Superstability of the generalized orthogonality equation on restricted domains

Chmieliński 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 Chmieliński 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 $||< f(x), f(y) >| - |< x, y >|| \leq ϕ(x,y)$ for an appropriate control function $ϕ(x,y)$ and for all $x, y \in D_{n}$, then $f$ satisfies the generalized orthogonality equation for any $x, y \in D_{n}$.

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