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

Let 3≤𝑛, and 3≤𝑘≤𝑛 be positive integers. Let 𝐴 be an algebra and let 𝑋 be an 𝐴-bimodule. A ℂ-linear mapping 𝑑∶𝐴→𝑋 is called a 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) for all 𝑎1,𝑎2,…,𝑎𝑛∈𝐴. The main purpose of this paper is to prove the generalized Hyers-Ulam stability of the generalized (𝑛,𝑘)-derivations.

Abstract licence: CC BY

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