New results about quadratic functional equation on semigroups - Akkaoui, Ahmed; Fadli, Brahim (Annales Mathematicae Silesianae, 2024)
Let S be a semigroup, let (H, +) be a uniquely 2-divisible, abelian group and let ϕ, ψ be two endomorphisms of S that need not be involutive. In this paper, we express the solutions f : S→H of the following quadratic functional equation f(xϕ(y)) + f(ψ(y)x) = 2f(x) + 2f(y), x,y ∈ S, in terms of bi-additive maps and solutions of the symmetrized additive Cauchy equation. Some applications of this result are presented.