Convergence and periodicity of solutions for a class of delay difference equations
type: Item
instance of: scholarly article
title: Convergence and periodicity of solutions for a class of delay difference equations
zbMATH Open document ID: 1074.39016
author: Huiyan Zhu
author: Xinyuan Liao
author: Lihong Huang
publication date: 2005-06-01T00:00:00Z
review text: The authors investigate convergence and asymptotic stability of solutions of the \(k^{th}\) order scalar difference equation: \[ x_n=a ~x_{n-1} + (1-a) f (x_{n-k}), \quad n=0,1,2,\dots \] where \(k\) is a positive integer, \(a\in (0,1)\), \(b,c \in [0,\infty)\) with \(b<c\), and \(f:\mathbb{R}\rightarrow \mathbb{R}\) is a piecewise constant function given by: \[ f(\xi)= \begin{cases} 1 &\text{if } \xi\in(b,c] \\ 0 &\text{if } \xi\in(-\infty,b]\cup (c,\infty).\end{cases} \] By imposing conditions on the magnitudes of the parameters \(a,~b,~c\), and restrictions on the initial conditions \(x_{-1},\dots,x_{-k}\) (such as all less than or equal to \(b\), all bigger than \(b\) and less than or equal to \(c\), or all greater than \(c\)), they establish two main results. One is concerned with the convergence of solutions to either 0 or 1. The other one is concerned with the existence of asymptotically stable periodic solution of a prescribed minimal period.
Mathematics Subject Classification ID: 39A11
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 2173419
zbMATH Keywords: delay difference equations
zbMATH Keywords: piecewise constant nonlinearity
zbMATH Keywords: convergence
zbMATH Keywords: asymptotic stability
zbMATH Keywords: periodic solution
zbMATH Keywords: prescribed minimal period
reviewed by: Raghib Abu-Saris
MaRDI profile type: Publication
full work available at URL: j.camwa.2004.07.005
OpenAlex ID: W2077804016
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DOI: 10.1016/J.CAMWA.2004.07.005
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 2173419