Superstability of the generalized orthogonality equation on restricted domains

Abstract

Chmielinski 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 Chmielinski by proving a theorem: Let D. be a suitable subset of R(n). If a function f : D(n) --> R(n) satisfies the inequality \\<f(x), f (y)>\ - \<x, y>\\ less than or equal to phi(x, y) for an appropriate control function phi(x, y) and for all x, y is an element of D(n), then f satisfies the generalized orthogonality equation for any x, y is an element of D(n).