Stability of abstract difference equations
type: Item
instance of: scholarly article
title: Stability of abstract difference equations
zbMATH Open document ID: 1031.39003
DOI: 10.1016/S0895-7177(02)00236-4
author: D. Massart
published in: Mathematical and Computer Modelling
publication date: 2003-10-14T00:00:00Z
review text: Let \(M\) be a closed subspace of \(\ell ^\infty (J,X)\), the space of all bounded sequences \(\{x_n\}_{n\in J}\subset X\) where \(J\in \{\mathbb Z^+,\mathbb Z\}\) and \(X\) is a complex Banach space. Consider the operator difference equation \(u(n+1)=Au(n)+\psi(n)\), where \(\psi \in \ell ^\infty (\mathbb Z,X)\) such that the restriction \(\psi |_J\) on \(J\) belongs to \(M\) and \(A\) is the generator of a \(C_0\)-semigroup of linear bounded operators \((T (t))_{t\geq 0}\) on \(X\) The author deduces that the zero equilibrium of the homogeneous equation \(u(n+1)=Au(n)\), \(n\in \mathbb Z^+\), is globally asymptotically stable if and only if every solution is bounded.
Mathematics Subject Classification ID: 39A11
zbMATH DE Number: 1992908
zbMATH Keywords: almost periodic functions
zbMATH Keywords: abstract difference equations
zbMATH Keywords: bounded solution
zbMATH Keywords: stability
zbMATH Keywords: global asymptotic stability
zbMATH Keywords: Banach space
zbMATH Keywords: operator difference equation
reviewed by: Fozi Dannan
MaRDI profile type: Publication
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full work available at URL: s0895-7177(02)00236-4
OpenAlex ID: W2019917506
description: scientific article; zbMATH DE number 1992908