1. bookVolume 36 (2022): Issue 1 (March 2022)
Journal Details
License
Format
Journal
eISSN
2391-4238
First Published
01 Jan 1985
Publication timeframe
2 times per year
Languages
English
access type Open Access

A Variant of D’Alembert’s Functional Equation on Semigroups with Endomorphisms

Published Online: 22 Mar 2022
Volume & Issue: Volume 36 (2022) - Issue 1 (March 2022)
Page range: 1 - 14
Received: 01 May 2021
Accepted: 24 Feb 2022
Journal Details
License
Format
Journal
eISSN
2391-4238
First Published
01 Jan 1985
Publication timeframe
2 times per year
Languages
English
Abstract

Let S be a semigroup, and let φ, ψ: SS be two endomorphisms (which are not necessarily involutive). Our main goal in this paper is to solve the following generalized variant of d’Alembert’s functional equation f(xϕ(y))+f(ψ(y)x)=2f(x)f(y),x,yS, f\left( {x\varphi \left( y \right)} \right) + f\left( {\psi \left( y \right)x} \right) = 2f\left( x \right)f\left( y \right),\,\,\,\,\,\,x,y\, \in \,S, where f : S → ℂ is the unknown function by expressing its solutions in terms of multiplicative functions. Some consequences of this result are presented.

Keywords

MSC 2010

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