuous and satisfies the following inequality
for any and .
Then the map is a nonnegative continuous convex function on K.
Suppose K is a cone. The map is continuous and satisfies the following inequality
for any and .
Then the map is a nonnegative continuous concave function on K.
Lemma 2.1  Assume , then the system
has a unique solution for ,
If we denote , then we have the following lemma.
Lemma 2.2  Let , then
1) , for any ;
2i) , for any ,
For positive real numbers , we define the following convex sets:
Lemma 2.3  (Arzela-Ascoli theorem) Let be a operator, then is sequentially compact in if and only if is uniformly bounded and equicontinuous.
Lemma 2.4  (Krasnoselskii fixed point theorem) Let E be a real Banach space. is a cone. Suppose are nonnegative continuous convex functions on K. is a nonnegative continuous concave function on K. is a nonnegative continuous function on K, which satisfied and for positive numbers of , we have
Let be a completely continuous operator. There exist positive numbers of and satisfing the following conditions:
1) , and , for all ;
2) , for , and ;
3) , and , for , ;
then T has at least three fixed points such that
, as ; .
3. The Existence of Three Positive Solutions
We define the norm
Define the cone by
Lemma 3.1. Let be the operator defined by
Then is completely continuous.
Proof From the fact that f is nonnegative continuous function and Lemma 2.2, we know that . Let , from Lemma 2.2, we have
thus . According to the Arzela-Ascoli theorem, we prove that T is a completely continuous operator.
For convenience, we note that
Theorem 3.1. Suppose there exist such that
then the system (1.4) has at least three positive points and satisfying
, for .
Proof For , we have
which also implies that
So we show that (1.7) of the Lemma 2.4 holds.
If , we have .
And for any . From assumption (H1), we have , therefore,
Let , it is easy to prove , , hence
If , then
From assumption (H2), we have .
It can be divided into two situations:
Therefore, we have for , that is to say, condition (i) of Lemma 2.4 is satisfied.
Since , we have
Thus condition (ii) of Lemma 2.4 is satisfied.
Obviously, , so . We assume and hold.
From assumption (H3). we have
Thus condition (iii) of Lemma 2.4 is also satisfied. From the above facts, the proof of Theorem 3.1 is completed.
Example 4.1 Consider the following boundary value problem
By the precise calculation, we have
All the conditions of theorem 3.1 are satisfied, so there are at least three positive solutions for the system.
In this paper, applying the fixed point theorem on the cone, we investigate the existence of positive solutions for a class of third-order three-point boundary value problem, which is a more general system. We obtain that the boundary value problem has at least three positive solutions.
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