# Fixed points of contractions approximating 1-Lipschitz maps

## TL;DR

The paper investigates how fixed points of contractions approximating 1-Lipschitz maps behave as they converge to the original map, providing insights into elementary proof techniques for fixed point theorems.

## Contribution

It offers an analysis of the convergence of fixed points of $\lambda$-contractions approximating 1-Lipschitz maps, enhancing understanding of elementary fixed point proofs.

## Key findings

- Fixed points of $\lambda$-contractions converge to the fixed point of the 1-Lipschitz map.
- Provides conditions under which convergence of fixed points occurs.
- Strengthens elementary approaches to fixed point theorems for non-expansive maps.

## Abstract

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 $f$.