Stability and Perturbed Metric-Preserved Mappings with Universal Theorem

Abstract

If Y is Gateaux smooth, strictly convex and admitting the Kadec- Klee property, then we has the following sharp estimate ∥Tf(x)-x∥ ≤2ε, for all x∈X. Let X, Z be two real Banach spaces and ε ≥ 0, we show that if there is a mapping ƒ: X→ Z with ƒ(0)=0 satisfying |∥f(x)-f(y)∥-∥x-y∥|≤ε for all x,y∈X, then we can define a linear surjective isometry U:X^*→Z^*∕N for some closed subspace N of Z^* by an invariant mean of X. There is a linear surjective operator T: Y→ X of norm one such that ∥Tf(x)-x∥≤2ε,for all x∈X ; when the 𝜀-isometry ƒ is surjective, it is equivalent to Omladič - Šemrl Theorem: There is a surjective linear isometry U:X→Y so that ∥f(x)-Ux∥≤2ε,for all x∈X.