NEARLY TERNARY DERIVATIONS

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 $D_1, D_2, D_3: A \rightarrow X$ such that $D_1(ab) = D_2(a)b + aD_3(b)$ for all $a, b \in A$. Our aim is to establish the stability of ternary derivations associated with the extended Jensen functional equation \[ qf (\frac{\sum_{k=1}^q x_k} {q}) = \sum_{k = 1}^{q} f(x_k) \] for all $x_1, \cdots, x_q \in A$, where $q \gt 1$ is a fixed positive integer.