Countably Many Positive Solutions for Nonlinear Singular n-Point Boundary Value Problems

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.


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.

( ) [ ] [ )
has countable many singularities in 1 0, 2 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.
. Suppose that there exists a completely continuous operator ( ) x P a α ∈ ∂ . Then T has at least three fixed points ( ) 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.

The Preliminary Lemmas
In this paper, we will use the following space is a non-increasing and nonnegative concave function on For convenience, let us list some conditions.
and on any subinterval of J and when u is bounded, has a unique solution The proof is easy, so we omit it.
Then we have (2.4) and the concavity of ( ) u t , we can easily get the following lemma.

Au t Au t t s a s f u s s t s a s f u s s t a s f u s s t t a s f u s s sa s f u s s
At last, by (2.5), ( ) 2 H , the Lebesgue dominated convergence theorem and continuity of f , we know A is continuous. Then by the Arzela-Ascoli theorem, we can get that : is completely continuous.

Main Results
Let ( ) In the following, we let ( ) ( ) ( ) Then it is easy to see ρ η > . The main result of this paper is as follows. , and Proof. From the definition of A , (2.7) and Lemma 2.4, it is easy to see that Next we show all the conditions of Lemma 1.2 hold. For any u K ∈ , it is easy to see Again from ( ) u t u ≤ , and Lemma (2.2) we can get that 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.

Example
Now we consider an example to illustrate our results.