Affiliation:
1. Department of Mathematics, Faculty of Sciences , Ibn Tofail University , , Kenitra , Morocco
2. Department of Mathematics, Faculty of Sciences , Chouaib Doukkali University , , El Jadida , Morocco
Abstract
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
∈
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.
Reference15 articles.
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3. [3] A.L. Cauchy, Cours d’Analyse de L’École Royale Polytechnique. Première Partie: Analyse Algébrique, De L’Imprimerie Royale, Paris, 1821.
4. [4] A. Chahbi, B. Fadli, and S. Kabbaj, A generalization of the symmetrized multiplicative Cauchy equation, Acta Math. Hungar. 149 (2016), no. 1, 170–176.
5. [5] J. d’Alembert, Addition au Mémoire sur la courbe que forme une corde tendue mise en vibration, Hist. Acad. Berlin 1750 (1750), 355–360.
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