New Results about Quadratic Functional Equation on Semigroups

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, f\left( {x\varphi \left( y \right)} \right) + f\left( {\psi \left( y \right)x} \right) = 2f\left( x \right) + 2f\left( y \right), \;\;\;\;x,y \in S, in terms of bi-additive maps and solutions of the symmetrized additive Cauchy equation. Some applications of this result are presented.