Generalized subdifferentials: a Baire categorical approach
Authors:
Jonathan M. Borwein, Warren B. Moors and Xianfu Wang
Journal:
Trans. Amer. Math. Soc. 353 (2001), 38753893
MSC (1991):
Primary 49J52, 54E52
Published electronically:
May 14, 2001
MathSciNet review:
1837212
Fulltext PDF Free Access
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Additional Information
Abstract: We use Baire categorical arguments to construct pathological locally Lipschitz functions. The origins of this approach can be traced back to Banach and Mazurkiewicz (1931) who independently used similar categorical arguments to show that ``almost every continuous realvalued function defined on [0,1] is nowhere differentiable". As with the results of Banach and Mazurkiewicz, it appears that it is easier to show that almost every function possesses a certain property than to construct a single concrete example. Among the most striking results contained in this paper are: Almost every 1Lipschitz function defined on a Banach space has a Clarke subdifferential mapping that is identically equal to the dual ball; if is a family of maximal cyclically monotone operators defined on a Banach space then there exists a realvalued locally Lipschitz function such that for each ; in a separable Banach space each nonempty weakcompact convex subset in the dual space is identically equal to the approximate subdifferential mapping of some Lipschitz function and for locally Lipschitz functions defined on separable spaces the notions of strong and weak integrability coincide.
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Additional Information
Jonathan M. Borwein
Affiliation:
Centre for Experimental and Constructive Mathematics, Department of Mathematics and Statistics, Simon Fraser University, Burnaby, B.C. V5A1S6, Canada
Email:
jborwein@cecm.sfu.ca
Warren B. Moors
Affiliation:
Department of Mathematics, The University of Waikato, Private bag 3105 Hamilton, New Zealand
Email:
moors@math.waikato.ac.nz
Xianfu Wang
Affiliation:
Centre for Experimental and Constructive Mathematics, Department of Mathematics and Statistics, Simon Fraser University, Burnaby, B.C. V5A1S6, Canada
Email:
xwang@cecm.sfu.ca
DOI:
http://dx.doi.org/10.1090/S0002994701028203
PII:
S 00029947(01)028203
Keywords:
Subdifferentials,
differentiability,
Baire category,
upper semicontinuous setvalued map,
$T$Lipschitz function
Received by editor(s):
March 24, 1999
Received by editor(s) in revised form:
February 25, 2000
Published electronically:
May 14, 2001
Additional Notes:
Research of the first author was supported by NSERC and the Shrum endowment of Simon Fraser University
Research of the second author was supported by a Marsden fund grant, VUW 703, administered by the Royal Society of New Zealand
Article copyright:
© Copyright 2001
American Mathematical Society
