Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 937670, 12 pages

http://dx.doi.org/10.1155/2012/937670

## Positive Solutions for a Class of Third-Order Three-Point Boundary Value Problem

School of Mathematical Sciences, Jiangsu Normal University, Xuzhou, Jiangsu 221116, China

Received 25 November 2011; Accepted 8 February 2012

Academic Editor: Yong Zhou

Copyright © 2012 Xiaojie Lin and Zhengmin Fu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We investigate the problem of existence of positive solutions for the nonlinear third-order three-point boundary value problem , , , , where is a positive parameter, , , , are continuous. Using a specially constructed cone, the fixed point index theorems and Leray-Schauder degree, this work shows the existence and multiplicities of positive solutions for the nonlinear third-order boundary value problem. Some examples are given to demonstrate the main results.

#### 1. Introduction

This paper deals with the following third-order nonlinear boundary value problem: Third-order boundary value problems arise in a variety of different areas of applied mathematics and physics. In the few years, there has been increasing interest in studying certain third-order boundary value problems for nonlinear differential equation and have received much attention. To identify a few, we refer the reader to [1–6].

Recently, El-Shahed [1] discussed the following third-order two-point boundary value problem: The methods employed in [1] are Kransnoselskii's fixed-point theorem of cone.

In later work, by placing restrictions on the nonlinear term , Sun [2] studied the following boundary value problems and obtained the three solution via leggett-williams fixed point theorem:

The upper and lower solution is a powerful tool for proving existence for boundary value problems, Ma [7] studied the multiplicity of positive solutions of three-point boundary value problem of second-order ordinary differential equations. Du et al. [5] investigated a class of third-order nonlinear problem.

Motivated by the work of the above papers, the purpose of this article is to study the existence of solution for boundary value problem (1.1) using a new technique (different from the proof of [1, 2, 7]) and we get a new existence result. The tools are based on the fixed point index theorems and Leray-Schauder degree.

The paper is organized as follows: Section 2 states some definitions and some lemmas which are important to obtain our main result. Section 3 is devoted to the existence result of BVP (1.1). Section 4 gives some examples to illustrate our main results.

#### 2. Preliminary

*Definition 2.1. *Let be a real Banach space. A nonempty closed convex set is called a cone of if it satisfies the following two conditions:(1), implies ;(2), implies .

*Definition 2.2. *An operator is called completely continuous if it is continuous and maps bounded sets into precompact sets.

Lemma 2.3. *Let , then the following boundary value problem:
**
has the unique solution
**
where
*

*Proof. *From (2.1), we have
In particular,
Combining this with boundary conditions (2.2), we conclude that
Therefore, BVP (2.1)-(2.2) has a unique solution:
The proof is completed.

Lemma 2.4. *For all , one has .*

Lemma 2.5. *for all , one has
**
where , and statisfies .*

*Proof. *For ,
For ,
For ,
For ,
Thus,
Therefore,
The proof is completed.

Lemma 2.6. *If and , then the unique solutionof the BVP (2.1)-(2.2) is nonnegative and satisfies
*

*Proof. *Let , it is obvious that it is nonnegative. For any , by (2.3) and Lemma 2.5, it follows that
and thus,
On the other hand, (2.3) and Lemma 2.5 imply, for any ,
Therefore,
This completes the proof.

Let with the usual normal .

Define the cone by

Define an operator by

By Lemma 2.3, BVP (1.1) has a positive solution if and only if is a fixed point of .

Lemma 2.7. *Assume that . Then, is completely continuous.*

*Proof. *Firstly, it is easy to check that is well defined. By Lemma 2.6, we know that .

Let be any boundary subset of , then there exists , , for all . Therefore, we have
So is boundary. Moreover, for any , we have
By the continuity of and , we have and are boundary on , which means that there exists a constant , depending only on such that
and thus for any ,
Therefore, we can get is equicontinuity. Thirdly, we prove that is continuous. Let as , . Then, the continuity of , we can get
Then, . Therefore, is continuous.The operator is completely continuous by an application of the Ascoli-Arzela theorem. This completes the proof.

Lemma 2.8 (see [7, 8]). *Let be a real Banach space and let be a cone in . For , define . Assume is a completely continuous operator such that for .*(1)*If for , then
*(2)*If for , then
*

#### 3. Main Results

Theorem 3.1. * Assume that *(A1)* is a positive parameter, and ;*(A2)* is continuous;*(A3)* is continuous;*(A4)*When is sufficiently small, (1.1) has at least one positive solution, whereas for is sufficiently large, (1.1) has no positive solution.*

* Proof. *If , then
For any number , let , and set
Then, for any , we have
Thus, Lemma 2.8 implies
Since , there is , such that , for , where is chosen so that
Let , and set
If , then
Therefore,
which implies that
for . An application of Lemma 2.8 again shows that
Since we can adjust so that , it follows the additivity of the fixed-point index that
Thus, has a fixed point in which is the desired positive solution of (1.1).

We verify that BVP of (1.1) has no positive solution for large enough.

Otherwise, there exist , with , such that for any positive integer , the BVP,
has a positive solution . By (2.22), we have
Thus,
Since , for , there exists , such that , for , which implies that
Let be large enough that , then
Choose so that which is a contradiction. The proof is completed.

Theorem 3.2. * Assume that *(B1)* is a positive parameter; and ;*(B2)* is continuous and there exists such that ; *(B3)* is continuous;*(B4)*;*(B5)*there exists , for , such that , where , then there exists , such that, for , BVP (1.1) has at least two positive solutions and .*

*Proof. *Let , then for , Lemma 2.7 implies that is completely continuous. Considering (B4), there exists such that , for , where .

So, for , we have from (2.4)

Consequently, for , we have , by Lemma 2.8,
Now considering (B4), there exists , for , such that . Letting , then
Choose
So for , from (3.18) and (3.19), we have
That is, by Lemma 2.8,
On the other hand, for , (2.3) and (2.4) yield that
Furthermore, for , from (2.3) and (2.4), we obtain
where . Let and , then is continuous, and from the analysis above, we obtain for :
Therefore, for , we have . Hence, by the normality property and the homotopy invariance property of the fixed point index, we obtain
Consequently, by the solution property of the fixed point index, has a fixed point and . By Lemma 2.4, it follows that is a solution to BVP (1.1), and
On the other hand, from (3.18) and (3.19) together with the additivity of the fixed point index, we get

Hence, by the solution property of the fixed point index, has a fixed point and . By Lemma 2.3, it follows that is also a solution to BVP (1.1), and . The proof is completed.

#### 4. Examples

*Example 4.1. *We consider the following third-order boundary value problems:
here is continuous, is continuous. By direct calculations, we obtain that for. Therefore, by Theorem 3.1, there exists at least one solution for BVP (4.1), whereas for large enough, (4.1) has no solution.

*Example 4.2. *Consider the following third-order ordinary differential equation:
where
is continuous, is continuous. Here, , . Choose , when , by Theorem 3.2, there exist at least two solutions for BVP (4.1).

#### Acknowledgment

This project is sponsored by the Natural Science Foundation of China (11101349, 11071205), the NFS of Jiangsu Province (BK2011042), the NSF of Education Department of Jiangsu Province (11KJB11003), and Jiangsu Government Scholarship Program.

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