FUZZY STABILITY OF THE M-DIMENSIONAL ADDITIVE FUNCTIONAL EQUATION

In this article, we investigate the stability and hyperstability of the following additive functional equation:\begin{equation*}f\left(\sum_{i=1}^{m} \xi_i\right) = \sum_{i=1}^{m} f(\xi_i)\end{equation*}within the framework of fuzzy normed vector spaces. Motivated by the concept of Hyers-Ulam stability and its generalizations, we adopt a fixed point alternative method to analyze the behavior of functions that approximately satisfy this equation. Using appropriate control functions, we derive sufficient conditions ensuring the existence and uniqueness of additive mappings that closely approximate the given function in a fuzzy sense. We also establish hyperstability results under natural asymptotic assumptions on the control functions. The results presented here extend and refine earlier stability studies of additive functional equations, by embedding them in the context of fuzzy analysis and non-classical norm structures. Several corollaries are provided to demonstrate the applicability of our main theorems.