1. Introduction
The multi-point boundary value problems arising from applied mathematics and physics have received a great deal of attention in the literature (for instance, [1] -[4] and references therein). But, by so far, few results are about the existence of more than five solutions. To the author’s knowledge, there are very few papers concerned with the existence of countable positive solutions for multiple point BVPS (for instance, [5] and references therein). In [5] , the authors discussed the existence of countable positive solutions of n-point boundary value problems for a p-Laplace operator on the half-line. Directly inspired by [5] , in this paper, by using a fixed-point theorem, we study the existence of countable positive solutions of the following n-point boundary value problems.
(1.1)
(1.2)
where
,
,
,
.
and
has countable many singularities in
.
This kind of problem arises in the study of a number of chemotherapy, population dynamics, ecology, industrial robotics and physics phenomena. Moreover, many problems in optimal control system, neural network (for example in BAM neural network) and information systems for computational science and engineering (especially in Internet-based computing) can be established as differential equation models with boundary condition (see, for instance, [6] and references therein).
At the end of this section, we state some definitions and lemmas which will be used in Section 2 and Section 3.
Definition 1.1 A map
is said to be a nonnegative, continuous, concave function on a cone
of a real Banach space
, if
is continuous, and
![](https://www.scirp.org/html/htmlimages\10-7402112x\bda51b55-d123-4108-add8-631a3d19701e.png)
for all
and
.
Definition 1.2 Given a nonnegative continuous function
on a cone
, for each
, we define the set ![](https://www.scirp.org/html/htmlimages\10-7402112x\55daeccc-febd-4aa5-a987-0343c8286405.png)
Lemma 1.1 [7] Let
be a Banach space and
be a cone in
. Let
, be three increasing, nonnegative and continuous functions on
, satisfying for some
and
such that
![](https://www.scirp.org/html/htmlimages\10-7402112x\cb1197ea-bae6-47a4-b0fe-ef6c29c21443.png)
for all
. Suppose that there exists a completely continuous operator
and
such that 1)
, for
.
2)
, for
.
3)
, and
, for
.
Then
has at least three fixed points
such that
![](https://www.scirp.org/html/htmlimages\10-7402112x\522f9555-b6d8-450a-bbcb-9f127419cbdb.png)
This paper is organized as follows: The preliminary lemmas are in Section 2. The main results are given in Section 3. Finally, in Section 4, we give an example to demonstrate our results.
2. The Preliminary Lemmas
In this paper, we will use the following space
and
is a Banach space with the norm
. Let
, we define a cone
by
.
For convenience, let us list some conditions.
![](https://www.scirp.org/html/htmlimages\10-7402112x\95785ebf-8fb5-4a51-828e-e0a4d1353906.png)
and on any subinterval of
and when
is bounded,
is bounded on
.
There exists a sequence
such that
,
,
,
, and
.
Lemma 2.1. Let
,
and
on
, then the boundary value problem
(2.1)
(2.2)
has a unique solution
![](https://www.scirp.org/html/htmlimages\10-7402112x\14ae9439-4299-4cdb-a45d-66727fe1d601.png)
Proof. The proof is easy, so we omit it.
By
, we know
is decreasing and concave on
. Then we have
(2.3)
(2.4)
From (2.3), (2.4) and the concavity of
, we can easily get the following lemma.
Lemma 2.2. Let
, if
and
, then the unique solution
of (2.1)-(2.2) satisfies
and
, where
.
For
, we define an operator
by
(2.5)
For
, then
, by
, we know
is bounded on
.
So there exists
, such that
. (2.6)
It is easy to see that
is decreasing and concave on
. Then for
, we have
, that is
. (2.7)
From
, (2.3) and (2.6), we have
(2.8)
From (2.7), (2.8), we can get the following lemma.
Lemma 2.3. Suppose
and
are satisfied. Then
is bounded.
Lemma 2.4. Assume
,
are satisfied, then
is completely continuous.
Proof. From Lemma 2.2, we know
is bounded. If
is a bounded subset of
, then
is uniformly bounded on
.
