Hierarchies of Difference Boundary Value Problems

  • Sonja Currie1Email author and

    Affiliated with

    • AnneD Love1

      Affiliated with

      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 http://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 http://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 http://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 http://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 http://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 http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ1_HTML.gif
      (1.1)

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq7_HTML.gif represents a weight function and http://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 http://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, http://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, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq11_HTML.gif say. The case of http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq14_HTML.gif -dependence, less http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq15_HTML.gif -dependence, or the same amount of http://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 http://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 http://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
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq20_HTML.gif , where the values of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq21_HTML.gif and http://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, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq23_HTML.gif is defined for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq24_HTML.gif .

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

      where, throughout this paper, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq27_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq28_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq29_HTML.gif . Whether or not http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ4_HTML.gif
      (2.2)
      where for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq31_HTML.gif
      http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ6_HTML.gif
        (2.4)
         
      where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq33_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq34_HTML.gif .
      1. (II)
        If
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ7_HTML.gif
        (2.5)
         
      then
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq35_HTML.gif has http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq36_HTML.gif zeros http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq37_HTML.gif has http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq38_HTML.gif poles. Also http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq39_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq40_HTML.gif so http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq41_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq42_HTML.gif . Thus, if http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq44_HTML.gif , http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ9_HTML.gif
        (2.7)
         
      then
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ10_HTML.gif
      (2.8)

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

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

      http://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;

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq56_HTML.gif indicates the value of http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ11_HTML.gif
      (2.9)

      3. General http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq59_HTML.gif obeying general http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq61_HTML.gif obeying various types of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq63_HTML.gif obeys the boundary condition
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ12_HTML.gif
      (3.1)
      then the domain of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq64_HTML.gif may be extended from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq65_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq66_HTML.gif by forcing the condition
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ13_HTML.gif
      (3.2)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ14_HTML.gif
      (3.3)

      with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq68_HTML.gif , together with (3.2) gives
      http://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
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ17_HTML.gif
      (3.6)
      Now (2.1), with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq69_HTML.gif , gives
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq70_HTML.gif results in
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ20_HTML.gif
      (3.9)
      Using (1.1), with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq71_HTML.gif , together with (3.1), gives
      http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq72_HTML.gif yields
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ23_HTML.gif
      (3.12)
      and hence
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ24_HTML.gif
      (3.13)

      Thus http://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 http://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 http://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 http://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, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq77_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq78_HTML.gif . Under the mapping (2.1), http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq79_HTML.gif obeying (3.1) transforms to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq80_HTML.gif obeying (3.2) as follows.

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

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

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

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

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq92_HTML.gif , that is, http://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), http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq96_HTML.gif in Lemma 3.1. Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq97_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq98_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq99_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq100_HTML.gif then
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ29_HTML.gif
      (3.18)
      But
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ30_HTML.gif
      (3.19)
      thus
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ31_HTML.gif
      (3.20)
      Now http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq101_HTML.gif has the expansion
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ32_HTML.gif
      (3.21)

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq102_HTML.gif and the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq103_HTML.gif 's correspond to where http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq106_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq107_HTML.gif is positive for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq108_HTML.gif , that is,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ33_HTML.gif
      (3.22)
      If http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ34_HTML.gif
      (3.23)

      are the poles of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq110_HTML.gif , that is, the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq111_HTML.gif 's and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq112_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq113_HTML.gif for http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq116_HTML.gif 's, thus in (3.21), http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq117_HTML.gif .

      We now examine the form of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq118_HTML.gif in (3.21). As http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq119_HTML.gif it follows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq120_HTML.gif . Thus
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ35_HTML.gif
      (3.24)
      Therefore
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ37_HTML.gif
      (3.26)
      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ38_HTML.gif
      (3.27)
      Then since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq121_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq122_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq123_HTML.gif we have that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq124_HTML.gif and clearly if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq125_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq126_HTML.gif giving (3.14), that is,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ39_HTML.gif
      (3.28)
      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq127_HTML.gif then we want http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ40_HTML.gif
      (3.29)
      which means that either,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ41_HTML.gif
      (3.30)
      giving that, since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq129_HTML.gif ,
      http://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,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ43_HTML.gif
      (3.32)
      giving that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ44_HTML.gif
      (3.33)

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

      Thus, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq131_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq132_HTML.gif , that is, given http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq133_HTML.gif , the ratio http://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 http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ45_HTML.gif
      (3.34)

      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq136_HTML.gif obeys (3.1), for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq137_HTML.gif , then http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq139_HTML.gif 's http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq140_HTML.gif is equal to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq141_HTML.gif and since http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq143_HTML.gif (specified later) it follows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq144_HTML.gif , as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq145_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq146_HTML.gif .

