Existence and uniqueness of local solutions of functional equations
type: Item
instance of: scholarly article
title: Existence and uniqueness of local solutions of functional equations
zbMATH Open document ID: 0623.39002
DOI: 10.1007/BF01083505
author: L. P. Kuchko
publication date: 1986-01-01T00:00:00Z
review text: The author considers local \(C^{\infty}\) solutions \(\phi\) of the following iterative functional equation \((1)\;\phi (f(x))=g(x)\phi (x)+h(x),\) where f is a real \(C^{\infty}\) selfmapping of a neighbourhood of the origin and \(g: {\mathbb{R}}\to {\mathcal C}^{m^ 2}\), \(h: {\mathbb{R}}\to {\mathcal C}^ m\) are also \(C^{\infty}\) mappings. One of the results obtained reads: If F is a local diffeomorphism satisfying either \(| f'(0)| \neq 1\) or \(| f'(0)| =1\) and \((f\circ f)(x)=x+\hat f(x),\hat f(x)\neq 0\), then for every formal solution \(\phi\) of (1) there exists a local \(C^{\infty}\) solution of (1) whose Taylor series at the origin coincides with \(\phi\). No proofs are given.
Mathematics Subject Classification ID: 39B12
Mathematics Subject Classification ID: 26E10
zbMATH DE Number: 4011092
zbMATH Keywords: local C infinity solutions
zbMATH Keywords: iterative functional equation
zbMATH Keywords: Taylor series
reviewed by: Bogdan A. Choczewski
MaRDI profile type: Publication
cites work: LINEAR FUNCTIONAL EQUATIONS
cites work: Differentiable solutions to degenerate systems of ordinary equations
full work available at URL: https://doi.org/10.1007/bf01083505
OpenAlex ID: W2047610465
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 4011092