A Note on a Beam Equation with Nonlinear Boundary Conditions

Boundary Value Problems20102011:376782

DOI: 10.1155/2011/376782

Received: 14 May 2010

Accepted: 31 July 2010

Published: 12 August 2010

Abstract

We present new results on the existence of multiple positive solutions of a fourth-order differential equation subject to nonlocal and nonlinear boundary conditions that models a particular stationary state of an elastic beam with nonlinear controllers. Our results are based on classical fixed point index theory. We improve and complement previous results in the literature. This is illustrated in some examples.

1. Introduction

The fourth-order differential equation
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ1_HTML.gif
(1.1)
arises naturally in the study of the displacement http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq1_HTML.gif of an elastic beam when we suppose that, along its length, a load is added to cause deformations. This classical problem has been widely studied under a variety of boundary conditions (BCs) that describe the controls at the ends of the beam. In particular, Gupta [1] studied, along other sets of local homogeneous BCs, the problem
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ2_HTML.gif
(1.2)

that models a bar with the left end being simply supported (hinged) and the right end being sliding clamped. This problem, and its generalizations, has been studied previously by Davies and coauthors [2], Graef and Henderson [3] and Yao [4].

Multipoint versions of this problem do have a physical interpretation. For example, the four-point boundary conditions
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ3_HTML.gif
(1.3)

model a bar where the displacement http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq2_HTML.gif and the bending moment http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq3_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq4_HTML.gif are zero, and there are relations, not necessarily linear, between the shearing force http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq5_HTML.gif and the angular attitude http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq6_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq7_HTML.gif and the displacement http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq8_HTML.gif in two other points of the beam.

In this paper we establish new results on the existence of positive solutions for the fourth-order differential equation (1.1) subject to the following nonlocal nonlinear boundary conditions:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ4_HTML.gif
(1.4)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ5_HTML.gif
(1.5)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ6_HTML.gif
(1.6)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq9_HTML.gif are nonnegative continuous functions and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq10_HTML.gif are linear functionals given by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ7_HTML.gif
(1.7)

involving Riemann-Stieltjes integrals.

The conditions (1.5)-(1.6) cover a variety of cases and include, as special cases when http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq11_HTML.gif , multipoint and integral boundary conditions. These are widely studied objects in the case of fourth-order BVPs; see, for example, [514]. BCs of nonlinear type also have been studied before in the case of fourth-order equations; see, for example, [1520] and references therein.

The study of positive solutions of BVPs that involve Stieltjes integrals has been done, in the case of positive measures, in [2124]. Signed measures were used in [12, 25]; here, as in [21, 22], due to some inequalities involved in our theory, the functionals http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq12_HTML.gif are assumed to be given by positive measures.

A standard methodology to solve (1.1) subject to local BCs is to find the corresponding Green's function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq13_HTML.gif and to rewrite the BVP as a Hammerstein integral equation of the form
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ8_HTML.gif
(1.8)
However, for nonlocal and nonlinear BCs the form of Green's functions can become very complicated. In the case of linear, nonlocal BCs, an elegant approach is due to Webb and Infante [12], where a unified method is given to study a large class of problems. This is done via an auxiliary perturbed Hammerstein integral equation of the form
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ9_HTML.gif
(1.9)

with suitable functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq14_HTML.gif .

Infante studied in [26, 27] the case of one nonlinear BC and in [21] a thermostat model with two nonlinear controllers. The approach used in [21] relied on an extension of the results of [25], valid for equations of the type
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ10_HTML.gif
(1.10)

and gives a simple general method to avoid long technical calculations.

In our paper the approach of [21] is applied to BVP (1.1)–(1.6): we rewrite this BVP as a perturbed Hammerstein integral equation, and we prove the existence of multiple positive solutions under a suitable oscillatory behavior of the nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq15_HTML.gif . We observe that our results are new even for the local BCs, when http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq16_HTML.gif . We illustrate our theory with some examples. We also point out that this approach may be utilized for other sets of nonlinear BCs that have a physical interpretation, this is done in the last section.

