Komlós’ theorem and the fixed point property for affine mappings

Domínguez Benavides, Tomás; Japón, Maria

doi:10.1090/proc/14201

Komlós’ theorem and the fixed point property for affine mappings
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Proc. Amer. Math. Soc. 146 (2018), 5311-5322 Request permission

Abstract:

Assume that $X$ is a Banach space of measurable functions for which Komlós’ 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 given proving that our fixed point theorem is optimal in terms of the Lipschitz constants and the coefficient $t(C)$.

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