A quadratic functional equation with involutive automorphisms on semigroups
type: Item
instance of: scholarly article
title: A quadratic functional equation with involutive automorphisms on semigroups
zbMATH Open document ID: 1490.39031
author: Y. Aissi
author: Driss Zeglami
author: A. Mouzoun
published in: Boletín de la Sociedad Matemática Mexicana. Third Series
publication date: 2022-03-18T00:00:00Z
review text: The authors consider a generalization of the classical quadratic functional equation in the following form: \[ f(x\sigma(y))+f(\tau(y)x)=2f(x)+2f(y),\qquad x,y\in S, \] where \(S\) is a semigroup (not necessarily abelian), \(H\) is an abelian uniquely 2-divisible group, \(\sigma,\tau\colon S\to S\) are involutive automorphisms of \(S\) and \(f\colon S\to H\) is the unknown function. This equation has been considered in many previous works, for particular cases or under additional assumptions, e.g., for mappings defined on abelian semigroups. Some related functional equations have been also widely studied. The main result in this paper is the general solution of the above functional equation. Precisely, it is proved that \(f\) satisfies the above equation if and only if it is of the form \[ f(x)=Q_1(x,\sigma(x))+Q_2(x,x)+\psi(x)+a(x),\qquad x\in S, \] where \(Q_1,Q_2\colon S\times S\to H\) are bi-additive and invariant with respect to \(\sigma\) and \(\tau\) mappings such that \(Q_i(\tau(y),x)=-Q_i(x,\sigma(x))\) for \(x,y\in S\) (\(i=1,2\)), \(a\colon S\to H\) is an additive mapping such that \(a\circ \sigma=a\circ\tau=-a\) and \(\psi\colon S\to H\) satisfies \[ \psi(xy)+\psi(yx)=2\psi(x)+2\psi(y),\qquad x,y\in S \] and \[ \psi(\tau(x)y)-\psi(\sigma(x)y)=4a(x),\qquad x\in S. \] Many of the known previous results about special cases of the considered functional equation are thus particular cases of the general description given in this paper.
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 7493122
zbMATH Keywords: functional equation
zbMATH Keywords: quadratic
zbMATH Keywords: additive function
zbMATH Keywords: symmetrized additive Cauchy equation
zbMATH Keywords: semigroup
zbMATH Keywords: automorphism
reviewed by: Jacek Chmieliński
MaRDI profile type: Publication
full work available at URL: s40590-022-00413-y
OpenAlex ID: W4212886478
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DOI: 10.1007/S40590-022-00413-Y
description: scientific article; zbMATH DE number 7493122