On approximate ternary m-derivations and $\sigma$-homomorphisms

TL;DR

This paper studies the stability of various derivations and homomorphisms in ternary algebra modules using fixed point methods, focusing on a specific functional equation for different values of m.

Contribution

It introduces ternary modules over ternary algebras and proves stability results for derivations and homomorphisms via fixed point techniques for a particular functional equation.

Findings

Established stability and super-stability of derivations and homomorphisms

Applied fixed point methods to functional equations in ternary structures

Extended results to multiple cases of the parameter m

Abstract

In this paper we introduce ternary modules over ternary algebras and using fixed point methods, we prove the stability and super-stability of ternary additive, quadratic, cubic and quartic derivations and $\sigma$-homomorphisms in such structures for the functional equation \begin{equation*} \begin{split} &\quad f(ax+y)+f(ax-y)= a^{m-2}[f(x+y)+f(x-y)]\\&+2(a^2-1)[a^{m-2}f(x)+\frac{(m-2)(1-(m-2)^2)}{6}f(y)]. \end{split} \end{equation*} for each $m=1,2,3,4$.