A Variant of d’Alembert Functional Equation on Monoids

In this paper, we determine the complex-valued solutions of the functional equation f (xσ(y)) + f (τ (y)x) = 2 f (x) f (y) for all x, y ∈ M, where M is a monoid, σ : M → M is an involutive automorphism and τ : M → M is an involutive anti-automorphism. The solutions are expressed in terms of multiplicative functions, and characters of 2-dimensional irreducible repre-sentations of M.