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
![](https://www.scirp.org/html/htmlimages\8-5300768x\420d4b23-3302-4c5d-90d6-435bb07aed5d.png)
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
![](https://www.scirp.org/html/htmlimages\8-5300768x\98dbc60d-0880-48e5-bdb4-c3f99d9798aa.png)
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 ![](https://www.scirp.org/html/htmlimages\8-5300768x\86d5a3ef-bec4-4ca1-ad25-e20549258f59.png)
![](https://www.scirp.org/html/htmlimages\8-5300768x\a8fb886d-1d49-42c0-9fdf-4ffb18af90b7.png)
satisfy
![](https://www.scirp.org/html/htmlimages\8-5300768x\11521f13-3870-4674-9981-cc7869d3085d.png)
where
![](https://www.scirp.org/html/htmlimages\8-5300768x\e375e5f0-6ba5-47e2-958e-5da7e92666ee.png)
, 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
![](https://www.scirp.org/html/htmlimages\8-5300768x\7cd25338-1162-47af-957a-96cc0cb8601a.png)
in
, then
,
. If
, then
implies
,
, and
![](https://www.scirp.org/html/htmlimages\8-5300768x\0a29398a-7f97-4f8e-92fa-3597ac5d4826.png)
3. Uniqueness Positive Solution of Problem (1.1)
This section discusses the problem
![](https://www.scirp.org/html/htmlimages\8-5300768x\68e282b3-0b8e-4948-b5a0-21b0210b4cfe.png)
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
![](https://www.scirp.org/html/htmlimages\8-5300768x\516961e3-a4b8-43d8-9626-d33ea5d7887a.png)
and
![](https://www.scirp.org/html/htmlimages\8-5300768x\d8ce36e0-fb9e-4736-8de9-9748a16b014b.png)
by
and
, respectively. It is well known that
and
can be written by
![](https://www.scirp.org/html/htmlimages\8-5300768x\43714f32-5920-47e3-8947-2dd875bacb1d.png)
where
and
![](https://www.scirp.org/html/htmlimages\8-5300768x\48c7d4b4-8316-4da7-a459-cdd1c2a0f4b8.png)
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
![](https://www.scirp.org/html/htmlimages\8-5300768x\b123339a-1596-4323-81af-1b45e9f6449a.png)
Following from Lemma 3.1, it is easy to see that
(a) ![](https://www.scirp.org/html/htmlimages\8-5300768x\842f152a-9843-4865-a716-b14513d6ca85.png)
![](https://www.scirp.org/html/htmlimages\8-5300768x\7a4a9092-0eb4-40cd-9c13-9c4c1b13310f.png)
(b) ![](https://www.scirp.org/html/htmlimages\8-5300768x\e7ea6c4b-ae29-4ea3-a27c-258f5feae5d6.png)
![](https://www.scirp.org/html/htmlimages\8-5300768x\4991e4cb-a623-4c8f-867a-e8d6a2cfcc6d.png)
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
![](https://www.scirp.org/html/htmlimages\8-5300768x\f292cf36-7e58-4a2e-95ad-76f28f6af222.png)
where
. Let
![](https://www.scirp.org/html/htmlimages\8-5300768x\0a90dd06-4f30-480a-9a16-69779b48c5a9.png)
Thus by (3.3) one has
![](https://www.scirp.org/html/htmlimages\8-5300768x\acb1c9e4-a9ce-4a33-af08-f6010e8f2dd2.png)
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
![](https://www.scirp.org/html/htmlimages\8-5300768x\3e60fd6b-7dc5-41d1-895a-5d12f0007112.png)
Proof. Since (3.5) holds, let
, one has
![](https://www.scirp.org/html/htmlimages\8-5300768x\623a103b-38b7-4760-ba2f-cfb599ffb7d6.png)
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
![](https://www.scirp.org/html/htmlimages\8-5300768x\6146c3d1-aaba-40bd-8a2b-3c5511a7dc23.png)
For any
, we define
(3.13)
First we show that
. Let
, from (3.10) and (3.11) we have
![](https://www.scirp.org/html/htmlimages\8-5300768x\9d9bdcc9-0ea2-4ae2-95d6-93ad384f52a3.png)
and from (3.9) we have
(3.14)
Then from (3.4) and (3.13) we have
![](https://www.scirp.org/html/htmlimages\8-5300768x\795c846f-6fac-43cd-9c62-86c0918a9cb9.png)
On the other hand, for any
, from (3.9) and (3.10), we have
(3.15)
Thus, from (3.15), we have
![](https://www.scirp.org/html/htmlimages\8-5300768x\b9d3ff69-a7a5-4b9b-abca-2c5ccf512004.png)
So,
is well defined and ![](https://www.scirp.org/html/htmlimages\8-5300768x\9b75995a-011f-4a7c-ab93-185a6c41e10f.png)
Next, for any
, one has
![](https://www.scirp.org/html/htmlimages\8-5300768x\7c4be74d-d3c0-497a-88f7-6a881451f3ef.png)
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
![](https://www.scirp.org/html/htmlimages\8-5300768x\65c07297-ae85-4d93-ae7c-44dde2b2411e.png)
This completes the proof.
Example. Consider the following singular fourth-order boundary value problem:
![](https://www.scirp.org/html/htmlimages\8-5300768x\59cfc077-7c63-4107-8715-2bd3d3d24915.png)
where
, satisfies
.
Let
![](https://www.scirp.org/html/htmlimages\8-5300768x\147d7578-cb96-4d0f-a0f4-3d3f255761b4.png)
Thus
and for any
,
,
![](https://www.scirp.org/html/htmlimages\8-5300768x\16c5438d-26ec-48fb-a3a3-8e373502e395.png)
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).