Orthogonal stability of functional equations with the fixed point alternative

Abstract

In this paper, we investigate the orthogonal stability of functional equations in orthogonality modules over a unital Banach algebra. Using a fixed point method, we prove the Hyers-Ulam stability of the orthogonally Jensen additive functional equation 2f (x + y /2) = f (x) + f (y), the orthogonally Jensen quadratic functional equation 2f (x + y /2) + 2f (x - y /2) = f (x) + f (y), the orthogonally cubic functional equation f (2x + y) + f (2x - y) = 2f (x + y) + 2f (x - y) + 12f (x), and the orthogonally quartic functional equation f (2x + y) + f (2x - y) = 4f (x + y) + 4f (x - y) + 24f (x) - 6f (y) for all x, y with x perpendicular to y, where perpendicular to is the orthogonality in the sense of Ratz.