On stability of nonlinear non-surjective ε-isometries of Banach spaces

Let X, Y   be two Banach spaces, ε⩾0ε⩾0, and let f:X→Yf:X→Y be an ε  -isometry with f(0)=0f(0)=0. In this paper, we show first that for every x⁎∈X⁎x⁎∈X⁎, there exists ϕ∈Y⁎ϕ∈Y⁎ with ‖ϕ‖=‖x⁎‖≡r‖ϕ‖=‖x⁎‖≡r such that|〈ϕ,f(x)〉−〈x⁎,x〉|⩽4εr,for all x∈X. Making use of it, we prove that if Y   is reflexive and if E⊂YE⊂Y [the annihilator of the subspace F⊂Y⁎F⊂Y⁎ consisting of all functionals bounded on co¯(f(X),−f(X))] is α-complemented in Y  , then there is a bounded linear operator T:Y→XT:Y→X with ‖T‖⩽α‖T‖⩽α such that‖Tf(x)−x‖⩽4ε,for all x∈X. If, in addition, Y is Gateaux smooth, strictly convex and admitting the Kadec–Klee property (in particular, locally uniformly convex), then we have the following sharp estimate‖Tf(x)−x‖⩽2ε,for all x∈X.