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

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‖≤ε(x,y) for all x,y∈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 Lq(Ω,,μ) (1＜q＜ ∞) space or Y is a Hilbert space. Furthermore, the stability of approximate isometries for the case that (x,y)=‖x‖p ‖y‖p or (x,y)=‖x-y‖p for p≠1 is investigated.

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

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. Foundation item: The Science and Art Foundation of Central South University. Biography of the author: XIANG Shu-huang, doctor, professor, born in 1966, majoring in numerical analysis and functional analysis.