- Research Article
- Open Access

# Existence and Uniqueness of Solutions for Higher-Order Three-Point Boundary Value Problems

- Minghe Pei
^{1}and - SungKag Chang
^{2}Email author

**2009**:362983

https://doi.org/10.1155/2009/362983

© M. Pei and S. K. Chang. 2009

**Received:**5 February 2009**Accepted:**14 July 2009**Published:**19 August 2009

## Abstract

We are concerned with the higher-order nonlinear three-point boundary value problems: with the three point boundary conditions ; where is continuous, are continuous, and are arbitrary given constants. The existence and uniqueness results are obtained by using the method of upper and lower solutions together with Leray-Schauder degree theory. We give two examples to demonstrate our result.

## Keywords

- Boundary Condition
- Continuous Function
- Unique Solution
- Ordinary Differential Equation
- Functional Equation

## 1. Introduction

Higher-order boundary value problems were discussed in many papers in recent years; for instance, see [1–22] and references therein. However, most of all the boundary conditions in the above-mentioned references are for two-point boundary conditions [2–11, 14, 17–22], and three-point boundary conditions are rarely seen [1, 12, 13, 16, 18]. Furthermore works for nonlinear three point boundary conditions are quite rare in literatures.

where , is a continuous function, are continuous functions, and are arbitrary given constants. The tools we mainly used are the method of upper and lower solutions and Leray-Schauder degree theory.

Note that for the cases of or in the boundary conditions (1.2), our theorems hold also true. However, for brevity we exclude such cases in this paper.

## 2. Preliminary

In this section, we present some definitions and lemmas that are needed to our main results.

Definition 2.1.

Definition 2.2.

Lemma 2.3 (see [10]).

satisfies

Lemma 2.4.

has only the trivial solution.

Proof.

which is a contradiction. Hence BVP (2.6), (2.7) has only the trivial solution.

## 3. Main Results

We may now formulate and prove our main results on the existence and uniqueness of solutions for -order three point boundary value problem (1.1), (1.2).

Theorem 3.1.

Assume that

(iii) is continuous on , and is nonincreasing in and nondecreasing in on ;

(iv) is continuous on , and nonincreasing in and nondecreasing in on

Proof.

where , .

In the following, we will complete the proof in four steps.

Step 1.

independently of .

There are three cases to consider.

Case 1 ( ).

Case 2 ( ).

Case 3 ( ).

are obtained by integration.

Step 2.

independently of .

By Step 1 and Lemma 2.3, there exists such that for . Since and do not depend on , the estimate on is also independent of .

Step 3.

Show that for , BVP (3.5), (3.6) has at least one solution .

Consider the set

has at least one solution in .

Step 4.

Show that is a solution of BVP (1.1), (1.2).

Now there are three cases to consider.

Case 1 ( ).

Case 2 ( ).

Case 3 ( ).

Hence is a solution of BVP (1.1), (1.2) and satisfies (3.3).

Now we give a uniqueness theorem by assuming additionally the differentiability for functions , and , and a kind of estimating condition in Theorem 3.1.

Theorem 3.2.

Assume that

(ii) and its first-order partial derivatives in are continuous on , on , on and satisfy the Nagumo condition on

(v)there exists a function such that on and

Then BVP (1.1), (1.2) has a unique solution satisfying (3.3).

Proof.

The existence of a solution for BVP (1.1), (1.2) satisfying (3.3) follows from Theorem 3.1.

where for each ,

which contradicts to (3.54). Thus . Similarly we can show that . Consequently .

which is a contradiction.

A similar contradiction can be obtained if . Hence on . By (3.55), we obtain on . This completes the proof of the theorem.

Next we give two examples to demonstrate the application of Theorem 3.2.

Example 3.3.

Example 3.4.

## Authors’ Affiliations

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