# Uniqueness and Parameter Dependence of Positive Solution of Fourth-Order Nonhomogeneous BVPs

- Jian-Ping Sun
^{1}Email author and - Xiao-Yun Wang
^{1}

**2010**:106962

https://doi.org/10.1155/2010/106962

© Jian-Ping Sun and Xiao-Yun Wang. 2010

**Received: **23 February 2010

**Accepted: **11 July 2010

**Published: **28 July 2010

## Abstract

We investigate the following fourth-order four-point nonhomogeneous Sturm-Liouville boundary value problem: , , , where and are nonnegative parameters. Some sufficient conditions are given for the existence and uniqueness of a positive solution. The dependence of the solution on the parameters is also studied.

## Keywords

## 1. Introduction

where . By means of the upper and lower solution method and Schauder fixed point theorem, some criteria on the existence of positive solutions to the BVP (1.1)–(1.3) were established. Bai et al. [2] obtained the existence of solutions for the BVP (1.1)–(1.3) by using a nonlinear alternative of Leray-Schauder type. For other related results, one can refer to [3–5] and the references therein.

where and are nonnegative parameters. They derived some conditions for the above BVP to have a unique solution and then studied the dependence of this solution on the parameters and . Sun [11] discussed the existence and nonexistence of positive solutions to a class of third-order three-point nonhomogeneous BVP. The authors in [12] studied the multiplicity of positive solutions for some fourth-order two-point nonhomogeneous BVP by using a fixed point theorem of cone expansion/compression type. For more recent results on higher-order BVPs with nonhomogeneous boundary conditions, one can see [13–16].

where and are nonnegative parameters. Under the following assumptions:

(A1) and are nonnegative constants with , , , , and

(A2) is continuous and monotone increasing in for every ;

we prove the uniqueness of positive solution for the BVP (1.5)–(1.7) and study the dependence of this solution on the parameters .

## 2. Preliminary Lemmas

First, we recall some fundamental definitions.

Definition 2.1.

Let be a Banach space with norm . Then

(1) a nonempty closed convex set is said to be a cone if for all and , where is the zero element of

(2) every cone in defines a partial ordering in by

(3) a cone is said to be normal if there exists such that implies that

(4) a cone is said to be solid if the interior of is nonempty.

Definition 2.2.

Next, we state a fixed point theorem, which is our main tool.

Lemma 2.3 (see [17]).

Assume that is a normal solid cone in a real Banach space and is a -concave increasing operator. Then has a unique fixed point in

The following two lemmas are crucial to our main results.

Lemma 2.4.

Proof.

Lemma 2.5.

Assume that (A1) holds. Then

## 3. Main Result

For convenience, we denote and . In the remainder of this paper, the following notations will be used:

(1) if at least one of approaches ;

(3) if for and at least one of them is strict.

Let . Then is a Banach space, where is defined as usual by the sup norm.

Our main result is the following theorem.

Theorem 3.1.

Assume that (A1)–(A3) hold. Then the BVP (1.5)–(1.7) has a unique positive solution for any , where . Furthermore, such a solution satisfies the following properties:

Proof.

then it is not difficult to verify that is a positive solution of the BVP (1.5)–(1.7) if and only if is a fixed point of .

Now, we will prove that has a unique fixed point by using Lemma 2.3.

First, in view of Lemma 2.5, we know that

Next, we claim that is a -concave operator.

Finally, we assert that is an increasing operator.

which indicates that is increasing.

Therefore, it follows from Lemma 2.3 that has a unique fixed point which is the unique positive solution of the BVP (1.5)–(1.7). The first part of the theorem is proved.

In the rest of the proof, we will prove that such a positive solution satisfies properties (P1), (P2), and (P3).

which together with for implies (P1).

which indicates that is strictly increasing in .

Hence, (P3) holds.

## Declarations

### Acknowledgment

Supported by the National Natural Science Foundation of China (10801068).

