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 $f : X\rightarrow Y$, where $X$ is a vector space and $Y$ is a complete random normed space, we consider the following functional equation
 \begin{eqnarray} \sum_{i,j=1}^n f(x_i - x_j ) = 2n  \sum_{i=1}^nf(x_i) \end{eqnarray}  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.