- Research Article
- Open Access

# 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

By using the Krasnoselskii's fixed point theorem and operator spectral theorem, the existence of positive solutions for the nonlocal fourth-order boundary value problem with variable parameter , , , is considered, where is a parameter, and , .

## Keywords

- Variable Parameter
- Point Theorem
- Fixed Point Theorem
- Lower Solution
- Fixed Point Theory

## 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:

(A1) and ,

(A2) , , , , , .

where , is a parameter.

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:

(H1) and ,

(H2) , , , and , .

## 2. The Preliminary Lemmas

Lemma 2.1 (see [16]).

if and only if

Let , and , for . , , , , .

It is easy to show that are norms on .

Lemma 2.2 (see [16]).

and ( ) is a Banach space.

Lemma 2.3 (see [5]).

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

(i) , for , ; , for , ,

(ii) , for , ,

where , ; , .

Computations yield the following results.

Lemma 2.4 (see [3]).

(i)when , ,

(ii)when , ,

(iii)when , .

Lemma 2.5 (see [16]).

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

(i) ,

(ii) , .

By Lemmas 2.4 and 2.5, .

Take , by Lemma 2.5, .

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

(i) and , , or

(ii) and ,

holds. Then, has a fixed point in .

## 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:

(i) , ,

(ii) , .

Proof.

and so , .

and so .

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

(i) , ,

(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

(i) , ,

(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

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