Functional equations which are exorbitant
type: Item
instance of: scholarly article
title: Functional equations which are exorbitant
zbMATH Open document ID: 0753.39001
DOI: 10.1007/BF01292932
author: Walter Benz
published in: Archiv der Mathematik
publication date: 1992-09-27T00:00:00Z
review text: The author calls a functional equation or a system of functional equations, containing the unknown functions \(f_ j:A_ j\to B_ j\) \((j=1,\ldots,m)\), exorbitant if the cardinality of the set of all solutions equals that of the set \(\{(g_ 1,\ldots,g_ m)\mid g_ j:A_ j\to B_ j\) \((j=1,\ldots,m)\}\) of all functions mapping \(A_ j\) into \(B_ j\) \((j=1,\ldots,m)\). He shows about several classes of functional equations that all equations in them are exorbitant, including the Cauchy equation \(f(x+y)=f(x)+f(y)\) and the Gołąb-Schnizel equation \(f(x+yf(x))=f(x)f(y)\). The last theorem in the paper states that a functional equation in one unknown real function, which is satisfied by all idempotent \((g(g(x))=g(x))\) additive functions, is exorbitant.
reviewed by: János Aczél
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 62267
zbMATH Keywords: systems of functional equations
zbMATH Keywords: solutions
zbMATH Keywords: cardinality
zbMATH Keywords: Hamel bases
zbMATH Keywords: additive
zbMATH Keywords: idempotent functions
zbMATH Keywords: Forte-Kampe de Fériet equations
zbMATH Keywords: Aczél-Benz equation
zbMATH Keywords: Fenyö equation
zbMATH Keywords: Cauchy equation
zbMATH Keywords: Gołąb- Schnizel equation
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01292932
OpenAlex ID: W1969200651
description: scientific article; zbMATH DE number 62267