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Fixed Points of Nonexpansive Maps on Closed, Bounded, Convex Sets in l1

Everest, Thomas (2013) Fixed Points of Nonexpansive Maps on Closed, Bounded, Convex Sets in l1. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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In 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.


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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Everest, Thomastme3@pitt.eduTME3
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairLennard, Christopherlennard@pitt.eduLENNARD
Committee MemberBeatrous, Frankbeatrous@pitt.eduBEATROUS
Committee MemberCaginalp, Gunduzcaginalp@pitt.eduCAGINALP
Committee MemberDowling,
Committee MemberTurett,
Date: 30 June 2013
Date Type: Publication
Defense Date: 16 November 2012
Approval Date: 30 June 2013
Submission Date: 12 April 2013
Access Restriction: 5 year -- Restrict access to University of Pittsburgh for a period of 5 years.
Number of Pages: 140
Institution: University of Pittsburgh
Schools and Programs: Dietrich School of Arts and Sciences > Mathematics
Degree: PhD - Doctor of Philosophy
Thesis Type: Doctoral Dissertation
Refereed: Yes
Uncontrolled Keywords: convex set; nonexpansive map; asymptotically nonexpansive map; closed, convex hull; simplex
Date Deposited: 30 Jun 2013 19:20
Last Modified: 30 Jun 2018 05:15


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