Boundary Value Problems
Volume 2011 (2011), Article ID 743135, 27 pages
Research Article

Hierarchies of Difference Boundary Value Problems

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

Received 25 November 2010; Accepted 11 January 2011

Academic Editor: Olimpio Miyagaki

Copyright © 2011 Sonja Currie and Anne D. Love. 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.


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 𝜆 -dependent boundary conditions at the end points, where 𝜆 is the eigenparameter. We now consider general 𝜆 -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 𝜆 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.