Stability of quadratic functional equation in modular spaces

TL;DR

This paper investigates the stability of a specific quadratic functional equation within modular spaces and certain Banach spaces, extending the understanding of functional stability in these mathematical contexts.

Contribution

It provides new results on the Hyers-Ulam stability of a complex quadratic functional equation in modular and Banach spaces, including cases with and without the -condition.

Findings

Established stability results in modular spaces

Extended stability analysis to -homogeneous Banach spaces

Analyzed the impact of the -condition on stability

Abstract

In this paper, we study the Hyers-Ulam stability of the following equation \begin{multline*} \phi(x+y-z)+\phi(x+z-y)+\phi(y+z-x)=\phi (x-y)+\phi(x-z)+\phi(z-y) +\phi(x)+\phi(y) +\phi(z) \end{multline*} in modular space, with or without $\Delta_2$-condition, and in $\beta$-homogeneous Banach space.