Open Access

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
https://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 https://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
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq2_HTML.gif and the bending moment https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq3_HTML.gif at https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq5_HTML.gif and the angular attitude https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq6_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq7_HTML.gif and the displacement https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ4_HTML.gif
(1.4)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ5_HTML.gif
(1.5)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ6_HTML.gif
(1.6)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq9_HTML.gif are nonnegative continuous functions and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq10_HTML.gif are linear functionals given by
https://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 https://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 https://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 https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ9_HTML.gif
(1.9)

with suitable functions https://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
https://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 https://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 https://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
https://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
https://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 https://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,
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ14_HTML.gif
(2.4)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ15_HTML.gif
(2.5)
Therefore the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq18_HTML.gif in (2.2) is given by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq19_HTML.gif . Now, since
https://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:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq20_HTML.gif , a function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq21_HTML.gif , and a constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq22_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ19_HTML.gif
(2.9)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq23_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq24_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq25_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq26_HTML.gif , a natural choice could be
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq28_HTML.gif .

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

Therefore, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq38_HTML.gif , we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq39_HTML.gif with respect to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq40_HTML.gif is non-positive for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq41_HTML.gif , that is, the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq42_HTML.gif is a non-increasing function of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq43_HTML.gif . Therefore, for an arbitrary https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq44_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ22_HTML.gif
(2.12)
where
https://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)
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ25_HTML.gif
(2.15)
that are
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq45_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ27_HTML.gif
(2.17)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq46_HTML.gif , and therefore
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ28_HTML.gif
(2.18)
with
https://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
https://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 https://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 https://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:

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

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

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq64_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ32_HTML.gif
(2.22)

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

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq67_HTML.gif are positive bounded linear functionals on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq68_HTML.gif given by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq69_HTML.gif ;

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq71_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq72_HTML.gif and https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ35_HTML.gif
(2.25)
for an arbitrary https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq74_HTML.gif and
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq75_HTML.gif leaves https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq77_HTML.gif hold, then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq78_HTML.gif maps https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq79_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq80_HTML.gif . Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq81_HTML.gif is a compact map.

Proof.

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

Hence we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq85_HTML.gif . Moreover, the map https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq87_HTML.gif is well known, and since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq88_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq89_HTML.gif are continuous, the perturbation https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq92_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ40_HTML.gif
(2.30)

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

We employ the following numbers:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ41_HTML.gif
(2.31)
and we note
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq95_HTML.gif on the set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq96_HTML.gif .

Lemma 2.2.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq97_HTML.gif hold. Assume that there exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq98_HTML.gif such that
https://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, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq99_HTML.gif , is https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq101_HTML.gif on the set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq102_HTML.gif .

Lemma 2.3.

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

Then https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq107_HTML.gif hold. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq108_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq112_HTML.gif with https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq114_HTML.gif and (2.33) is satisfied for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq115_HTML.gif ;

there exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq117_HTML.gif with https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq119_HTML.gif and (2.34) is satisfied for https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq123_HTML.gif , with https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq125_HTML.gif , (2.33) is satisfied for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq126_HTML.gif , and (2.34) is satisfied for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq127_HTML.gif ;

there exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq129_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq130_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq132_HTML.gif , (2.34) is satisfied for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq133_HTML.gif , and (2.33) is satisfied for https://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
https://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 https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ46_HTML.gif
(3.2)

We seek the "optimal" https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq136_HTML.gif for which https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq138_HTML.gif is a positive, nondecreasing function of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq139_HTML.gif , thus
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ47_HTML.gif
(3.3)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq140_HTML.gif is a nondecreasing function of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq141_HTML.gif , we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq142_HTML.gif . Thus the "optimal" interval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq143_HTML.gif , for which https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq144_HTML.gif is a minimum, is the interval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq145_HTML.gif ; this gives https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq146_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ49_HTML.gif
(3.5)

For example, with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq148_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq150_HTML.gif if there exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq151_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq152_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq153_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq156_HTML.gif . In [2] the choice https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq157_HTML.gif gives the values https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq158_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq159_HTML.gif which replace our constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq160_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq161_HTML.gif in the growth conditions of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq162_HTML.gif . The growth conditions of the nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq163_HTML.gif in Theorem https://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 https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ50_HTML.gif
(3.6)
with the BCs
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ51_HTML.gif
(3.7)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq167_HTML.gif and, as in [22], for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq168_HTML.gif
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq169_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq170_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq171_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq172_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ53_HTML.gif
(3.9)
We now fix https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq173_HTML.gif , https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ54_HTML.gif
(3.10)
Moreover we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq175_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq176_HTML.gif . Since https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq178_HTML.gif . A nonlinearity that easily verifies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq179_HTML.gif , for example, is the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq180_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq181_HTML.gif and every https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ56_HTML.gif
(3.12)
with the BCs
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ57_HTML.gif
(3.13)
with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq183_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq184_HTML.gif , https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ58_HTML.gif
(3.14)
and for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq186_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ59_HTML.gif
(3.15)
The Condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq187_HTML.gif becomes
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ60_HTML.gif
(3.16)
We now fix https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq188_HTML.gif . This choice leads to
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq189_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ62_HTML.gif
(4.1)
or
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ63_HTML.gif
(4.2)
or
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ64_HTML.gif
(4.3)
or
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_Equ65_HTML.gif
(4.4)
or
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq192_HTML.gif and the constants https://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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

the cone https://static-content.springer.com/image/art%3A10.1155%2F2011%2F376782/MediaObjects/13661_2010_Article_36_IEq222_HTML.gif , given by the constant https://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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