Hyers-Ulam-Rassias stability of approximate isometries on restricted domains

Abstract

Let X and Y be real Banach spaces. The stability of Hyers-Ulam-Rassias approximate isometries on restricted domains S (unbounded or bounded) for into mapping f: S → Y satisfying | ‖ f(x) − f(y) ‖ − ‖ x − y ‖ | ⩽ εφ (x, y) for all x, y ε S is studied in case that the target space Y is uniformly convex Banach space of the modulus of convexity of power type q ⩾ 2 or Y is the L^q (Ω, Σ, μ) (1<q<+∞) space or Y is a Hilbert space. Furthermore, the stability of approximate isometries for the case that φ (x, y)=‖ x ‖ ^p + ‖ y^p or φ(x, y)=‖ x − y ‖ ^p for p≠1 is investigated.

Keywords

Hyers-Ulam-Rassias stability / isometry / uniformly convex space / Hilbert space