On the second part of Hilbert's fifth problem
type: Item
instance of: scholarly article
title: On the second part of Hilbert's fifth problem
zbMATH Open document ID: 0766.39007
DOI: 10.1007/BF02571809
author: Mario Bonk
published in: Mathematische Zeitschrift
publication date: 1993-03-03T00:00:00Z
full work available at URL: https://eudml.org/doc/174414
review text: The following system of functional equations \[ \begin{aligned} \varphi(u+v)\varphi(u-v) &= \varphi(u)^ 2 f(v)^ 2-\varphi(v)^ 2 f(u)^ 2 \tag{1}\\ f(u+v)f(u-v) &= f(u)^ 2 f(v)^ 2-c^ 2\varphi(u)^ 2\varphi(v)^ 2\tag{2}\end{aligned} \] has been solved by N. H. Abel, under suitable differentiability assumptions on \(\varphi:\mathbb{C}\to \mathbb{C}\) and \(f:\mathbb{C}\to\mathbb{C}\). Similarly H. Haruki found the general solution of the first above equation. In this paper the author solves the system (1) and (2) replacing Abel's differentiability assumptions by weaker assumptions. He finds the general solutions \(\varphi: \mathbb{R}^ n\to\mathbb{C}\) and \(f: \mathbb{R}^ n\to\mathbb{C}\) of the system (1) and (2) where only the continuity of \(\varphi\), \(\varphi\not\equiv 0\), is supposed. If \(\varphi=0\) then the system (1)-- (2) reduces to \(f(u+v)f(u-v)=f(u)^ 2f(v)^ 2\), \(u,v\in\mathbb{R}^ n\). The continuous solutions \(f:\mathbb{R}^ n\to\mathbb{C}\) of this functional equation are determined. The paper is connected with recent work of \textit{J. Aczél} [The state of the second part of Hilbert's fifth problem, Bull. Am. Math. Soc., New Ser. 20, No. 2, 153-163 (1989; Zbl 0676.39004)].
Mathematics Subject Classification ID: 39B62
Mathematics Subject Classification ID: 39B32
zbMATH DE Number: 126394
zbMATH Keywords: system of functional equations
zbMATH Keywords: continuous solutions
zbMATH Keywords: Hilbert's fifth problem
reviewed by: Ilie Corovei
MaRDI profile type: Publication
cites work: The state of the second part of Hilbert’s fifth problem
cites work: Q3992445
cites work: Q4161928
cites work: Q3253970
cites work: Q5514020
cites work: Q5664012
Recommended article: The multiplicative solutions of the parallelogram equation
Recommended article: Solution of some nonlinear complex vector functional equations
Recommended article: Q4530584
Recommended article: On an exponential-cosine functional equation
Recommended article: Q3378883
Recommended article: The continuous solution of a functional equation of Abel
Recommended article: Q4385546
Recommended article: Contribution to the study of two functional equations
Recommended article: Q4869668
Recommended article: Q4796676
description: scientific article; zbMATH DE number 126394