Stability of additive-quadratic functional equation in modular space

TL;DR

This paper establishes the stability of a specific additive-quadratic functional equation within modular spaces that satisfy certain conditions, extending the understanding of functional stability in these mathematical structures.

Contribution

It proves the generalized Hyers-Ulam stability of a particular functional equation in modular spaces with the Fatou property or Δ₂-condition, using the direct method.

Findings

Proves stability of the functional equation in modular spaces.

Extends stability results to spaces satisfying Fatou property or Δ₂-condition.

Uses the direct method for the proof.

Abstract

Using the direct method, we prove the generalised Hyers-Ulam stability of the following functional equation \begin{equation} \phi(x+y, z+w)+\phi(x-y, z-w)-2 \phi(x, z)-2 \phi(x, w)=0 \end{equation} in modular space satisfying the Fatou property or $\Delta_2$-condition.