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


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.


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

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Published : 2022-03-22

AkkaouiA., El FatiniM., & FadliB. (2022). A variant of d’Alembert’s functional equation on semigroups with endomorphisms. Annales Mathematicae Silesianae, 36(1), 1-14. Retrieved from https://www.journals.us.edu.pl/index.php/AMSIL/article/view/13456

Ahmed Akkaoui  ahmed.maths78@gmail.com
Department of Mathematics, Faculty of Sciences, Ibn Tofail University, Morocco  Morocco
Mohamed El Fatini 
Department of Mathematics, Faculty of Sciences, Ibn Tofail University, Morocco  Morocco
Brahim Fadli 
Department of Mathematics, Faculty of Sciences, Chouaib Doukkali University, Morocco  Morocco

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