Existence Theory for Single Positive Solution to Fourth-Order Boundary Value Problems ()
1. Introduction
Fourth-order differential equations play an important role in various fields of science and engineering. With the help of boundary value conditions, we can describe the natural phenomena and mathematical model more accurately. Therefore, the fourth-order differential equations have received much attention and the theory and application have been greatly developed (see [1] -[4] and their references). Most of the results told us that the equations had at least single and multiple positive solutions. In papers [1] -[3] , the authors obtained some newest results for the singular fourth-order boundary value problems. But there is no result on the uniqueness of solution in them.
In this paper, we consider the following singular fourth-order boundary value problem:
(1.1)
Throughout this paper, we always suppose that
Moreover, 
may be singular at
, 
, or
.
Equation (1.1) is often referred to as the deformation for an elastic beam under a variety of boundary conditions. A brief discussion of the physical interpretation under some boundary conditions associated with the linear beam equation can be found in Zill and Cullen [5] . In this article, we consider the existence and uniqueness of positive solutions for fourth-order singular boundary value problems by using mixed monotone method.
2. Preliminary
Let P be a normal cone of a Banach space E, and 
with
,
. Define
Now we give a definition(see [7] ).
Definition 2.1. Assume
. A is said to be mixed monotone if 
is nondecreasing in x and nonincreasing in y, i.e. if 
implies 
for any
, and 
implies 
for any
. 
is said to be a fixed point of A if
.
Theorem 2.1. Suppose that 
is a mixed monotone operator and 
a constant
, 
, such that
(2.1)
Then A has a unique fixed point
. Moreover, for any 
satisfy
where
, r is a constant from
.
Theorem 2.2. (See [7] ): Suppose that 
is a mixed monotone operator and 
a constant 
such that (2.1) holds. If 
is a unique solution of equation
in
, then
,
. If
, then 
implies
, 
, and
3. Uniqueness Positive Solution of Problem (1.1)
This section discusses the problem
Throughout this section, we assume that
(3.1)
where
(3.2)
Let 
and
,
. We denote the Green’s functions for the following boundary value problems
and
by 
and
, respectively. It is well known that 
and 
can be written by
where 
and
Lemma 3.1. Suppose that 
holds, then the Green’s function
, possesses the following properties:
1) 
is increasing and
,
.
2) 
is decreasing and
,
.
3)
.
4)
.
5) 
is a positive constant. Moreover,
.
6) 
is continuous and symmetrical over Q.
7) 
has continuously partial derivative over
,
.
8) For each fixed
, 
satisfies 
for
,
. Moreover, 
for
.
9) 
has discontinuous point of the first kind at 
and
Following from Lemma 3.1, it is easy to see that
(a) 
(b) 
Suppose that x is a positive solution of (1.1). Then
(3.3)
By using (3.3) and (a), we see that for every positive solution x one has
where
. Let
Thus by (3.3) one has
by (a) one has
(3.4)
Let
. Obviously, P is a normal cone of Banach space
.
Theorem 3.1. Suppose that there exists 
such that
(3.5)
for any 
and
, and 
satisfies
(3.6)
Then (1.1) has a unique positive solution
. And moreover, 
implies
,
. If
, then
Proof. Since (3.5) holds, let
, one has
then
(3.7)
Let
. The above inequality is
(3.8)
From (3.5), (3.7) and (3.8), one has
(3.9)
Similarly, from (3.5), one has
(3.10)
Let
, 
, one has
(3.11)
Let
. It is clear that
, and now let
(3.12)
where 
is chosen such that
For any
, we define
(3.13)
First we show that
. Let
, from (3.10) and (3.11) we have
and from (3.9) we have
(3.14)
Then from (3.4) and (3.13) we have
On the other hand, for any
, from (3.9) and (3.10), we have
(3.15)
Thus, from (3.15), we have
So, 
is well defined and 
Next, for any
, one has
So the conditions of Theorems 2.1 and 2.2 hold. Therefore there exists a unique 
such that
. It is easy to check that 
is a unique positive solution of (1.1) for given
. MoreoverTheorem 2.2 means that if 
then
, 
, and if
, then
This completes the proof.
Example. Consider the following singular fourth-order boundary value problem:
where
, satisfies
.
Let
Thus 
and for any 
, 
,
Now Theorem 3.1 guarantees that the above equation has a positive solution.
Funding
Project was supported by Heilongjiang Province Education Department Natural Science Research Item, China (12541076).