Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 835893, 12 pages
http://dx.doi.org/10.1155/2012/835893
Research Article

Stability of a Bi-Additive Functional Equation in Banach Modules Over a 𝐶 ⋆ -Algebra

Won-Gil Park1 and Jae-Hyeong Bae2

1Department of Mathematics Education, College of Education, Mokwon University, Daejeon 302-729, Republic of Korea
2Graduate School of Education, Kyung Hee University, Yongin 446-701, Republic of Korea

Received 6 April 2012; Accepted 30 May 2012

Academic Editor: Baodong Zheng

Copyright © 2012 Won-Gil Park and Jae-Hyeong Bae. 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 solve the bi-additive functional equation 𝑓 ( 𝑥 + 𝑦 , 𝑧 − 𝑤 ) + 𝑓 ( 𝑥 − 𝑦 , 𝑧 + 𝑤 ) = 2 𝑓 ( 𝑥 , 𝑧 ) − 2 𝑓 ( 𝑦 , 𝑤 ) and prove that every bi-additive Borel function is bilinear. And we investigate the stability of a bi-additive functional equation in Banach modules over a unital 𝐶 ⋆ -algebra.