For any
,
, without loss generality, we may assume
, by (2.5), (2.6),
, we have
![](https://www.scirp.org/html/htmlimages\10-7402112x\35afc137-64f6-40ce-9f5d-cfcbeda96a1c.png)
uniformly as
.
So
is equi-continuous on
.
At last, by (2.5),
, the Lebesgue dominated convergence theorem and continuity of
, we know
is continuous. Then by the Arzela-Ascoli theorem, we can get that
is completely continuous.
3. Main Results
Let
,
and
be three nonnegative, decreasing and continuous functions with
![](https://www.scirp.org/html/htmlimages\10-7402112x\49d1a1b7-a8d7-49c0-9d22-d827fbca5b9c.png)
Obviously, for
we have
.
In the following, we let
![](https://www.scirp.org/html/htmlimages\10-7402112x\67bd890e-57a9-41f4-b32c-b64328da26b9.png)
Then it is easy to see
.
The main result of this paper is as follows.
Theorem 3.1. Assume that
hold. Let
be such that
,
be such that
and![](https://www.scirp.org/html/htmlimages\10-7402112x\bfa5a5f8-9f34-430b-b07e-e8a039363b03.png)
.
.
Furthermore for each natural number
we assume that
satisfies:
for all ![](https://www.scirp.org/html/htmlimages\10-7402112x\536c2d27-b95a-4b0e-9977-d697a8aa38d0.png)
for all ![](https://www.scirp.org/html/htmlimages\10-7402112x\1e9e3751-b4a7-4ab1-80cc-25941526ef4a.png)
for all
.
Then the BVP (1.1)-(1.2) has at least three infinite families of positive solutions
with
,
,
, for
.
Proof. From the definition of
, (2.7) and Lemma 2.4, it is easy to see that
, for
is completely continuous.
Next we show all the conditions of Lemma 1.2 hold.
For any
, it is easy to see
. From Lemma 2.2, we have
, so
(3.1)
First, we choose
, then we have
. From
and (3.1), we can get
, for
. Then with
, it implies that
, for
.
So ![](https://www.scirp.org/html/htmlimages\10-7402112x\56d85205-1ea4-45fa-bc6f-eb915e30be51.png)
Therefore, the first condition of Lemma 1.2 satisfies.
Next, we select
. Then
, we have
, for
.
Again from
, and Lemma (2.2) we can get that
![](https://www.scirp.org/html/htmlimages\10-7402112x\d61e9d10-6894-42c5-87ca-a0ebfe2be0f9.png)
Then
, for
. By
, we have
, for
.
So, there has
![](https://www.scirp.org/html/htmlimages\10-7402112x\5225cef2-ff7d-434d-8f56-ab6b4629c0a1.png)
This implies the second condition of Lemma 1.2 is satisfied.
Finally, we only need to show the third condition of Lemma 1.2 is also satisfied.
We select
, for
. Obviously,
, hence
is nonempty.
, we have
. Also from
and Lemma (2.4), we can get
, for
. Then from
, we have
.
So
.
Then all the conditions of Lemma 1.2 are satisfied. From Lemma 1.2, we get the conclusion in Theorem 3.1.
4. Example
Now we consider an example to illustrate our results.
Example 4.1. Consider the boundary value problem
, (4.1)
, (4.2)
Then the BVP (4.1)-(4.2) can be regarded as a BVP of the form (1.1)-(1.2) in
. In this situation,
.
Let
,
,
.
Consider the function
, where
![](https://www.scirp.org/html/htmlimages\10-7402112x\df91c3e9-de85-49b0-9743-6c02e5c521a1.png)
![](https://www.scirp.org/html/htmlimages\10-7402112x\182c5295-ec58-4166-8f36-31da30a43d3a.png)
It is easy to know
satisfies.
Let
,
,
be such that
,
be such that
, and
.
This with
implies that
,
,
.
Let ![](https://www.scirp.org/html/htmlimages\10-7402112x\14dadc1f-8cf8-4472-930a-32189ceaaa4d.png)
Obviously,
are satisfied, and it is easy to prove that
is also satisfied. So all the conditions of Theorem 3.1 are satisfied, thus the BVP (4.1)-(4.2) has at least three infinite families of positive solutions
satisfying
,
,
, for
.
NOTES
*Corresponding author.