Open Access

Erratum to: Hierarchies of difference boundary value problems

Boundary Value Problems20122012:66

DOI: 10.1186/1687-2770-2012-66

Received: 25 May 2012

Accepted: 30 May 2012

Published: 28 June 2012

Erratum to: Boundary value problems, Volume 2011, Article ID 743135

  1. (1)

    The following paragraph needs to be inserted immediately after Theorem 4.2:

     
It is important to note that the spectral parameter in the original boundary value problems given in cases (1)-(9) of Table 1 for Theorem 4.2 must first, without loss of generality, be shifted so as to ensure that all the eigenvalues are greater than zero. Similarly, for cases (10)-(12) of Table 1 for Theorem 4.2, the spectral parameter must be shifted so that the original boundary value problem has the least eigenvalue 0. Having made these shifts we then take z ( n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq1_HTML.gif to be a solution to (1.1) for λ = λ 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq2_HTML.gif, i.e., throughout the paper we set λ 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq3_HTML.gif.
  1. (2)

    In Corollary 4.4 and its proof, there were typographical errors as well as notation that was not apparent. These should read as follows:

     
Corollary 4.4 If λ 1 , , λ s + l + m + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq4_HTML.gifare the eigenvalues of any one of the original boundary value problems (1)-(9), in Theorem 4.2, with corresponding eigenfunctions u 1 ( n ) , , u s + l + m + 1 ( n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq5_HTML.gif, then
  1. (i)

    λ 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq6_HTML.gif, λ 1 , , λ s + l + m + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq4_HTML.gif are the eigenvalues of the corresponding transformed boundary value problems (1)-(3), in Theorem 4.2, with corresponding eigenfunctions 1 z ( n 1 ) c ( n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq7_HTML.gif, u 1 ( n ) , , u s + l + m + 1 ( n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq8_HTML.gif;

     
  2. (ii)

    λ 1 , , λ s + l + m + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq4_HTML.gif are the eigenvalues of the corresponding transformed boundary value problems (4)-(9), in Theorem 4.2, with corresponding eigenfunctions u 1 ( n ) , , u s + l + m + 1 ( n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq9_HTML.gif.

     

Also, if λ 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq6_HTML.gif, λ 1 , , λ s + l + m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq10_HTML.gifare the eigenvalues of any one of the original boundary value problems (10)-(12), in Theorem 4.2, with corresponding eigenfunctions z ( n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq1_HTML.gif, u 1 ( n ) , , u s + l + m ( n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq11_HTML.gif, then λ 1 , , λ s + l + m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq10_HTML.gifare the eigenvalues of the corresponding transformed boundary value problems (10)-(12), in Theorem 4.2, with corresponding eigenfunctions u 1 ( n ) , , u s + l + m ( n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq12_HTML.gif.

Proof By Theorems 2.1, 3.2, 3.3, 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 λ 1 , , λ s + l + m + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq4_HTML.gif are the eigenvalues of one of the original boundary value problems, (1)-(9), with eigenfunctions u 1 ( n ) , , u s + l + m + 1 ( n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq13_HTML.gif, then:
  1. (i)

    1 z ( n 1 ) c ( n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq7_HTML.gif, u 1 ( n ) , , u s + l + m + 1 ( n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq13_HTML.gif are the eigenfunctions of the corresponding transformed boundary value problem, (1)-(3), with eigenvalues λ 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq6_HTML.gif, λ 1 , , λ s + l + m + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq4_HTML.gif. Since the transformed boundary value problems, (1)-(3), have s + l + m + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq14_HTML.gif eigenvalues, it follows that λ 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq6_HTML.gif, λ 1 , , λ s + l + m + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq15_HTML.gif constitute all the eigenvalues of the transformed boundary value problem;

     
  2. (ii)

    u 1 ( n ) , , u s + l + m + 1 ( n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq13_HTML.gif are the eigenfunctions of the corresponding transformed boundary value problem, (4)-(9), with eigenvalues λ 1 , , λ s + l + m + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq4_HTML.gif. Since the transformed boundary value problems, (4)-(9), have s + l + m + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq16_HTML.gif eigenvalues, it follows that λ 1 , , λ s + l + m + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq4_HTML.gif constitute all the eigenvalues of the transformed boundary value problem.

     

Also, again by Theorems 2.1, 3.2, 3.3, 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 λ 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq6_HTML.gif, λ 1 , , λ s + l + m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq17_HTML.gif are the eigenvalues of one of the original boundary value problems, (10)-(12), with eigenfunctions z ( n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq1_HTML.gif, u 1 ( n ) , , u s + l + m ( n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq18_HTML.gif, then u 1 ( n ) , , u s + l + m ( n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq11_HTML.gif are the eigenfunctions of the corresponding transformed boundary value problem, (10)-(12), with eigenvalues λ 1 , , λ s + l + m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq10_HTML.gif. Since the transformed boundary value problems, (10)-(12), have s + l + m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq19_HTML.gif eigenvalues, it follows that λ 1 , , λ s + l + m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-66/MediaObjects/13661_2012_Article_169_IEq10_HTML.gif constitute all the eigenvalues of the transformed boundary value problem. □

Declarations

Acknowledgements

SC was supported by NRF grant no. IFR2011040100017.

Authors’ Affiliations

(1)
School of Mathematics, University of the Witwatersrand

Copyright

© Currie and Love; licensee Springer 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.