Functional equations with an anti-endomorphism for functions with multidimensional codomains - Ouhabi, Ayoub; Zeglami, Driss; Ayoubi, Mohamed (Annales Mathematicae Silesianae, 2024)
Let S be a semigroup, ℍ be the skew field of quaternions, and ψ: S→S be an anti-endomorphism. We determine the general solution of the functional equation g(xy) - g(xψ(y)) = 2g(x)g(y), x,y ∈ S, where g: S→ℂ is the unknown function. And when S = M is a monoid, we solve the functional equation g(xy) + g(xψ(y)) = 2g(x)g(y), x,y ∈ M, where g: M→ℍ is the unknown function.