On the global asymptotic stability of difference equations satisfying a Markus-Yamabe condition
type: Item
instance of: scholarly article
title: On the global asymptotic stability of difference equations satisfying a Markus-Yamabe condition
zbMATH Open document ID: 1308.39015
author: Anna Cima
author: Armengol Gasull
author: Francesc Mañosas
published in: Publicacions Matemàtiques
publication date: 2014-12-19T00:00:00Z
review text: Consider the \(n\)-dimensional reccurrence \[ x_{k+1} = F(x_k) \] with \(F:\mathbb R^n \to \mathbb R^n\) being a \(C^1\) map. The following result is proved. Theorem. Let \(F:\mathbb R^n \to \mathbb R^n\) be given by \[ F(x_1,\dots,x_n) = col(x_2,\dots,x_n,f(x_1,\dots, x_n)) \] with \(f:\mathbb R^n \to \mathbb R^n\) a \(C^1\) function. Then the following are equivalent (\(H_1\)) \(F\) is such that \(\rho(|DF(x)|)<1\), \(\forall x\in \mathbb R^n\); (\(H_2\)) \(\sum_1^n |D_i(f)(x)|<1\), \(\forall x\in \mathbb R^n\). If one of them holds and \(F\) has a fixed point then this fixed point is globally asymptotically stable. Here \(DF(x)=(D_j(F_i)(x))\) is the Jacobian matrix of \(F\) at \(x\).
reviewed by: Vladimir Răsvan
Mathematics Subject Classification ID: 39A30
Mathematics Subject Classification ID: 37C05
Mathematics Subject Classification ID: 37C75
zbMATH DE Number: 6381929
zbMATH Keywords: global asymptotic stability
zbMATH Keywords: Lyapunov function
zbMATH Keywords: difference equation
zbMATH Keywords: Markus-Yamabe condition
zbMATH Keywords: recurrence
MaRDI profile type: Publication
DOI: 10.5565/PUBLMAT_EXTRA14_09
description: scientific article; zbMATH DE number 6381929