Bull. Korean Math. Soc. 2007 Vol. 44, No. 4, 851-860 Published online December 1, 2007

Jaiok Roh and Hui Joung Shin Hallym University, Chungnam National University

Abstract : In this paper, we prove that a function satisfying the following inequality $$ {\parallel f(x) + 2f(y)+ 2f(z)\parallel} \le {\parallel 2 f(\frac{x}{2}+y+z)\parallel} + \epsilon ({\parallel x\parallel}^r \cdot {\parallel y\parallel}^r \cdot {\parallel z\parallel}^r) $$ for all $x, y, z \in X$ and for $\epsilon \ge 0$, is Cauchy additive. Moreover, we will investigate for the stability in Banach spaces.