Fixed points of contractions approximating 1-Lipschitz maps

Maxime Zavidovique – GJM, Volume 4, Issue 2 (2019), 56-61.

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. This sheds a new light on results linked with weak KAM theory and infinite horizon optimal control obtained in [2],[3].


Received: May 15, 2019
Accepted: November 4, 2019


Maxime Zavidovique
Sorbonne Université - Campus Pierre et Marie Curie,
4, place Jussieu, Boite Courrier 247,
75252 Paris Cedex 05, France.


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