Open Access

Hierarchies of Difference Boundary Value Problems

Boundary Value Problems20112011:743135

DOI: 10.1155/2011/743135

Received: 25 November 2010

Accepted: 11 January 2011

Published: 18 January 2011

Abstract

This paper generalises the work done in Currie and Love (2010), where we studied the effect of applying two Crum-type transformations to a weighted second-order difference equation with various combinations of Dirichlet, non-Dirichlet, and affine https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq1_HTML.gif -dependent boundary conditions at the end points, where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq2_HTML.gif is the eigenparameter. We now consider general https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq3_HTML.gif -dependent boundary conditions. In particular we show, using one of the Crum-type transformations, that it is possible to go up and down a hierarchy of boundary value problems keeping the form of the second-order difference equation constant but possibly increasing or decreasing the dependence on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq4_HTML.gif of the boundary conditions at each step. In addition, we show that the transformed boundary value problem either gains or loses an eigenvalue, or the number of eigenvalues remains the same as we step up or down the hierarchy.

1. Introduction

Our interest in this topic arose from the work done on transformations and factorisations of continuous Sturm-Liouville boundary value problems by Binding et al. [1] and Browne and Nillsen [2], notably. We make use of analogous ideas to those discussed in [35] to study difference equations in order to contribute to the development of the theory of discrete spectral problems.

Numerous efforts to develop hierarchies exist in the literature, however, they are not specifically aimed at difference equations per se and generally not for three-term recurrence relations. Ding et al., [6], derived a hierarchy of nonlinear differential-difference equations by starting with a two-parameter discrete spectral problem, as did Luo and Fan [7], whose hierarchy possessed bi-Hamiltonian structures. Clarkson et al.'s, [8], interest in hierarchies lay in the derivation of infinite sequences of systems of difference equations by using the B https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq5_HTML.gif cklund transformation for the equations in the second Painlev https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq6_HTML.gif equation hierarchy. Wu and Geng, [9], showed early on that the hierarchy of differential-difference equations possesses Hamiltonian structures while a Darboux transformation for the discrete spectral problem is shown to exist.

In this paper, we consider a weighted second-order difference equation of the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq7_HTML.gif represents a weight function and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq8_HTML.gif a potential function.

Our aim is to extend the results obtained in [10, 11] by establishing a hierarchy of difference boundary value problems. A key tool in our analysis will be the Crum-type transformation (2.1). In [10], it was shown that (2.1) leaves the form of the difference equation (1.1) unchanged. For us, the effect of (2.1) on the boundary conditions will be crucial. We consider https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq9_HTML.gif (eigenparameter)-dependent boundary conditions at the end points. In particular, the eigenparameter dependence at the initial end point will be given by a positive Nevanlinna function, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq10_HTML.gif say, and at the terminal end point by a negative Nevanlinna function, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq11_HTML.gif say. The case of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq12_HTML.gif was covered in [10] and the the case of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq13_HTML.gif constant was studied in [11]. Applying transformation (2.1) to the boundary conditions results in a so-called transformed boundary value problem, where either the new boundary conditions have more https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq14_HTML.gif -dependence, less https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq15_HTML.gif -dependence, or the same amount of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq16_HTML.gif -dependence as the original boundary conditions. Consequently the transformed boundary value problem has either one more eigenvalue, one less eigenvalue, or the same number of eigenvalues as the original boundary value problem. Thus, it is possible to construct a chain, or hierarchy, of difference boundary value problems where the successive links in the chain are obtained by applying the variations of (2.1) given in this paper. For instance, it is possible to go from a boundary value problem with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq17_HTML.gif -dependent boundary conditions to a boundary value problem with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq18_HTML.gif -independent boundary conditions or vice versa simply by applying the correct variation of (2.1) an appropriate number of times. Moreover, at each step, we can precisely track the eigenvalues that have been lost or gained. Hence, this paper provides a significant development in the theory of three-term difference boundary value problems in regard to singularities and asymptotics in the hierarchy structure. For similar results in the continuous case, see [12].

There is an obvious connection between the three-term difference equation and orthogonal polynomials. In fact, the three-term recurrence relation satisfied by orthogonal polynomials is perhaps the most important information for the constructive and computational use of orthogonal polynomials [13].

