Perturbations for linear difference equations
type: Item
instance of: scholarly article
title: Perturbations for linear difference equations
zbMATH Open document ID: 1072.39004
author: Tewfik Sari
author: Tahia Zerizer
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-04-21T00:00:00Z
review text: The authors study the following boundary value problem \[ \varepsilon y_{k+2} + a_k y_{k+1} + b_k y_k = f_k \quad (0\leq k\leq N-2),\quad y_0=\alpha,\quad y_N=\beta, \] where \(a_k,b_k\), and \(f_k\) are given sequences of real numbers, \(\varepsilon\) is a small parameter, and \(\alpha,\beta\) are real constants. They prove that a unique solution of the above problem exists which is given by \(y_k(\varepsilon)=\sum\limits_{j=0}^\infty \varepsilon^j y_k^{(j)}\) for \(0\leq k\leq N\), and satisfies \[ \left| y_k(\varepsilon)-\sum_{j=0}^n \varepsilon^j y_k^{(j)}\right| \leq C{(| \varepsilon| /\varepsilon_0)^{n+1}\over 1-| \varepsilon| /\varepsilon_0}, \] where the constants \(C\) and \(\varepsilon_0\) depend only on the sequences \(a_k\) and \(b_k\), and \(y_k^{(0)}\) and \(y_k^{(j)}\) are the solutions of \[ a_ky_{k+1}^{(0)}+b_ky_k^{(0)}=f_k\quad (0\leq k\leq N-2),\quad y_0^{(0)}=\alpha,\quad y_N^{(0)}=\beta \] and \[ a_ky_{k+1}^{(j)}+b_ky_k^{(j)}=-y_{k+2}^{(j-1)}\quad (0\leq k\leq N-2),\quad y_0^{(j)}=0,\quad y_N^{(j)}=0, \] respectively.
reviewed by: Oleksiy Karlovych
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 2159771
zbMATH Keywords: linear second order difference equation
zbMATH Keywords: perturbation
zbMATH Keywords: boundary value problem
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.10.012
OpenAlex ID: W1964811032
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DOI: 10.1016/J.JMAA.2004.10.012
description: scientific article; zbMATH DE number 2159771