Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 346561, 13 pages
http://dx.doi.org/10.1155/2012/346561
Research Article

On the Stability of an 𝑚 -Variables Functional Equation in Random Normed Spaces via Fixed Point Method

A. Ebadian,1 M. Eshaghi Gordji,2 H. Khodaei,2 R. Saadati,3 and Gh. Sadeghi4

1Department of Mathematics, Payame Noor University, Tehran, Iran
2Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran
3Department of Mathematics, Iran University of Science and Technology, Behshahr, Iran
4Department of Mathematics and Computer Sciences, Tarbiat Moallem University Sabzevar, Sabzevar, P.O. Box 397, Iran

Received 17 September 2011; Revised 5 January 2012; Accepted 29 January 2012

Academic Editor: Seenith Sivasundaram

Copyright © 2012 A. Ebadian 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

At first we find the solution of the functional equation 𝐷 𝑓 ( 𝑥 1 , … , 𝑥 𝑚 ) ∶ = ∑ 𝑚 𝑘 = 2 ( ∑ 𝑘 𝑖 1 = 2 ∑ 𝑘 + 1 𝑖 2 = 𝑖 1 + 1 ⋯ ∑ 𝑚 𝑖 𝑚 − 𝑘 + 1 = 𝑖 𝑚 − 𝑘 + 1 ) 𝑓 ( ∑ 𝑚 𝑖 = 1 , 𝑖 ≠ 𝑖 1 , … , 𝑖 𝑚 − 𝑘 + 1 𝑥 𝑖 − ∑ 𝑚 − 𝑘 + 1 𝑟 = 1 𝑥 𝑖 𝑟 ) + 𝑓 ( ∑ 𝑚 𝑖 = 1 𝑥 𝑖 ) − 2 𝑚 − 1 𝑓 ( 𝑥 1 ) = 0 , where 𝑚 ≥ 2 is an integer number. Then, we obtain the generalized Hyers-Ulam-Rassias stability in random normed spaces via the fixed point method for the above functional equation.