Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 423231, 23 pages
doi:10.1155/2010/423231
Research Article

A Fixed Point Approach to the Stability of Quintic and Sextic Functional Equations in Quasi- 𝛽 -Normed Spaces

Tian Zhou Xu,1 John Michael Rassias,2 Matina John Rassias,2 and Wan Xin Xu3

1Department of Mathematics, School of Science, Beijing Institute of Technology, Beijing 100081, China
2Pedagogical Department E. E., Section of Mathematics and Informatics, National and Kapodistrian University of Athens, 4 Agamemnonos Street, Aghia Paraskevi, Athens 15342, Greece
3School of Communication and Information Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China

Received 19 September 2010; Accepted 31 December 2010

Academic Editor: Vijay Gupta

Copyright © 2011 Tian Zhou Xu 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

We achieve the general solution of the quintic functional equation 𝑓 ( 𝑥 + 3 𝑦 ) − 5 𝑓 ( 𝑥 + 2 𝑦 ) + 1 0 𝑓 ( 𝑥 + 𝑦 ) − 1 0 𝑓 ( 𝑥 ) + 5 𝑓 ( 𝑥 − 𝑦 ) − 𝑓 ( 𝑥 − 2 𝑦 ) = 1 2 0 𝑓 ( 𝑦 ) and the sextic functional equation 𝑓 ( 𝑥 + 3 𝑦 ) − 6 𝑓 ( 𝑥 + 2 𝑦 ) + 1 5 𝑓 ( 𝑥 + 𝑦 ) − 2 0 𝑓 ( 𝑥 ) + 1 5 𝑓 ( 𝑥 − 𝑦 ) − 6 𝑓 ( 𝑥 − 2 𝑦 ) + 𝑓 ( 𝑥 − 3 𝑦 ) = 7 2 0 𝑓 ( 𝑦 ) . Moreover, we prove the stability of the quintic and sextic functional equations in quasi- 𝛽 -normed spaces via fixed point method.