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

## Keywords:

random normed space, generalized Hyers-Ulam stability, quadratic functional equation, inner product## Abstract

Th.M. Rassias introduced the following equalityÂ \begin{eqnarray*}\sum_{i,j=1}^n \|x_i - x_j \|^2 = 2nÂ \sum_{i=1}^n\|x_i\|^2, \qquad \sum_{i=1}^n x_i =0Â end{eqnarray*}Â Â for a fixed integerÂ $n \ge 3$.Â For a mapping $f : X\rightarrow Y$, where $X$ is a vector space and $Y$ is a complete random normed space, we considerÂ the following functional equation

Â \begin{eqnarray}Â \sum_{i,j=1}^n f(x_i - x_j ) = 2nÂ \sum_{i=1}^nf(x_i)Â \end{eqnarray}Â forall $x_1, \cdots, x_{n} \in X$ with $\sum_{i=1}^n x_i =0$. In thisÂ paper, we prove the generalized Hyers-Ulam stability of theÂ functional equation {\rm (0.1)} related to an inner product space.

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## Published

2012-11-07

## Issue

## Section

Approximation Theory

## License

Upon acceptance of an article by the journal, the author(s) accept(s) the transfer of copyright of the article to *European Journal of Pure and Applied Mathematics.*

*European Journal of Pure and Applied Mathematics will be Copyright Holder.*

## How to Cite

*European Journal of Pure and Applied Mathematics*,

*5*(4), 540-553. https://ejpam.com/index.php/ejpam/article/view/847