Fixed Points for Nonexpansive Mappings and Normal Structure

The most known and important metric fixed point theorem is the Banach fixed point theorem, also called the contractive mapping principle, which assures that every contraction from a complete metric space into itself has a unique fixed point. We recall that a mapping T from a metric space (X, d) into itself is said to be a contraction if there exists k ∈ [0,1) such that d(Tx, Ty) ≤ kd(x, y) for every x, y ∈ X. This theorem appeared in explicit form in Banach’s Thesis in 1922 [Bn] where it was used to establish the existence of a solution for an integral equation. The simplicity of its proof and the possibility of attaining the fixed point by using successive approximations have made this theorem a very useful tool in Analysis and in Applied Mathematics.