- Research Article
- Open Access

# The Existence of Countably Many Positive Solutions for Nonlinear th-Order Three-Point Boundary Value Problems

- Yude Ji
^{1}Email author and - Yanping Guo
^{1}

**Received:**5 July 2009**Accepted:**30 October 2009**Published:**4 November 2009

## Abstract

We consider the existence of countably many positive solutions for nonlinear th-order three-point boundary value problem , , , , , where , for some and has countably many singularities in . The associated Green's function for the th-order three-point boundary value problem is first given, and growth conditions are imposed on nonlinearity which yield the existence of countably many positive solutions by using the Krasnosel'skii fixed point theorem and Leggett-Williams fixed point theorem for operators on a cone.

## Keywords

- Banach Space
- Unique Solution
- Ordinary Differential Equation
- Functional Equation
- Open Subset

## 1. Introduction

The existence of positive solutions for nonlinear second-order and higher-order multipoint boundary value problems has been studied by several authors, for example, see [1–12] and the references therein. However, there are a few papers dealing with the existence of positive solutions for the th-order multipoint boundary value problems with infinitely many singularities. Hao et al. [13] discussed the existence and multiplicity of positive solutions for the following th-order nonlinear singular boundary value problems:

where , , may be singular at and/or . Hao et al. established the existence of at least two positive solution for the boundary value problems if is either superlinear or sublinear by applying the Krasnosel'skii-Guo theorem on cone expansion and compression.

In [14], Kaufmann and Kosmatov showed that there exist countably many positive solutions for the two-point boundary value problems with infinitely many singularities of following form:

where for some and has countably many singularities in .

In [15], Ji and Guo proved the existence of countably many positive solutions for the th-order ordinary differential equation

with one of the following -point boundary conditions:

where , ( ), , ( , ), , for some and has countably many singularities in .

Motivated by the result of [13–15], in this paper we are interested in the existence of countably many positive solutions for nonlinear th-order three-point boundary value problem

where , , , , , ( , ), , for some and has countably many singularities in . We show that the problem (1.5) has countably many solutions if and satisfy some suitable conditions. Our approach is based on the Krasnosel'skii fixed point theorem and Leggett-Williams fixed point theorem in cones.

Suppose that the following conditions are satisfied.

There exists a sequence such that , , , and for all

There exists such that for all .

Assuming that satisfies the conditions - (we cite [15, Example ] to verify existence of ) and imposing growth conditions on the nonlinearity , it will be shown that problem (1.5) has infinitely many solutions.

The paper is organized as follows. In Section 2, we provide some necessary background material such as the Krasnosel'skii fixed-point theorem and Leggett-Williams fixed point theorem in cones. In Section 3, the associated Green's function for the th-order three-point boundary value problem is first given and we also look at some properties of the Green's function associated with problem (1.5). In Section 4, we prove the existence of countably many positive solutions for problem (1.5) under suitable conditions on and . In Section 5, we give two simple examples to illustrate the applications of obtained results.

## 2. Preliminary Results

Definition 2.1.

Let be a Banach space over . A nonempty convex closed set is said to be a cone provided that

(i) for all and for all ;

(ii) implies .

Definition 2.2.

for all and

Definition 2.3.

The following Krasnosel'skii fixed point theorem and Leggett-Williams fixed point theorem play an important role in this paper.

Theorem 2.4 ([16], Krasnosel'skii fixed point theorem).

is a completely continuous operator such that, either

(i) , and , or

(ii) , and .

Then has a fixed point in .

Theorem 2.5 ([17], Leggett-Williams fixed point theorem).

Let be a completely continuous operator and let be a nonnegative continuous concave functional on such that for all . Suppose there exist such that

, and for ,

for ,

for , with .

In order to establish some of the norm inequalities in Theorems 2.4 and 2.5 we will need Holder's inequality. We use standard notation of for the space of measurable functions such that

where the integral is understood in the Lebesgue sense. The norm on , is defined by

Theorem 2.6 ([18], Holder's inequality).

## 3. Preliminary Lemmas

To prove the main results, we need the following lemmas.

