Characterization and the stability of a system of multi-radical mappings related to the additive mapping

In the current investigation, we define $s$-multi-radical mappings, characterize the structure of such mappings and then obtain an equation for describing them. In fact, we find a necessary and sufficient condition for a multiple mapping to be $s$-multi-radical. We also deal with the Hyers-Ulam stability in the spirit of Gavruta for an $s$-multi-radical equation by applying the so-called direct (Hyers) method in the setting of 2-Banach spaces. For a typical case, by means of a norm, induced from a 2-norm of $\mathbb R^m$, we investigate the stability of a mapping $f:\mathbb R^{mn}\longrightarrow \mathbb R^{m}$ by a known fixed point method.