On multiplicatively-additive iteration groups
type: Item
instance of: scholarly article
title: On multiplicatively-additive iteration groups
zbMATH Open document ID: 1412.39021
author: Marek Cezary Zdun
published in: Aequationes Mathematicae
publication date: 2019-03-13T00:00:00Z
review text: The main motivation of the paper is an economical model describing the so-called impatience. Impatience is a special order on a sequences space of preferences of consumption outcomes. The main problem of the paper is to find the continuous solutions of the functional equation \[ F\bigl((t,a),F\bigl((s, b\bigr), x)\bigr)=F\bigl((ts,tb+a),x\bigr), \qquad t,s > 0,\quad a,b\in\mathbb{R},\quad x\in I, \] where $I$ is a nonempty open interval. It turns out, that every sequence of consumption outcomes can be represented by an increasing homeomorphism with one fixed point and the problem of the existence of impatience order can be reduced to the determination of suitable properties of semigroups of increasing homeomorphisms of $\mathbb{R}$ satisfying some extra conditions. \par The author gives a construction of large families of homeomorphisms semigroups possessing at most one fixed point satisfying the extra conditions. These semigroups are conjugate to the family of all affine functions. Their construction relates to the determination of two parameter families of homeomorphisms satisfying the functional equation above.
reviewed by: Mihály Bessenyei
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 39B72
Mathematics Subject Classification ID: 26A18
Mathematics Subject Classification ID: 91B74
zbMATH DE Number: 7035780
zbMATH Keywords: translation equation
zbMATH Keywords: affine function
zbMATH Keywords: iteration group
zbMATH Keywords: Schröder functional equation
zbMATH Keywords: Abel functional equation
MaRDI profile type: Publication
full work available at URL: s00010-018-0622-z
OpenAlex ID: W2903294413
cites work: Q5514010
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cites work: Q5592596
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Wikidata QID: Q128900788
DOI: 10.1007/S00010-018-0622-Z
description: scientific article; zbMATH DE number 7035780