Open Access

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

Fixed Point Theory and Applications20112011:604046

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

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 , .

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., [115] and references therein). The multipoint boundary value problem is in fact a special case of the boundary value problem with integral boundary conditions.

Recently, Bai [16] studied the existence of positive solutions of nonlocal fourth-order boundary value problem
(1.1)

under the assumption:

(A1) and ,

(A2) , , , , , .

In this paper, we study the above generalizing form with variable parameters BVP
(1.2)

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

Set , and
(2.1)
By (H1), (H2), we get , . Denote by the Green's function of the problem
(2.2)
and the Green's function of the problem
(2.3)
Then, carefully calculation yield
(2.4)

Lemma 2.1 (see [16]).

Suppose that (A1), (A2) hold. Then, for any , solves the problem
(2.5)

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 , ; , .

Denote
(2.6)

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, .

Define
(2.7)

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

Suppose that , , , , , , and , are defined as in Section 2, we introduce some notations as follows:
(3.1)

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.

For any , consider the following BVP:
(3.2)
It is easy to see that the above question is equivalent to the following question:
(3.3)
For any , let . Obviously, the operator is linear. By Lemma 2.2, for all , , . Hence , and so . On the other hand, is a solution of (3.3) if and only if satisfies , that is,
(3.4)
Owing to and , the operator maps into . From (by Lemma 2.6) together with and condition , applying operator spectral theorem, we have that the exists and is bounded. Let , then (3.4) is equivalent to . By the Neumann expansion formula, can be expressed by
(3.5)
The complete continuity of with the continuity of yields that the operator is completely continuous. For all , let , then , and . So, we have , . Hence,
(3.6)

and so , .

Assume that for all , , , let , by (3.6) we have , and so , . Thus by induction, it follows that , for all , , . By (3.5), for all , we have
(3.7)

and so .

On the other hand, for all , we have
(3.8)
(3.9)
(3.10)
For any , define . By (H1) and (H2), we have that is continuous. It is easy to see that being a positive solution of BVP(1.2) is equivalent to being a nonzero solution equation as follows:
(3.11)
Let . Obviously, is completely continuous. We next show that the operator has a nonzero fixed point in . Let
(3.12)

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

For , by (2.7), there is , . Hence, by (3.7), , , . By proof of Lemma 2.5 in [16],
(3.13)
By (3.7) and (3.10),
(3.14)
Thus .
  1. (i)
    Since , by the definition of , there exists such that
    (3.15)
     
Let , one has
(3.16)
So, by (3.10), we get
(3.17)
Hence, for ,
(3.18)
On the other hand, since , there exists such that
(3.19)
Choose , let . For , , there is . Thus,
(3.20)
Hence, for ,
(3.21)
By the use of the Krasnoselskii's fixed point theorem, we know there exists such that , namely, is a solution of (1.2) and satisfied , , .
  1. (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.

Consider the following boundary value problem
(3.22)
In this problem, we know that , , , , then we can get , , , , , , . Further more, we obtain , , then , , so
(3.23)

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

(1)
Department of Mathematics, Northwest Normal University

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© Xiaoling Han et al. 2011

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