A logical analysis of the generalized Banach contractions principle
Résumé
Let (X,d) be a complete metric space, m a natural number, and w a real with 0<= w < 1. A g-contraction is a mapping T: X->X such that for all x,y in X there is an i in [1,m] with d(T^ix, T^iy) < w^i d(x,y)$. The generalized Banach contractions principle states that each g-contraction has a fixed point. We show that this principle is a consequence of Ramsey's theorem for pairs over, roughly, RCA_0 + \Sigma^0_2-IA.