      Now
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ46_HTML.gif
      (3.35)
      and as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq147_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ47_HTML.gif
      (3.36)
      Thus http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq148_HTML.gif is a removable singularity. Alternatively,
      http://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 http://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, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq151_HTML.gif 's http://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
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ50_HTML.gif
      (3.39)

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

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

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

      Thus, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq169_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq170_HTML.gif , that is, given http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq171_HTML.gif , the ratio http://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 http://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,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ60_HTML.gif
      (3.49)
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Fig1_HTML.jpg
      Figure 1

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq175_HTML.gif dependence by introducing a nonzero http://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 http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq181_HTML.gif obeying the boundary condition
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ61_HTML.gif
      (3.50)

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

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

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

      Proof.

      Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq194_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq196_HTML.gif , together with (3.50) gives
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ64_HTML.gif
      (3.53)
      Also (2.1), at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq197_HTML.gif , is
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ65_HTML.gif
      (3.54)
      Substituting in for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq198_HTML.gif from (1.1), with http://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
      http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq200_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq201_HTML.gif . Then (3.56) becomes
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq202_HTML.gif so
      http://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,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ70_HTML.gif
      (3.59)
      has the expansion
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ71_HTML.gif
      (3.60)

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq203_HTML.gif corresponds to http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq206_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq207_HTML.gif is positive for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq208_HTML.gif , that is,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ72_HTML.gif
      (3.61)
      If http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ73_HTML.gif
      (3.62)

      are the poles of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq210_HTML.gif , that is, the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq211_HTML.gif 's and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq212_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq213_HTML.gif for http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq216_HTML.gif 's, thus in (3.60), http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq217_HTML.gif .

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

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

      Using (3.58) we now obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ75_HTML.gif
      (3.64)
      Note that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq222_HTML.gif . Let
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ76_HTML.gif
      (3.65)
      then
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ77_HTML.gif
      (3.66)
      Now http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq223_HTML.gif since if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq224_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq225_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq226_HTML.gif but http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq227_HTML.gif and http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq229_HTML.gif does obey (3.50) for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq230_HTML.gif then http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq232_HTML.gif 's, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq233_HTML.gif is equal to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq234_HTML.gif and since http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq236_HTML.gif (specified later) it follows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq237_HTML.gif , as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq238_HTML.gif for all http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ79_HTML.gif
      (3.68)
      and as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq240_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ80_HTML.gif
      (3.69)
      Thus http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq242_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq243_HTML.gif to illustrate that the singularity at http://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 http://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 http://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
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ82_HTML.gif
      (3.71)

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

      We now examine the form of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq250_HTML.gif in (3.70). As http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq251_HTML.gif , we have that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq252_HTML.gif , thus
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ83_HTML.gif
      (3.72)
      Hence, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq253_HTML.gif . So, from (3.58) with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq254_HTML.gif , we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ84_HTML.gif
      (3.73)
      where, as before,
      http://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
      http://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.
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Fig2_HTML.jpg
      Figure 2

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

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

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

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

      Proof.

      Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq270_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq273_HTML.gif , gives
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ90_HTML.gif
      (3.79)
      From (1.1), with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq274_HTML.gif , we can substitute in for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq275_HTML.gif in the above equation to get
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ92_HTML.gif
      (3.81)
      But http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq276_HTML.gif obeys (1.1) at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq277_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq278_HTML.gif , so that (3.81) becomes
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ93_HTML.gif
      (3.82)
      Also, for http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ94_HTML.gif
      (3.83)
      Therefore,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ95_HTML.gif
      (3.84)
      Let http://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
      http://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 http://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 http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq283_HTML.gif has expansion
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ98_HTML.gif
      (3.87)

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq284_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq286_HTML.gif . The graph of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq288_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq289_HTML.gif is positive for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq290_HTML.gif , that is
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ99_HTML.gif
      (3.88)
      If http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ100_HTML.gif
      (3.89)

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

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

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

      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq309_HTML.gif does obey (3.76) for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq310_HTML.gif then http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq312_HTML.gif 's, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq313_HTML.gif is equal to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq314_HTML.gif and since http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq316_HTML.gif , as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq317_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq318_HTML.gif .

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

      Thus http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq321_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq322_HTML.gif to illustrate that the singularity at http://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 http://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 http://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
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ106_HTML.gif
      (3.95)

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

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

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq336_HTML.gif together with boundary conditions
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ109_HTML.gif
      (4.1)
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq338_HTML.gif .)

      Proof.

