Asymptotic behavior of generalized quadratic mappings

Document Type : Research Paper

Authors

1 Department of Mathematics, Chungnam National University 99 Daehangno, Yuseong-gu, Daejeon 305-764, Korea

2 Department of Mathematics, Chungnam National University, 99 Daehangno, Yuseong-gu, Daejeon 34134, Korea

Abstract

We show in this paper that a mapping $f$ satisfies
the following functional equation
\begin{eqnarray*}
\biguplus_{x_2,\cdots,x_{d+1}}^{d}f(x_1) = 2^{d} \sum_{i=1}^{d+1}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