Stability of Homomorphisms and Derivations on C^*-Ternary Rings Associated to an Euler–Lagrange Type Additive Mapping

Abstract.

Let X, Y be Banach spaces and let $r_1, \ldots, r_n\, \in {\mathbb{R}}$ be given. We prove the Hyers–Ulam–Rassias stability of the following functional equation in Banach spaces:

$$ \sum\limits_{{j = 1}}^{n} {f\left( { - r_{j} x_{{j}} + \,\sum\limits_{\begin{subarray}{l} 1 \le i \le n \\ \quad i \ne j \end{subarray} } {r_{i} x_{i} } } \right)\, + \,\sum\limits_{{i = 1}}^{n} {r_{i} f\left( {x_{i} } \right)} \, = \,\sum\limits_{{1 \le i<j \le n}} {f\left( {r_{i} x_{{i\,}} + \,r_{j} x_{j} } \right)} }\quad (0.1)$$

We show that if $\sum^{n}_{i=1} r_i \neq \frac{n^{2}-3n}{2}$ and r _i , r _j ≠ 0 for some 1 ≤ i < j ≤ n and a mapping f : X → Y satisfies the functional equation (0.1), then the mapping f : X → Y is Cauchy additive. As an application, we investigate homomorphisms and derivations between C^*-ternary rings.