This paper investigates a fixed-point theorem within the framework of quasi-(m, β)-normed spaces, extending the results of Brzdek 5 and El-Fassi 6 to a more generalized setting. The extension is utilized to study the hyperstability of Apollonius-type functional equations, emphasizing the role of inequalities in stability analysis. By employing advanced fixed-point methods, we provide a comprehensive framework that highlights the interplay between inequalities and stability phenomena in functional equations. These findings contribute to the growing body of research on inequalities in functional analysis and their applications in diverse mathematical contexts.
Jakhar et al. (Thu,) studied this question.