Stability of Banach spaces via nonlinear ε-isometries

In this paper, we prove that the existence of an ε-isometry from a separable Banach space X into Y (the James space or a reflexive space) implies the existence of a linear isometry from X into Y. Then we present a set valued mapping version lemma on non-surjective ε-isometries of Banach spaces. Using the above results, we also discuss the rotundity and smoothness of Banach spaces under the perturbation by ε-isometries.