Advances in Difference Equations
Volume 2009 (2009), Article ID 273165, 22 pages
doi:10.1155/2009/273165
Research Article

Stability of a Generalized Euler-Lagrange Type Additive Mapping and Homomorphisms in C∗-Algebras

Abbas Najati1 and Choonkil Park2

1Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, 56199-11367, Iran
2Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, 133-791, South Korea

Received 17 June 2009; Revised 30 July 2009; Accepted 4 August 2009

Academic Editor: Patricia J. Y. Wong

Copyright © 2009 Abbas Najati and Choonkil Park. 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

Let X,Y be Banach modules over a C∗-algebra and let r1,…,rn∈ℝ be given. We prove the generalized Hyers-Ulam stability of the following functional equation in Banach modules over a unital C∗-algebra: ∑j=1nf(−rjxj+∑1≤i≤n,i≠jrixi)+2∑i=1nrif(xi)=nf(∑i=1nrixi). We show that if ∑i=1nri≠0, ri,rj≠0 for some 1≤i<j≤n and a mapping f:X→Y satisfies the functional equation mentioned above then the mapping f:X→Y is Cauchy additive. As an application, we investigate homomorphisms in unital C∗-algebras.