# Stability of Euler-Lagrange type cubic functional equations in quasi-Banach spaces

## TL;DR

This paper investigates the stability of a specific cubic functional equation of Euler-Lagrange type within quasi-Banach spaces, extending the understanding of functional stability in non-normed settings using fixed point methods.

## Contribution

It establishes the generalized Hyers-Ulam stability of the cubic functional equation in quasi-Banach spaces via the alternative fixed point theorem, a novel approach in this context.

## Key findings

- Proves stability results for the cubic functional equation in quasi-Banach spaces.
- Employs fixed point theorem to establish stability, extending classical methods.
- Provides conditions under which the functional equation is stable in quasi-normed spaces.

## Abstract

In this paper, we study the generalized Hyers-Ulam stability of Euler-Lagrange type cubic functional equation of the form \begin{align*} 2mf(x + my) + 2f(mx - y) = (m^3 + m)[f(x+ y) + f(x - y)] + 2(m^4 - 1)f(y) \end{align*} for all $x,y \in X$, where $m$ is a fixed scalar such that $m \neq 0,1$, and $f$ is a map from a quasi-normed space $X$ to a quasi-Banach space $Y$ over the same field with $X$ by applying the alternative fixed point theorem.