Superstability of the generalized orthogonality equation on restricted domains

Abstract

Chmieli\'{n}ski has proved in the paper [4] the superstability of the generalized orthogonality equation $|< f(x), f(y) >| = |< x, y >|$. In this paper, we will extend the result of Chmieli\'{n}ski by proving a theorem: Let $D_{n}$ be a suitable subset of $ \R^n$. If a function $f\hbox{:} D_{n} \to \R^n$ satisfies the inequality $||< f(x), f(y) >| - |< x, y >|| \leq \phi(x,y)$ for an appropriate control function $\phi(x,y)$ and for all $x, y \in D_{n}$, then $f$ satisfies the generalized orthogonality equation for any $x, y \in D_{n}$.