Difference equations and operators and results concerning their existence and construction of their solutions have been discussed in [14, 15]. Difference equations arise in numerous settings and have applications in diverse areas such as quantum field theory, combinatorics, mathematical physics and biology, dynamical systems, economics, statistics, electrical circuit analysis, computer visualization, and many other fields. They are especially useful where recursive computations are required. In particular see [16] [9, Introduction] for three physical applications of the difference equation (1.1), namely, the vibrating string, electrical network theory and Markov processes, in birth and death processes and random walks.

It should be noted that G. Teschl's work, [17, Chapter  11], on spectral and inverse spectral theory of Jacobi operators, provides an alternative factorisation, to that of [10], of a second-order difference equation, where the factors are adjoints of one another.

This paper is structured as follows.

In Section 2, all the necsessary results from [10] are recalled, in particular how (1.1) transforms under (2.1). In addition, we also recap some important properties of Nevanlinna functions.

The focus of Section 3 is to show exactly the effect that (2.1) has on boundary conditions of the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ2_HTML.gif
(1.2)

We give explicitly the new boundary conditions which are obeyed, from which it can be seen whether the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq19_HTML.gif -dependence has increased, decreased, or remained the same.

Lastly, in Section 4, we compare the spectrum of the original boundary value problem with that of the transformed boundary value problem and show under which conditions the transformed boundary value problem has one more eigenvalue, one less eigenvalue, or the same number of eigenvalues as the original boundary value problem.

2. Preliminaries

In [10], we considered (1.1) for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq20_HTML.gif , where the values of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq21_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq22_HTML.gif are given by boundary conditions, that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq23_HTML.gif is defined for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq24_HTML.gif .

Let the mapping https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq25_HTML.gif be defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ3_HTML.gif
(2.1)

where, throughout this paper, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq26_HTML.gif is a solution to (1.1) for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq27_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq28_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq29_HTML.gif . Whether or not https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq30_HTML.gif obeys the various given boundary conditions (to be specified later) is of vital importance in obtaining the results that follow.

From [10], we have the following theorem.

Theorem 2.1.

Under the mapping (2.1), (1.1) transforms to
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ4_HTML.gif
(2.2)
where for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq31_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ5_HTML.gif
(2.3)
We now recall some properties of Nevanlinna functions.
  1. (I)
    The inverse of a positive Nevanlinna function is a negative Nevanlinna function, that is
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ6_HTML.gif
    (2.4)
     
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq32_HTML.gif are positive Nevanlinna functions. This follows directly from the fact that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq33_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq34_HTML.gif .
  1. (II)
    If
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ7_HTML.gif
    (2.5)
     
then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ8_HTML.gif
(2.6)
This follows by (I) together with the fact that since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq35_HTML.gif has https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq36_HTML.gif zeros https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq37_HTML.gif has https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq38_HTML.gif poles. Also https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq39_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq40_HTML.gif so https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq41_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq42_HTML.gif . Thus, if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq43_HTML.gif is a positive Nevanlinna function of the form (2.5), then for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq44_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq45_HTML.gif is a negative Nevanlinna function of the same form.
  1. (III)
    If
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ9_HTML.gif
    (2.7)
     
then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ10_HTML.gif
(2.8)

since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq46_HTML.gif has https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq47_HTML.gif zeros so https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq48_HTML.gif has https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq49_HTML.gif poles and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq50_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq51_HTML.gif so https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq52_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq53_HTML.gif .

For the remainder of the paper, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq54_HTML.gif will denote a Nevanlinna function where

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq55_HTML.gif is the number of terms in the sum;

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq56_HTML.gif indicates the value of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq57_HTML.gif at which the boundary condition is imposed and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ11_HTML.gif
(2.9)

3. General https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq58_HTML.gif -Dependent Boundary Conditions

In this section, we show how https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq59_HTML.gif obeying general https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq60_HTML.gif -dependent boundary conditions transforms, under (2.1), to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq61_HTML.gif obeying various types of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq62_HTML.gif -dependent boundary conditions. The exact form of these boundary conditions is obtained by considering the number of zeros and poles (singularities) of the various Nevanlinna functions under discussion and these correlations are illustrated in the different graphs depicted in this section.

