Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 713675, 14 pages
Research Article

Fixed Points, Inner Product Spaces, and Functional Equations

Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Republic of Korea

Received 1 February 2010; Revised 30 May 2010; Accepted 5 July 2010

Academic Editor: Marlène Frigon

Copyright © 2010 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.


Rassias introduced the following equality i,j=1nxi-xj2=2ni=1nxi2, i=1nxi=0, for a fixed integer n3. Let V,W be real vector spaces. It is shown that, if a mapping f:VW satisfies the following functional equation i,j=1nf(xi-xj)=2ni=1nf(xi) for all x1,,xnV with i=1nxi=0, which is defined by the above equality, then the mapping f:VW is realized as the sum of an additive mapping and a quadratic mapping. Using the fixed point method, we prove the generalized Hyers-Ulam stability of the above functional equation in real Banach spaces.