Nezir, Veysel
(2012)
Fixed Point Properties for c₀-like 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(Unpublished)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 non-weakly 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 non-weakly 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.) c₀-summing 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 c₀-summing basic sequences turn out to be L-scaled a.i. c₀-summing basic sequences. In Chapter 4, we work on Lorentz-Marcinkiewicz 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 (w-FPP) 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 w-FPP for any w, but fails the FPP for n.e. mappings. In Chapter 5, we show that any closed non-reflexive 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 semi-strongly 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 semi-strongly asymptotically nonexpansive mappings.
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Details
| Item Type: |
University of Pittsburgh ETD
|
| Status: |
Unpublished |
| Creators/Authors: |
|
| ETD Committee: |
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| 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;
non-weakly compact; closed, bounded, convex set;
asymptotically isometric c₀-summing basic sequence; c_0-summing basic sequence; closed, convex hull, Lorentz-Marcinkiewicz Spaces, Riesz Angle, strongly asymptotically nonexpansive map, semi-strongly asymptotically nonexpansive map |
| Date Deposited: |
28 Sep 2012 01:37 |
| Last Modified: |
15 Nov 2016 14:00 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/13140 |
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