Hyers-Ulam Stability of an $n$-Apollonius type Quadratic Mapping

Abstract

Let $X$ and $Y$ be linear spaces. It is shown that for a fixed positive integer $n\geq2,$ if a mapping $Q:X\to Y$ satisfies the following functional equation \begin{equation}\label{A} \sum_{i=1}^{n}Q(z-x_{i})=\frac{1}{n}\sum_{\substack{1\le i,j\le n\\ j<i}}Q(x_{i}-x_{j})+nQ\bigg(z-\frac{1}{n}\sum_{i=1}^{n}x_{i}\bigg) \end{equation} for all $z, x_1, \dots, x_n\in X,$ then the mapping $Q: X \to Y$ is a {\it quadratic mapping of Apollonius type} and a quadratic mapping. We moreover prove the Hyers-Ulam stability of the functional equation $(\ref{A})$ in Banach spaces.