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 𝜃, 𝜙 be ring automorphisms of 𝐴. An additive mapping 𝐻∶𝐴→𝐴 is called a (𝜃,𝜙)-derivation if 𝐻(𝑥𝑦)=𝐻(𝑥)𝜃(𝑦)+𝜙(𝑥)𝐻(𝑦) for all 𝑥,𝑦∈𝐴. Moreover, an additive mapping 𝐹∶𝐴→𝐴 is said to be a generalized (𝜃,𝜙)-derivation if there exists a (𝜃,𝜙)-derivation 𝐻∶𝐴→𝐴 such that 𝐹(𝑥𝑦)=𝐹(𝑥)𝜃(𝑦)+𝜙(𝑥)𝐻(𝑦) for all 𝑥,𝑦∈𝐴. In this paper, we investigate the superstability of generalized (𝜃,𝜙)-derivations in non-Archimedean algebras by using a version of fixed point theorem via Cauchy’s functional equation.