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

and Applied Analysis 3 Generalized derivations first appeared in the context of operator algebras 7 . Later, these were introduced in the framework of pure algebra 8, 9 . Definition 1.1. LetA be an algebra and let X be anA-bimodule. A linear mapping d : A → X is called i derivation if d ab d a b ad b , for all a, b ∈ A; ii generalized derivation if there exists a derivation in the usual sense δ : A → X such that d ab ad b δ a b, for all a, b ∈ A. Every right multiplier i.e., a linear map h on A satisfying h ab ah b , for all a, b ∈ A is a generalized derivation. Definition 1.2. Let n ≥ 2, k ≥ 3 be positive integers. Let A be an algebra and let X be an A-bimodule. A C-linear mapping d : A → X is called i n-derivation if d a1a2 · · ·an d a1 a2 · · ·an a1d a2 a3 · · ·an · · · a1 · · ·an−1d an 1.9 for all a1, a2, . . . , an ∈ A; ii generalized n, k -derivation if there exists a k − 1 -derivation δ : A → X such that d a1a2 · · ·an δ a1 a2 · · ·an a1δ a2 a3 · · ·an · · · a1a2 · · ·ak−2δ ak−1 ak · · ·an a1a2 · · ·ak−1d ak ak 1 · · ·an a1a2 · · ·akd ak 1 ak 2 · · ·an a1a2 · · ·ak 1d ak 2 ak 3 · · ·an · · · a1 · · ·an−1d an 1.10 for all a1, a2, . . . , an ∈ A. By Definition 1.2, we see that a generalized 2, 3 -derivation is a generalized derivation. For instance, let A be a Banach algebra. Then we take