Lemma 3.1.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq63_HTML.gif obeys the boundary condition
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ12_HTML.gif
(3.1)
then the domain of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq64_HTML.gif may be extended from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq65_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq66_HTML.gif by forcing the condition
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ13_HTML.gif
(3.2)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ14_HTML.gif
(3.3)

with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq67_HTML.gif .

Proof.

The transformed equation (2.2), for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq68_HTML.gif , together with (3.2) gives
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ15_HTML.gif
(3.4)
Also the mapping (2.1), together with (3.1), yields
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ16_HTML.gif
(3.5)
Substituting (3.5) into (3.4), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ17_HTML.gif
(3.6)
Now (2.1), with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq69_HTML.gif , gives
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ18_HTML.gif
(3.7)
which when substituted into (3.6) and dividing through by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq70_HTML.gif results in
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ19_HTML.gif
(3.8)
This may be rewritten as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ20_HTML.gif
(3.9)
Using (1.1), with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq71_HTML.gif , together with (3.1), gives
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ21_HTML.gif
(3.10)
Subtracting (3.10) from (3.9) results in
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ22_HTML.gif
(3.11)
Rearranging the above equation and dividing through by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq72_HTML.gif yields
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ23_HTML.gif
(3.12)
and hence
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ24_HTML.gif
(3.13)

Thus https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq73_HTML.gif obeys the equation on the extended domain.

The remainder of this section illustrates why it is so important to distinguish between the two cases of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq74_HTML.gif obeying or not obeying the boundary conditions.

Theorem 3.2.

Consider https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq75_HTML.gif obeying the boundary condition (3.1) where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq76_HTML.gif is a positive Nevanlinna function, that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq77_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq78_HTML.gif . Under the mapping (2.1), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq79_HTML.gif obeying (3.1) transforms to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq80_HTML.gif obeying (3.2) as follows.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq82_HTML.gif does not obey (3.1) then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq83_HTML.gif obeys

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ25_HTML.gif
(3.14)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ26_HTML.gif
(3.15)

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq87_HTML.gif does obey (3.1) for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq88_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq89_HTML.gif obeys

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ27_HTML.gif
(3.16)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ28_HTML.gif
(3.17)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq92_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq93_HTML.gif are positive Nevanlinna functions.

In (A) and (B), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq94_HTML.gif is not possible.

Proof.

The fact that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq95_HTML.gif is by construction, see Lemma 3.1. We now examine the form of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq96_HTML.gif in Lemma 3.1. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq97_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq98_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq99_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq100_HTML.gif then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ29_HTML.gif
(3.18)
But
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ30_HTML.gif
(3.19)
thus
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ31_HTML.gif
(3.20)
Now https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq101_HTML.gif has the expansion
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ32_HTML.gif
(3.21)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq102_HTML.gif and the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq103_HTML.gif 's correspond to where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq104_HTML.gif , that is, the singularities of (3.20).

Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq105_HTML.gif is a positive Nevanlinna function it has a graph of the form shown in Figure 1.

Clearly, the gradient of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq106_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq107_HTML.gif is positive for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq108_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ33_HTML.gif
(3.22)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq109_HTML.gif does not obey (3.1), then the zeros of
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ34_HTML.gif
(3.23)

are the poles of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq110_HTML.gif , that is, the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq111_HTML.gif 's and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq112_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq113_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq114_HTML.gif . It is evident, from Figure 1, that the number of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq115_HTML.gif 's is equal to the number of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq116_HTML.gif 's, thus in (3.21), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq117_HTML.gif .

We now examine the form of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq118_HTML.gif in (3.21). As https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq119_HTML.gif it follows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq120_HTML.gif . Thus
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ35_HTML.gif
(3.24)
Therefore
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ36_HTML.gif
(3.25)
Hence, substituting into (3.20) gives
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ37_HTML.gif
(3.26)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ38_HTML.gif
(3.27)
Then since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq121_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq122_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq123_HTML.gif we have that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq124_HTML.gif and clearly if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq125_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq126_HTML.gif giving (3.14), that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ39_HTML.gif
(3.28)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq127_HTML.gif then we want https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq128_HTML.gif so that we have a positive Nevanlinna function, that is
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ40_HTML.gif
(3.29)
which means that either,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ41_HTML.gif
(3.30)
giving that, since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq129_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ42_HTML.gif
(3.31)
which is as shown in Figure 1, or,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ43_HTML.gif
(3.32)
giving that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ44_HTML.gif
(3.33)

but this means that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq130_HTML.gif which is not possible.

