Asymptotic behavior of generalized quadratic mappings

We show in this paper that a mapping f satisfiesthe following functional equation\begin{eqnarray*}\biguplus_{x_۲,\cdots,x_{d+۱}}^{d}f(x_۱) = ۲^{d} \sum_{i=۱}^{d+۱}f(x_i),\end{eqnarray*}if and only if it is quadratic. In addition, we investigate generalized Hyers-Ulam stability problem for the equation, and thus obtain an asymptotic property of quadratic mappings as applications.

Keywords:

stability , Parallel polyhedron equality , Generalized quadratic mappings