Nearly generalized Jordan derivations

Abstract

Let $A$ be an algebra and let $X$ be an $A$-bimodule. A $\mathbb{C}-$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.