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

Let X, Y be two Banach spaces, ε 0, and let f : X→Y be an ε-isometry with f (0) = 0. In this paper, we show first that for every x ∗ ∈ X ∗, there exists φ ∈ Y ∗ with 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 ⊂ Y [the annihilator of the subspace 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 →X with 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.