Analytic solutions of a nonlinear iterative equation near neutral fixed points and poles
type: Item
instance of: scholarly article
title: Analytic solutions of a nonlinear iterative equation near neutral fixed points and poles
zbMATH Open document ID: 1030.39025
DOI: 10.1016/S0022-247X(03)00363-9
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-09-04T00:00:00Z
review text: This is a continuation of earlier papers, particularly by the first author. Let \(f^m(x)\) indicate the \(m\)th iterate of \(f(x)\). The authors wish to study analytic solutions to the functional equation \[ f^m(x)= G\left( \sum^{m-1}_{k=0} a_kf^k(x)\right) +F(x),\;m\geq 2,\;x\in\mathbb{C}. \] Here the \(a_k\in \mathbb{C}\) and \(G,F,f\) are complex-valued functions of a complex variable. They show that under certain conditions, this functional equation has an analytic solution of the form \(f(x)=\varphi (\varphi^{-1} (x)+\alpha)\) in a neighborhood of the origin where \(\varphi\) is an analytic solution of \[ \varphi(x+m \alpha)=G \left( \sum^{m-1}_{k=0} a_k\varphi (x+k\alpha) \right)+F \bigl(\varphi(x) \bigr). \] The conditions imposed are on the indeterminate constant \(\alpha\), and are of three different sorts, treated separately. In the end of these, the analytic solution \(\varphi\) is only shown to exist in a certain half-plane. The case where \(G\) and \(F\) have poles of order 1 at the origin is also treated.
Mathematics Subject Classification ID: 39B12
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 39B32
zbMATH DE Number: 1974978
zbMATH Keywords: iteration
zbMATH Keywords: functional equation
zbMATH Keywords: analytic solution
zbMATH Keywords: neutral fixed point
zbMATH Keywords: Diophantine condition
zbMATH Keywords: poles
author: Jianguo Si
author: Weinian Zhang
reviewed by: Sanford L. Segal
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1974978