Moslehian, Mohammad Sal Almost derivations on \(C^*\)-ternary rings. (English) Zbl 1132.39026 Bull. Belg. Math. Soc. - Simon Stevin 14, No. 1, 135-142 (2007). The author presents generalized stability results, in the spirit of P. Găvruţa [J. Math. Anal. Appl. 184, No. 3, 431–436 (1994; Zbl 0818.46043)], for derivations in \(C^*\)-ternary rings. In particular, the following result is obtained: Let \({\mathcal A}\) denote a \(C^*\)-ternary ring, \( \varepsilon > 0\), \( 0 \leq p < 1\), and suppose that \( f : {\mathcal A} \to {\mathcal A} \) fulfils \( f(0) = 0 \) and \[ \begin{split} \| f(\mu x + \mu y + [u \, v \, w]) - \mu f(x) - \mu f(y) - [f(u) \, v \, w] - [u \, f(v) \, w] - [u \, v \, f(w)] \| \\ \leq \varepsilon ( {\| x \|}^p + {\| y \|}^p + {\| u \|}^p + {\| v \|}^p + {\| w \|}^p ) \end{split} \]for every \( x,y,u,v,w \in {\mathcal A} \) and every complex number \( \mu \) with \( | \mu | = 1 \). Then \(f\) is a derivation. Reviewer: Zoltán Boros (Debrecen) Cited in 4 ReviewsCited in 16 Documents MSC: 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges 46L05 General theory of \(C^*\)-algebras Keywords:generalized Hyers-Ulam-Aoki-Rassias stability; \(C^*\)-ternary ring; derivation; Cauchy functional equation Citations:Zbl 0818.46043 PDFBibTeX XMLCite \textit{M. S. Moslehian}, Bull. Belg. Math. Soc. - Simon Stevin 14, No. 1, 135--142 (2007; Zbl 1132.39026) Full Text: Euclid