# Existence of Positive Solutions for Nonlocal Fourth-Order Boundary Value Problem with Variable Parameter

- Xiaoling Han
^{1}Email author, - Hongliang Gao
^{1}and - Jia Xu
^{1}

**2011**:604046

https://doi.org/10.1155/2011/604046

© Xiaoling Han et al. 2011

**Received: **26 November 2010

**Accepted: **14 January 2011

**Published: **23 January 2011

## Abstract

## 1. Introduction

The existence of positive solutions for nonlinear fourth-order multipoint boundary value problems has been studied by many authors using nonlinear alternatives of Leray-Schauder, the fixed point theory, and the method of upper and lower solutions (see, e.g., [1–15] and references therein). The multipoint boundary value problem is in fact a special case of the boundary value problem with integral boundary conditions.

under the assumption:

Obviously, BVP(1.1) can be regarded as the special case of BVP(1.2) with . Since the parameters is variable, we cannot expect to transform directly BVP(1.2) into an integral equation as in [16]. We will apply the cone fixed point theory, combining with the operator spectra theorem to establish the existence of positive solutions of BVP(1.2). Our results generalize the main result in [16].

Let , and we assume that the following conditions hold throughout the paper:

## 2. The Preliminary Lemmas

Lemma 2.1 (see [16]).

It is easy to show that are norms on .

Lemma 2.2 (see [16]).

Lemma 2.3 (see [5]).

Assume that (A1), (A2) hold. Then,

Computations yield the following results.

Lemma 2.4 (see [3]).

Lemma 2.5 (see [16]).

Suppose that (A1), (A2) hold and , , are given as above. Then,

Lemma 2.6.

is completely continuous, and .

Proof.

It is similar to Lemma 6 of [3], so we omit it.

Lemma 2.7 (see [17]).

Let E be a Banach space, a cone, and , be two bounded open sets of with . Suppose that is a completely continuous operator such that either

## 3. The Main Results

Theorem 3.1.

Assume that (H1), (H2) hold and . Then BVP(1.2) has at least one positive solution if one of the following cases holds:

Proof.

It is easy to know that is a cone in , . Now, we show .

- (ii)
The proof is similar to (i), so we omit it.

Corollary 3.2.

Assume that (H1), (H2) hold, and . Then that (1.2) has at least two positive solution, if satisfy

(ii)There exists such that , for , .

Proof.

By the proof of Theorem 3.1, we know that (1) from the condition , there exists , such that , , (2) from the condition , there exists , , such that , , (3) from the condition (ii), there exists , , such that , . By the use of Krasnoselskii's fixed point theorem, it is easy to know that (1.2) has at least two positive solutions.

Corollary 3.3.

Assume (H1), (H2) hold, and . Then problem (1.2) has at least two positive solution, if satisfy

(ii)There exists such that , for , .

Proof.

The proof is similar to Corollary 3.2, so we omit it.

Example 3.4.

Thus, , , , and satisfy the conditions of Theorem 3.1, and there exists at least a positive solution of the above problem.

## Declarations

### Acknowledgments

This work is sponsored by the NSFC (no. 11061030), NSFC (no. 11026060), and nwnu-kjcxgc-03-69, 03-61.

## Authors’ Affiliations

## References

- Bai Z:
**The method of lower and upper solutions for a bending of an elastic beam equation.***Journal of Mathematical Analysis and Applications*2000,**248**(1):195–202. 10.1006/jmaa.2000.6887MATHMathSciNetView ArticleGoogle Scholar - Bai Z:
**The upper and lower solution method for some fourth-order boundary value problems.***Nonlinear Analysis. Theory, Methods & Applications*2007,**67**(6):1704–1709. 10.1016/j.na.2006.08.009MATHMathSciNetView ArticleGoogle Scholar - Chai G:
**Existence of positive solutions for fourth-order boundary value problem with variable parameters.***Nonlinear Analysis. Theory, Methods & Applications*2007,**66**(4):870–880. 10.1016/j.na.2005.12.028MATHMathSciNetView ArticleGoogle Scholar - Feng H, Ji D, Ge W:
**Existence and uniqueness of solutions for a fourth-order boundary value problem.***Nonlinear Analysis. Theory, Methods & Applications*2009,**70**(10):3561–3566. 10.1016/j.na.2008.07.013MATHMathSciNetView ArticleGoogle Scholar - Li Y:
**Positive solutions of fourth-order boundary value problems with two parameters.***Journal of Mathematical Analysis and Applications*2003,**281**(2):477–484. 10.1016/S0022-247X(03)00131-8MATHMathSciNetView ArticleGoogle Scholar - Li Y:
**Positive solutions of fourth-order periodic boundary value problems.***Nonlinear Analysis. Theory, Methods & Applications*2003,**54**(6):1069–1078. 10.1016/S0362-546X(03)00127-5MATHMathSciNetView ArticleGoogle Scholar - Liu X-L, Li W-T:
**Existence and multiplicity of solutions for fourth-order boundary value problems with three parameters.***Mathematical and Computer Modelling*2007,**46**(3–4):525–534. 10.1016/j.mcm.2006.11.018MATHMathSciNetView ArticleGoogle Scholar - Ma H:
**Symmetric positive solutions for nonlocal boundary value problems of fourth order.***Nonlinear Analysis. Theory, Methods & Applications*2008,**68**(3):645–651. 10.1016/j.na.2006.11.026MATHMathSciNetView ArticleGoogle Scholar - Ma R:
**Existence of positive solutions of a fourth-order boundary value problem.***Applied Mathematics and Computation*2005,**168**(2):1219–1231. 10.1016/j.amc.2004.10.014MATHMathSciNetView ArticleGoogle Scholar - Pang C, Dong W, Wei Z:
**Multiple solutions for fourth-order boundary value problem.***Journal of Mathematical Analysis and Applications*2006,**314**(2):464–476. 10.1016/j.jmaa.2005.04.008MATHMathSciNetView ArticleGoogle Scholar - Wei Z, Pang C:
**Positive solutions and multiplicity of fourth-order m-point boundary value problems with two parameters.***Nonlinear Analysis. Theory, Methods & Applications*2007,**67**(5):1586–1598. 10.1016/j.na.2006.08.001MATHMathSciNetView ArticleGoogle Scholar - Yang Y, Zhang J:
**Existence of solutions for some fourth-order boundary value problems with parameters.***Nonlinear Analysis. Theory, Methods & Applications*2008,**69**(4):1364–1375. 10.1016/j.na.2007.06.035MATHMathSciNetView ArticleGoogle Scholar - Yao QL, Bai ZB:
**Existence of positive solutions of a BVP for .***Chinese Annals of Mathematics. Series A*1999,**20**(5):575–578.MATHMathSciNetGoogle Scholar - Yao Q:
**Local existence of multiple positive solutions to a singular cantilever beam equation.***Journal of Mathematical Analysis and Applications*2010,**363**(1):138–154. 10.1016/j.jmaa.2009.07.043MATHMathSciNetView ArticleGoogle Scholar - Zhao J, Ge W:
**Positive solutions for a higher-order four-point boundary value problem with a p-Laplacian.***Computers & Mathematics with Applications*2009,**58**(6):1103–1112. 10.1016/j.camwa.2009.04.022MATHMathSciNetView ArticleGoogle Scholar - Bai Z:
**Positive solutions of some nonlocal fourth-order boundary value problem.***Applied Mathematics and Computation*2010,**215**(12):4191–4197. 10.1016/j.amc.2009.12.040MATHMathSciNetView ArticleGoogle 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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