Nezir, Veysel
(2012)
Fixed Point Properties for c₀like Spaces.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
In 1981, Maurey proved that every weakly compact, convex subset C of c₀ is such that every nonexpansive (n.e.) mapping T:C→C has a fixed point; i.e., C has the fixed point property (FPP). Dowling, Lennard, and Turett proved the converse of Maurey's result by showing each closed bounded convex nonweakly compact subset C of c₀ fails FPP for n.e. mappings. However, in general the mapping failing to have a fixed point is not affine. In Chapter 2 and Chapter 3, we prove that for certain classes of closed bounded convex nonweakly compact subsets C of c₀, there exists an affine nonexpansive mapping T:C→C that fails to have a fixed point. Our result depends on our main theorem: if a Banach space contains an asymptotically isometric (a.i.) c₀summing basic sequence (xᵢ)i∈ℕ, then the closed convex hull of the sequence fails the FPP for affine nonexpansive mappings. In fact, in Chapter 3, we show that very large classes of c₀summing basic sequences turn out to be Lscaled a.i. c₀summing basic sequences. In Chapter 4, we work on LorentzMarcinkiewicz spaces and explore the FPP for lw,∞⁰ spaces. Using Borwein and Sims' technique we prove for certain classes of weight sequence w that X := lw,∞⁰ has the weak fixed point property (wFPP) by using the Riesz angle concept. Furthermore, we find a formula for the Riesz angle of X for any weight sequence. Next, we show that X has the wFPP for any w, but fails the FPP for n.e. mappings. In Chapter 5, we show that any closed nonreflexive vector subspace Y of lw,∞⁰ contains an isomorphic copy of c₀ and so Y fails the FPP for strongly asymptotically nonexpansive maps. Also, we show that l¹ cannot be renormed to have the FPP for semistrongly asymptotically nonexpansive maps, and that c₀ cannot be renormed to have the FPP for strongly asymptotically nonexpansive maps. Finally, we show that reflexivity for Banach lattices is equivalent to the FPP for affine semistrongly asymptotically nonexpansive mappings.
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Item Type: 
University of Pittsburgh ETD

Status: 
Unpublished 
Creators/Authors: 

ETD Committee: 

Date: 
27 September 2012 
Date Type: 
Publication 
Defense Date: 
27 July 2012 
Approval Date: 
27 September 2012 
Submission Date: 
25 July 2012 
Access Restriction: 
1 year  Restrict access to University of Pittsburgh for a period of 1 year. 
Number of Pages: 
157 
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: 
nonexpansive mapping; affine mapping; fixed point property;
nonweakly compact; closed, bounded, convex set;
asymptotically isometric c₀summing basic sequence; c_0summing basic sequence; closed, convex hull, LorentzMarcinkiewicz Spaces, Riesz Angle, strongly asymptotically nonexpansive map, semistrongly asymptotically nonexpansive map 
Date Deposited: 
28 Sep 2012 01:37 
Last Modified: 
15 Nov 2016 14:00 
URI: 
http://dscholarship.pitt.edu/id/eprint/13140 
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