Stability and Superstability of Generalized ( θ , ϕ )-Derivations in Non-Archimedean Algebras: Fixed Point Theorem via the Additive Cauchy Functional Equation

Abstract

Let $A$ be an algebra, and let $\theta$, $\phi$ be ring automorphisms of $A$. An additive mapping $H:A\to A$ is called a $(\theta ,\phi )$-derivation if $H\left(xy\right)=H\left(x\right)\theta \left(y\right)+\phi \left(x\right)H\left(y\right)$ for all $x,y\in A$. Moreover, an additive mapping $F:A\to A$ is said to be a generalized $(\theta ,\phi )$-derivation if there exists a $(\theta ,\phi )$-derivation $H:A\to A$ such that $F\left(xy\right)=F\left(x\right)\theta \left(y\right)+\phi \left(x\right)H\left(y\right)$ for all $x,y\in A$. In this paper, we investigate the superstability of generalized $(\theta ,\phi )$-derivations in non-Archimedean algebras by using a version of fixed point theorem via Cauchy’s functional equation.