Hyperstability of some functional equations in modular spaces

TL;DR

This paper explores the hyperstability of certain functional equations within modular spaces, extending the concept of Ulam stability to these equations and providing new stability results.

Contribution

It introduces new hyperstability results for functional equations in modular spaces, expanding the understanding of stability phenomena in this mathematical context.

Findings

Established hyperstability conditions for specific functional equations

Extended Ulam stability concepts to modular spaces

Provided new stability theorems for equations involving modular functions

Abstract

In this paper, we investigate some hyperstability results, inspired by the concept of Ulam stability, for the following functional equations: \begin{equation} \varphi(x+y)+\varphi(x-y)=2\varphi(x)+2\varphi(y) \end{equation} \begin{equation} \varphi(ax+by) = A\varphi(x)+B\varphi(y)+C \end{equation} \begin{equation}\label{eqnd} f\left(\sum_{i=1}^{m}x_{i}\right)+\sum_{1\leq i<j\leq m}f\big(x_{i}-x_{j}\big)=m\sum_{i=1}^{m}f(x_{i}) \end{equation} in modular spaces.