# FIXED POINTS AND DUALITY OF CLOSED CONVEX SETS IN BANACH SPACES

Popescu, Roxana-Irina (2018) FIXED POINTS AND DUALITY OF CLOSED CONVEX SETS IN BANACH SPACES. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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## Abstract

In the first chapter we construct a new example of an affine norm continuous mapping on a closed, convex, non-weakly 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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## Details

Item Type: University of Pittsburgh ETD
Status: Unpublished
Creators/Authors:
Popescu, Roxana-Irinarop42@pitt.edurop42
ETD Committee:
Committee MemberLennard, Christopher J.lennard@pitt.edu
Committee MemberDowling, Patrick N.dowlinpn@miamioh.edu
Committee MemberGartside, Paulpmg20@pitt.edu
Committee MemberHajłasz, Piotrhajlasz@pitt.edu
Committee MemberJapón Pineda, Maria A.japon@us.es
Committee MemberTurett, Barryturett@oakland.edu
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