Inner product spaces and the stability of a quadratic mapping with fixed point alternative

Abstract

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equation\begin{eqnarray*} n f\left(\sum_{i=1}^{n} x_i\right) &+& \sum_{i=1}^{n} f\left(n x_i - \sum_{j=1}^{n} x_j\right) \\ &=&\frac{n^2 +n}{2} \sum_{i=1}^{n}f(x_i) + \frac{n^2 -n}{2} \sum_{i=1}^{n}f(-x_i) \qquad (n\ge 2) \nonumber \end{eqnarray*} in real Banach spaces.