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Lipschitz functions with maximal Clarke subdifferentials are staunch

Published online by Cambridge University Press:  17 April 2009

Jonathan M. Borwein
Affiliation:
Faculty of Computer Science, Dalhousie University 6050 University Avenue, Halifax, NS, Canada, B3H 1W5, e-mail: jborwein@cs.dal.ca
Xianfu Wang
Affiliation:
Department of Mathematics and Statistics, UBC Okanagan, 3333 University Way, Kelowna, BC., Canada, V1V 1V7, e-mail: Shawn.Wang@ubc.ca
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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.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Borwein, J.M. and Wang, X., ‘Lipschitz functions with maximal subdifferentials are generic’, Proc. Amer. Math. Soc. 128 (2000), 32213229.CrossRefGoogle Scholar
[2]Borwein, J.M., Moors, W.B. and Wang, X., ‘Generalized subdifferentials: a Baire categorical approach’, Trans. Amer. Math. Soc. 353 (2001), 38753893.CrossRefGoogle Scholar
[3]Clarke, F.H., Optimization and nonsmooth analysis (Wiley Interscience, New York, 1983).Google Scholar
[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.CrossRefGoogle Scholar
[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.CrossRefGoogle Scholar
[6]Zajicek, L., ‘Small non-σ-porous sets in topologically complete metric spaces’, Colloq. Math. 77 (1998), 293304.CrossRefGoogle Scholar
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