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Averaging and fixed points in Banach spaces.

Gallagher, Torrey (2016) Averaging and fixed points in Banach spaces. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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We use various averaging techniques to obtain results in different aspects of functional analysis
and Banach space theory, particularly in fixed point theory.
Specifically, in the second chapter, we discuss the class of so-called mean nonexpansive
maps, introduced in 2007 by Goebel and Japon Pineda, and we prove that mean isometries
must be isometries in the usual sense. We further generalize this class of mappings to
what we call the affine combination maps, give many examples, and study some preliminary
properties of this class.
In the third chapter, we extend Browder's and Opial's famous Demiclosedness Principles
to the class of mean nonexpansive mappings in the setting of uniformly convex spaces and
spaces satisfying Opial's property. Using this new demiclosedness principle, we prove that
the iterates of a mean nonexpansive map converge weakly to a fixed point in the presence of
asymptotic regularity at a point.
In the fourth chapter, we investigate the geometry and fixed point properties of some
equivalent renormings of the classical Banach space c0. In doing so, we prove that all norms
on `1 which have a certain form must fail to contain asymptotically isometric copies of c0.


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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Gallagher, Torreytmg34@pitt.eduTMG340000-0002-7941-8869
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairLennard,
Committee MemberDowling,
Committee MemberTonge,
Committee MemberDeBlois,
Committee MemberGartside,
Date: 29 September 2016
Date Type: Publication
Defense Date: 15 July 2016
Approval Date: 29 September 2016
Submission Date: 21 July 2016
Access Restriction: 2 year -- Restrict access to University of Pittsburgh for a period of 2 years.
Number of Pages: 95
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; functional analysis; demiclosedness; Opial; mean nonexpansive
Date Deposited: 30 Sep 2016 00:46
Last Modified: 29 Sep 2018 05:15


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