- Research Article
- Open Access

# Multiple Positive Solutions for th Order Multipoint Boundary Value Problem

- Yaohong Li
^{1, 2}Email author and - Zhongli Wei
^{2, 3}

**Received:**22 January 2010**Accepted:**3 June 2010**Published:**20 June 2010

## Abstract

We study the existence of multiple positive solutions for th-order multipoint boundary value problem. , , , , where , , . We obtained the existence of multiple positive solutions by applying the fixed point theorems of cone expansion and compression of norm type and Leggett-Williams fixed-point theorem. The results obtained in this paper are different from those in the literature.

## Keywords

- Fixed Point Theorem
- Existence Condition
- Real Banach Space
- Norm Type
- Point Index

## 1. Introduction

where

By using the fixed point theorems of cone expansion and compression of norm type, Yang and Wei in [2] also obtained the existence of at least one positive solutions for the BVP (1.1) if . This work is motivated by Ma (see [3]). This method is simpler than that of [1]. In addition, Eloe and Ahmad in [4] had solved successfully the existence of positive solutions to the BVP (1.1) if = 1. Hao et al. in [5] had discussed the existence of at least two positive solutions for the BVP (1.1) by applying the Krasonse'skii-Guo fixed point theorem on cone expansion and compression if = 1. However, there are few papers dealing with the existence of multiple positive solutions for th-order multipoint boundary value problem.

In this paper, we study the existence of at least two positive solutions associated with the BVP (1.1) by applying the fixed point theorems of cone expansion and compression of norm type if and the existence of at least three positive solutions for BVP (1.1) by using Leggett-Williams fixed-point theorem. The results obtained in this paper are different from those in the literature and essentially improve and generalize some well-known results (see [1–8]).

The rest of the paper is organized as follows. In Section 2, we present several lemmas. In Section 3, we give some preliminaries and the fixed point theorems of cone expansion and compression of norm type. The existence of at least two positive solutions for the BVP (1.1) is formulated and proved in Section 4. In Section 5, we give Leggett-Williams fixed-point theorem and obtain the existence of at least three positive solutions for the BVP (1.1).

## 2. Several Lemmas

Definition 2.1.

A function is said to be a position of the BVP (1.1) if satisfies the following:

() ;

() for and satisfies boundary value conditions (1.1);

() hold for

Lemma 2.2 (see [1]).

Lemma 2.3 (see [1]).

We omit the proof Lemma 2.3 here and you can see the detail of Theorem in [1].

Lemma 2.4 (see [2]).

Let , and ; the unique solution of the BVP (2.2)

where is defined by Lemma 2.3, .

## 3. Preliminaries

Obviously, is a positive cone in , where is from Lemma 2.3.

For convenience, we make the following assumptions:

() is continuous and does not vanish identically, for ;

() is continuous;

()

where is defined by (2.4).

From Lemmas 2.2–2.4, we have the following result.

Lemma 3.1 (see [2]).

Suppose that are satisfied, then is a completely continuous operator, , and the fixed points of the operator in are the positive solutions of the BVP (1.1).

Problem 1.

The aim of the following section is to establish some simple criteria for the existence of multiple positive solutions of the BVP (1.1), which gives a positive answer to the questions stated above. The key tool in our approach is the following fixed point theorem, which is a useful method to prove the existence of positive solutions for differential equations, for example [2–5, 9].

Suppose that is a real Banach space and is cone in , and let be two bounded open sets in such that . Let operator be completely continuous. Suppose that one of two conditions holds:

(i) for all for all

(ii) for all for all .

then has at least one fixed point in

## 4. The Existence of Two Positive Solutions

Theorem 4.1.

Suppose that the conditions are satisfied and the following assumptions hold:

() ;

() ;

()There exists a constant such that .

Proof.

The proof is complete.

Corollary 4.2.

Suppose that the conditions are satisfied and the following assumptions hold:

() ;

() ;

()there exists a constant such that

Proof.

Theorem 4.3.

Suppose that the conditions are satisfied and the following assumptions hold:

() ;

() ;

()there exists a constant such that .

Proof.

It follows from the condition that there exists such that for and we consider two cases.

Case i.

Case ii.

The proof is complete.

Corollary 4.4.

Suppose that the conditions are satisfied and the following assumptions hold:

() ;

() ;

()there exists a constant such that .

The proof of Corollary 4.4 is similar to that of Corollary 4.2; so we omit it.

## 5. The Existence of Three Positive Solutions

Lemma 5.1 (see [12]).

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

(i) and for

(ii) for

(iii) for with

Now, we establish the existence conditions of three positive solutions for the BVP (1.1).

Theorem 5.2.

Suppose that hold and there exist numbers and with such that the following conditions are satisfied:

Then the boundary value problem (1.1) has at least three positive solutions.

Proof.

Then it is easy to check that is a nonnegative continuous concave functional on with for and is completely continuous.

Hence (5.10) show that if holds, then there exists a number such that maps into .

Hence, for .

The proof is complete.

## Declarations

### Acknowledgments

The authors are grateful to the referee's valuable comments and suggestions. The project is supported by the Natural Science Foundation of Anhui Province (KJ2010B226), The Excellent Youth Foundation of Anhui Province Office of Education (2009SQRZ169), and the Natural Science Foundation of Suzhou University (2009yzk17)

## Authors’ Affiliations

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