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Abstract and Applied Analysis
Volume 2008 (2008), Article ID 628178, 8 pages
http://dx.doi.org/10.1155/2008/628178
Research Article

On the Stability of Quadratic Functional Equations

1Department of Mathematics, Daejin University, Kyeonggi 487-711, South Korea
2Department of Mathematics Education, Pusan National University, Pusan 609-735, South Korea
3Department of Mathematics, Hanyang University, Seoul 133-791, South Korea

Received 5 October 2007; Revised 27 November 2007; Accepted 4 January 2008

Academic Editor: Paul Eloe

Copyright Β© 2008 Jung Rye Lee 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 𝑋,π‘Œ be vector spaces and π‘˜ a fixed positive integer. It is shown that a mapping 𝑓(π‘˜π‘₯+𝑦)+𝑓(π‘˜π‘₯βˆ’π‘¦)=2π‘˜2𝑓(π‘₯)+2𝑓(𝑦) for all π‘₯,π‘¦βˆˆπ‘‹ if and only if the mapping π‘“βˆΆπ‘‹β†’π‘Œ satisfies 𝑓(π‘₯+𝑦)+𝑓(π‘₯βˆ’π‘¦)=2𝑓(π‘₯)+2𝑓(𝑦) for all π‘₯,π‘¦βˆˆπ‘‹. Furthermore, the Hyers-Ulam-Rassias stability of the above functional equation in Banach spaces is proven.