A logical analysis of the generalized Banach contractions principle

Let (X , d) be a complete metric space, m ∈ N \ {0}, and γ ∈ R with 0 ≤ γ < 1. A g-contraction is a mapping T : X −→ X such that for all x, y ∈ X there is an i ∈ [1,m] with d(T ix,T iy) <R γid(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, RCA0 + Σ2-IA. 2010 Mathematics Subject Classification 03B30, 03F60, 47H10 (primary); 03F35 (secondary)