Thus, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq131_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq132_HTML.gif , that is, given https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq133_HTML.gif , the ratio https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq134_HTML.gif must be chosen suitably to ensure that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq135_HTML.gif is a positive Nevanlinna function as required. Hence we obtain (3.15), that is
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ45_HTML.gif
(3.34)

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq136_HTML.gif obeys (3.1), for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq137_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq138_HTML.gif . Thus in Figure 1, one of the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq139_HTML.gif 's https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq140_HTML.gif is equal to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq141_HTML.gif and since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq142_HTML.gif is less than the least eigenvalue of the boundary value problem (1.1), (3.1) together with a boundary condition at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq143_HTML.gif (specified later) it follows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq144_HTML.gif , as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq145_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq146_HTML.gif .

Now
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ46_HTML.gif
(3.35)
and as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq147_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ47_HTML.gif
(3.36)
Thus https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq148_HTML.gif is a removable singularity. Alternatively,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ48_HTML.gif
(3.37)

which illustrates that the singularity at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq149_HTML.gif is removable.

We now have that the number of nonremovable singularities, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq150_HTML.gif , in (3.20) is one less than the number of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq151_HTML.gif 's https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq152_HTML.gif , see Figure 1. Thus (3.21) becomes
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ49_HTML.gif
(3.38)
which may be rewritten as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ50_HTML.gif
(3.39)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq153_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq154_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq155_HTML.gif .

We now examine the form of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq156_HTML.gif in (3.39). As https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq157_HTML.gif , we have that, as before, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq158_HTML.gif . Thus
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ51_HTML.gif
(3.40)
Hence, from (3.20),
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ52_HTML.gif
(3.41)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ53_HTML.gif
(3.42)
Then since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq159_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq160_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq161_HTML.gif we have that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq162_HTML.gif and clearly if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq163_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq164_HTML.gif giving (3.16), that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ54_HTML.gif
(3.43)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq165_HTML.gif then we need https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq166_HTML.gif so that we have a positive Nevanlinna function, that is
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ55_HTML.gif
(3.44)
which means that either
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ56_HTML.gif
(3.45)
giving that, since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq167_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ57_HTML.gif
(3.46)
which is as shown in Figure 1, or,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ58_HTML.gif
(3.47)
giving that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ59_HTML.gif
(3.48)

but this means that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq168_HTML.gif which is not possible.

Thus, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq169_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq170_HTML.gif , that is, given https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq171_HTML.gif , the ratio https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq172_HTML.gif must be chosen suitably to ensure that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq173_HTML.gif is a positive Nevanlinna function as required. Hence, we obtain (3.17), that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ60_HTML.gif
(3.49)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Fig1_HTML.jpg
Figure 1

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq174_HTML.gif .

In the theorem below, we increase the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq175_HTML.gif dependence by introducing a nonzero https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq176_HTML.gif term in the original boundary condition. As in Theorem 3.2, the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq177_HTML.gif dependence of the transformed boundary condition depends on whether or not https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq178_HTML.gif obeys the given boundary condition. In addition, to ensure that the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq179_HTML.gif dependence of the transformed boundary condition is given by a positive Nevanlinna function it is necessary that the transformed boundary condition is imposed at 0 and 1 as opposed to −1 and 0. Thus the interval under consideration shrinks by one unit at the initial end point. By routine calculation it can be shown that the form of the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq180_HTML.gif dependence of the transformed boundary condition, if imposed at −1 and 0, is neither a positive Nevalinna function nor a negative Nevanlinna function.

Theorem 3.3.

Consider https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq181_HTML.gif obeying the boundary condition
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ61_HTML.gif
(3.50)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq182_HTML.gif is a positive Nevanlinna function, that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq183_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq184_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq185_HTML.gif . Under the mapping (2.1), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq186_HTML.gif obeying (3.50) transforms to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq187_HTML.gif obeying the following.

(1) If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq188_HTML.gif does not obey (3.50) then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq189_HTML.gif obeys
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ62_HTML.gif
(3.51)
(2) If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq190_HTML.gif does obey (3.50), for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq191_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq192_HTML.gif obeys
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ63_HTML.gif
(3.52)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq193_HTML.gif .

Proof.

Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq194_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq195_HTML.gif are defined we do not need to extend the domain in order to impose the boundary conditions (3.51) or (3.52).

