Analytic solutions of an iterative functional equation
type: Item
instance of: scholarly article
title: Analytic solutions of an iterative functional equation
zbMATH Open document ID: 1063.39018
author: Wenrong Li
author: Sui Sun Cheng
published in: Aequationes Mathematicae
publication date: 2004-11-05T00:00:00Z
review text: The authors give a construction of an analytic solution \(f\) of the functional equation \[ f(p(z)+bf(z))=h(z), \] where \(b\) is a nonzero complex number and \(p\), \(h\) are given complex functions of a complex variable. Moreover, it is assumed that \(p\), \(h\) are analytic in a neighbourhood of the origin, \(p(0)=h(0)=0\), \(p'(0)=r\), \(h'(0)=s\neq 0\) and a root \(\beta\) of the equation \(z^2- rz -bs=0\) satisfies some additional conditions. Under these assumptions it is proved that for every comlex number \(\eta \neq 0\) the auxiliary functional equation \[ g(\beta^{2} z) - p(g(\beta z)) = b h(g(z)) \] has a solution which is analytic in a neighbourhood of the origin and \(g(0)=0\), \(g'(0)=\eta\). Then the function given by the formula \[ f(z)=\frac{1}{b} (g(\beta g^{-1} (z)) - p(z)) \] is a solution of the first equation which is analytic in a neighbourhood of the origin.
reviewed by: Zbigniew Leśniak
Mathematics Subject Classification ID: 39B12
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 39B32
zbMATH DE Number: 2113892
zbMATH Keywords: iterative functional equations
zbMATH Keywords: analytic solutions
MaRDI profile type: Publication
full work available at URL: s00010-004-2726-x
OpenAlex ID: W1511504945
DOI: 10.1007/S00010-004-2726-X
description: scientific article; zbMATH DE number 2113892