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

Akkaoui, Ahmed; El Fatini, Mohamed; Fadli, Brahim

doi:10.2478/amsil-2022-0004

Abstract

Let S be a semigroup, and let φ, ψ: S → S 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 $f (x ϕ (y)) + f (ψ (y) x) = 2 f (x) f (y), x, y \in S,$ 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.

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

Abstract

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