ABSTRACT

In this paper, a fixed-point theorem has been used to investigate the existence of countable positive solutions of n-point boundary value problem. As an application, we also give an example to demonstrate our results.

Keywords:Boundary Value Problem, Existence of Positive Solutions, Fixed-Point Theorem

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

for all and.

Definition 1.2 Given a nonnegative continuous function on a cone, for each, we define the set

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

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

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.

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

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

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

Obviously, for we have.

In the following, we let

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..

Furthermore for each natural number we assume that satisfies:

for all

for all

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

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

Then, for. By, we have, for.

So, there has

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

It is easy to know satisfies.

Let, , be such that , be such that, and .

This with implies that, ,.

Let

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.

References

NOTES

^*Corresponding author.