Document Type
Lecture
Publication Date
10-16-2003
Abstract
Over the past few years, we have studied fixed-point-free nonexpansive mappings �� on closed, bounded, convex subsets �� of a Banach space ��. For example, we showed that in every non-reflexive subspace �� of the function space ��¹[0,1], there exists a closed, bounded, convex set �� and a nonexpansive map �� on �� that is fixed-point-free. This is the converse of a theorem of B. Maurey (1981). At about that time, Maurey also showed that every weakly compact, convex subset �� of the sequence space ��₀ is such that every nonexpansive ��: �� → �� has a fixed point. Recently, we proved the converse to this theorem; i.e., every closed, bounded, convex, non-weakly compact subset �� of ��₀ supports a nonexpansive map ��: �� → �� that fails to have a fixed point in ��. Thus, weakly compact, convex subsets �� of ��₀ are precisely those for which every nonexpansive ��: �� → �� has a fixed point.
Relational Format
presentation
Recommended Citation
Lennard, Chris, "Fixed point free nonexpansive mappings in Banach spaces" (2003). Analysis Seminar. 61.
https://egrove.olemiss.edu/math_analysis/61
