## Abstract

We obtain in terms of additive and multi-additive functions the solutions *f*, *h*: *S* → *H* of the functional equation

$$\begin{array}{}{\displaystyle \sum _{\lambda \in \mathrm{\Phi}}f(x+\lambda y+{a}_{\lambda})=Nf(x)+h(y),\phantom{\rule{1em}{0ex}}x,y\in S,}\end{array}$$

where (*S*, +) is an abelian monoid, Φ is a finite group of automorphisms of *S*, *N* = | Φ | designates the number of its elements, {*a*_{λ}, λ ∈ Φ} are arbitrary elements of *S* and (*H*, +) is an abelian group. In addition, some applications are given. This equation provides a joint generalization of many functional equations such as Cauchy’s, Jensen’s, Łukasik’s, quadratic or Φ-quadratic equations.

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.