Stability of Bi-additive s-Functional Inequalities and Quasi-multipliers

Abstract

Park et al. (Rocky Mt J Math 49, 593–607 (2019)) solved the following bi-additive s-functional inequalities: ∥f(x+y,z−w)+f(x−y,z+w)−2f(x,z)+2f(y,w)∥≤∥∥s(2f(x+y2,z−w)+2f(x−y2,z+w)−2f(x,z)+2f(y,w))∥∥, (1) ∥∥2f(x+y2,z−w)+2f(x−y2,z+w)−2f(x,z)+2f(y,w)∥∥≤∥s(f(x+y,z−w)+f(x−y,z+w)−2f(x,z)+2f(y,w))∥, (2) where s is a fixed nonzero complex number with |s| < 1. Using the direct method, we prove the Hyers–Ulam stability of quasi-multipliers on Banach algebras, associated with the bi-additive s-functional inequalities (1) and (2).