Copyright © 2010 Yeol Je Cho 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 and the generalized stability result for the following functional equation in random normed spaces (in the sense of Sherstnev) under arbitrary t-norms: f(x+ky)+f(x−ky)=k2[f(x+y)+f(x−y)]+(2(k2−1)/k2(k−2))f(kx)−((k3−k2−k+1)/2(k−2))f(2x)+f˜(2y)−8f˜(y), where f˜(y):=f(y)−f(−y) for any fixed integer k with k≠0,  ±1,2.