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

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

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq5_HTML.gif is Yang's modulus constant, http://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 http://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, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq9_HTML.gif and a width of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq10_HTML.gif Also, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq11_HTML.gif is loading at http://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

http://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

http://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 http://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

http://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

http://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

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

where http://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:

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

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

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

where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq20_HTML.gif are constants and
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq21_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq22_HTML.gif . In addition, let
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ10_HTML.gif
(2.3)

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

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

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

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

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

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

then http://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

http://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 http://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

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

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

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

Let

http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq39_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq40_HTML.gif solves the operator equation

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

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

http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq48_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ17_HTML.gif
(3.7)

Proof.

Let
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq49_HTML.gif . In addition, let
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ19_HTML.gif
(3.9)
For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq50_HTML.gif , by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq51_HTML.gif , there is
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ20_HTML.gif
(3.10)
For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq52_HTML.gif , because http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq53_HTML.gif , so http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq54_HTML.gif , that is to say http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq55_HTML.gif concave on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq56_HTML.gif , it follows that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ21_HTML.gif
(3.11)
Combined with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq57_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq58_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq59_HTML.gif , there is
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ22_HTML.gif
(3.12)
For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq60_HTML.gif , by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq61_HTML.gif , there is
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ23_HTML.gif
(3.13)
For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq62_HTML.gif , by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq63_HTML.gif , there is
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq64_HTML.gif such that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq66_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ25_HTML.gif
(3.15)
that is,
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq67_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq68_HTML.gif and the following assumptions hold

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

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

http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq75_HTML.gif such that
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq76_HTML.gif , the concavity of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq77_HTML.gif implies that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq78_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq79_HTML.gif . By http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq80_HTML.gif , there is
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_Equ28_HTML.gif
(3.18)
For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq81_HTML.gif , by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F393259/MediaObjects/13661_2009_Article_844_IEq82_HTML.gif , there is
http://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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