Journal of Applied Mathematics (Jan 2011)
Stability and Superstability of Generalized (π, π)-Derivations in Non-Archimedean Algebras: Fixed Point Theorem via the Additive Cauchy Functional Equation
Abstract
Let π΄ be an algebra, and let π, π be ring automorphisms of π΄. An additive mapping π»βΆπ΄βπ΄ is called a (π,π)-derivation if π»(π₯π¦)=π»(π₯)π(π¦)+π(π₯)π»(π¦) for all π₯,π¦βπ΄. Moreover, an additive mapping πΉβΆπ΄βπ΄ is said to be a generalized (π,π)-derivation if there exists a (π,π)-derivation π»βΆπ΄βπ΄ such that πΉ(π₯π¦)=πΉ(π₯)π(π¦)+π(π₯)π»(π¦) for all π₯,π¦βπ΄. In this paper, we investigate the superstability of generalized (π,π)-derivations in non-Archimedean algebras by using a version of fixed point theorem via Cauchyβs functional equation.