Nearly generalized Jordan derivations

TL;DR

This paper investigates the stability properties of generalized Jordan derivations in algebraic structures, establishing conditions under which approximate derivations are close to exact ones.

Contribution

It proves the Hyers-Ulam-Rassias stability and superstability of generalized Jordan derivations, extending understanding of their behavior in algebraic systems.

Findings

Established Hyers-Ulam-Rassias stability for generalized Jordan derivations

Proved superstability under certain conditions

Extended stability results to broader classes of derivations

Abstract

Let $A$ be an algebra and let $X$ be an $A$-bimodule. A $\Bbb C-$linear mapping $d:A \to X$ is called a generalized Jordan derivation if there exists a Jordan derivation (in the usual sense) $\delta:A \to X$ such that $d(a^2)=ad(a)+\delta(a)a$ for all $a \in A.$ The main purpose of this paper to prove the Hyers-Ulam-Rassias stability and superstability of the generalized Jordan derivations.