Stability of an (A,Q)-functional equation in a Banach space

Abstract

In this paper, we study an (A,Q)-functional equationf(x + y,z + w) + f(x + y,z - w) + f(x - y,z + w) + f(x - y,z - w) = 4f(x,z) + 4f(x,w) in a Banach space and determine its general solutions (in certain groups). Moreover, we verify some stability results of the (A,Q)-functional equation as a consequence of a fixed point theorem in a complete metric space investigated by Brzd?k et al. [A fixed point approach to stability of functional equations, Nonlinear Anal. 74 (2011) 6728-6732].