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Zbl 0881.49009
Mordukhovich, Boris S.; Shao, Yongheng
Nonsmooth sequential analysis in Asplund spaces. (English)
[J] Trans. Am. Math. Soc. 348, No.4, 1235-1280 (1996). ISSN 0002-9947; ISSN 1088-6850/e

Based on the Kruger-Mordukhovich limiting normal cone and the associated subdifferential notion, the authors develop a general differentiation theory for nonsmooth functions in infinite-dimensional spaces. Though these constructions are nonconvex and not topological closed in general, they turn out to be smaller than the counterparts of Clarke and Ioffe and admit the description of comprehensive calculus rules, especially in Asplund spaces.\par The paper is divided in 9 sections. After an introduction in the first section, section 2 contains the basic constructions of general normal cones, coderivations of multifunctions and subdifferentials of real-valued functions. Shorter representations of these objects in Asplund spaces are demonstrated. The extremal principles for systems of closed sets in section 3 provide an interesting approach for the following generalized differential calculus. As a conclusion of these results, a nonconvex analogue of the well-known Bishop-Phelps theorem is pointed out. Sections 4-7 are devoted to the calculus rules in Asplund spaces. The authors prove sum rules, scalarization formulas, generalized chain rules and rules for products, quotients, maxima and minima of functions. Especially the subdifferential rules for generalized marginal functions are essential elements of the paper. Using a Zagrodny type approximate mean value theorem which is proved in section 8, the authors give some characterizations of the Lipschitz continuity and some exact relationships to other normal cones and subdifferential notions. It is shown that (in Asplund spaces) the basic constructions of the paper are smaller than the corresponding objects of Clarke and Ioffe but also (section 9) than other constructions satisfying some natural requirements.
[J.Thierfelder (Ilmenau)]

MSC 2000:: *49J52 Nonsmooth analysis (other weak concepts of optimality)
46B20 Geometry and structure of normed spaces
58C20 Generalized differentiation theory on manifolds
26E15 Calculus of functions on infinite-dimensional spaces
47A60 Functional calculus of operators

Keywords: limiting normal cones; generalized subdifferentials; subdifferential calculus; Asplund spaces; coderivations; multifunctions; generalized differential calculus

Cited in: Zbl 1140.49013 Zbl 1109.46043 Zbl 1089.49027

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