On hyperstability of additive mappings onto banach spaces


Let (X; +) be an Abelian group and E be a Banach space. Suppose that f: X → E is a surjective map satisfying the inequality |||f(x) - ||f(y)||-||f(x - y)|||≤ε min{||f(x) - f (y)||p, ||f (x - y)||p} for some ε > 0, p > 1 and for all x, y ε X. We prove that f is an additive map. However, this result does not hold for 0 < p ≤ 1. As an application, we show that if f is a surjective map from a Banach space E onto a Banach space F so that for some ε > 0 and p > 1 |||f(x) - f(y)|| - ||f (u) - f(v)||| ≤ ε min{||f(x) - f(y)||p, ||f(u) - f(v)||p} whenever ||x - y|| = ||u - v||, then f preserves equality of distance. Moreover, if dim E ≤ 2, there exists a constant K ≠ 0 such that K f is an ane isometry. This improves a result of Vogt ['Maps which preserve equality of distance', Studia Math. 45 (1973) 43-48].

Publication Title

Bulletin of the Australian Mathematical Society