- Research Article
- Open Access

# Monotone Positive Solution of Nonlinear Third-Order BVP with Integral Boundary Conditions

- Jian-Ping Sun
^{1}Email author and - Hai-Bao Li
^{1}

**2010**:874959

https://doi.org/10.1155/2010/874959

Â© The Author(s) Jian-Ping Sun and Hai-Bao Li. 2010

**Received:**7 September 2010**Accepted:**31 October 2010**Published:**2 November 2010

## Abstract

This paper is concerned with the following third-order boundary value problem with integral boundary conditions , where and . By using the Guo-Krasnoselskii fixed-point theorem, some sufficient conditions are obtained for the existence and nonexistence of monotone positive solution to the above problem.

## Keywords

- Banach Space
- Unique Solution
- Dominate Convergence Theorem
- Convergent Subsequence
- 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].

For second-order or fourth-order BVPs with integral boundary conditions, one can refer to [20â€“24].

Throughout this paper, we always assume that and . Some sufficient conditions are established for the existence and nonexistence of monotone positive solution to the BVP (1.3). Here, a solution of the BVP (1.3) is said to be monotone and positive if , and for . Our main tool is the following Guo-Krasnoselskii fixed-point theorem [25].

Theorem 1.1.

Let be a Banach space and let be a cone in . Assume that and are bounded open subsets of such that , and let be a completely continuous operator such that either

(1) for and for , or

(2) for and for .

Then has a fixed point in .

## 2. Preliminaries

For convenience, we denote .

Lemma 2.1.

Proof.

Lemma 2.2 (see [12]).

Lemma 2.3 (see [26]).

In the remainder of this paper, we always assume that , and .

Lemma 2.4.

If and for , then the unique solution of the BVP (2.1) satisfies

(1) , ,

(2) , and , where .

Proof.

which indicates that for .

Obviously, if is a fixed point of , then is a monotone nonnegative solution of the BVP (1.3).

Lemma 2.5.

is completely continuous.

Proof.

First, by Lemma 2.4, we know that .

which implies that is equicontinuous. Again, by Arzela-Ascoli theorem, we know that has a convergent subsequence in . Therefore, has a convergent subsequence in . Thus, we have shown that is a compact operator.

which indicates that is continuous. Therefore, is completely continuous.

## 3. Main Results

Theorem 3.1.

If , then the BVP (1.3) has at least one monotone positive solution.

Proof.

Therefore, it follows from (3.6), (3.10), and Theorem 1.1 that the operator has one fixed point , which is a monotone positive solution of the BVP (1.3).

Theorem 3.2.

If , then the BVP (1.3) has at least one monotone positive solution.

Proof.

The proof is similar to that of Theorem 3.1 and is therefore omitted.

Theorem 3.3.

If for and , then the BVP (1.3) has no monotone positive solution.

Proof.

This is a contradiction. Therefore, the BVP (1.3) has no monotone positive solution.

Similarly, we can prove the following theorem.

Theorem 3.4.

If for and , then the BVP (1.3) has no monotone positive solution.

Example 3.5.

So, it follows from Theorem 3.1 that the BVP (3.14) has at least one monotone positive solution.

## Declarations

### Acknowledgment

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

## Authorsâ€™ Affiliations

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