Generalized Hyers-Ulam Stability ofGeneralized (N,K)-Derivations

Abstract

Let $3\le n$, and $3\le k\le n$ be positive integers. Let $A$ be an algebra and let $X$ be an $A$-bimodule. A $ℂ$-linearmapping $d:A\to X$ is called a generalized $\left(n,k\right)$-derivation ifthere exists a $\left(k-1\right)$-derivation $\delta :A\to X$ such that$d\left(a_1{a}_2\cdots a_n\right)=\delta \left(a_1\right)a_2\cdots a_n+a_1\delta \left(a_2\right)a_3\cdots a_n+\cdots +a_1{a}_2\cdots a_{k-2}\delta \left(a_{k-1}\right)a_k\cdots a_n+a_1{a}_2\cdots a_{k-1}d\left(a_k\right)a_{k+1}\cdots a_n+a_1{a}_2\cdots a_{k}d\left(a_{k+1}\right)a_{k+2}\cdots a_n+a_1{a}_2\cdots a_{k+1}d\left(a_{k+2}\right)a_{k+3}\cdots a_n+\cdots +a_1\cdots a_{\mathrm{n}-1}d\left(a_n\right)$ for all $a_1,a_2,\dots ,a_n\in A$. The main purpose of this paper is to prove the generalized Hyers-Ulam stability of the generalized $\left(n,k\right)$-derivations.