Journal of Inequalities and Applications
Volume 2009 (2009), Article ID 718020, 10 pages
doi:10.1155/2009/718020
Research Article

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

Huai-Xin Cao,1 Ji-Rong Lv,1 and J. M. Rassias2

1College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, China
2Pedagogical Department, Section of Mathematics and Informatics, National and Capodistrian University of Athens, Athens 15342, Greece

Received 23 January 2009; Revised 2 March 2009; Accepted 3 July 2009

Academic Editor: Jozsef Szabados

Copyright © 2009 Huai-Xin Cao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We 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 Δf,g1, Δf,g2, Δf,g3, and Δf,g4 are defined and it is proved that if ∥Δf,g1(x,y,z,w)∥ (resp., ∥Δf,g3(x,y,z,w,α,β)∥) is dominated by φ(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 ∥Δf,g2(x,y,z,w)∥ (resp., ∥Δf,g4(x,y,z,w,α,β)∥) is dominated by φ(x,y,z,w), then f is a generalized (resp., linear) module-𝒜 derivation and g is a (resp., linear) module-X derivation.