A Variant of D’Alembert’s Functional Equation on Semigroups with Endomorphisms

Let S be a semigroup, and let φ, ψ: S → S be two endomorphisms (which are not necessarily involutive). Our main goal in this paper is to solve the following generalized variant of d’Alembert’s functional equation f(xϕ(y))+f(ψ(y)x)=2f(x)f(y), x,y ∈ S, f\left( {x\varphi \left( y \right)} \right) + f\left( {\psi \left( y \right)x} \right) = 2f\left( x \right)f\left( y \right),\,\,\,\,\,\,x,y\, \in \,S, where f : S → ℂ is the unknown function by expressing its solutions in terms of multiplicative functions. Some consequences of this result are presented.