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Transactions of the American Mathematical Society

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Generalized subdifferentials: a Baire categorical approach

Authors: Jonathan M. Borwein, Warren B. Moors and Xianfu Wang
Journal: Trans. Amer. Math. Soc. 353 (2001), 3875-3893
MSC (1991): Primary 49J52, 54E52
Published electronically: May 14, 2001
MathSciNet review: 1837212
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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 real-valued 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 1-Lipschitz function defined on a Banach space has a Clarke subdifferential mapping that is identically equal to the dual ball; if $\{T_{1}, T_{2},\ldots, T_{n}\}$ is a family of maximal cyclically monotone operators defined on a Banach space $X$ then there exists a real-valued locally Lipschitz function $g$such that $\partial_{0}g(x)=\mbox{co}\{T_{1}(x),T_{2}(x),\ldots, T_{n}(x)\}$for each $x\in X$; in a separable Banach space each non-empty weak$^{*}$compact 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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  • 1. J. M. Borwein, Minimal CUSCOS and subgradients of Lipschitz functions, in Fixed Point Theory and its Applications, Pitman Research Notes 252 (1991), 57-81. MR 92j:46077
  • 2. J. M. Borwein and S. Fitzpatrick, Characterization of Clarke subgradients among one-dimensional multifunctions, in Proc. of the Optimization Miniconference II, edited by B. M. Glover and V. Jeyakumar, (1995), 61-73.
  • 3. J. M. Borwein and A. Ioffe, Proximal analysis in smooth spaces, Set-Valued Anal. 4 (1996), 1-24. MR 96m:49028
  • 4. J. M. Borwein and W. B. Moors, Null sets and essentially smooth Lipschitz functions, SIAM J. Optim. 8 (1998), 309-323. MR 99g:49013
  • 5. J. M. Borwein and W. B. Moors, Separable determination of integrability and minimality of the Clarke subdifferential mapping, Proc. Amer. Math. Soc. 128 (2000), 215-221. MR 2000e:49025
  • 6. J. M. Borwein and W. B. Moors, Essentially smooth Lipschitz functions, J. Funct. Anal. 149 (1997), 305-351. MR 98i:58028
  • 7. J. M. Borwein and W. B. Moors, Y. Shao, Subgradient representation of multi-functions, J. Austral. Math. Soc. Ser. B. 40 (1998), 1-13. MR 2001b:49020
  • 8. J. M. Borwein, W. B. Moors and X. Wang, Lipschitz functions with prescribed derivatives and subderivatives, Nonlinear Anal. 29 (1997), 53-64. MR 98j:49019
  • 9. J. M. Borwein and D. Preiss, A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. Amer. Math. Soc. 303 (1987), 517-527. MR 88k:49013
  • 10. A. M. Bruckner, J. B. Bruckner, B. S. Thomson, Real Analysis, Prentice-Hall, Inc. 1997.
  • 11. F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley Interscience, New York, 1983. MR 85m:49002
  • 12. B. Dacorogna and P. Marcellini, General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases, Acta Math. 178 (1997), 1-37. MR 98d:35029
  • 13. B. Dacorogna and P. Marcellini, Dirichlet problem for nonlinear first order partial differential equations, Optimization methods in partial differential equations (South Hadley, MA, 1996), 43-57, Contemp. Math. 209, Amer. Math. Soc., Providence, RI, 1997. MR 98h:35037
  • 14. Marián J. Fabian, Gâteaux Differentiability of Convex Functions and Topology: Weak Asplund Spaces, Wiley Interscience, New York, 1997. MR 98h:46009
  • 15. J. R. Giles and W. B. Moors, A continuity property related to Kuratowski's index of non-compactness, its relevance to the drop property and its implications for differentiability, J. Math. Anal. and Appl. 178 (1993), 247-268. MR 94m:46022
  • 16. J. R. Giles and S. Sciffer, Locally Lipschitz functions are generically pseudo-regular on separable Banach spaces, Bull. Austral. Math. Soc. 47 (1993), 205-212. MR 94a:58022
  • 17. A. D. Ioffe, Approximate subdifferentials and applications II, Mathematika 33 (1986), 111-128. MR 87k:49028
  • 18. A. D. Ioffe, Approximate subdifferentials and applications III. The metric theory, Mathematika 36 (1989), 1-38. MR 90g:49012
  • 19. E. Michael, A note on closed maps and compact sets, Israel J. Math. 2 (1964), 173-176. MR 31:1659
  • 20. R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Mathematics, 1364 Springer-Verlag Berlin Heidelberg, 1993. MR 94f:46055
  • 21. D. Preiss, Differentiability of Lipschitz functions on Banach spaces, J. Funct. Anal. 91 (1990), 312-345. MR 91g:46051
  • 22. D. Preiss, R. R. Phelps and I. Namioka, Smooth Banach spaces, weak Asplund spaces and monotone or usco mappings, Israel J. Math. 72 (1990), 257-279. MR 92h:46021
  • 23. D. Preiss and J. Tiser, Points of non-differentiability of typical Lipschitz functions, Real Anal. Exchange 20 (1995), 219-226. MR 95m:26006
  • 24. R. T. Rockafellar, The Theory of Subgradients and Its Applications to Problems of Optimization: Convex and Nonconvex Functions, Heldermann-Verlag, Berlin, 1981. MR 83b:90126
  • 25. R. T. Rockafellar, R. J-B. Wets, Variational Analysis, Springer-Verlag, Berlin, 1998. MR 98m:49001
  • 26. Xianfu Wang, Fine and pathological properties of subdifferentials, Ph. D. Thesis, Simon Fraser University, 1999.

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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

Warren B. Moors
Affiliation: Department of Mathematics, The University of Waikato, Private bag 3105 Hamilton, New Zealand

Xianfu Wang
Affiliation: Centre for Experimental and Constructive Mathematics, Department of Mathematics and Statistics, Simon Fraser University, Burnaby, B.C. V5A1S6, Canada

Keywords: Subdifferentials, differentiability, Baire category, upper semi--continuous set--valued 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

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