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Nonexpansive Mappings in Fixed-Point Theory and Entropy

Stawski, Adam (2024) Nonexpansive Mappings in Fixed-Point Theory and Entropy. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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Abstract

Fixed-point theory studies the structures on a space X that provide every sufficiently nice self-mapping f:C\to C, on a sufficiently nice subset C\subset X, with a fixed point. Here we present two counterexamples. The positive face of the unit sphere of C(K)^* fails the fixed-point-property with a contractive map, if K is an infinite, compact Hausdorff space. We also show that if the continuum hypothesis is assumed, then the unit ball of certain ideals in C(\mathbb{N}^*) fails the fixed-point-property for nonexpansive maps, where \mathbb{N}^*=\beta\mathbb{N}\setminus\mathbb{N}, the Stone-\u{C}ech remainder space. We then consider an \ell^1-extension of the classical Shannon entropy functional for finite, discrete probability spaces, and we present an L^1([0,1])-analogue. In each case, the set of elements with finite entropy is a non-closed subspace, which can be equipped with a natural topological vector space structure. In the \ell^1-case, we show that entropy can be used to characterize the closed subspaces of \ell^1 that fail the fixed-point-property.

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Details

Item Type: University of Pittsburgh ETD
Status: Unpublished
Creators/Authors:
CreatorsEmailPitt UsernameORCID
Stawski, Adamacs192@pitt.eduacs192
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairLennard, Christopherlennard@pitt.edu
Committee MemberHajlasz, Piotrhajlasz@pitt.edu
Committee MemberManfredi, Juanmanfredi@pitt.edu
Committee MemberDowling, Patrickdowlinpn@miamioh.edu
Date: 10 January 2024
Date Type: Publication
Defense Date: 10 November 2023
Approval Date: 10 January 2024
Submission Date: 19 November 2023
Access Restriction: 2 year -- Restrict access to University of Pittsburgh for a period of 2 years.
Number of Pages: 80
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, entropy
Date Deposited: 10 Jan 2024 13:56
Last Modified: 10 Jan 2024 13:56
URI: http://d-scholarship.pitt.edu/id/eprint/45547

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