The mapping (2.1), at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq196_HTML.gif , together with (3.50) gives
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ64_HTML.gif
(3.53)
Also (2.1), at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq197_HTML.gif , is
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ65_HTML.gif
(3.54)
Substituting in for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq198_HTML.gif from (1.1), with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq199_HTML.gif , and using (3.50), we obtain that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ66_HTML.gif
(3.55)
From (3.53) and (3.55), it now follows that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ67_HTML.gif
(3.56)
As in Theorem 3.2, let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq200_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq201_HTML.gif . Then (3.56) becomes
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ68_HTML.gif
(3.57)
From Theorem 3.2, we have that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq202_HTML.gif so
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ69_HTML.gif
(3.58)
Also, as in Theorem 3.2,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ70_HTML.gif
(3.59)
has the expansion
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ71_HTML.gif
(3.60)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq203_HTML.gif corresponds to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq204_HTML.gif , that is, the singularities of (3.59). Now https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq205_HTML.gif is a positive Nevanlinna function with graph given in Figure 2.

Clearly, the gradient of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq206_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq207_HTML.gif is positive for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq208_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ72_HTML.gif
(3.61)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq209_HTML.gif does not obey (3.50) then the zeros of
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ73_HTML.gif
(3.62)

are the poles of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq210_HTML.gif , that is, the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq211_HTML.gif 's and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq212_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq213_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq214_HTML.gif . It is evident, from Figure 2, that the number of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq215_HTML.gif 's is one more than the number of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq216_HTML.gif 's, thus in (3.60), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq217_HTML.gif .

We now examine the form of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq218_HTML.gif in (3.60). As https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq219_HTML.gif it follows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq220_HTML.gif , thus
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ74_HTML.gif
(3.63)

Hence, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq221_HTML.gif .

Using (3.58) we now obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ75_HTML.gif
(3.64)
Note that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq222_HTML.gif . Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ76_HTML.gif
(3.65)
then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ77_HTML.gif
(3.66)
Now https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq223_HTML.gif since if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq224_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq225_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq226_HTML.gif but https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq227_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq228_HTML.gif so this is not possible. Therefore by Section 2, Nevanlinna result (II), we have that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ78_HTML.gif
(3.67)

that is, (3.51) holds.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq229_HTML.gif does obey (3.50) for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq230_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq231_HTML.gif . Thus, in Figure 2, one of the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq232_HTML.gif 's, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq233_HTML.gif is equal to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq234_HTML.gif and since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq235_HTML.gif is less than the least eigenvalue of the boundary value problem (1.1), (3.50) together with a boundary condition at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq236_HTML.gif (specified later) it follows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq237_HTML.gif , as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq238_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq239_HTML.gif .

Now (3.59) can be written as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ79_HTML.gif
(3.68)
and as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq240_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ80_HTML.gif
(3.69)
Thus https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq241_HTML.gif is a removable singularity. Alternatively, we could substitute in for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq242_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq243_HTML.gif to illustrate that the singularity at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq244_HTML.gif is removable, see Theorem 3.2. Hence the number of nonremovable https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq245_HTML.gif 's is the same as the number of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq246_HTML.gif 's, see Figure 2. So (3.60) becomes
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ81_HTML.gif
(3.70)
which may be rewritten as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ82_HTML.gif
(3.71)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq247_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq248_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq249_HTML.gif .

We now examine the form of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq250_HTML.gif in (3.70). As https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq251_HTML.gif , we have that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq252_HTML.gif , thus
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ83_HTML.gif
(3.72)
Hence, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq253_HTML.gif . So, from (3.58) with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq254_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ84_HTML.gif
(3.73)
where, as before,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ85_HTML.gif
(3.74)
Thus, by Section 2, Nevanlinna result (II), we have that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ86_HTML.gif
(3.75)
that is, (3.52) holds.
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Fig2_HTML.jpg
Figure 2

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq255_HTML.gif .

In Theorem 3.4, we impose a boundary condition at the terminal end point and show how it is transformed according to whether or not https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq256_HTML.gif obeys the given boundary condition.

Theorem 3.4.

Consider https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq257_HTML.gif obeying the boundary condition at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq258_HTML.gif given by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ87_HTML.gif
(3.76)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq259_HTML.gif is a negative Nevanlinna function, that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq260_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq261_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq262_HTML.gif . Under the mapping (2.1), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq263_HTML.gif obeying (3.76) transforms to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq264_HTML.gif obeying the following.

