Cubic Derivations on Banach Algebras

TL;DR

This paper investigates the stability of cubic derivations on Banach algebras, employing direct and fixed point methods to establish conditions under which these mappings are stable or superstable.

Contribution

It introduces new stability results for cubic derivations on Banach algebras using both direct and fixed point methods, expanding understanding of their behavior.

Findings

Established stability of cubic derivations via direct method

Proved superstability of cubic derivations using fixed point approach

Provided conditions ensuring stability and superstability

Abstract

Let $A$ be a Banach algebra and $X$ be a Banach $A$-bimodule. A mapping $D :A\longrightarrow X$ is a cubic derivation if $D$ is a cubic homogeneous mapping, that is $D$ is cubic and $D(\lambda a)={\lambda}^3 D(a)$ for any complex number $\lambda$ and all $a\in A$, and $D(ab)=D(a)\cdot b^3 +a^3\cdot D(b)$ for all $a,b\in A$. In this paper, we prove the stability of a cubic derivation with direct method. We also employ a fixed point method to establish of the stability and the superstability for cubic derivations.