Popescu, RoxanaIrina
(2018)
FIXED POINTS AND DUALITY OF CLOSED CONVEX SETS IN BANACH SPACES.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
In the first chapter we construct a new example of an affine norm continuous mapping on a closed, convex, nonweakly compact set $C$ that cannot be extended to a continuous linear map on the entire space $X$. Although often used in the field of the fixed point theory, most of the examples known in the literature are restrictions of continuous, linear mappings from $X$ to $C$.
The second chapter continues with a main focus on the notion of the affine dual of a closed, convex, bounded set. Using a theorem of M. Krein and D. Milman from Studia Mathematica 1940, one can show that certain spaces like $c_0$ and $L^1{[0,1]}$ are not dual spaces. However, it turns out that we can see them as affine dual spaces.
In the third part of this thesis we provide a new proof that compactness in $\ell_1$ for closed, bounded, convex sets is equivalent with the fixed point property for cascading nonexpansive mappings. We also prove an analogue of this result in $L^1{[0,1]}$.
The last part is dedicated to the study of the stability constant of the weak$^*$fixed point property for the dual of separable Lindenstrauss spaces. Initiated in 1980 and 1982 by P. Soardi and T.C. Lim for the space $c_0$, we will now find a precise formula in the general case of an arbitrary predual of $\ell_1$ that depends only on a geometrical property of the unit ball of $\ell_1$ with respect to the predual considered. Therefore, this formula establishes a quantitative result in terms of geometric properties.
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Item Type: 
University of Pittsburgh ETD

Status: 
Unpublished 
Creators/Authors: 
Creators  Email  Pitt Username  ORCID 

Popescu, RoxanaIrina  rop42@pitt.edu  rop42  

ETD Committee: 

Date: 
26 September 2018 
Date Type: 
Publication 
Defense Date: 
30 July 2018 
Approval Date: 
26 September 2018 
Submission Date: 
29 July 2018 
Access Restriction: 
1 year  Restrict access to University of Pittsburgh for a period of 1 year. 
Number of Pages: 
92 
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: 
fixed point theory, Lindenstrauss spaces 
Date Deposited: 
26 Sep 2018 23:03 
Last Modified: 
26 Sep 2018 23:03 
URI: 
http://dscholarship.pitt.edu/id/eprint/35056 
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