Cubic Derivations on Banach Algebras

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.