On an equation of linear iteration
type: Item
instance of: scholarly article
title: On an equation of linear iteration
zbMATH Open document ID: 0872.39010
DOI: 10.1007/BF01833285
author: Witold Jarczyk
published in: Aequationes Mathematicae
publication date: 1997-10-09T00:00:00Z
full work available at URL: https://eudml.org/doc/137648
review text: Given positive numbers \(a_1,\dots, a_k\) and positive integers \(n_1,\dots, n_k\) consider the equation \[ \sum_{i=1}^k a_if^{n_i}(x)=x.\tag \(*\) \] Let \(c\) be the only positive real number satisfying \(\sum_{i=1}^k a_ic^{n_i}=1\). Assuming that the greatest common divisor of \(n_1,\dots,n_k\) equals 1, \(D\subset(-\infty,0)\) or \(D\subset (0,+\infty)\), and \(f:D\to D\) is a solution of \((*)\), the author proves that \(cd\subset D\) and \(f(x)=cx\) for \(x\in D\). Reviewer's remark: This result was generalized by \textit{J. Tabor} and \textit{J. Tabor} [Result. Math. 27, No. 3-4, 412-421 (1995; Zbl 0831.39006)]. They answered also in negative three of the five questions formulated in the paper under review. Cf. also the abstracts of talks in Aequationes Math. 51, pp. 159, 163-164, 170 (1996)].
Mathematics Subject Classification ID: 39B12
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 26A18
zbMATH DE Number: 902694
zbMATH Keywords: iterative functional equation
zbMATH Keywords: iterates
zbMATH Keywords: general solution
reviewed by: Karol Baron
MaRDI profile type: Publication
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cites work: Q3854084
cites work: Q4002346
cites work: Q5592596
cites work: Q4040882
cites work: Non-Negative Measurable Solutions of Difference Equations
cites work: On a functional equation involving iterates of a bijection on the unit interval
cites work: On the functional equation \(f(p+qx+rf(x))=a+bx+cf(x)\)
cites work: A functional equation involving f and $f^{-1}$.
cites work: Q3781512
cites work: Q3814148
cites work: Discussion on the differentiable solutions of the iterated equation
OpenAlex ID: W2043245544
description: scientific article; zbMATH DE number 902694