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Nonsmooth sequential analysis in Asplund spaces. (English) Zbl 0881.49009

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.

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.

##### MSC:
 49J52 Nonsmooth analysis (other weak concepts of optimality) 46B20 Geometry and structure of normed linear spaces 58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds 26E15 Calculus of real functions on infinite-dimensional spaces 47A60 Functional calculus of operators