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Lipschitz functions with maximal Clarke subdifferentials are staunch

  • Jonathan M. Borwein (a1) and Xianfu Wang (a2)

In a recent paper we have shown that most non-expansive Lipschitz functions (in the sense of Baire's category) have a maximal Clarke subdifferential. In the present paper, we show that in a separable Banach space the set of non-expansive Lipschitz functions with a maximal Clarke subdifferential is not only generic, but also staunch in the space of non-expansive functions.

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[1]Borwein, J.M. and Wang, X., ‘Lipschitz functions with maximal subdifferentials are generic’, Proc. Amer. Math. Soc. 128 (2000), 32213229.
[2]Borwein, J.M., Moors, W.B. and Wang, X., ‘Generalized subdifferentials: a Baire categorical approach’, Trans. Amer. Math. Soc. 353 (2001), 38753893.
[3]Clarke, F.H., Optimization and nonsmooth analysis (Wiley Interscience, New York, 1983).
[4]Giles, J.R. and Sciffer, S., ‘Locally Lipschitz functions are generically pseudo-regular on separable Banach spaces’, Bull. Austral. Math. Soc. 47 (1993), 205212.
[5]Reich, S., Zaslavski, A.J., The set of noncontractive mappings is σ-porous in the space of all non-expansive mappings, C. R. Acad. Sci. Paris 333 (2001), 539544.
[6]Zajicek, L., ‘Small non-σ-porous sets in topologically complete metric spaces’, Colloq. Math. 77 (1998), 293304.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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