Extremal characterizations of asplund spaces

Authors:
Boris S. Mordukhovich and Yongheng Shao

Journal:
Proc. Amer. Math. Soc. **124** (1996), 197-205

MSC (1991):
Primary 46B20; Secondary 49J52

DOI:
https://doi.org/10.1090/S0002-9939-96-03049-3

MathSciNet review:
1291788

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove new characterizations of Asplund spaces through certain extremal principles in nonsmooth analysis and optimization. The latter principles provide necessary conditions for extremal points of set systems in terms of Fréchet normals and -normals.

**1**E. Asplund,*Fréchet differentiability of convex functions*, Acta Math.**121**(1968), 31--47. MR**37:6754****2**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****3**J. M. Borwein and H. M. Strojwas,*Proximal analysis and boundaries of closed sets in Banach spaces, part*II:*Applications*, Canad. J. Math.**39**(1987), 428--472. MR**88f:46034****4**R. Deville, G. Godefroy, and V. Zizler,*Smoothness and renorming in Banach spaces*, Pitman Monographs Surveys Pure Appl. Math., vol. 64, Wiley, New York, 1993. MR**94d:46012****5**I. Ekeland,*On the variational principle*, J. Math. Anal. Appl.**47**(1974), 324--358. MR**49:11344****6**M. Fabian,*Subdifferentials, local -supports and Asplund spaces*, J. London Math. Soc.**34**(1986), 568--576. MR**88a:90216****7**------,*Subdifferentiability and trustworthiness in the light of a new variational principle of Borweiss and Preiss*, Acta Univ. Carolinae**30**(1989), 51--56. MR**91c:49024****8**A. D. Ioffe,*On subdifferentiability spaces*, Ann. New York Acad. Sci.**410**(1983), 107--119. MR**86g:90124****9**------,*Proximal analysis and approximate subdifferentials*, J. London Math. Soc.**41**(1990), 175--192.**10**A. Y. Kruger,*Generalized differentials of nonsmooth functions and necessary conditions for an extremum*, Siberian Math. J.**26**(1985), 370--379. MR**86j:49038****11**A. Y. Kruger and B. S. Mordukhovich,*Extremal points and Euler equations in nonsmooth optimization*, Dokl. Akad. Nauk BSSR**24**(1980), 684--687. (Russian) MR**82b:90127****12**B. S. Mordukhovich,*Maximum principle in the problem of time optimal control with nonsmooth constraints*, J. Appl. Math. Mech.**40**(1976), 960--969. MR**58:7284****13**------,*Metric approximations and necessary optimality conditins for general classes of nonsmooth extremal problems*, Soviet Math. Dokl.**22**(1980), 526--530.**14**------,*Approximation methods in problems of optimization and control*, Nauka, Moscow, 1988. (Russian)**15**------,*Generalized differential calculus for nonsmooth and set-valued mappings*, J. Math. Anal. Appl.**184**(1994), 250--288. CMP**94:11****16**B. S. Mordukhovich and Y. Shao,*Nonsmooth sequential analysis in Asplund spaces*, Trans. Amer. Math. Soc. (to appear).**17**R. R. Phelps,*Convex functions, monotone operators and differentiability*, 2nd ed., Lecture Notes in Math., vol. 1364, Springer-Verlag, Berlin, 1993. MR**94f:46055**

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Additional Information

**Boris S. Mordukhovich**

Affiliation:
Department of Mathematics, Wayne State University, Detroit, Michigan 48202

Email:
boris@math.wayne.edu

**Yongheng Shao**

Affiliation:
Department of Mathematics, Wayne State University, Detroit, Michigan 48202

DOI:
https://doi.org/10.1090/S0002-9939-96-03049-3

Received by editor(s):
February 22, 1994

Received by editor(s) in revised form:
August 1, 1994

Additional Notes:
This research was partially supported by the National Science Foundation under grants DMS-9206989 and DMS-9404128

Communicated by:
Dale Alspach

Article copyright:
© Copyright 1996
American Mathematical Society