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

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq341_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq343_HTML.gif , together with (4.5) results in
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ114_HTML.gif
      (4.6)

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

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

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

      Since http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_Equ117_HTML.gif
      (4.9)

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq351_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq353_HTML.gif , of order http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq355_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq356_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq357_HTML.gif , http://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 http://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 http://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 http://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 http://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)

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

       

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq365_HTML.gif

      2

      (3.1) with http://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)

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

       

      http://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 http://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)

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

       

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq372_HTML.gif

      4

      (3.1) with http://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)

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

       

      http://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 http://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)

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

       

      http://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)

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

       

      http://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 http://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)

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

       

      http://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 http://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)

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

       

      http://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)

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

       

      http://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 http://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)

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

       

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq392_HTML.gif

      11

      (3.1) with http://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)

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

       

      http://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 http://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)

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

       

      http://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 http://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 http://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, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq401_HTML.gif .

         
      2. (b)

        If http://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 http://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, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq404_HTML.gif .

         

      Corollary 4.4.

      If http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq406_HTML.gif then

      (i) http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq408_HTML.gif ;

      (ii) http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq410_HTML.gif .

      Also, if http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq412_HTML.gif then http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq416_HTML.gif then

      (i) http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq419_HTML.gif eigenvalues it follows that http://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) http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq423_HTML.gif eigenvalues it follows that http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq426_HTML.gif then http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F743135/MediaObjects/13661_2010_Article_56_IEq429_HTML.gif eigenvalues it follows that http://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

      References

      1. Binding PA, Browne PJ, Watson BA: Spectral isomorphisms between generalized Sturm-Liouville problems. In Linear Operators and Matrices, Operator Theory: Advances and Applications. Volume 130. Birkhäuser, Basel, Switzerland; 2002:135-152.View Article
      2. Browne PJ, Nillsen RV: On difference operators and their factorization. Canadian Journal of Mathematics 1983, 35(5):873-897. 10.4153/CJM-1983-050-2MATHMathSciNetView Article
      3. Binding PA, Browne PJ, Watson BA: Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter. II. Journal of Computational and Applied Mathematics 2002, 148(1):147-168. 10.1016/S0377-0427(02)00579-4MATHMathSciNetView Article
      4. Binding PA, Browne PJ, Watson BA: Sturm-Liouville problems with reducible boundary conditions. Proceedings of the Edinburgh Mathematical Society. Series II 2006, 49(3):593-608. 10.1017/S0013091505000131MATHMathSciNetView Article
      5. Binding PA, Browne PJ, Watson BA: Transformations between Sturm-Liouville problems with eigenvalue dependent and independent boundary conditions. Bulletin of the London Mathematical Society 2001, 33(6):749-757. 10.1112/S0024609301008177MATHMathSciNetView Article
      6. Ding H-Y, Sun Y-P, Xue F-C: A hierarchy of differential-difference equations, conservation laws and new integrable coupling system. Communications in Nonlinear Science and Numerical Simulation 2010, 15(8):2037-2043. 10.1016/j.cnsns.2009.08.022MATHMathSciNetView Article
      7. Luo L, Fan E-G: A hierarchy of differential-difference equations and their integrable couplings. Chinese Physics Letters 2007, 24(6):1444-1447. 10.1088/0256-307X/24/6/005MathSciNetView Article
      8. Clarkson PA, Hone ANW, Joshi N: Hierarchies of difference equations and Bäcklund transformations. Journal of Nonlinear Mathematical Physics 2003, 10(supplement 2):13-26. 10.2991/jnmp.2003.10.s2.2MathSciNetView Article
      9. Wu Y, Geng X: A new hierarchy of integrable differential-difference equations and Darboux transformation. Journal of Physics A 1998, 31(38):L677-L684. 10.1088/0305-4470/31/38/004MATHMathSciNetView Article
      10. Currie S, Love AD: Transformations of difference equations I. Advances in Difference Equations 2010, 2010:-22.
      11. Currie S, Love AD: Transformations of difference equations II. Advances in Difference Equations 2010, 2010:-23.
      12. Binding PA, Browne PJ, Watson BA: Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter. I. Proceedings of the Edinburgh Mathematical Society. Series II 2002, 45(3):631-645.MATHMathSciNetView Article
      13. Gautschi W: Orthogonal Polynomials: Computation and Approximation, Numerical Mathematics and Scientific Computation. Oxford University Press, New York, NY, USA; 2004:x+301.
      14. Miller KS: Linear Difference Equations. W. A. Benjamin, New York, NY, USA; 1968:x+105.MATH
      15. Miller KS: An Introduction to the Calculus of Finite Differences and Difference Equations. Dover, New York, NY, USA; 1966:viii+167.MATH
      16. Atkinson FV: Discrete and Continuous Boundary Problems, Mathematics in Science and Engineering. Volume 8. Academic Press, New York, NY, USA; 1964:xiv+570.
      17. Teschl G: Jacobi Operators and Completely Integrable Nonlinear Lattices, Mathematical Surveys and Monographs. Volume 72. American Mathematical Society, Providence, RI, USA; 2000:xvii+351.

      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.