Invariants and Liapunov functions for nonautonomous systems
type: Item
instance of: scholarly article
title: Invariants and Liapunov functions for nonautonomous systems
zbMATH Open document ID: 1001.39024
DOI: 10.1016/S0898-1221(01)00163-8
publication date: 2002-06-13T00:00:00Z
review text: Let \(T_n:M\to M\) \((M\subset \mathbb{R}^k)\) be a sequence of continuous self-mappings. If there exists an unbounded function \(U:\mathbb{N}\times C\to \mathbb{R}\) such that for every \(x\in M\) and some \(\alpha(x)\in\mathbb{R}\) \[ T_{n-1}\circ\cdots\circ T_0x\in C\Rightarrow U(n,T_{n-1}\circ\cdots\circ T_0x)\leq \alpha(x), \] then every solution of the recurrence \[ x_{n+1}=T_nx_n, n\in \mathbb{N}, \] eventually contained in \(C\), is boudned. Moreover, under some additional assumptions every such solution has the \(\omega\)-limit: \(\omega(x_0,T_n)\subset H\), where \(H\) is an appropriate set. The proofs are based on the concept of Lyapunov functions for nonautonomous discrete dynamical systems. Some illustrative examples are included.
reviewed by: Dobiesław Bobrowski
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 39A11
Mathematics Subject Classification ID: 37B25
zbMATH DE Number: 1755260
zbMATH Keywords: nonautonomous difference equation
zbMATH Keywords: invariant
zbMATH Keywords: stability
zbMATH Keywords: Lyapunov functions
zbMATH Keywords: discrete dynamical systems
author: T. Nesemann
MaRDI profile type: Publication
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full work available at URL: s0898-1221(01)00163-8
OpenAlex ID: W1994546333
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 1755260