## Authors’ Affiliations

## References

- Chen S, Ni W, Wang C:
**Positive solution of fourth order ordinary differential equation with four-point boundary conditions.***Applied Mathematics Letters*2006,**19**(2):161-168. 10.1016/j.aml.2005.05.002MathSciNetView ArticleMATHGoogle Scholar - Bai C, Yang D, Zhu H:
**Existence of solutions for fourth order differential equation with four-point boundary conditions.***Applied Mathematics Letters*2007,**20**(11):1131-1136. 10.1016/j.aml.2006.11.013MathSciNetView ArticleMATHGoogle Scholar - Graef JR, Yang B:
**Positive solutions of a nonlinear fourth order boundary value problem.***Communications on Applied Nonlinear Analysis*2007,**14**(1):61-73.MathSciNetMATHGoogle Scholar - Wu H, Zhang J:
**Positive solutions of higher-order four-point boundary value problem with****-Laplacian operator.***Journal of Computational and Applied Mathematics*2010,**233**(11):2757-2766. 10.1016/j.cam.2009.06.040MathSciNetView ArticleMATHGoogle Scholar - Zhao J, Ge W:
**Positive solutions for a higher-order four-point boundary value problem with a****-Laplacian.***Computers & Mathematics with Applications*2009,**58**(6):1103-1112. 10.1016/j.camwa.2009.04.022MathSciNetView ArticleMATHGoogle Scholar - Ma R:
**Positive solutions for second-order three-point boundary value problems.***Applied Mathematics Letters*2001,**14**(1):1-5. 10.1016/S0893-9659(00)00102-6MathSciNetView ArticleMATHGoogle Scholar - Ma R:
**Positive solutions for nonhomogeneous****-point boundary value problems.***Computers & Mathematics with Applications*2004,**47**(4-5):689-698. 10.1016/S0898-1221(04)90056-9MathSciNetView ArticleMATHGoogle Scholar - Kong L, Kong Q:
**Second-order boundary value problems with nonhomogeneous boundary conditions. I.***Mathematische Nachrichten*2005,**278**(1-2):173-193. 10.1002/mana.200410234MathSciNetView ArticleMATHGoogle Scholar - Kong L, Kong Q:
**Second-order boundary value problems with nonhomogeneous boundary conditions. II.***Journal of Mathematical Analysis and Applications*2007,**330**(2):1393-1411. 10.1016/j.jmaa.2006.08.064MathSciNetView ArticleMATHGoogle Scholar - Kong L, Kong Q:
**Uniqueness and parameter dependence of solutions of second-order boundary value problems.***Applied Mathematics Letters*2009,**22**(11):1633-1638. 10.1016/j.aml.2009.05.009MathSciNetView 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 - do Ó JM, Lorca S, Ubilla P:
**Multiplicity of solutions for a class of non-homogeneous fourth-order boundary value problems.***Applied Mathematics Letters*2008,**21**(3):279-286. 10.1016/j.aml.2007.02.025MathSciNetView ArticleMATHGoogle Scholar - Graef JR, Kong L, Kong Q, Wong JSW:
**Higher order multi-point boundary value problems with sign-changing nonlinearities and nonhomogeneous boundary conditions.***Electronic Journal of Qualitative Theory of Differential Equations*2010,**2010**(28):1-40.MathSciNetView ArticleMATHGoogle Scholar - Kong L, Kong Q:
**Higher order boundary value problems with nonhomogeneous boundary conditions.***Nonlinear Analysis: Theory, Methods & Applications*2010,**72**(1):240-261. 10.1016/j.na.2009.06.050MathSciNetView ArticleMATHGoogle Scholar - Kong L, Piao D, Wang L:
**Positive solutions for third order boundary value problems with****-Laplacian.***Results in Mathematics*2009,**55**(1-2):111-128. 10.1007/s00025-009-0383-zMathSciNetView ArticleMATHGoogle Scholar - Kong L, Wong JSW:
**Positive solutions for higher order multi-point boundary value problems with nonhomogeneous boundary conditions.***Journal of Mathematical Analysis and Applications*2010,**367**(2):588-611. 10.1016/j.jmaa.2010.01.063MathSciNetView 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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