2. The Boundary Value Problem

We begin by considering the homogeneous BVP
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ11_HTML.gif
(2.1)
of which we seek an equivalent integral formulation of the form
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ12_HTML.gif
(2.2)
Due to the nature of these particular BCs, the Green's function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq17_HTML.gif can be constructed (as in [4]) by means of an auxiliary second-order BVP, namely,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ13_HTML.gif
(2.3)
The solutions of the BVP (2.3) can be written in the form
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ14_HTML.gif
(2.4)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ15_HTML.gif
(2.5)
Therefore the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq18_HTML.gif in (2.2) is given by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ16_HTML.gif
(2.6)
In order to use the approach of [21, 25, 28], we need to use some monotonicity properties of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq19_HTML.gif . Now, since
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ17_HTML.gif
(2.7)
we obtain the following formulation for the Green's function:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ18_HTML.gif
(2.8)
We now look for a suitable interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq20_HTML.gif , a function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq21_HTML.gif , and a constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq22_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ19_HTML.gif
(2.9)
Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq23_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq24_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq25_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq26_HTML.gif , a natural choice could be
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ20_HTML.gif
(2.10)

here we look for a better http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq27_HTML.gif , since this enables us to weaken the growth requirements on the nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq28_HTML.gif .

An upper bound for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq29_HTML.gif is obtained by finding http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq30_HTML.gif for each fixed http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq31_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq32_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq33_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq34_HTML.gif is a nondecreasing function of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq35_HTML.gif that attains its maximum, for each fixed http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq36_HTML.gif , when http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq37_HTML.gif .

Therefore, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq38_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ21_HTML.gif
(2.11)
Now, one can see that the derivative of the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq39_HTML.gif with respect to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq40_HTML.gif is non-positive for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq41_HTML.gif , that is, the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq42_HTML.gif is a non-increasing function of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq43_HTML.gif . Therefore, for an arbitrary http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq44_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ22_HTML.gif
(2.12)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ23_HTML.gif
(2.13)
We now turn our attention to the BVP (1.1)–(1.6)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ24_HTML.gif
(2.14)

and we show that we can study this problem by means of a perturbation of the Hammerstein integral equation (2.2).

In order to do this, we look for the (unique) solutions of the linear problems
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ25_HTML.gif
(2.15)
that are
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ26_HTML.gif
(2.16)
We observe that, for an arbitrary http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq45_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ27_HTML.gif
(2.17)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq46_HTML.gif , and therefore
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ28_HTML.gif
(2.18)
with
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ29_HTML.gif
(2.19)
By a solution of the BVP (1.1)–(1.6) we mean a solution of the perturbed integral equation
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ30_HTML.gif
(2.20)

and we work in a suitable cone in the Banach space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq47_HTML.gif of continuous functions defined on the interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq48_HTML.gif endowed with the usual supremum norm.

Our assumptions are the following:

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq50_HTML.gif satisfies Carathéodory conditions, that is, for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq51_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq52_HTML.gif is measurable, for almost every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq53_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq54_HTML.gif is continuous, and for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq55_HTML.gif , there exists an http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq56_HTML.gif -function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq57_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ31_HTML.gif
(2.21)

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq59_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq60_HTML.gif almost everywhere, and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq61_HTML.gif ;

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq63_HTML.gif are positive continuous functions such that there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq64_HTML.gif with
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ32_HTML.gif
(2.22)

for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq65_HTML.gif ;

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq67_HTML.gif are positive bounded linear functionals on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq68_HTML.gif given by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ33_HTML.gif
(2.23)

involving Stieltjes integrals with positive measures http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq69_HTML.gif ;

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq71_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq72_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq73_HTML.gif .

It follows from this last hypothesis that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ34_HTML.gif
(2.24)
The above hypotheses enable us to utilize the cone
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ35_HTML.gif
(2.25)
for an arbitrary http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq74_HTML.gif and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ36_HTML.gif
(2.26)

and to use the classical fixed point index for compact maps (see e.g., [29] or [30]).

We observe, as in [21], that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq75_HTML.gif leaves http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq76_HTML.gif invariant and is compact. We give the proof in the Carathéodory case for completeness.

Lemma 2.1.

If the hypotheses http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq77_HTML.gif hold, then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq78_HTML.gif maps http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq79_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq80_HTML.gif . Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq81_HTML.gif is a compact map.

Proof.

