On the Stability of an Additive and Quadratic Functional Equation

Abstract

In Park et al. (J. Chungcheong Math. Soc. 21:455–466, 2008) considered the following Jensen additive and quadratic type functional equation $$2 f \biggl(\frac{x+y}{2} \biggr) + f \biggl( \frac{x-y}{2} \biggr ) + f \biggl(\frac{y-x}{2} \biggr) = f(x) + f(y) . $$ In this paper, we investigate the following additive and quadratic functional equation $$ 2 f(x+y) + f(x-y) + f(y-x) = 3f(x) + f(-x) + 3f(y) + f(-y) . $$ (34.1) Furthermore, we prove the generalized Hyers–Ulam stability of the functional equation (34.1) in Banach spaces.