A Class of Functional Equations of Type d’Alembert on Monoids

Belaid Bouikhalene () and Elhoucien Elqorachi ()
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Belaid Bouikhalene: Sultan Moulay Slimane University, Laboratory LIMATI, Polydisciplinary Faculty
Elhoucien Elqorachi: Ibn Zohr University, Department of Mathematics, Faculty of Sciences

A chapter in Frontiers in Functional Equations and Analytic Inequalities, 2019, pp 219-235 from Springer

Abstract: Abstract Recently, the solutions of the functional equation f(xy) − f(σ(y)x) = g(x)h(y) obtained, where σ is an involutive automorphism and f, g, h are complex-valued functions, in the setting of a group G and a monoid S. Our main goal is to determine the general complex-valued solutions of the following version of this equation, viz. f(xy) − μ(y)f(σ(y)x) = g(x)h(y) where μ : G → ℂ $$\mu : G\longrightarrow \mathbb {C}$$ is a multiplicative function such that μ(xσ(x)) = 1 for all x ∈ G. As an application we find the complex-valued solutions (f, g, h) on groups of equation f(xy) + μ(y)g(σ(y)x) = h(x)h(y) on monoids.

Date: 2019
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DOI: 10.1007/978-3-030-28950-8_13

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