Abstract and Applied Analysis
Volume 2009 (2009), Article ID 907121, 11 pages
doi:10.1155/2009/907121
Research Article

Functional Equations Related to Inner Product Spaces

Choonkil Park,1 Won-Gil Park,2 and Abbas Najati3

1Department of Mathematics, Hanyang University, Seoul 133-791, South Korea
2National Institute for Mathematical Sciences, Daejeon 305-340, South Korea
3Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil 51664, Iran

Received 23 March 2009; Accepted 25 May 2009

Academic Editor: John Rassias

Copyright © 2009 Choonkil Park 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

Let V,W be real vector spaces. It is shown that an odd mapping f:V→W satisfies ∑i−12nf(xi−1/2n∑j=12nxj)=∑i=12nf(xi)−2nf(1/2n∑i=12nxi) for all x1,…,x2n∈V if and only if the odd mapping f:V→W is Cauchy additive. Furthermore, we prove the generalized Hyers-Ulam stability of the above functional equation in real Banach spaces.