A Hyperstability Approach to Generalized Functional Equations on Involutive Semigroups

Abstract

In this paper, we examine the hyperstability of a general functional equation involving involutions on semigroups, where  and  is an arbitrary semigroup equipped with involutive mappings  and  . The functional equation studied in this paper emerges naturally from the theory of additive and quadratic equations, particularly as a hybrid form combining structural properties of both. Such equations typically appear when analyzing mappings that preserve symmetrical or involutive relationships within algebraic structures, especially in semigroups endowed with additional symmetries. By employing a technique inspired by Maksa and Páles, we derive sufficient asymptotic conditions ensuring hyperstability. Additionally, we extend the result to an inhomogeneous variant incorporating a perturbation term . These findings contribute to the broader understanding of stability phenomena in functional equations on algebraic structures.