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Nearly ternary derivations. (English) Zbl 1141.39024
Authors’ abstract: Let \(A\) be a normed algebra and \(X\) a normed \(A\)-bimodule. By a ternary derivation we mean a triple \( (D_1, D_2, D_3)\) of linear mappings \(D_1, D_ 2, D_3 : A \to X\) such that \( D_1 (ab) = D_2 (a) b + a D_3 (b)\) for all \( a,b \in A\). Our aim is to establish the stability of ternary derivation associated with the extended Jensen functional equation \[ qf \left( { \sum_{k=1}^q x_k } \over q \right) = \sum_{k=1}^q f (x_k) \] for all \(x_1,\dots, x_q \in A\), where \( q> 1\) is a fixed positive integer.

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.
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
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