Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 672531, 15 pages
http://dx.doi.org/10.1155/2012/672531
Research Article

Hyers-Ulam-Rassias RNS Approximation of Euler-Lagrange-Type Additive Mappings

H. Azadi Kenary,1 H. Rezaei,1 A. Ebadian,2 and A. R. Zohdi3

1Department of Mathematics, College of Sciences, Yasouj University, Yasouj 75914-353, Iran
2Department of Mathematics, Payame Noor University, Tehran, Iran
3Department of Mathematics, Marvdasht Branch, Islamic Azad University, Marvdasht, Iran

Received 24 December 2011; Revised 5 March 2012; Accepted 19 March 2012

Academic Editor: Tadeusz Kaczorek

Copyright © 2012 H. Azadi Kenary 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 generalized Hyers-Ulam (or Hyers-Ulam-Rassias) stability of the following functional equation ∑ 𝑚 𝑗 = 1 𝑓 ( − 𝑟 𝑗 𝑥 𝑗 + ∑ 1 ≤ 𝑖 ≤ 𝑚 , 𝑖 ≠ 𝑗 𝑟 𝑖 𝑥 𝑖 ) + 2 ∑ 𝑚 𝑖 = 1 𝑟 𝑖 𝑓 ( 𝑥 𝑖 ) = 𝑚 𝑓 ( ∑ 𝑚 𝑖 = 1 𝑟 𝑖 𝑥 𝑖 ) where 𝑟 1 , … , 𝑟 𝑚 ∈ ℝ , proved in Banach modules over a unital 𝐶 ∗ -algebra. It was shown that if ∑ 𝑚 𝑖 = 1 𝑟 𝑖 ≠ 0 , 𝑟 𝑖 , 𝑟 𝑗 ≠ 0 for some 1 ≤ 𝑖 < 𝑗 ≤ 𝑚 and a mapping 𝑓 ∶ 𝑋 → 𝑌 satisfies the above mentioned functional equation then the mapping 𝑓 ∶ 𝑋 → 𝑌 is Cauchy additive. In this paper we prove the Hyers-Ulam-Rassias stability of the above mentioned functional equation in random normed spaces (briefly RNS).