Stability of basic sequences via nonlinear ε-isometries

In this paper, Let X, Y be two real Banach spaces and ε ≥ 0. A mapping f: X → Y is said to be a standard ε-isometry provided f(0) = 0 and (1) $$\parallel f\left( x \right) - f\left( y \right)\parallel - \parallel x - y\parallel | \leqslant \varepsilon $$ ‖ f ( x ) − f ( y ) ‖ − ‖ x − y ‖ | ≤ ε for all x, y ∈ X. If ε = 0, then it is simply called a standard isometry. We prove a sufficient and necessary condition for which {f(xn)}n≥1 is a basic sequence of Y equivalent to {xn}n≥1 whenever {xn}n≥1 is a basic sequence in X and f: X → Y is a nonlinear standard isometry. As a corollary we obtain the stability of basic sequences under the perturbation by nonlinear and non-surjective standard ε-isometries.