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

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 increment 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.

Automated summary

In this study, the direct method (Hyers approach) is used to examine the refined stability of additive, quartic, and sextic functional equations in modular spaces with and without the increment 2-condition. The direct approach is also employed to discuss Ulam stability in 2-Banach spaces. Ultimately, appropriate counter-examples are utilized to demonstrate that stability of the above equations does not hold in a particular scenario.