Journal of Inequalities and Applications
Volume 2008 (2008), Article ID 902187, 11 pages
doi:10.1155/2008/902187
Research Article
On the Stability of Cubic Mappings and Quadratic Mappings in Random Normed Spaces
1Youngs Researchers Club and Department of Basic Sciences, Islamic Azad University, Ayatollah Amoli Branch, P.O. Box 678, Amol, Iran
2Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, South Korea
3Department of Mechanical Engineering, Islamic Azad University, Ayatollah Amoli Branch, P.O. Box 678, Amol, Iran
4Department of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Avenue, Tehran 15914, Iran
5Faculty of Sciences, University of Shomal, P.O. Box 731, Amol, Iran
Received 27 August 2008; Revised 23 October 2008; Accepted 24 November 2008
Academic Editor: Wing-Sum Cheung
Copyright © 2008 E. Baktash 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
Recently, the stability of the cubic functional equation
f(2x+y)+f(2x−y)=2f(x+y)+2f(x−y)+12f(x) in fuzzy normed spaces was proved in earlier work; and the stability of the additive functional equations
f(x+y)=f(x)+f(y), 2f((x+y)/2)=f(x)+f(y)
in random normed spaces was proved as well.
In this paper, we prove the stability of the cubic functional equation
f(2x+y)+f(2x−y)=2f(x+y)+2f(x−y)+12f(x)
in random normed spaces by an alternative proof which provides a better estimation. Finally, we prove the stability of the quartic functional equation
f(2x+y)+f(2x−y)=4f(x+y)+4f(x−y)+24f(x)−6f(y)
in random normed spaces.