APPROXIMATE DERIVATIONS MAPPING INTO THE RADICALS OF BANACH ALGEBRAS

Abstract

In the present paper, we investigate the situations so that the generalized Hyers-Ulam-Rassias stability for functional equations $f(x^2) = f(x)x + xf(x)$ and $f(xy) = f(x)y + xf(y)$ is satisfied. As a result we obtain that every linear mapping on a commutative Banach algebra which is an $\varepsilon$-approximate derivation maps the algebra into its radical.