Random Stability of a Functional Equation Related to An Inner Product Space

Abstract

Th.M. Rassias introduced the following equality \begin{eqnarray*}
\sum_{i,j=1}^n \|x_i - x_j \|^2 = 2n  \sum_{i=1}^n\|x_i\|^2, \qquad \sum_{i=1}^n x_i =0 end{eqnarray*}  for a fixed integer $n\ge 3$. For a mapping $f:X\to Y$, where $X$ is a vector space and $Y$ is a complete random normed space, we consider the following functional equation
 $$\begin{array}{r}Â\sum _{i,j=1}^{n}f(x_i-x_j)=2nÂ\sum _{i=1}^{n}f(x_i)Â\end{array}$$  forall $x_1,\cdots ,x_n\in X$ with $\sum _{i=1}^n{x}_i=0$. In this paper, we prove the generalized Hyers-Ulam stability of the functional equation {\rm (0.1)} related to an inner product space.