Pompeiu’s functional equations between unital algebras

Let \(\mathcal {A}\) and \(\mathcal {B}\) be unital algebras, that need not be abelian, over fields \(\mathbb {K}\) and \(\mathbb {K^{\prime }}\) respectively, let \(\alpha ,a,b,c\in \mathbb {K},\) \(\beta \in \mathbb {K^{\prime }}\) and \( \lambda \in \mathbb {C}\). The present work aims to determine the general solution \(\Phi :\mathcal {A}\rightarrow \mathcal {B}\) of the functional equation

$$\begin{aligned} \Phi (x+y+\alpha xy)=\Phi (x)+\Phi (y)+\beta \Phi (x)\Phi (y),\ x,y\in \mathcal {A}, \end{aligned}$$

and to describe the solutions \(\Phi :\mathcal {A}\rightarrow M_{2}(\mathbb {C} ) \) of the functional equation

$$\begin{aligned} \Phi (ax+by+cxy)=\Phi (x)+\Phi (y)+\lambda \Phi (x)\Phi (y),\ x,y\in \mathcal {A}. \end{aligned}$$

We also show that, when \((\mathcal {A},\cdot )\) (as a semigroup) is commutative and regular (for instance when \(\dim \mathcal {A}=1\)), the explicit forms of the solutions of the last equation can be given.