A Class of Functional Equations of Type d’Alembert on Monoids Belaid Bouikhalene and Elhoucien Elqorachi 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 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. 1 Introduction We recall that a semigroup S is a non-empty set equipped with an associative operation. We write the operation multiplicatively. A monoid is a semigroup S with identity element that we denote e. A function μ : S −→ C is said to be multiplicative if μ(xy) = μ(x)μ(y) for all x,y ∈ S. Let S be a semigroup and σ : S −→ G a homomorphism involutive, that is σ(xy) = σ(x)σ(y) and σ(σ(x)) = x for all x,y ∈ S. Recently, Stetkær [21] obtained the complex-valued solutions of the following variant of d’Alembert’s functional equation f (xy) + f(σ(y)x) = 2f(x)f(y), x,y ∈ S. (1) B. Bouikhalene () Laboratory LIMATI, Polydisciplinary Faculty, Sultan Moulay Slimane University, Beni Mellal, Morocco e-mail: b.bouikhalene@usms.ma E. Elqorachi Department of Mathematics, Faculty of Sciences, Ibn Zohr University, Agadir, Morocco © Springer Nature Switzerland AG 2019 G. A. Anastassiou and J. M. Rassias (eds.), Frontiers in Functional Equations and Analytic Inequalities, https://doi.org/10.1007/978-3-030-28950-8_13 219