The stability of quadratic alpha-functional equations

In this paper, we investigate the quadratic alpha-functional equation 2f (x) + 2 f (y) - f (x y) + alpha(-2) f (alpha(x + y)), (1) 2f (x) + 2f (y) = f (x + y) + alpha(-2) f (alpha(x-y)), (2) where alpha is a fixed nonzero real or complex number with alpha(-1) not equal +/-root 3. Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the quadratic alpha-functional equations (1) and (2) in Banach spaces. (C) 2016 All rights reserved.S. Yun was supported by Hanshin University Research Grant