Random stability of a functional equation related to an inner product space

Abstract

In [14], Th.M. Rassias introduced the following equality$sumlimits_{i,j=1}^n parallel x_i - x_j parallel^2=2n sumlimits_{i=1}^n parallel x_i parallel^2, sumlimits_{i=1}^n x_i = 0$for a fixed integer n ≥ 3. For a mapping f : X → Y , where X is a vector space and Y is a complete random normed space, we consider the following functional equation $sumlimits_{i,j=1}^n f(x_i - x_j) = 2n sumlimits_{i=1}^n f(x_i)$ (1)for all $x_1, . . . , x_n in X$ with $sumlimits_{i=1)^n x_i = 0$. In this paper, we prove the Hyers-Ulam stability of the functional equation (1) related to an inner product space.