A variant of d'Alembert's functional equation on semigroups with endomorphisms

Ahmed Akkaoui; Mohamed El Fatini; Brahim Fadli

Published: 2022-03-22

Vol. 36 No. 1 (2022)

A variant of d'Alembert's functional equation on semigroups with endomorphisms

Ahmed Akkaoui

, Mohamed El Fatini , Brahim Fadli

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(xϕ(y))+f(ψ(y)x) = 2f(x)f(y), x,y∈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:

functional equation , d’Alembert , semigroup , multiplicative function , endomorphism

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Akkaoui, A., El Fatini, M., & Fadli, B. (2022). A variant of d’Alembert’s functional equation on semigroups with endomorphisms. Annales Mathematicae Silesianae, 36(1), 1–14. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/13456

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Ahmed Akkaoui

ahmed.maths78@gmail.com
https://orcid.org/0000-0001-8468-3408

Affiliation:

Department of Mathematics, Faculty of Sciences, Ibn Tofail University, Morocco Morocco

Mohamed El Fatini

Affiliation:

Department of Mathematics, Faculty of Sciences, Ibn Tofail University, Morocco Morocco

Brahim Fadli

Affiliation:

Department of Mathematics, Faculty of Sciences, Chouaib Doukkali University, Morocco Morocco

Vol. 36 No. 1 (2022)
Published: 2022-03-30

ISSN: 0860-2107

eISSN: 2391-4238

10.1515/amsil

Publisher

University of Silesia Press

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