Disconjugacy and transformations for symplectic systems
type: Item
instance of: scholarly article
title: Disconjugacy and transformations for symplectic systems
zbMATH Open document ID: 0894.39005
published in: Rocky Mountain Journal of Mathematics
publication date: 1998-08-10T00:00:00Z
full work available at URL: CONT27-3.html
review text: Let \(I\) be the \(n\times n\) identity matrix, and let \(J=\left (\begin{smallmatrix} 0 & I \\ -I & 0 \end{smallmatrix} \right)\). The \(2n\times 2n\) matrix \(S=\left( \begin{smallmatrix} A & B \\ C & D \end{smallmatrix} \right)\) is called symplectic if \(S^TJS=J\) holds, and the system \(z_{k+1} =S_kz_k\), \(0\leq k\leq N\), is symplectic if the matrix \(S\) is symplectic. A solution \(z= {x\choose u}\) to the system has a generalized zero in \([k,k+1]\) if \(x_k\neq 0\), \(x_{k+1} \in\operatorname{Im} B_k\) and \(x^T_k B^+_kx_{k+1} =0\), where \(B^+_k\) denotes the Moore-Penrose inverse of the matrix \(B_k\). The system is called disconjugate on \({\mathcal J}=[0,N]\cap\mathbb{Z}\) if no solution of the system has more than one generalized zero (if \(x_0=0\) no zero). Theorem 1 shows a lot of equivalent facts about disconjugacy, possession of generalized zeros and other similar properties of the solution to the symplectic system. Similar results are obtained relative to the reciprocal symplectic system \(z_{k+1}= S^{-1}_k z_k\). The problems of eventually disconjugate solutions and disconjugacy preserving transformations are considered as well.
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 1143464
zbMATH Keywords: Sturm-Liouville difference equation
zbMATH Keywords: Hamiltonian difference equation
zbMATH Keywords: symplectic system
zbMATH Keywords: disconjugacy
zbMATH Keywords: reciprocal system
author: Martin Bohner
author: Ondřej Došlý
reviewed by: Dobiesław Bobrowski
MaRDI profile type: Publication
OpenAlex ID: W1975153123
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DOI: 10.1216/RMJM/1181071889
description: scientific article; zbMATH DE number 1143464