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Fixed Point Properties for c₀-like Spaces

Nezir, Veysel (2012) Fixed Point Properties for c₀-like Spaces. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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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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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Nezir, Veyselven1@pitt.eduVEN1
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairLennard, Christopherlennard@pitt.eduLENNARD
Committee MemberBeatrous, Frankbeatrous@pitt.eduBEATROUS
Committee MemberDowling,
Committee MemberGartside, Paulgartside@math.pitt.eduPMG20
Committee MemberTurett,
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


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