advanced approaches to approximating cubic and radical cubic functional equations in gβ_-normed spaces

This article investigates the approximation of cubic and radical cubic functional equations in gβ-normed and gβ_-normed vector spaces. we define these spaces and employ the hyers-ulam-rassias stability methods to establish the stability of these functional equations. this study illuminates the stability properties of these equations in gβ-normed and gβ_- normed spaces, providing useful insights into their behavior and mathematical properties.