Take http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq82_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq83_HTML.gif . Then we have, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq84_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ37_HTML.gif
(2.27)
therefore
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ38_HTML.gif
(2.28)
Then we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ39_HTML.gif
(2.29)

Hence we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq85_HTML.gif . Moreover, the map http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq86_HTML.gif is compact since it is a sum of two compact maps: the compactness of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq87_HTML.gif is well known, and since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq88_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq89_HTML.gif are continuous, the perturbation http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq90_HTML.gif maps bounded sets into bounded subsets of a 1-dimensional space.

For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq91_HTML.gif , we use, as in [23, 31], the following bounded open subsets of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq92_HTML.gif :
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ40_HTML.gif
(2.30)

Note that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq93_HTML.gif .

We employ the following numbers:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ41_HTML.gif
(2.31)
and we note
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ42_HTML.gif
(2.32)

The proofs of the following results can be immediately deduced from the analogous results in [21], where the proofs involve a careful analysis of fixed point index and utilize order-preserving matrices. The only difference here is that we allow nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq94_HTML.gif to be Carathéodory instead of continuous. The lines of proof are not effected and therefore the proofs are omitted.

Firstly we give conditions which imply that the fixed point index is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq95_HTML.gif on the set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq96_HTML.gif .

Lemma 2.2.

Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq97_HTML.gif hold. Assume that there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq98_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ43_HTML.gif
(2.33)

Then the fixed point index, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq99_HTML.gif , is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq100_HTML.gif .

Now, we give conditions which imply that the fixed point index is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq101_HTML.gif on the set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq102_HTML.gif .

Lemma 2.3.

Suppose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq103_HTML.gif hold. Assume that there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq104_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ44_HTML.gif
(2.34)

Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq105_HTML.gif .

The two lemmas above lead to the following result on existence of one or two positive solutions for the integral equation (2.20). Note that, if the nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq106_HTML.gif has a suitable oscillatory behavior, it is possible to state, with the same arguments as in [23], a theorem on the existence of more than two positive solutions.

Theorem 2.4.

Suppose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq107_HTML.gif hold. Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq108_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq109_HTML.gif be as in (2.26). Then (2.20) has one positive solution in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq110_HTML.gif if either of the following conditions holds:

there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq112_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq113_HTML.gif such that (2.34) is satisfied for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq114_HTML.gif and (2.33) is satisfied for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq115_HTML.gif ;

there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq117_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq118_HTML.gif such that (2.33) is satisfied for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq119_HTML.gif and (2.34) is satisfied for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq120_HTML.gif .

Equation (2.20) has at least two positive solutions in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq121_HTML.gif if one of the following conditions hold.

there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq123_HTML.gif , with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq124_HTML.gif , such that (2.34) is satisfied for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq125_HTML.gif , (2.33) is satisfied for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq126_HTML.gif , and (2.34) is satisfied for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq127_HTML.gif ;

there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq129_HTML.gif , with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq130_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq131_HTML.gif , such that (2.33) is satisfied for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq132_HTML.gif , (2.34) is satisfied for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq133_HTML.gif , and (2.33) is satisfied for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq134_HTML.gif .

3. Optimal Constants and Examples

Consider the differential equation
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ45_HTML.gif
(3.1)

with the BCs (1.4)–(1.6).

The value http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq135_HTML.gif is given by direct calculation as follows:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ46_HTML.gif
(3.2)

We seek the "optimal" http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq136_HTML.gif for which http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq137_HTML.gif is a minimum. This type of problems has been tackled in the past, for example, in the second-order case for heat-flow problems in [32] and for beam equations (under different BCs) in [9, 12, 13].

The kernel http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq138_HTML.gif is a positive, nondecreasing function of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq139_HTML.gif , thus
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ47_HTML.gif
(3.3)
Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq140_HTML.gif is a nondecreasing function of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq141_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ48_HTML.gif
(3.4)

Such maximum is attained at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq142_HTML.gif . Thus the "optimal" interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq143_HTML.gif , for which http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq144_HTML.gif is a minimum, is the interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq145_HTML.gif ; this gives http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq146_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq147_HTML.gif .

Remark 3.1.

From Theorem 2.4, it is possible to state results for the existence of several nonnegative solutions for the homogeneous BVP
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ49_HTML.gif
(3.5)

For example, with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq148_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq149_HTML.gif , the BVP (3.5) has at least two positive solutions in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq150_HTML.gif if there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq151_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq152_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq153_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq154_HTML.gif .