(I) If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq265_HTML.gif does not obey (3.76) then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq266_HTML.gif obeys
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ88_HTML.gif
(3.77)
(II) If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq267_HTML.gif does obey (3.76) then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq268_HTML.gif obeys
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ89_HTML.gif
(3.78)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq269_HTML.gif .

Proof.

Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq270_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq271_HTML.gif are defined we do not need to extend the domain of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq272_HTML.gif in order to impose the boundary conditions (3.77) or (3.78).

The mapping (2.1), at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq273_HTML.gif , gives
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ90_HTML.gif
(3.79)
From (1.1), with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq274_HTML.gif , we can substitute in for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq275_HTML.gif in the above equation to get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ91_HTML.gif
(3.80)
Using (3.76), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ92_HTML.gif
(3.81)
But https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq276_HTML.gif obeys (1.1) at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq277_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq278_HTML.gif , so that (3.81) becomes
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ93_HTML.gif
(3.82)
Also, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq279_HTML.gif , (2.1) together with (3.76) yields
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ94_HTML.gif
(3.83)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ95_HTML.gif
(3.84)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq280_HTML.gif , then (3.84) may be rewritten as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ96_HTML.gif
(3.85)
By Section 2, Nevanlinna result (I), since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq281_HTML.gif is a negative Nevanlinna function it follows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq282_HTML.gif is a positive Nevanlinna function, which has the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ97_HTML.gif
(3.86)

by Section 2, Nevanlinna result (III).

As before https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq283_HTML.gif has expansion
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ98_HTML.gif
(3.87)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq284_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq285_HTML.gif , corresponds to the singularities of (3.85), that is, where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq286_HTML.gif . The graph of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq287_HTML.gif is as shown in Figure 3.

As before, the gradient of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq288_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq289_HTML.gif is positive for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq290_HTML.gif , that is
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ99_HTML.gif
(3.88)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq291_HTML.gif does not obey (3.76) then the zeros of
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ100_HTML.gif
(3.89)

are the poles of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq292_HTML.gif , that is, the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq293_HTML.gif 's and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq294_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq295_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq296_HTML.gif . Clearly, from Figure 3, the number of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq297_HTML.gif 's is the same as the the number of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq298_HTML.gif 's, thus in (3.87), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq299_HTML.gif .

Next, we examine the form of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq300_HTML.gif in (3.87). As https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq301_HTML.gif it follows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq302_HTML.gif . Thus
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ101_HTML.gif
(3.90)
Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq303_HTML.gif . Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ102_HTML.gif
(3.91)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq304_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq305_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq306_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq307_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq308_HTML.gif , which is precisely (3.77).

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq309_HTML.gif does obey (3.76) for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq310_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq311_HTML.gif . Thus in Figure 3, one of the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq312_HTML.gif 's, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq313_HTML.gif is equal to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq314_HTML.gif and since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq315_HTML.gif is less than the least eigenvalue of the boundary value problem (1.1), (3.76) together with a boundary condition at −1 (as given in Theorems 3.2 or 3.3) it follows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq316_HTML.gif , as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq317_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq318_HTML.gif .

Now
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ103_HTML.gif
(3.92)
and as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq319_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ104_HTML.gif
(3.93)

Thus https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq320_HTML.gif is a removable singularity. Again, alternatively, we could have substituted in for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq321_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq322_HTML.gif to illustrate that the singularity at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq323_HTML.gif is removable, see Theorem 3.2. Hence the number of nonremovable https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq324_HTML.gif 's is one less than the number of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq325_HTML.gif 's, see Figure 3.

So (3.87) becomes
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ105_HTML.gif
(3.94)
which may be rewritten as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ106_HTML.gif
(3.95)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq326_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq327_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq328_HTML.gif .

Now as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq329_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ107_HTML.gif
(3.96)
So, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ108_HTML.gif
(3.97)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq330_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq331_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq332_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq333_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq334_HTML.gif , that is, we obtain (3.78).
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Fig3_HTML.jpg
Figure 3

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq335_HTML.gif .

4. Comparison of the Spectra

In this section, we investigate how the spectrum of the original boundary value problem compares to the spectrum of the transformed boundary value problem. This is done by considering the degree of the eigenparameter polynomial for the various eigenconditions.

