Additive s-functional inequalities

Abstract

We solve the following additive s-functional inequality ‖f((k + 1)x − y) − f(kx − y) − f(x)‖ ≤ ‖s(f(x + y) − f(x) − f(y))‖ (0.1) ‖f(x + y) − f(x) − f(y)‖ ≤ ‖s(f((k + 1)x − y) − f(kx − y) − f(x))‖ (0.2) where k is an integer greater than 1 and s is acomplex number with |s| ˂ 1. Furthermore, we prove the Hyers-Ulam stability of the additive s-functional inequalities (0.1) and (0.2) in complex Banach spaces.