These results are new and improve and complement the previous ones. Gupta [1] and Yao [4] studied the problem with more general nonlinearity but established existence results only. Davies and co-authors [2] and Graef and Henderson [3] obtain the existence of multiple positive solutions for a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq155_HTML.gif -order differential equation subject to our boundary conditions in the case http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq156_HTML.gif . In [2] the choice http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq157_HTML.gif gives the values http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq158_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq159_HTML.gif which replace our constants http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq160_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq161_HTML.gif in the growth conditions of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq162_HTML.gif . The growth conditions of the nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq163_HTML.gif in Theorem http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq164_HTML.gif in [3] cannot be compared with ours, but we do not require the restriction http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq165_HTML.gif .

The next examples illustrate the applicability of our results. Firstly we consider, as an illustrative example, the case of a nonlinear http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq166_HTML.gif -point problem.

Example 3.2.

Consider the differential equation
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ50_HTML.gif
(3.6)
with the BCs
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ51_HTML.gif
(3.7)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq167_HTML.gif and, as in [22], for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq168_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ52_HTML.gif
(3.8)
In this case we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq169_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq170_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq171_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq172_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ53_HTML.gif
(3.9)
We now fix http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq173_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq174_HTML.gif and show that all the constants that appear in (2.33) and in (2.34) can be computed. This choice leads to
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ54_HTML.gif
(3.10)
Moreover we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ55_HTML.gif
(3.11)

and conditions (2.34) and (2.33) read http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq175_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq176_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq177_HTML.gif holds, from Theorem 2.4 it follows that this BVP has a nontrivial solution in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq178_HTML.gif . A nonlinearity that easily verifies http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq179_HTML.gif , for example, is the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq180_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq181_HTML.gif and every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq182_HTML.gif .

We now give an example with continuously distributed positive measures.

Example 3.3.

Consider the differential equation
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ56_HTML.gif
(3.12)
with the BCs
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ57_HTML.gif
(3.13)
with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq183_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq184_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq185_HTML.gif , as in the previous example. In this case, we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ58_HTML.gif
(3.14)
and for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq186_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ59_HTML.gif
(3.15)
The Condition http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq187_HTML.gif becomes
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ60_HTML.gif
(3.16)
We now fix http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq188_HTML.gif . This choice leads to
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ61_HTML.gif
(3.17)

and conditions (2.34) and (2.33) read http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq189_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq190_HTML.gif .

4. Other Nonlinear BCs

So far we have discussed in detail the case of the BCs (1.4)–(1.6), but the same approach may be applied to (1.1) subject to the nonlinear BCs
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ62_HTML.gif
(4.1)
or
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ63_HTML.gif
(4.2)
or
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ64_HTML.gif
(4.3)
or
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ65_HTML.gif
(4.4)
or
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ66_HTML.gif
(4.5)

As in [12], where a different set of BCs were investigated, we point out that these nonlocal boundary conditions can be interpreted as feedback controls: for example, the BCs (4.1) can be seen as a control on the displacement at the left end and a device handling the shear force at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq191_HTML.gif .

Table 1 illustrates how the choice of the BCs affects the functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq192_HTML.gif and the constants http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq193_HTML.gif .
Table 1

Table 1

 

BCs (1.4)–(1.6)

BCs (4.1)

BCs (4.2)

BCs (4.3)

BCs (4.4)

BCs (4.5)

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq194_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq195_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq196_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq197_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq198_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq199_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq200_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq201_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq202_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq203_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq204_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq205_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq206_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq207_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq208_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq209_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq210_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq211_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq212_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq213_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq214_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq215_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq216_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq217_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq218_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq219_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq220_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq221_HTML.gif

Since one can see that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ67_HTML.gif
(4.6)

the cone http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq222_HTML.gif , given by the constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq223_HTML.gif , varies according to the nonhomogeneous BCs considered. This affects also, in a natural manner, conditions (2.33) and (2.34).

Declarations

Acknowledgments

The author would like to thank professor Salvatore Lopez of the Faculty of Engineering, University of Calabria, for shedding some light on the physical interpretation of this problem. The author wishes to thank the anonymous referees for their constructive comments.

Authors’ Affiliations

(1)
Dipartimento di Matematica, Università della Calabria

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© Paolamaria Pietramala. 2011

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