Superstability of the Generalized Orthogonality Equation on Restricted Domains

[摘要] ChmieliÅ„ski has proved in the paper [4] the superstability of the generalized orthogonality equation $|langle f(x),f(y)angle|=|langle x,yangle|$. In this paper, we will extend the result of ChmieliÅ„ski by proving a theorem: Let $D_n$ be a suitable subset of $mathbb{R}^n$. If a function $f: D_n→mathbb{R}^n$ satisfies the inequality $||langle f(x),f(y)angle|-|langle x,yangle||≤varphi(x,y)$ for an appropriate control function $varphi(x,y)$ and for all $x,y in D_n$, then 𝑓 satisfies the generalized orthogonality equation for any $x,y in D_n$.

[学科分类] 数学（综合）

[关键词] Superstability;generalized orthogonality equation.