Asymptotic behavior of generalized quadratic mappings

Abstract

We show in this paper that a mapping $f$ satisfiesthe following functional equationbegin{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.