- Research Article
- Open Access

# Solutions to a Three-Point Boundary Value Problem

- Jin Liang
^{1}Email author and - Zhi-Wei Lv
^{2, 3}

**2011**:894135

https://doi.org/10.1155/2011/894135

© J. Liang and Z.-W. Lv. 2011

**Received:**25 November 2010**Accepted:**19 January 2011**Published:**9 February 2011

## Abstract

By using the fixed-point index theory and Leggett-Williams fixed-point theorem, we study the existence of multiple solutions to the three-point boundary value problem , ; ; , where , are constants, is a parameter, and , are given functions. New existence theorems are obtained, which extend and complement some existing results. Examples are also given to illustrate our results.

## Keywords

- Boundary Condition
- Differential Equation
- Banach Space
- Partial Differential Equation
- Unique Solution

## 1. Introduction

It is known that when differential equations are required to satisfy boundary conditions at more than one value of the independent variable, the resulting problem is called a multipoint boundary value problem, and a typical distinction between initial value problems and multipoint boundary value problems is that in the former case one is able to obtain the solutions depend only on the initial values, while in the latter case, the boundary conditions at the starting point do not determine a unique solution to start with, and some random choices among the solutions that satisfy these starting boundary conditions are normally not to satisfy the boundary conditions at the other specified point(s). As it is noticed elsewhere (see, e.g., Agarwal [1], Bisplinghoff and Ashley [2], and Henderson [3]), multi point boundary value problem has deep physical and engineering background as well as realistic mathematical model. For the development of the research of multi point boundary value problems for differential equations in last decade, we refer the readers to, for example, [1, 4–9] and references therein.

where , , , and , are given functions. To the authors' knowledge, few results on third-order differential equations with inhomogeneous three-point boundary values can be found in the literature. Our purpose is to establish new existence theorems for (1.1) which extend and complement some existing results.

This paper is organized in the following way. In Section 2, we present some lemmas, which will be used in Section 3. The main results and proofs are given in Section 3. Finally, in Section 4, we give some examples to illustrate our results.

## 2. Lemmas

It is not hard to see Lemmas 2.1 and 2.2.

Lemma 2.1.

Lemma 2.2.

One has the following.

- (i)
.

- (ii)
.

- (iii)
.

Lemma 2.3.

Let be the unique solution of (1.1). Then is nonnegative and satisfies .

Proof.

- (i)

The proof is completed.

Lemma 2.4.

Proof.

Lemma 2.1 implies that (1.1) has a solution if and only if is a fixed point of .

From Lemmas 2.1 and 2.2 and the Ascoli-Arzela theorem, the following follow.

Lemma 2.5.

The operator defined in (2.17) is completely continuous and satisfies .

Theorem 2.6 (see [10]).

Let be a real Banach Space, let be a cone, and . Let operator be completely continuous and satisfy , . Then

(i)if , for all , then ,

(ii)if , for all , then .

Theorem 2.7 (see [8]).

Let be a completely continuous operator and a nonnegative continuous concave functional on such that for all . Suppose that there exist such that

- (a)
and for ,

- (b)
for ,

- (c)
for with .

## 3. Main Results

In this section, we give new existence theorem about two positive solutions or three positive solutions for (1.1).

Theorem 3.1.

Assume that

;

there exists a constant such that , for , and .

for small enough.

Proof.

which means that . Thus, , for all .

The proof of Theorem 3.1 is completed.

Theorem 3.2.

Assume that

;

there exists a constant such that , for and .

for small enough.

Proof.

So . Hence, , .

Case 1.

is unbounded.

So , and then .

Case 2.

is bounded.

which implies , and then .

The proof of Theorem 3.2 is completed.

Theorem 3.3.

for .

Proof.

Hence, . This means that .

Therefore, in Theorem 2.7 holds.

So, . This means that of Theorem 2.7 holds.

## 4. Examples

In this section, we give three examples to illustrate our results.

Example 4.1.

Thus, condition is satisfied.

Example 4.2.

Thus, condition is satisfied.

Example 4.3.

## Declarations

### Acknowledgments

This paper was supported partially by the NSF of China (10771202) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007035805).

## Authors’ Affiliations

## References

- Agarwal RP:
*Focal Boundary Value Problems for Differential and Difference Equations, Mathematics and Its Applications*.*Volume 436*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1998:x+289.View ArticleGoogle Scholar - Bisplinghoff RL, Ashley H:
*Principles of Aeroelasticity*. Dover Publications, Mineola, NY, USA; 2002.MATHGoogle Scholar - Henderson J (Ed):
*Boundary Value Problems for Functional-Differential Equations*. World Scientific, River Edge, NJ, USA; 1995:x+306.MATHGoogle Scholar - Anderson DR:
**Green's function for a third-order generalized right focal problem.***Journal of Mathematical Analysis and Applications*2003,**288**(1):1-14. 10.1016/S0022-247X(03)00132-XMathSciNetView ArticleMATHGoogle Scholar - Avery RI, Peterson AC:
**Three positive fixed points of nonlinear operators on ordered Banach spaces.***Computers & Mathematics with Applications*2001,**42**(3–5):313-322.MathSciNetView ArticleMATHGoogle Scholar - Boucherif A, Al-Malki N:
**Nonlinear three-point third-order boundary value problems.***Applied Mathematics and Computation*2007,**190**(2):1168-1177. 10.1016/j.amc.2007.02.039MathSciNetView ArticleMATHGoogle Scholar - Karakostas GL, Mavridis KG, Tsamatos PC:
**Triple solutions for a nonlocal functional boundary value problem by Leggett-Williams theorem.***Applicable Analysis*2004,**83**(9):957-970. 10.1080/00036810410001724571MathSciNetView ArticleMATHGoogle Scholar - Leggett RW, Williams LR:
**Multiple positive fixed points of nonlinear operators on ordered Banach spaces.***Indiana University Mathematics Journal*1979,**28**(4):673-688. 10.1512/iumj.1979.28.28046MathSciNetView ArticleMATHGoogle Scholar - Sun Y:
**Positive solutions for third-order three-point nonhomogeneous boundary value problems.***Applied Mathematics Letters*2009,**22**(1):45-51. 10.1016/j.aml.2008.02.002MathSciNetView ArticleMATHGoogle Scholar - Guo DJ, Lakshmikantham V:
*Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering*.*Volume 5*. Academic Press, Boston, Mass, USA; 1988:viii+275.MATHGoogle Scholar

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