Radosław Łukasik

A special class of functional equations

Abstract We obtain in terms of additive and multi-additive functions the solutions f, h: S → H of the functional equation ∑λ∈Φf(x+λy+aλ)=Nf(x)+h(y),x,y∈S,

Some generalization of Cauchy’s and the quadratic functional equations

\begin{array}{} \displaystyle \sum\limits_{\lambda \in \Phi }f(x+\lambda y+a_{\lambda })=Nf(x)+h(y),\quad x,y\in S, \end{array}

Some generalization of the quadratic and Wilson’s functional equation

where (S, +) is an abelian monoid, Φ is a finite group of automorphisms of S, N = | Φ | designates the number of its elements, {aλ, 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.

Decomposition of two functions in the orthogonality equation

In this paper, we investigate the stability in the sense of Hyers-Ulam for a class of the following type functional equations:

A note on the orthogonality equation with two functions

\sum_{\lambda \in \Phi}{f(x+\lambda y+a_{\lambda})}=Nf(x)+h(y),\ x,y\in S

The Form of Multi-additive Symmetric Functions

where