tailieunhanh - Báo cáo hóa học: " Research Article Hierarchies of Difference Boundary Value Problems"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Hierarchies of Difference Boundary Value Problems | Hindawi Publishing Corporation Boundary Value Problems Volume 2011 Article ID 743135 27 pages doi 2011 743135 Research Article Hierarchies of Difference Boundary Value Problems Sonja Currie and Anne D. Love School of Mathematics University of the Witwatersrand Private Bag 3 Wits 2050 South Africa Correspondence should be addressed to Sonja Currie Received 25 November 2010 Accepted 11 January 2011 Academic Editor Olimpio Miyagaki Copyright 2011 S. Currie and A. 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 l-dependent boundary conditions at the end points where l is the eigenparameter. We now consider general l-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 l 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 3-5 to study difference equations in order to contribute to the development of the theory of discrete spectral problems. Numerous efforts to .