tailieunhanh - báo cáo hóa học:" Research Article Transformations of Difference Equations II"

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 Transformations of Difference Equations II | Hindawi Publishing Corporation Advances in Difference Equations Volume 2010 Article ID 623508 23 pages doi 2010 623508 Research Article Transformations of Difference Equations II 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 13 April 2010 Revised 30 July 2010 Accepted 6 September 2010 Academic Editor M. Cecchi Copyright 2010 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 is an extension of the work done by Currie and Love 2010 where we studied the effect of applying two Crum-type transformations to a weighted second-order difference equation with non-eigenparameter-dependent boundary conditions at the end points. In particular we now consider boundary conditions which depend affinely on the eigenparameter together with various combinations of Dirichlet and non-Dirichlet boundary conditions. The spectra of the resulting transformed boundary value problems are then compared to the spectra of the original boundary value problems. 1. Introduction This paper continues the work done in 1 where we considered a weighted second-order difference equation of the following form c n y n 1 - b n y n cfn - 1 y n - 1 -c n fy n with cfri 0 representing a weight function and bin a potential function. This paper is structured as follows. The relevant results from 1 which will be used throughout the remainder of this paper are briefly recapped in Section 2. In Section 3 we show how non-Dirichlet boundary conditions transform to affine 1-dependent boundary conditions. In addition we provide conditions which ensure that the linear function in 1 in the affine 1-dependent boundary conditions is a Nevanlinna or Herglotz function.