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Extremal characterizations of reflexive spaces

Part of: Existence theories

Published online by Cambridge University Press:  09 April 2009

Xianfu Wang
Xianfu Wang
Affiliation:
Mathematics, The Irving K. Barber School of Arts and SciencesUBC Okanagan3333 University WayKelowna BC VIV 1V7CanadaShawn.Wang@ubc.ca
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Abstract

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Assume that a Banach space has a Fréchet differentiable and locally uniformly convex norm. We show that the reflexive property of the Banach space is not only sufficient, but also a necessary condition for the fulfillment of the proximal extremal principle in nonsmooth analysis.

Keywords

subgradientsproximal extremal principleproximal fuzzy sum rulereflexive space.

MSC classification

Secondary: 49J52: Nonsmooth analysis
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 82 , Issue 3 , June 2007 , pp. 429 - 440
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Borwein, J. M. and Fitzpatrick, S., ‘Existence of nearest points in Banach spaces’, Canad. J. Math. 4 (1989), 702–720.CrossRefGoogle Scholar
[2]Borwein, J. M. and Giles, J., ‘The proximal normal formula in Banach space’, Trans. Amer. Math. Soc. 302 (1987), 371–381.CrossRefGoogle Scholar
[3]Borwein, J. M. and Strojwas, H. M., ‘Proximal analysis and boundaries of closed sets in Banach space. Part I: Theory’, Canad. J. Math. 38 (1986), 431–452; ‘Part II: Applications’, Canad. J. Math. 39 (1987), 428–472.CrossRefGoogle Scholar
[4]Borwein, J. M. and Zhu, Q. J., Techniques of variational analysis: an introduction (Springer, Berlin, 2005).Google Scholar
[5]Clarke, F. H., Ledyaev, Y., Stern, R. J. and Wolenski, P. R., Nonsmooth analysis and control theory. Graduate Texts in Mathematics 178 (Springer, New York, 1998).Google Scholar
[6]Fabian, M., ‘Subdifferentiability and trustworthiness in the light of the smooth variational principle of Borwein and Preiss’, Ada Univ. Carotin. Math. Phys. 30 (1989), 51–56.Google Scholar
[7]Fabian, M. and Mordukhovich, B. S., ‘Nonsmooth characterizations of Asplund spaces and smooth variational principles’, Set-Valued Anal. 6 (1998), 381–406.CrossRefGoogle Scholar
[8]Ioffe, A. D.. ‘Proximal analysis and approximate subdifferentials’, J. London Math. Soc. (2) 41 (1990), 175–192.CrossRefGoogle Scholar
[9]Kruger, A. Y. and Mordukhovich, B. S., ‘Extremal points and the Euler equation in nonsmooth optimization’, Dolk. Akad. Nauk BSSR 24 (1980), 684–687.Google Scholar
[10]Lau, K. S., ‘Almost Chebyshev subsets in reflexive Banach spaces’, Indiana Univ. Math. J. 27 (1978), 791–795.CrossRefGoogle Scholar
[11]Loewen, P. D., ‘A mean value theorem for Frechet subgradients’, Nonlinear Anal. 23 (1994), 1365–1381.CrossRefGoogle Scholar
[12]Mordukhovich, B. S., Variational analysis and generalized differentiation, I: Basic Theory, II: Applications, Grundlehren Series (Fundamental Principles of Mathematical Sciences) 330 and 331 (Springer, Berlin, 2005).Google Scholar
[13]Mordukhovich, B. S. and Shao, Y., ‘Extremal characterizations of Asplund spaces’, Proc. Amer. Math. Soc., 124 (1996), 197–205.CrossRefGoogle Scholar
[14]Phelps, R. R., Convex functions, monotone operators and differentiability, Lecture Notes in Mathematics 1364 (Springer, Berlin, 1993).Google Scholar