Fixed Points of Nonexpansive Maps on Closed, Bounded, Convex Sets in l1Everest, Thomas (2013) Fixed Points of Nonexpansive Maps on Closed, Bounded, Convex Sets in l1. Doctoral Dissertation, University of Pittsburgh. (Unpublished)
AbstractIn 1965, W.A. Kirk proved that all reflexive Banach spaces (X, ∥·∥) with normal structure are such that for all nonempty, closed, bounded, and convex subsets C ⊆ X, every nonexpansive map T:C→C has a fixed point, i.e. (X, ∥·∥) has the fixed point property for nonexpansive mappings (FPP(n.e.)). In 1979, K. Goebel and T. Kuczumow constructed “very irregular” closed, bounded, convex, non-weak∗-compact subsets K of l1, and showed that such K have the FPP(n.e.). We show that we may perturb the sets of Goebel and Kuczumow to construct a new and larger class of sets that have the FPP(n.e.). Ultimately, we would like to answer the following: which isomorphic l1-basic sequences (xn)n∈N are such that their closed convex hulls have the FPP(n.e.)? Theorem 2.2.1, Theorem 2.3.15, and Theorem 2.3.18 give new and interesting isomorphic l1-basic sequences in (l1, ∥·∥1) whose closed convex hulls have the FPP(n.e.). In 2003, W. Kaczor and S. Prus showed that under a certain assumption, the sets constructed by Goebel and Kuczumow have the fixed point property for asymptotically nonexpansive mappings and that this is equivalent to the sets having the fixed point property for mappings of asymptotically nonexpansive type. In the second part of this thesis, we prove a theorem (Theorem 3.4.1) that provides an estimate for the l1-distance of a point to a simplex. As a corollary, we prove an interesting special case of the theorem of Kaczor and Prus. We further calculate the best uniform-Lipschitz constant of the right shift R on one of the sets K of Goebel and Kuczumow. We also consider another closed, bounded, convex, non-weak∗-compact subset G of the positive face of the usual unit sphere S in l1. We show that, in contrast to the sets K above, G fails to have the fixed point property for asymptotically nonexpansive mappings. Share
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