Generalized Hyers-Ulam stability of generalized \((N,K)\)-derivations. (English) Zbl 1177.39032

Eshaghi Gordji, M.; Rassias, J. M.; Ghobadipour, N.

Abstr. Appl. Anal. 2009, Article ID 437931, 8 p. (2009).

Summary: Let \(3\leq n\), and \(3\leq k\leq n\) be positive integers. 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 \((n,k)\)-derivation if there exists a \((k-1)\)-derivation \(\delta :A\to X\) such that \(d(a_1a_2\dots a_n)=\delta(a_1)a_2\dots a_n+a_1\delta(a_2)a_3\dots a_n+\cdots +a_1a_2\dots a_{k-2}\delta(a_{k-1})a_k\dots a_n+a_1a_2\dots a_{k-1}d(a_k)a_{k+1}\cdots a_n+a_1a_2\dots a_kd(a_{k+1})a_{k+2}\cdots a_n+a_1a_2\cdots a_{k+1}d(a_{k+2})a_{k+3}\cdots a_n+\cdots +a_1\cdots a_{n-1}d(a_n)\) 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 \((n,k)\)-derivations.

MSC:

39B82	Stability, separation, extension, and related topics for functional equations
39B52	Functional equations for functions with more general domains and/or ranges
47B47	Commutators, derivations, elementary operators, etc.

Keywords:

bimodule; \((n,k)\)-derivation; Hyers-Ulam stability

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