Lemma 4.1.

Consider the boundary value problem given by (1.1) for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq336_HTML.gif together with boundary conditions
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ109_HTML.gif
(4.1)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ110_HTML.gif
(4.2)

Then the boundary value problem (1.1), (4.1), (4.2) has https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq337_HTML.gif eigenvalues. (Note that the number of unit intervals considered is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq338_HTML.gif .)

Proof.

From (1.1), with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq339_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ111_HTML.gif
(4.3)
Substituting in for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq340_HTML.gif from (4.1) yields
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ112_HTML.gif
(4.4)
which may be rewritten as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ113_HTML.gif
(4.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq341_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq342_HTML.gif are real constants.

Now (1.1), for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq343_HTML.gif , together with (4.5) results in
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ114_HTML.gif
(4.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq344_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq345_HTML.gif are real constants.

Thus, by induction,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ115_HTML.gif
(4.7)
for real constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq346_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq347_HTML.gif . Similarly
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ116_HTML.gif
(4.8)

for real constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq348_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq349_HTML.gif .

Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq350_HTML.gif , using boundary condition (4.2) we obtain the following eigencondition:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ117_HTML.gif
(4.9)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq351_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq352_HTML.gif , are real constants.

Thus, the numerator is a polynomial, in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq353_HTML.gif , of order https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq354_HTML.gif . Note that, none of the roots of this polynomial are given by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq355_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq356_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq357_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq358_HTML.gif since, from Figures 1 to 3, it is easy to see that none of the eigenvalues of the boundary value problem are equal to the poles of the boundary conditions. Also https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq359_HTML.gif is not a problem as the curve of the Nevanlinna function never intersects with the horizontal or oblique asymptote. This means that there are no common factors to cancel out. Hence the eigencondition has https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq360_HTML.gif roots giving that the boundary value problem has https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq361_HTML.gif eigenvalues.

As a direct consequence of Theorems 2.1, 3.2, 3.3, 3.4, and Lemma 4.1 we have the following theorem.

Theorem 4.2.

For the original boundary value problem we consider twelve cases, (see Table 1 in the Appendix), each of which has s+l+m+1 eigenvalues. The corresponding transformed boundary value problem for each of the twelve cases, together with the number of eigenvalues for that transformed boundary value problem, is given in Table 1 (see the appendix).
Table 1

Table 1

 

Original BVP: (1.1) with bc's

Trans. BVP: (2.2) with bc's

No. of evals of Trans. BVP

1

(3.1) with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq362_HTML.gif and (3.76)

(3.14) and (3.77)

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq363_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq364_HTML.gif does not obey (3.1) or (3.76)

 

That is, one extra eval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq365_HTML.gif

2

(3.1) with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq366_HTML.gif and (3.76)

(3.15) and (3.77)

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq367_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq368_HTML.gif does not obey (3.1) or (3.76)

 

That is, one extra eval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq369_HTML.gif

3

(3.50) and (3.76)

(3.51) and (3.77)

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq370_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq371_HTML.gif does not obey (3.50) or (3.76)

 

That is, one extra eval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq372_HTML.gif

4

(3.1) with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq373_HTML.gif and (3.76)

(3.16) and (3.77)

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq374_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq375_HTML.gif obeys (3.1) but not (3.76)

 

That is, same number of evals

5

(3.1) with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq376_HTML.gif and (3.76)

(3.17) and (3.77)

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq377_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq378_HTML.gif obeys (3.1) but not (3.76)

 

That is, same number of evals

6

(3.50) and (3.76)

(3.52) and (3.77)

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq379_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq380_HTML.gif obeys (3.50) but not (3.76)

 

That is, same number of evals

7

(3.1) with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq381_HTML.gif and (3.76)

(3.14) and (3.78)

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq382_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq383_HTML.gif obeys (3.76) but not (3.1)

 

That is, same number of evals

8

(3.1) with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq384_HTML.gif and (3.76)

(3.15) and (3.78)

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq385_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq386_HTML.gif obeys (3.76) but not (3.1)

 

That is, same number of evals

9

(3.50) and (3.76)

(3.51) and (3.78)

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq387_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq388_HTML.gif obeys (3.76) but not (3.1)

 

