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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq1_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq2_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq3_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq4_HTML.gif 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:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq5_HTML.gif is Yang's modulus constant, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq6_HTML.gif is moment of inertia with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq7_HTML.gif axes, determined completely by the beam's shape cross-section. Specially, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq8_HTML.gif if the cross-section is a rectangle with a height of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq9_HTML.gif and a width of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq10_HTML.gif Also, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq11_HTML.gif is loading at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq12_HTML.gif . 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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ2_HTML.gif
(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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ3_HTML.gif
(1.3)

under some growth conditions of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq13_HTML.gif 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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ4_HTML.gif
(1.4)

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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ5_HTML.gif
(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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ6_HTML.gif
(1.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq14_HTML.gif 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:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ7_HTML.gif
(1.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq15_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq16_HTML.gif be a Banach space and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq17_HTML.gif a cone. Suppose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq18_HTML.gif are two continuous nonnegative functionals satisfying

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ8_HTML.gif
(2.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq19_HTML.gif are two positive constants.

Lemma 2.1 (see [5]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq20_HTML.gif are constants and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ9_HTML.gif
(2.2)
are two open subsets in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq21_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq22_HTML.gif . In addition, let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ10_HTML.gif
(2.3)

Assume https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq23_HTML.gif is a completely continuous operator satisfying

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq25_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq27_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq29_HTML.gif

then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq30_HTML.gif has at least one fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq31_HTML.gif

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq32_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq33_HTML.gif be a Banach space, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq34_HTML.gif a cone. Define functionals

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ11_HTML.gif
(3.1)

then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq35_HTML.gif are two continuous nonnegative functionals such that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ12_HTML.gif
(3.2)

and (2.1) hold.

Denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq36_HTML.gif Green's function for boundary value problem

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ13_HTML.gif
(3.3)

Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq37_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq38_HTML.gif , and

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ14_HTML.gif
(3.4)

Let

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ15_HTML.gif
(3.5)

However, (1.7) has a solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq39_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq40_HTML.gif solves the operator equation

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ16_HTML.gif
(3.6)

It is well know that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq41_HTML.gif is completely continuous.

Theorem 3.1.

Suppose there are four constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq42_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq43_HTML.gif and the following assumptions hold:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq45_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq47_HTML.gif

Then, (1.7) has at least one positive solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq48_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ17_HTML.gif
(3.7)

Proof.

Let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ18_HTML.gif
(3.8)
be two bounded open subsets in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq49_HTML.gif . In addition, let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ19_HTML.gif
(3.9)
For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq50_HTML.gif , by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq51_HTML.gif , there is
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ20_HTML.gif
(3.10)
For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq52_HTML.gif , because https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq53_HTML.gif , so https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq54_HTML.gif , that is to say https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq55_HTML.gif concave on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq56_HTML.gif , it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ21_HTML.gif
(3.11)
Combined with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq57_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq58_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq59_HTML.gif , there is
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ22_HTML.gif
(3.12)
For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq60_HTML.gif , by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq61_HTML.gif , there is
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ23_HTML.gif
(3.13)
For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq62_HTML.gif , by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq63_HTML.gif , there is
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ24_HTML.gif
(3.14)
Now, Lemma 2.1 implies there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq64_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq65_HTML.gif , namely, (1.7) has at least one positive solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq66_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ25_HTML.gif
(3.15)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ26_HTML.gif
(3.16)

The proof is complete.

Theorem 3.2.

Suppose there are five constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq67_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq68_HTML.gif and the following assumptions hold

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq70_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq72_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq74_HTML.gif

Then, (1.7) has at least one positive solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq75_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ27_HTML.gif
(3.17)

Proof.

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

For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq76_HTML.gif , the concavity of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq77_HTML.gif implies that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq78_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq79_HTML.gif . By https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq80_HTML.gif , there is
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ28_HTML.gif
(3.18)
For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq81_HTML.gif , by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq82_HTML.gif , there is
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ29_HTML.gif
(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

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