Advanced Approaches to Approximating Cubic and Radical Cubic Functional Equations in G\beta-Normed Spaces

This article investigates the approximation of cubic and radical cubic functional equations in \( G \)-normed and \( G\beta \)-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 \( G \)-normed and \( G\beta \)-normed spaces, providing useful insights into their behavior and mathematical properties,During a noteworthy speech at the Mathematical Club of the University of Wisconsin in the autumn of ۱۹۴۰, Ulam \cite{Ulam} addressed a set of unanswered questions. This lecture marked the beginning of the development of functional equation stability theory. ``If the suppositions of the theorem holds approximately, can we claim that the corresponding theorem will also hold approximately?" Ulam asked when introducing the stability problem. The essence of the stability problem for functional equations lies in its fundamental question: ``If an approximate solution exists for a given functional equation, can this approximation effectively approach an exact solution for the same equation?" In cases where the response is positive, we designate the specific equation as possessing stability.

کلیدواژه ها:

Hyers-Ulam stability ، G\beta-Normed spaces ، functional equations