The Solution of a Class Functional Equations on Semi-Groups

Abstract

Let S be a semi-group, and let σ, τ ∈ Antihom(S, S) satisfy τ ◦τ = σ◦σ = id.
We show that any solutions f : S → C of the functional equation
f(xσ(y)) + f(τ(y)x) = 2f(x)f(y), x, y ∈ S,
has the form f = (m + m ◦ σ ◦ τ)/2, where m is a multiplicative function on.