Additive s-functional inequality and hom-derivations in Banach algebras

Abstract

In this paper, we introduce and solve the following additive s-functional inequality: 0.1 $$\begin{aligned} \left\| f\left( x+y\right) - f(x )- f(y)\right\| \le \Vert s (f(x-y)-f(x)-f(-y))\Vert , \end{aligned}$$ where s is a fixed nonzero complex number with $$|s|<1$$ . Using the fixed point method and the direct method, we prove the Hyers–Ulam stability of the additive s-functional inequality (0.1) in complex Banach spaces. Furthermore, we prove the Hyers–Ulam stability of hom-derivations in complex Banach algebras.