Investigating a Novel Stability Results of Generalized Alternate Cubic Functional Equations: Classical Method for Banach Spaces and Direct -Fixed Point Approaches for Fuzzy Normed Spaces

This paper explores novel stability results for generalized alternate cubic functional equation(Fun Eq) using two distinct analytical frameworks: the classical method for Banach spaces and the direct and fixed point approaches for fuzzy normed spaces. The study examines the stability behavior of the generalized alternate cubic functional equation, focusing on how small deviations from exact solutions influence the overall stability in different normed environments. In Banach spaces, the classical approach is applied to derive conditions for Hyers-Ulam stability, providing insight into the equation’s behavior under small perturbations. For fuzzy normed spaces, both direct and fixed point methods are employed to account for the inherent uncertainties and fuzziness in the normed structure, offering a more flexible stability analysis. The results obtained highlight the differences and advantages of each approach, contributing to the broader understanding of functional equations in both deterministic and fuzzy frameworks. These findings have potential applications in various mathematical and applied fields, where both precise and imprecise data structures are considered.