Koml\'os' Theorem and the Fixed Point Property for affine mappings

TL;DR

This paper explores fixed point properties for affine mappings in Banach spaces of measurable functions where Komlós' Theorem holds, establishing conditions under which fixed points exist and analyzing the influence of the coefficient t(C).

Contribution

It introduces a coefficient t(C) to characterize fixed point properties for affine Lipschitzian mappings in function spaces, providing new fixed point theorems and optimality results.

Findings

Fixed points exist for affine L-Lipschitzian mappings with L<2 on certain sets.

The fixed point property is preserved under small perturbations of the norm.

Failure of fixed point property occurs only when t(C)=2 in L1 spaces.

Abstract

Assume that $X$ is a Banach space of measurable functions for which Koml\'os' Theorem holds. We associate to any closed convex bounded subset $C$ of $X$ a coefficient $t(C)$ which attains its minimum value when $C$ is closed for the topology of convergence in measure and we prove some fixed point results for affine Lipschitzian mappings, depending on the value of $t(C)\in [1,2]$ and the value of the Lipschitz constants of the iterates. As a first consequence, for every $L<2$, we deduce the existence of fixed points for affine uniformly $L$-Lipschitzian mappings defined on the closed unit ball of $L_1[0,1]$. Our main theorem also provides a wide collection of convex closed bounded sets in $L^1([0,1])$ and in some other spaces of functions, which satisfy the fixed point property for affine nonexpansive mappings. Furthermore, this property is still preserved by equivalent renormings when the Banach-Mazur distance is small enough. In particular, we prove that the failure of the fixed point property for affine nonexpansive mappings in $L_1(\mu)$ can only occur in the extremal case $t(C)=2$. Examples are displayed proving that our fixed point theorem is optimal in terms of the Lipschitz constants and the coefficient $t(C)$.