Komlós’ theorem and the fixed point property for affine mappings

Abstract

Assume that X X is a Banach space of measurable functions for which Komlós’ Theorem holds. We associate to any closed convex bounded subset C C of X X a coefficient t ( C ) t(C) which attains its minimum value when C 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 ) ∈ [ 1 , 2 ] t(C)\in [1,2] and the value of the Lipschitz constants of the iterates. As a first consequence, for every L > 2 L>2 , we deduce the existence of fixed points for affine uniformly L L -Lipschitzian mappings defined on the closed unit ball of L 1 [ 0 , 1 ] L_1[0,1] . Our main theorem also provides a wide collection of convex closed bounded sets in L 1 ( [ 0 , 1 ] ) 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 ( μ ) L_1(\mu ) can only occur in the extremal case t ( C ) = 2 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 ) t(C) .