Tuyển tập các báo cáo nghiên cứu về sinh học được đăng trên tạp chí sinh học Journal of Biology đề tài: Research Article Symmetric Three-Term Recurrence Equations and Their Symplectic Structure | Hindawi Publishing Corporation Advances in Difference Equations Volume 2010 Article ID 626942 17 pages doi 2010 626942 Research Article Symmetric Three-Term Recurrence Equations and Their Symplectic Structure Roman Simon Hilscher1 and Vera Zeidan2 1 Department of Mathematics and Statistics Faculty of Science Masaryk University Kotlarska 2 61137 Brno Czech Republic 2 Department of Mathematics Michigan State University East Lansing MI 48824-1027 USA Correspondence should be addressed to Roman Simon Hilscher hilscher@ Received 11 March 2010 Accepted 1 May 2010 Academic Editor Martin Bohner Copyright 2010 R. Simon Hilscher and V. Zeidan. 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. We revive the study of the symmetric three-term recurrence equations. Our main result shows that these equations have a natural symplectic structure that is every symmetric three-term recurrence equation is a special discrete symplectic system. The assumptions on the coefficients in this paper are weaker and more natural than those in the current literature. In addition our result implies that symmetric three-term recurrence equations are completely equivalent with Jacobi difference equations arising in the discrete calculus of variations. Presented applications of this study include the Riccati equation and inequality detailed Sturmian separation and comparison theorems and the eigenvalue theory for these three-term recurrence and Jacobi equations. 1. Introduction In this paper we consider the symmetric three-term recurrence equation Sk 1Xk 2 - Tk 1Xk 1 SỊxk - 0 k 6 0 N - 1 z T where xk 6 Rn for k 6 0 N 1 z the real n X n matrices Sk and Tk are defined on 0 N Z with Tk being symmetric and Sk being invertible. The discrete intervals are defined by a b z a b n Z. Traditionally the recurrence equation T is .
