Fixed points and functional equations connected with derivations on Banach algebras
Let \(\mathcal{A}\) and \(\mathcal{B}\) be two algebras and \(1\leq m\leq 4\) be a fixed positive integer. A function \(f\colon \mathcal{A}\to \mathcal{B}\) is called \(m\)-derivation, if it satisfies both functional equations \[ \begin{aligned} f(ax+y)+f(ax-y)&=a^{m-2}\left[f(x+y)+f(x-y)\right]\\ &\quad+2(a^{2}-1)\left[a^{m-2}f(x)+\frac{1}{6}(m-2)(1-(m-2)^{2})f(y)\right]\end{aligned} \] and \[ f(xy)=x^{m}f(y)+f(x)y^{m} \] for all \(x,y\in \mathcal{A}\) and for some \(a\in \mathbb{R}\setminus \{-1,0,1\}\). In this paper, the authors prove the stability and hyperstability of \(m\)-derivations on Banach algebras using a fixed point method.
additive, quadratic, cubic, and quartic functional equation
scientific article; zbMATH DE number 6006178
scientific article