Additive Functional Inequalities and Partial Multipliers in Complex Banach Algebras

Abstract

In this paper, we solve the additive functional inequalities $$\displaystyle \begin{aligned} \begin{array}{rcl}{} \| f(x+y+z)-f(x+y)- f(z)\| \le \|s (f(x-y) + f(y-z )- f(x-z))\| \end{array} \end{aligned} $$ (1) and $$\displaystyle \begin{aligned} \begin{array}{rcl}{} \|f(x-y) + f(y-z )- f(x-z) \| \le \|s ( f(x+y-z) + f(x-y+z )- 2f(x) ) \| , \end{array} \end{aligned} $$ (2) where s is a fixed nonzero complex number with |s| < 1. Using the direct method, we prove the Hyers-Ulam stability of the additive functional inequalities (1) and (2) in complex Banach spaces. This is applied to investigate partial multipliers in Banach ∗-algebras, unital C∗-algebras, Lie C∗-algebras, JC∗-algebras, and C∗-ternary algebras, associated with the additive functional inequalities (1) and (2).