Refined stability of the additive, quartic and sextic functional equations with counter-examples

Abstract

<abstract><p>In this study, we utilize the direct method (Hyers approach) to examine the refined stability of the additive, quartic, and sextic functional equations in modular spaces with and without the $ \Delta _{2} $-condition. We also use the direct approach to discuss the Ulam stability in $ 2 $-Banach spaces. Ultimately, we ensure that stability of above equations does not hold in a particular scenario by utilizing appropriate counter-examples.</p></abstract>