- 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]).

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.

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

Step 1.

There are three cases to consider.

are obtained by integration.

Step 2.

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 .

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.

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.

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

## References

- Aftabizadeh AR, Gupta CP, Xu J-M: Existence and uniqueness theorems for three-point boundary value problems.
*SIAM Journal on Mathematical Analysis*1989, 20(3):716–726. 10.1137/0520049MATHMathSciNetView ArticleGoogle Scholar - Agarwal RP:
*Boundary Value Problems for Higher Order Differential Equations*. World Scientific, Teaneck, NJ, USA; 1986:xii+307.MATHView ArticleGoogle Scholar - Agarwal RP, Wong F-H: Existence of positive solutions for non-positive higher-order BVPs.
*Journal of Computational and Applied Mathematics*1998, 88(1):3–14. 10.1016/S0377-0427(97)00211-2MATHMathSciNetView ArticleGoogle Scholar - Agarwal RP, Grace SR, O'Regan D: Semipositone higher-order differential equations.
*Applied Mathematics Letters*2004, 17(2):201–207. 10.1016/S0893-9659(04)90033-XMATHMathSciNetView ArticleGoogle Scholar - Cabada A: The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems.
*Journal of Mathematical Analysis and Applications*1994, 185(2):302–320. 10.1006/jmaa.1994.1250MATHMathSciNetView ArticleGoogle Scholar - Cabada A, Grossinho MR, Minhós F: Extremal solutions for third-order nonlinear problems with upper and lower solutions in reversed order.
*Nonlinear Analysis: Theory, Methods & Applications*2005, 62(6):1109–1121. 10.1016/j.na.2005.04.023MATHMathSciNetView ArticleGoogle Scholar - Du Z, Ge W, Lin X: Existence of solutions for a class of third-order nonlinear boundary value problems.
*Journal of Mathematical Analysis and Applications*2004, 294(1):104–112. 10.1016/j.jmaa.2004.02.001MATHMathSciNetView ArticleGoogle Scholar - Feng Y, Liu S: Solvability of a third-order two-point boundary value problem.
*Applied Mathematics Letters*2005, 18(9):1034–1040. 10.1016/j.aml.2004.04.016MATHMathSciNetView ArticleGoogle Scholar - Grossinho MR, Minhós FM: Existence result for some third order separated boundary value problems.
*Nonlinear Analysis: Theory, Methods & Applications*2001, 47(4):2407–2418. 10.1016/S0362-546X(01)00364-9MATHMathSciNetView ArticleGoogle Scholar - Grossinho MR, Minhós FM: Upper and lower solutions for higher order boundary value problems.
*Nonlinear Studies*2005, 12(2):165–176.MATHMathSciNetGoogle Scholar - Grossinho MR, Minhós FM, Santos AI: Solvability of some third-order boundary value problems with asymmetric unbounded nonlinearities.
*Nonlinear Analysis: Theory, Methods & Applications*2005, 62(7):1235–1250. 10.1016/j.na.2005.04.029MATHMathSciNetView ArticleGoogle Scholar - Gupta CP, Lakshmikantham V: Existence and uniqueness theorems for a third-order three-point boundary value problem.
*Nonlinear Analysis: Theory, Methods & Applications*1991, 16(11):949–957. 10.1016/0362-546X(91)90099-MMATHMathSciNetView ArticleGoogle Scholar - Henderson J, Taunton RD: Solutions of boundary value problems by matching methods.
*Applicable Analysis*1993, 49(3–4):235–246. 10.1080/00036819308840175MATHMathSciNetView ArticleGoogle Scholar - Lin Y, Pei M: Positive solutions for two-point semipositone right focal eigenvalue problem.
*Boundary Value Problems*2007, 2007:-12.Google Scholar - Moorti VRG, Garner JB: Existence-uniqueness theorems for three-point boundary value problems for nth-order nonlinear differential equations.
*Journal of Differential Equations*1978, 29(2):205–213. 10.1016/0022-0396(78)90120-1MATHMathSciNetView ArticleGoogle Scholar - Murty KN, Rao YS: A theory for existence and uniqueness of solutions to three-point boundary value problems.
*Journal of Mathematical Analysis and Applications*1992, 167(1):43–48. 10.1016/0022-247X(92)90232-3MATHMathSciNetView ArticleGoogle Scholar - Pei M, Chang SK: Existence and uniqueness of solutions for third-order nonlinear boundary value problems.
*Journal of Mathematical Analysis and Applications*2007, 327(1):23–35. 10.1016/j.jmaa.2006.03.057MATHMathSciNetView ArticleGoogle Scholar - Shi Y, Pei M: Two-point and three-point boundary value problems for nth-order nonlinear differential equations.
*Applicable Analysis*2006, 85(12):1421–1432. 10.1080/00036810601066061MATHMathSciNetView ArticleGoogle Scholar - Wang L, Pei M: Existence and uniqueness for nonlinear third-order two-point boundary value problems.
*Dynamics of Continuous, Discrete & Impulsive Systems. Series A*2007, 14(3):321–332.MathSciNetGoogle Scholar - Wong F-H: An application of Schauder's fixed point theorem with respect to higher order BVPs.
*Proceedings of the American Mathematical Society*1998, 126(8):2389–2397. 10.1090/S0002-9939-98-04709-1MATHMathSciNetView ArticleGoogle Scholar - Yao Q, Feng Y: The existence of solution for a third-order two-point boundary value problem.
*Applied Mathematics Letters*2002, 15(2):227–232. 10.1016/S0893-9659(01)00122-7MATHMathSciNetView ArticleGoogle Scholar - Zhao WL: Existence and uniqueness of solutions for third order nonlinear boundary value problems.
*The Tohoku Mathematical Journal*1992, 44(4):545–555. 10.2748/tmj/1178227249MATHMathSciNetView ArticleGoogle Scholar

## 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.