- Research Article
- Open Access

# Existence Results of Three-Point Boundary Value Problems for Second-order Ordinary Differential Equations

- Sheng-Ping Wang
^{1}Email author and - Long-Yi Tsai
^{2}

**2011**:901796

https://doi.org/10.1155/2011/901796

© S.-P.Wang and L.-Y. Tsai. 2011

**Received:**19 May 2010**Accepted:**24 September 2010**Published:**28 September 2010

## Abstract

We establish existence results of the following three-point boundary value problems: , , and , where and . The approach applied in this paper is upper and lower solution method associated with basic degree theory or Schauder's fixed point theorem. We deal with this problem with the function which is Carathéodory or singular on its domain.

## Keywords

- Fixed Point Theorem
- Open Interval
- Lower Solution
- Nonlocal Boundary
- Linear Ordinary Differential Equation

## 1. Introduction

In the mathematical literature, a number of works have appeared on nonlocal boundary value problems, and one of the first of these was [1]. Il'in and Moiseev initiated the research of multipoint boundary value problems for second-order linear ordinary differential equations, see [2, 3], motivated by the study [4–6] of Bitsadze and Samarskii.

Recently, nonlinear multipoint boundary value problems have been receiving considerable attention, and have been studied extensively by using iteration scheme (e.g., [7]), fixed point theorems in cones (e.g., [8]), and the Leray-Schauder continuation theorem (e.g., [9]). We refer more detailed treatment to more interesting research [10, 11] and the references therein.

The theory of upper and lower solutions is also a powerful tool in studying boundary value problems. For the existence results of two-point boundary value problem, there already are lots of interesting works by applying this essential technique (see [12, 13]). Recently, it is shown that this method plays an important role in proving the existence of solutions for three-point boundary value problems (see [14–16]).

Last but not least, as the singular source term appearing in two-point problems, singular three-point boundary value problems also attract more attention (e.g., [17]).

In this paper, we will discuss the existence of solutions of some general types on three-point boundary value problems by using upper and lower solution method associated with basic degree theory or Schauder's fixed point theorem.

and prove the existence of -solution in Theorems 4.1 and 4.4. Some sufficient conditions for constructing upper and lower solutions are given in each section for applications.

## 2. Preliminaries

By direct computations, we get the following results.

- (i)

Lemma 2.2.

## 3. Carathéodory Case

In this section we first introduce the Carathéodory function as follows.

Definition 3.1.

A function defined on is called a Carathéodory function on if

(i)for almost every is continuous on ;

(ii)for any the function is measurable on ;

(iii)for any , there exists such that for any and for almost every with , we have .

We in this section assume that is a Carathéodory function and discuss the existence of -solution by assuming the existence of upper and lower solutions.

### 3.1. Existence of -Solutions

We first introduce the definitions of -upper and lower solutions as below.

Definition 3.2.

A function is called a -lower solution of problem (1.1) and (1.2) if it satisfies

Definition 3.3.

A function is called a -upper solution of problem (1.1) and (1.2) if it satisfies

Proposition 3.4.

Let and be respective -lower and upper solutions of problem (1.1) and (1.2) with on . If is a solution of problem (3.3) and (1.2), then , for any .

Proof.

Case 1.

This implies that the minimum of cannot occur at , a contradiction.

Case 2.

And we get a contradiction.

Case 3.

which is impossible.

Theorem 3.5.

Proof.

It is clear that is a closed, bounded and convex set in and one can show that is a completely continuous mapping by Arzelà-Ascoli theorem and Lebesgue dominated convergence theorem. By applying Schauder's fixed point theorem, we obtain that has a fixed point in which is a solution of problem (3.3) and (1.2). From Proposition 3.4, this fixed point is also a solution of problem (1.1) and (1.2). Hence, we complete the proof.

with the boundary condition (1.2).

Corollary 3.6.

Then, problem (3.18) and (1.2) has at least one solution.

Proof.

Hence, if is large enough, we can show that and , where , which implies that is a positive -upper solution. In the same way we construct a -lower solution on .

### 3.2. Nontangency Solution

In this subsection, we afford another stronger lower and upper solutions to get a strict inequality of the solution between them.

Definition 3.7.

A function is a strict -lower solution of problem (1.1) and (1.2), if it is not a solution of problem (1.1) and (1.2), , and for any , one of the following is satisfied:

Definition 3.8.

A function is a strict -upper solution of problem (1.1) and (1.2), if it is not a solution of problem (1.1) and (1.2), , and for any , one of the following is satisfied:

Remark 3.9.

Every strict -lower(upper) solution of problem (1.1) and (1.2) is a -lower(upper) solution.

Now we are going to show that the solution curve of problem (1.1) and (1.2) cannot be tangent to upper or lower solutions from below or above.

Proposition 3.10.

Let and be respective strict -lower and upper solutions of problem (1.1) and (1.2) with on . If is a solution of problem (1.1) and (1.2) with on , then , for any .

Proof.

exists. Hence, has minimum at , that is, .

Case 1.

Case 2.

And we get a contradiction.

Case 3.

If , repeat the same arguments in Case 3 of the proof of Proposition 3.4. Therefore, we obtain on . The inequality on can be proved by the similar arguments as above.

Theorem 3.11.

Proof.

This is a consequence of Theorem 3.5 and Proposition 3.10 and hence, we omits this proof.

## 4. Singular Case

In this section we give a more general existence result than Theorem 3.11 by assuming the existence of -lower and upper solutions. This makes us to deal with problem (1.1) and (1.2), where the function is singular at the end point and .

Theorem 4.1.

