Abstract and Applied Analysis
Volume 2012 (2012), Article ID 232630, 9 pages
http://dx.doi.org/10.1155/2012/232630
Research Article

Ulam Stability of a Quartic Functional Equation

Abasalt Bodaghi,1,2 Idham Arif Alias,2 and Mohammad Hosein Ghahramani1

1Department of Mathematics, Garmsar Branch, Islamic Azad University, Garmsar, Iran
2Institute for Mathematical Research, University Putra Malaysia, 43400 UPM, Serdang, Selangor Darul Ehsan, Malaysia

Received 11 January 2012; Revised 9 February 2012; Accepted 13 February 2012

Academic Editor: Nicole Brillouet-Belluot

Copyright © 2012 Abasalt Bodaghi 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

The oldest quartic functional equation was introduced by J. M. Rassias in (1999), and then was employed by other authors. The functional equation 𝑓 ( 2 𝑥 + 𝑦 ) + 𝑓 ( 2 𝑥 − 𝑦 ) = 4 𝑓 ( 𝑥 + 𝑦 ) + 4 𝑓 ( 𝑥 − 𝑦 ) + 2 4 𝑓 ( 𝑥 ) − 6 𝑓 ( 𝑦 ) is called a quartic functional equation, all of its solution is said to be a quartic function. In the current paper, the Hyers-Ulam stability and the superstability for quartic functional equations are established by using the fixed-point alternative theorem.