The continuous solution of a functional equation of Abel
type: Item
instance of: scholarly article
title: The continuous solution of a functional equation of Abel
zbMATH Open document ID: 0694.39004
DOI: 10.1007/BF01833940
published in: Aequationes Mathematicae
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137335
review text: In 1827, \textit{N. H. Abel} [J. Reine Angew. Math. 2, 386-394] obtained the differentiable solutions of the functional equation \(\phi (x)+\phi (y)=\psi (xf(y)+yf(x))\) on the reals. In 1900, Hilbert asked if this can be solved under weaker conditions than differentiability. In 1987, \textit{J. Aczél} drew attention to this question again and asked for its continuous solutions [Bull. Am. Math. Soc., New. Ser. 20, No.2, 153-163 (1989; Zbl 0676.39004)]. In this paper, the author nicely solves the equation under the assumptions that the functions are continuous, and that the equation holds for all x, y in an interval containing 0.
Mathematics Subject Classification ID: 39B62
zbMATH DE Number: 4136640
zbMATH Keywords: Abel's equation
zbMATH Keywords: Hilbert's fifth problem
zbMATH Keywords: continuous solutions
author: Maciej Sablik
MaRDI profile type: Publication
cites work: Q5514010
cites work: The state of the second part of Hilbert’s fifth problem
cites work: Equations fonctionnelles et recherche de sous-groupes
cites work: Local Boundedness and Continuity for a Functional Equation on Topological Spaces
cites work: Über ein Problem von Fenyö
description: scientific article; zbMATH DE number 4136640