On Fréchet subdifferentials.

*(English. Russian original)*Zbl 1039.49021The author gives a survey (with over 100 references) about the theory of Fréchet subdifferentiation in Banach spaces.

After the definitions of Fréchet subdifferentials (and superdifferentials) of real-valued functions, Fréchet normal cones of sets and Fréchet coderivatives of set-valued mappings, the most important properties of these notions are presented in the first section.

In the second section, relations between the well-known variational principles of Ekeland, Borwein/Preiss and Mordukhovich/Wang, the (weak and strong) fuzzy sum rule for lower semicontinuous functions in terms of Fréchet subdifferentials and the extremal principle for sets in terms of Fréchet normal cones are pointed out. The results are used for the characterizations of Asplund spaces.

The last section is devoted to extending the traditional (prime-space) extremality notions. A weaker definition of extremality (so-called extended extremality) is introduced for which known (dual-space) necessary conditions in terms of Fréchet subdifferentials become sufficient.

After the definitions of Fréchet subdifferentials (and superdifferentials) of real-valued functions, Fréchet normal cones of sets and Fréchet coderivatives of set-valued mappings, the most important properties of these notions are presented in the first section.

In the second section, relations between the well-known variational principles of Ekeland, Borwein/Preiss and Mordukhovich/Wang, the (weak and strong) fuzzy sum rule for lower semicontinuous functions in terms of Fréchet subdifferentials and the extremal principle for sets in terms of Fréchet normal cones are pointed out. The results are used for the characterizations of Asplund spaces.

The last section is devoted to extending the traditional (prime-space) extremality notions. A weaker definition of extremality (so-called extended extremality) is introduced for which known (dual-space) necessary conditions in terms of Fréchet subdifferentials become sufficient.

Reviewer: Jörg Thierfelder (Ilmenau)

##### MSC:

49J52 | Nonsmooth analysis |

49J50 | Fréchet and Gateaux differentiability in optimization |

49J53 | Set-valued and variational analysis |

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

46G05 | Derivatives of functions in infinite-dimensional spaces |

58C20 | Differentiation theory (Gateaux, Fréchet, etc.) on manifolds |