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Erratum to: Hierarchies of difference boundary value problems

Sonja Currie* and Anne D Love

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School of Mathematics, University of the Witwatersrand, Private Bag 3, WITS, 2050, Johannesburg, South Africa

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Boundary Value Problems 2012, 2012:66  doi:10.1186/1687-2770-2012-66

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/66


Received:25 May 2012
Accepted:30 May 2012
Published:28 June 2012

© 2012 Currie and Love; licensee Springer

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.

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

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M1">View MathML</a> to be a solution to (1.1) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M2">View MathML</a>, i.e., throughout the paper we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M3">View MathML</a>.

(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 4.4If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M4">View MathML</a>are the eigenvalues of any one of the original boundary value problems (1)-(9), in Theorem 4.2, with corresponding eigenfunctions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M5">View MathML</a>, then

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M4">View MathML</a>are the eigenvalues of the corresponding transformed boundary value problems (1)-(3), in Theorem 4.2, with corresponding eigenfunctions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M9">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M4">View MathML</a>are the eigenvalues of the corresponding transformed boundary value problems (4)-(9), in Theorem 4.2, with corresponding eigenfunctions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M11">View MathML</a>.

Also, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M13">View MathML</a>are the eigenvalues of any one of the original boundary value problems (10)-(12), in Theorem 4.2, with corresponding eigenfunctions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M15">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M13">View MathML</a>are the eigenvalues of the corresponding transformed boundary value problems (10)-(12), in Theorem 4.2, with corresponding eigenfunctions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M17">View MathML</a>.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M4">View MathML</a> are the eigenvalues of one of the original boundary value problems, (1)-(9), with eigenfunctions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M19">View MathML</a>, then:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M19">View MathML</a> are the eigenfunctions of the corresponding transformed boundary value problem, (1)-(3), with eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M4">View MathML</a>. Since the transformed boundary value problems, (1)-(3), have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M24">View MathML</a> eigenvalues, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M26">View MathML</a> constitute all the eigenvalues of the transformed boundary value problem;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M19">View MathML</a> are the eigenfunctions of the corresponding transformed boundary value problem, (4)-(9), with eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M4">View MathML</a>. Since the transformed boundary value problems, (4)-(9), have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M29">View MathML</a> eigenvalues, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M4">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M32">View MathML</a> are the eigenvalues of one of the original boundary value problems, (10)-(12), with eigenfunctions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M34">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M15">View MathML</a> are the eigenfunctions of the corresponding transformed boundary value problem, (10)-(12), with eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M13">View MathML</a>. Since the transformed boundary value problems, (10)-(12), have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M37">View MathML</a> eigenvalues, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/66/mathml/M13">View MathML</a> constitute all the eigenvalues of the transformed boundary value problem. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both SC and ADL worked jointly and separately on all aspects of this research.

Acknowledgements

SC was supported by NRF grant no. IFR2011040100017.