On the stability of derivations of higher order

Abstract

Derivations of order n as defined by L. Reich are additive and nonlinear functions $f : ℝ → ℝ$ with $f(1) = 0$ which satisfy the functional equation $δ_{a_1}$ ○ $δ_{a_2}$ ○ ... ○ $δ_{a_{(n+1)}}f = 0$ for all $a_1, a_2,..., a_{n+1} ϵ ℝ$, where $δ_af(x) := f(ax) - af(x)$. Here we prove several stability results concerning this (and similar) functional equations.