- Research Article
- Open Access

# Existence of Solution and Positive Solution for a Nonlinear Third-Order -Point BVP

- Jian-Ping Sun
^{1}Email author and - Fan-Xia Jin
^{1}

**2010**:126192

https://doi.org/10.1155/2010/126192

© J.-P. Sun and F.-X. Jin. 2010

**Received:**5 November 2010**Accepted:**14 December 2010**Published:**19 December 2010

## Abstract

In this paper, we are concerned with the following nonlinear third-order -point boundary value problem: , , , , . Some existence criteria of solution and positive solution are established by using the Schauder fixed point theorem. An example is also included to illustrate the importance of the results obtained.

## Keywords

- Convex Subset
- Fixed Point Theorem
- Point Equation
- Nonnegative Solution
- Curve Beam

## 1. Introduction

Third-order differential equations arise in a variety of different areas of applied mathematics and physics, for example, in the deflection of a curved beam having a constant or varying cross-section, a three-layer beam, electromagnetic waves, or gravity-driven flows and so on [1].

The main tool used was Mawhin's continuation theorem.

Throughout, we always assume that and . The purpose of this paper is to consider the local properties of on some bounded sets and establish some existence criteria of solution and positive solution for the BVP (1.3) by using the Schauder fixed point theorem. An example is also included to illustrate the importance of the results obtained.

## 2. Main Results

Lemma 2.1.

Proof.

The following theorem guarantees the existence of solution for the BVP (1.3).

Theorem 2.2.

Proof.

Let be equipped with the norm , where . Then is a Banach space.

On the basis of Theorem 2.2, we now give some existence results of nonnegative solution and positive solution for the BVP (1.3).

Theorem 2.3.

Proof.

Corollary 2.4.

Assume that all the conditions of Theorem 2.3 are fulfilled. Then the BVP (1.3) has one positive solution if one of the following conditions is satisfied:

(i) ;

(ii) ;

(iii) , .

Proof.

which shows that is a positive solution of the BVP (1.3).

Example 2.5.

where , .

A simple calculation shows that and . Thus, if we choose and , then all the conditions of Theorem 2.3 and (i) of Corollary 2.4 are fulfilled. It follows from Corollary 2.4 that the BVP (2.35) has a positive solution.

## Declarations

### Acknowledgment

This paper was supported by the National Natural Science Foundation of China (10801068).

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. 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.