Let and be -lower and upper solutions of problem (1.1) and (1.2) such that on and let satisfy the following conditions:

(i)for almost every is continuous on ;

(ii)for any the function is measurable on ;

Proof.

and is defined by (3.17). The rest arguments are similar to the proof of Theorem 3.5.

Remark 4.2.

where is a constant and , are given as (1.2).

Example 4.3.

Notice that in Theorem 4.1, one can only deal with the case that is singular at end points , . However, when is singular at , there is no hope to obtain the solutions directly from Theorem 4.1. We will establish the following theorem to deal with this case by constructing upper and lower solutions to solve this problem.

Theorem 4.4.

Assume

where is defined as in Lemma 2.1.

Remark 4.5 (see [12, Remark ]).

Assumption is equivalent to the assumption that there exists and a function such that:

Proof.

Step 1.

Step 2.

Step 3.

It follows from (4.25) and (4.27) that is a lower solution of ( ).

Step 4.

Step 5.

Step 6.

it remains only to check the continuity of at . This can be deduced from the continuity of and the fact that as .

Example 4.6.

## Authors’ Affiliations

## References

- Bitsadze AV, Samarskii AA: On some of the simplest generalizations of linear elliptic boundary-value problems.
*Doklady Akademii Nauk SSSR*1969, 185: 739–740.MathSciNetGoogle Scholar - Il'in VA, Moiseev EI: Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects.
*Differential Equations*1987, 23(7):803–810.MathSciNetMATHGoogle Scholar - Il'in VA, Moiseev EI: Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator.
*Differential Equations*1987, 23(7):979–987.MathSciNetMATHGoogle Scholar - Bitsadze AV: On the theory of nonlocal boundary value problems.
*Soviet Mathematics—Doklady*1984, 30(1):8–10.MATHGoogle Scholar - Bitsadze AV: On a class of conditionally solvable nonlocal boundary value problems for harmonic functions.
*Soviet Mathematics—Doklady*1985, 31(1):91–94.MATHGoogle Scholar - Bitsadze AV, Samarskii AA: On some simple generalizations of linear elliptic boundary problems.
*Soviet Mathematics—Doklady*1969, 10(2):398–400.MATHGoogle Scholar - Yao Q: Successive iteration and positive solution for nonlinear second-order three-point boundary value problems.
*Computers & Mathematics with Applications*2005, 50(3–4):433–444. 10.1016/j.camwa.2005.03.006View ArticleMathSciNetMATHGoogle Scholar - Yao Q: On the positive solutions of a second-order three-point boundary value problem with Caratheodory function.
*Southeast Asian Bulletin of Mathematics*2004, 28(3):577–585.MathSciNetMATHGoogle Scholar - Gupta CP, Trofimchuk SI: A sharper condition for the solvability of a three-point second order boundary value problem.
*Journal of Mathematical Analysis and Applications*1997, 205(2):586–597. 10.1006/jmaa.1997.5252View ArticleMathSciNetMATHGoogle Scholar - Ma R: Positive solutions of a nonlinear
*m*-point boundary value problem.*Computers & Mathematics with Applications*2001, 42(6–7):755–765. 10.1016/S0898-1221(01)00195-XView ArticleMathSciNetMATHGoogle Scholar - Thompson HB, Tisdell C: Three-point boundary value problems for second-order, ordinary, differential equations.
*Mathematical and Computer Modelling*2001, 34(3–4):311–318. 10.1016/S0895-7177(01)00063-2View ArticleMathSciNetMATHGoogle Scholar - De Coster C, Habets P: Upper and lower solutions in the theory of ODE boundary value problems: classical and recent results. In
*Nonlinear Analysis and Boundary Value Problems for Ordinary Differential Equations, CISM Courses and Lectures*.*Volume 371*. Edited by: Zanolin F. Springer, Vienna, Austria; 1993:1–78.Google Scholar - Lü H, O'Regan D, Agarwal RP: Upper and lower solutions for the singular
*p*-Laplacian with sign changing nonlinearities and nonlinear boundary data.*Journal of Computational and Applied Mathematics*2005, 181(2):442–466. 10.1016/j.cam.2004.11.037View ArticleMathSciNetMATHGoogle Scholar - Du Z, Xue C, Ge W: Multiple solutions for three-point boundary value problem with nonlinear terms depending on the first order derivative.
*Archiv der Mathematik*2005, 84(4):341–349. 10.1007/s00013-004-1196-7View ArticleMathSciNetMATHGoogle Scholar - Khan RA, Webb JRL: Existence of at least three solutions of a second-order three-point boundary value problem.
*Nonlinear Analysis. Theory, Methods & Applications*2006, 64(6):1356–1366. 10.1016/j.na.2005.06.040View ArticleMathSciNetMATHGoogle Scholar - Minghe P, Chang SK: The generalized quasilinearization method for second-order threepoint boundary value problems.
*Nonlinear Analysis. Theory, Methods & Applications*2008, 68(9):2779–2790. 10.1016/j.na.2007.02.025View ArticleMathSciNetMATHGoogle Scholar - Qu WB, Zhang ZX, Wu JD: Positive solutions to a singular second order three-point boundary value problem.
*Applied Mathematics and Mechanics*2002, 23(7):854–866. 10.1007/BF02456982View ArticleMathSciNetMATHGoogle Scholar - De Coster C, Habets P:
*Two-Point Boundary Value Problems: Lower and Upper Solutions*. Springer, Berlin, Germany; 1984.Google Scholar - Habets P, Zanolin F: Upper and lower solutions for a generalized Emden-Fowler equation.
*Journal of Mathematical Analysis and Applications*1994, 181(3):684–700. 10.1006/jmaa.1994.1052View ArticleMathSciNetMATHGoogle Scholar

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