Quadratic $f$-hom-ders in Banach Algebra Related to System of Quadratic Functional Equations

Abstract

Mirzavaziri and Moslehian [13] introduced the concept of f-derivations and Sripattanet
et al. [18] introduced a quadratic hom-der in Banach algebras. In this paper, we solve the system of quadratic  functional equations
\begin{align*}
\left \{
\begin{array}{c}
f(x+y)+f(x-y)=g(x)+ g(y), \\
g\left(\frac{x+y}{2}\right) + g\left(\frac{x-y}{2}\right)= f(x) + f(y)
\end{array}
\right.
\end{align*}

Using Mirzavaziri and Moslehian’s idea and Sripattanet et al.’s idea, we define a quadratic f-hom-
der in Banach algebras, and we investigate the Hyers-Ulam stability of quadratic f-hom-ders in Banach algebras.