Comment on "On the stability of quadratic double centralizers on Banach algebras" [M. Eshaghi Gordji, A. Bodaghi, J. Comput. Anal. Appl. 13 (2011), 724-729]

Abstract

Eshaghi Gordji and Bodaghi [2] proved the Hyers-Ulam stability of quadratic double centralizers on Banach algebras for the system of the functional equationsf (kx + y) + f (kx - y) = 2k(2) f (x) + 2f (y) & f (xy) = f (x)yfor a fixed integer k greater than 1. One can easily show that all the quadratic double centralizers (L, R) in the results are (0, 0). The results are trivial. In this paper, we correct the results.Using the direct method, we prove the Hyers-Ulam stability of quadratic double centralizers on Banach algebras for the system of the functional equationsf (kx + y) + f (kx - y) = 2k(2) f (x) + 2f (y) & f (xy) = f (x)y(2)for a fixed integer k greater than 1.