Open Access

Positive Solutions for Some Beam Equation Boundary Value Problems

Boundary Value Problems20092009:393259

DOI: 10.1155/2009/393259

Received: 2 September 2009

Accepted: 1 November 2009

Published: 15 November 2009

Abstract

A new fixed point theorem in a cone is applied to obtain the existence of positive solutions of some fourth-order beam equation boundary value problems with dependence on the first-order derivative where is continuous.

1. Introduction

It is well known that beam is one of the basic structures in architecture. It is greatly used in the designing of bridge and construction. Recently, scientists bring forward the theory of combined beams. That is to say, we can bind up some stratified structure copings into one global combined beam with rock bolts. The deformations of an elastic beam in equilibrium state, whose two ends are simply supported, can be described by following equation of deflection curve:

(1.1)

where is Yang's modulus constant, is moment of inertia with respect to axes, determined completely by the beam's shape cross-section. Specially, if the cross-section is a rectangle with a height of and a width of Also, is loading at . If the loading of beam considered is in relation to deflection and rate of change of deflection, we need to research the more general equation

(1.2)

According to the forms of supporting, various boundary conditions should be considered. Solving corresponding boundary value problems, one can obtain the expression of deflection curve. It is the key in design of constants of beams and rock bolts.

Owing to its importance in physics and engineering, the existence of solutions to this problem has been studied by many authors, see [110]. However, in practice, only its positive solution is significant. In [1, 9, 11, 12], Aftabizadeh, Del Pino and Manásevich, Gupta, and Pao showed the existence of positive solution for

(1.3)

under some growth conditions of and a nonresonance condition involving a two-parameter linear eigenvalue problem. All of these results are based on the Leray-Schauder continuation method and topological degree.

The lower and upper solution method has been studied for the fourth-order problem by several authors [2, 3, 7, 8, 13, 14]. However, all of these authors consider only an equation of the form

(1.4)

with diverse kind of boundary conditions. In [10], Ehme et al. gave some sufficient conditions for the existence of a solution of

(1.5)

with some quite general nonlinear boundary conditions by using the lower and upper solution method. The conditions assume the existence of a strong upper and lower solution pair.

Recently, Krasnosel'skii's fixed point theorem in a cone has much application in studying the existence and multiplicity of positive solutions for differential equation boundary value problems, see [3, 6]. With this fixed point theorem, Bai and Wang [6] discussed the existence, uniqueness, multiplicity, and infinitely many positive solutions for the equation of the form

(1.6)

where is a constant.

In this paper, via a new fixed point theorem in a cone and concavity of function, we show the existence of positive solutions for the following problem:

(1.7)

where is continuous.

We point out that positive solutions of (1.7) are concave and this concavity provides lower bounds on positive concave functions of their maximum, which can be used in defining a cone on which a positive operator is defined, to which a new fixed point theorem in a cone due to Bai and Ge [5] can be applied to obtain positive solutions.

2. Fixed Point Theorem in a Cone

Let be a Banach space and a cone. Suppose are two continuous nonnegative functionals satisfying

(2.1)

where are two positive constants.

Lemma 2.1 (see [5]).

Let are constants and
(2.2)
are two open subsets in such that . In addition, let
(2.3)

Assume is a completely continuous operator satisfying

then has at least one fixed point in

3. Existence of Positive Solutions

In this section, we are concerned with the existence of positive solutions for the fourth-order two-point boundary value problem (1.7).

Let with be a Banach space, a cone. Define functionals

(3.1)

then are two continuous nonnegative functionals such that

(3.2)

and (2.1) hold.

Denote by Green's function for boundary value problem

(3.3)

Then , for , and

(3.4)

Let

(3.5)

However, (1.7) has a solution if and only if solves the operator equation

(3.6)

It is well know that is completely continuous.

Theorem 3.1.

Suppose there are four constants such that and the following assumptions hold:

Then, (1.7) has at least one positive solution such that
(3.7)

Proof.

Let
(3.8)
be two bounded open subsets in . In addition, let
(3.9)
For , by , there is
(3.10)
For , because , so , that is to say concave on , it follows that
(3.11)
Combined with and , for , there is
(3.12)
For , by , there is
(3.13)
For , by , there is
(3.14)
Now, Lemma 2.1 implies there exists such that , namely, (1.7) has at least one positive solution such that
(3.15)
that is,
(3.16)

The proof is complete.

Theorem 3.2.

Suppose there are five constants such that and the following assumptions hold

Then, (1.7) has at least one positive solution such that
(3.17)

Proof.

We just need notice the following difference to the proof of Theorem 3.1.

For , the concavity of implies that for . By , there is
(3.18)
For , by , there is
(3.19)

The rest of the proof is similar to Theorem 3.1 and the proof is complete.

Authors’ Affiliations

(1)
Department of Civil Engineering, Hohai University
(2)
Zaozhuang Coal Mining Group Co., Ltd
(3)
Graduate School, Hohai University

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Copyright

© J. Liu and W. Xu. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.