Additive s-functional inequalities and derivations on Banach algebras

Abstract

In this paper, we introduce the following new additive s-functional inequalities kf(x − y) + f(y) + f(−x)k ≤ ks (f(x + y) − f(x) − f(y)) k, (0.1) kf(x + y) − f(x) − f(y)k ≤ ks (f(x − y) + f(y) + f(−x)) k, (0.2) where s is a fixed complex number with |s| < 1, and prove the Hyers-Ulam stability of linear derivations on Banach algebras associated to the additive s-functional inequalities (0.1) and (0.2).