Fixed points of contractions approximating 1-Lipschitz maps

A $1$-Lipschitz map $f$ from a convex compact set to itself has fixed points. This consequence of Brouwer's or Schauder's fixed point theorem has more elementary proofs by approximating $f$ by $\lambda$-contractions, $f_\lambda$. We study the convergence of the fixed points of those contractions as they converge to $f$.