Lemma 3.1 (see [15]).

Lemma 3.2 (see [15]).

Lemma 3.3 (see [15]).

The Green's function defined by (3.4) satisfies that

(i) is continuous on ;

(ii) for all and there exists a constant for any such that

Lemma 3.4.

Proof.

Lemma 3.5.

where is defined by (3.4).

We omit the proof as it is immediate from Lemma 3.4 and (3.4).

Lemma 3.6.

Suppose , the Green's function defined by (3.14) satisfies that

(i) is continuous on ;

(ii) for all and there exists a constant for any such that

Proof.

Next, we prove that (3.15) holds.

for all , where , .

Obviously, is a solution of (1.5) if and only if is a fixed point of operator .

Theorems 2.4 and 2.5 require the operator to be completely continuous and cone preserving. If is continuous and compact, then it is completely continuous. The next lemma shows that for and that is continuous and compact.

Lemma 3.7.

The operator is completely continuous and for each .

Proof.

Fix . Since for all , and since for all , then for all .

Clearly operator (3.21) is continuous. By the Arzela-Ascoli theorem is compact. Hence, the operator is completely continuous and the proof is complete.

## 4. Main Results

In this section we present that problem (1.5) has countably many solutions if and satisfy some suitable conditions.

For convenience, we denote

Theorem 4.1.

where , . Furthermore, for each natural number , assume that satisfies the following two growth conditions:

for all ,

for all .

Then problem (1.5) has countably many positive solutions such that for each

Proof.

It is obvious that . Therefore, by Theorem 2.4, the operator has at least one fixed point such that . Since was arbitrary, Theorem 4.1 is completed.

Let is defined by Theorem 4.1. We define the nonnegative continuous concave functionals on by

We observe here that, for each , .

For convenience, we denote

Theorem 4.2.

where , . Furthermore, for each natural number , assume that satisfies the following growth conditions:

for all ,

for all ,

for all .

for each

Proof.

We note first that is completely continuous operator. If , then from properties of , , and by Lemma 3.7, . Consequently, .

Therefore, . Standard applications of Arzela-Ascoli theorem imply that is completely continuous operator.

In a completely analogous argument, condition implies that condition of Theorem 2.5 is satisfied.

Therefore, condition of Theorem 2.5 is satisfied.

Finally, we show that condition of Theorem 2.5 is also satisfied.

for each Thus, Theorem 4.2 is completed.

## 5. Example

In this section, we cite an example (see [15]) to verify existence of , and two simple examples are presented to illustrate the applications for obtained conclusion of Theorems 4.1 and 4.2.

Example 5.1.

We notice that , , , .

If we take , , , then , and = = min , =

Then all the conditions of Theorem 4.1 are satisfied. Therefore, by Theorem 4.1 we know that problem (5.1) has countably many positive solutions such that for each

Example 5.2.

We notice that , , , .

If we take = , = , = , = then , and = , = , =

Then all the conditions of Theorem 4.2 are satisfied. Therefore, by Theorem 4.2 we know that problem (5.7) has countably many positive solutions such that

for each

Remark 5.3.

In [8–12], the existence of solutions for local or nonlocal boundary value problems of higher-order nonlinear ordinary (fractional) differential equations that has been treated did not discuss problems with singularities. In [13], the singularity only allowed to appear at and/or , the existence and multiplicity of positive solutions were asserted under suitable conditions on . Although, [14, 15] seem to have considered the existence of countably many positive solutions for the second-order and higher-order boundary value problems with infinitely many singularities in . However, in [15], only the boundary conditions or have been considered. It is clear that the boundary conditions of Examples 5.1 and 5.2 are and . Hence, we generalize second-order and higher-order multipoint boundary value problem.

## Declarations

### Acknowledgments

The project is supported by the Natural Science Foundation of Hebei Province (A2009000664), the Foundation of Hebei Education Department (2008153), the Foundation of Hebei University of Science and Technology (XL2006040), and the National Natural Science Foundation of PR China (10971045).

## Authors’ Affiliations

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