Hyers–Ulam stability of the additive $s$-functional inequality and hom-derivations in Banach algebras

Phakdi Charoensawan, Raweerote Suparatulatorn

Abstract

In this work, we solve the following additive $s$-functional inequality: \begin{eqnarray}\label{0.1}  \|f (x+y)-f(x)-f(y)\|\leq\|s(f(x-y)-f(x)-f(-y))\|, \end{eqnarray} where $s$ is a fixed nonzero complex number with $|s| < 1$. We prove the Hyers–Ulam stability of the additive $s$-functional inequality \eqref{0.1} in complex Banach spaces by using the fixed point method and the direct method. Moreover, we prove the Hyers–Ulam stability of hom-derivations in complex Banach algebras.