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

Generalized Hyers-Ulam Stability of Generalized (๐‘,๐พ)-Derivations

Let 3โ‰ค๐‘›, and 3โ‰ค๐‘˜โ‰ค๐‘› be positive integers. Let ๐ด be an algebra and let ๐‘‹ be an ๐ด-bimodule. A โ„‚-linear mapping ๐‘‘โˆถ๐ดโ†’๐‘‹ is called a generalized (๐‘›,๐‘˜)-derivation if there exists a (๐‘˜โˆ’1)-derivation ๐›ฟโˆถ๐ดโ†’๐‘‹ such that ๐‘‘(๐‘Ž1๐‘Ž2โ‹ฏ๐‘Ž๐‘›)=๐›ฟ(๐‘Ž1)๐‘Ž2โ‹ฏ๐‘Ž๐‘›+๐‘Ž1๐›ฟ(๐‘Ž2)๐‘Ž3โ‹ฏ๐‘Ž๐‘›+โ‹ฏ+๐‘Ž1๐‘Ž2โ‹ฏ๐‘Ž๐‘˜โˆ’2๐›ฟ(๐‘Ž๐‘˜โˆ’1)๐‘Ž๐‘˜โ‹ฏ๐‘Ž๐‘›+๐‘Ž1๐‘Ž2โ‹ฏ๐‘Ž๐‘˜โˆ’1๐‘‘(๐‘Ž๐‘˜)๐‘Ž๐‘˜+1โ‹ฏ๐‘Ž๐‘›+๐‘Ž1๐‘Ž2โ‹ฏ๐‘Ž๐‘˜๐‘‘(๐‘Ž๐‘˜+...

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Bibliographic Details
Main Authors: M. Eshaghi Gordji, J. M. Rassias, N. Ghobadipour
Format: Article
Language:English
Published: Wiley 2009-01-01
Series:Abstract and Applied Analysis
Online Access:http://dx.doi.org/10.1155/2009/437931
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Summary:Let 3โ‰ค๐‘›, and 3โ‰ค๐‘˜โ‰ค๐‘› be positive integers. Let ๐ด be an algebra and let ๐‘‹ be an ๐ด-bimodule. A โ„‚-linear mapping ๐‘‘โˆถ๐ดโ†’๐‘‹ is called a generalized (๐‘›,๐‘˜)-derivation if there exists a (๐‘˜โˆ’1)-derivation ๐›ฟโˆถ๐ดโ†’๐‘‹ such that ๐‘‘(๐‘Ž1๐‘Ž2โ‹ฏ๐‘Ž๐‘›)=๐›ฟ(๐‘Ž1)๐‘Ž2โ‹ฏ๐‘Ž๐‘›+๐‘Ž1๐›ฟ(๐‘Ž2)๐‘Ž3โ‹ฏ๐‘Ž๐‘›+โ‹ฏ+๐‘Ž1๐‘Ž2โ‹ฏ๐‘Ž๐‘˜โˆ’2๐›ฟ(๐‘Ž๐‘˜โˆ’1)๐‘Ž๐‘˜โ‹ฏ๐‘Ž๐‘›+๐‘Ž1๐‘Ž2โ‹ฏ๐‘Ž๐‘˜โˆ’1๐‘‘(๐‘Ž๐‘˜)๐‘Ž๐‘˜+1โ‹ฏ๐‘Ž๐‘›+๐‘Ž1๐‘Ž2โ‹ฏ๐‘Ž๐‘˜๐‘‘(๐‘Ž๐‘˜+1)๐‘Ž๐‘˜+2โ‹ฏ๐‘Ž๐‘›+๐‘Ž1๐‘Ž2โ‹ฏ๐‘Ž๐‘˜+1๐‘‘(๐‘Ž๐‘˜+2)๐‘Ž๐‘˜+3โ‹ฏ๐‘Ž๐‘›+โ‹ฏ+๐‘Ž1โ‹ฏ๐‘Ž๐‘›โˆ’1๐‘‘(๐‘Ž๐‘›) for all ๐‘Ž1,๐‘Ž2,โ€ฆ,๐‘Ž๐‘›โˆˆ๐ด. The main purpose of this paper is to prove the generalized Hyers-Ulam stability of the generalized (๐‘›,๐‘˜)-derivations.
ISSN:1085-3375
1687-0409

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