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**Topological fixed point theory of multivalued mappings.
2nd ed.**
*(English)*
Zbl 1107.55001

Topological Fixed Point Theory and Its Applications 4. Dordrecht: Springer (ISBN 1-4020-4665-0/hbk; 1-4020-4666-9/ebook). x, 538 p. (2006).

This is the second edition of the book Topological Fixed Point Theory of Multivalued Mappings. The review of the first edition was done by R. F. Brown, see [Topological fixed point theory of multivalued mappings. Mathematics and its Applications (Dordrecht). 495. Dordrecht: Kluwer Academic Publishers. (1999; Zbl 0937.55001)], and most of it still applies. This new editon contains 7 chapters, and the seventh is new. It is called Recent results and it follows the same style as the previous ones.

Although the first 6 chapters were already in the first edition, several changes and general improvements were made, and the new chapter is a nice contribution. It provides a fair complete update of the development of the subject in the past 6 years. It consists of 10 subsections which are: 76: Periodic invariants; the Euler-Poincaré characteristic; 77: The coincidence Nielsen number; 78: Fixed point of symmetric product mappings; 79: The category of weighted maps; 80: Darbo homology functor and its applications to fixed point problems; 81: More about spheric mappings; 82: A coincidence index involving Fredholm operators; 83: Fixed points of monotone-type multivalued operators; 84: Multivalued Poincaré operators; 85:Multivalued fractals. Some of the subsections generalize classical topics, already known for functions, to multivalued functions. Others are devoted to further results on multivalued functions. This last chapter is less self-contained than the others, which is not surprising.

In the new chapter the reader should watch out for some misprints, especially in subsection 82: A coincidence index involving Fredholm operators. One of the problems pointed out in the review of the first edition still remains, namely, the lack of an adequate index.

Although the first 6 chapters were already in the first edition, several changes and general improvements were made, and the new chapter is a nice contribution. It provides a fair complete update of the development of the subject in the past 6 years. It consists of 10 subsections which are: 76: Periodic invariants; the Euler-Poincaré characteristic; 77: The coincidence Nielsen number; 78: Fixed point of symmetric product mappings; 79: The category of weighted maps; 80: Darbo homology functor and its applications to fixed point problems; 81: More about spheric mappings; 82: A coincidence index involving Fredholm operators; 83: Fixed points of monotone-type multivalued operators; 84: Multivalued Poincaré operators; 85:Multivalued fractals. Some of the subsections generalize classical topics, already known for functions, to multivalued functions. Others are devoted to further results on multivalued functions. This last chapter is less self-contained than the others, which is not surprising.

In the new chapter the reader should watch out for some misprints, especially in subsection 82: A coincidence index involving Fredholm operators. One of the problems pointed out in the review of the first edition still remains, namely, the lack of an adequate index.

Reviewer: Daciberg Gonçalves (São Paulo)

### MSC:

55M20 | Fixed points and coincidences in algebraic topology |

54H25 | Fixed-point and coincidence theorems (topological aspects) |

47H10 | Fixed-point theorems |