That is, same number of evals

10

(3.1) with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq389_HTML.gif and (3.76)

(3.16) and (3.78)

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq390_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq391_HTML.gif obeys both (3.1) and (3.76)

 

That is, one less eval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq392_HTML.gif

11

(3.1) with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq393_HTML.gif and (3.76)

(3.17) and (3.78)

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq394_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq395_HTML.gif obeys both (3.1) and (3.76)

 

That is, one less eval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq396_HTML.gif

12

(3.50) and (3.76)

(3.52) and (3.78)

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq397_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq398_HTML.gif obeys both (3.50) and (3.76)

 

That is, one less eval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq399_HTML.gif

Remark 4.3.

To summarise we have the following.
  1. (a)

    If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq400_HTML.gif obeys the boundary conditions at both ends the transformed boundary value problem will have one less eigenvalue than the original boundary value problem, namely, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq401_HTML.gif .

     
  2. (b)

    If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq402_HTML.gif obeys the boundary condition at one end only the transformed boundary value problem will have the same eigenvalues as the original boundary value problem.

     
  3. (c)

    If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq403_HTML.gif does not obey any of the boundary conditions the transformed boundary value problem will have one more eigenvalue than the original boundary value problem, namely, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq404_HTML.gif .

     

Corollary 4.4.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq405_HTML.gif are the eigenvalues of any one of the original boundary value problems (1)–(9), in Theorem 4.2, with corresponding eigenfunctions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq406_HTML.gif then

(i) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq407_HTML.gif are the eigenvalues of the corresponding transformed boundary value problems (1)–(3), in Theorem 4.2, with corresponding eigenfunctions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq408_HTML.gif ;

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq409_HTML.gif are the eigenvalues of the corresponding transformed boundary value problems (4)–(9), in Theorem 4.2, with corresponding eigenfunctions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq410_HTML.gif .

Also, if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq411_HTML.gif are the eigenvalues of any one of the original boundary value problems (10)–(12), in Theorem 4.2, with corresponding eigenfunctions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq412_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq413_HTML.gif are the eigenvalues of the corresponding transformed boundary value problems (10)–(12), in Theorem 4.2, with corresponding eigenfunctions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq414_HTML.gif .

Proof.

By Theorems 2.1, 3.2, 3.3, and 3.4, we have that (2.1) transforms eigenfunctions of the original boundary value problems (1)–(9) to eigenfunctions of the corresponding transformed boundary value problems. In particular, if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq415_HTML.gif are the eigenvalues of one of the original boundary value problems, (1)–(9), with eigenfunctions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq416_HTML.gif then

(i) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq417_HTML.gif are the eigenfunctions of the corresponding transformed boundary value problem, (1)–(3), with eigenvalues https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq418_HTML.gif . Since the transformed boundary value problems, (1)–(3), have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq419_HTML.gif eigenvalues it follows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq420_HTML.gif constitute all the eigenvalues of the transformed boundary value problem;

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq421_HTML.gif are the eigenfunctions of the corresponding transformed boundary value problem, (4)–(9), with eigenvalues https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq422_HTML.gif . Since the transformed boundary value problems, (4)–(9), have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq423_HTML.gif eigenvalues it follows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq424_HTML.gif constitute all the eigenvalues of the transformed boundary value problem.

Also, again by Theorems 2.1, 3.2, 3.3, and 3.4, we have that (2.1) transforms eigenfunctions of the original boundary value problems (10)–(12) to eigenfunctions of the corresponding transformed boundary value problems. In particular, if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq425_HTML.gif are the eigenvalues of one of the original boundary value problems, (10)–(12), with eigenfunctions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq426_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq427_HTML.gif are the eigenfunctions of the corresponding transformed boundary value problem, (10)–(12), with eigenvalues https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq428_HTML.gif . Since the transformed boundary value problems, (10)–(12), have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq429_HTML.gif eigenvalues it follows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq430_HTML.gif constitute all the eigenvalues of the transformed boundary value problem.

Declarations

Acknowledgments

The authors would like to thank Professor Bruce A. Watson for his useful input. S. Currie is supported by NRF Grant nos. TTK2007040500005 and FA2007041200006.

Authors’ Affiliations

(1)
School of Mathematics, University of the Witwatersrand

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Copyright

© S. Currie and A. D. Love. 2011

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