Superstability for Generalized Module Left Derivations and Generalized Module Derivations on a Banach Module (I)

discuss the superstability of generalized module left derivations and generalized module derivations on a Banach module. Let A be a Banach algebra and X a Banach A-module, f : X -> X and g : A -> A. The mappings Delta(1)(f,g), Delta(2)(f,g), Delta(3)(f,g), and Delta(4)(f,g) are defined and it is proved that if parallel to Delta(1)(f,g) (x, y, z, w)parallel to (resp., parallel to Delta(3)(f,g) (x, y, z, w, alpha, beta)parallel to) is dominated by phi(x, y, z, w), then f is a generalized (resp., linear) module-A left derivation and g is a (resp., linear) module-X left derivation. It is also shown that if parallel to Delta(2)(f,g) (x, y, z, w)parallel to (resp., parallel to Delta(4)(f,g) (x, y, z, w, alpha, beta)parallel to) is dominated by phi(x, y, z, w), then f is a generalized (resp., linear) module-A derivation and g is a (resp., linear) module-X derivation. Copyright (c) 2009 Huai-Xin Cao et al.