Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Komlós' Theorem and the Fixed Point Property for affine mappings (1709.03333v1)

Published 11 Sep 2017 in math.FA

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 $L1([0,1])$ 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 $t(C)$.

Summary

We haven't generated a summary for this paper yet.