Existence of Multiple Positive Solutions for Third-Order Three-Point Boundary Value Problem ()
1. Introduction
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 [1] [2] [3] [4] [5] . Many works on the third-order boundary value problems have been established. In [6] [7] [8] [9] [10] , 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 [11] obtained some existence results for positive solutions for the following system:
(1.1)
where
is continuous, f is nonnegative for
and
.
Moreover, Yao [12] considered the following system:
(1.2)
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. [2] considered the existence of a positive solution to the third-order three-point boundary value problem as follows
(1.3)
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
(1.4)
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 [13] . 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.
2. Preliminaries
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 [2] Assume
, then the system
(1.5)
has a unique solution
for
,
where
(1.6)
If we denote
, then we have the following lemma.
Lemma 2.2 [2] Let
, then
1)
, for any
;
2i)
, for any
,
where
.
For positive real numbers
, we define the following convex sets:
Lemma 2.3 [14] (Arzela-Ascoli theorem) Let
be a operator, then
is sequentially compact in
if and only if
is uniformly bounded and equicontinuous.
Lemma 2.4 [5] (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
(1.7)
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
Suppose
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
so
and
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
(H1)
,
(H2)
,
(H3)
,
then the system (1.4) has at least three positive points
and
satisfying
;
;
;
, for
.
Proof For
, we have
so
Since
which also implies that
therefore,
So we show that (1.7) of the Lemma 2.4 holds.
If
, we have
.
And
for any
. From assumption (H1), we have
, therefore,
hence
.
Let
, it is easy to prove
,
, hence
.
If
, then
From assumption (H2), we have
.
It can be divided into two situations:
(i)
,
(ii)
,
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.
4. Example
Example 4.1 Consider the following boundary value problem
where,
where
.
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.
5. Conclusion
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.