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**Fixed point theory and best approximation: The KKM-map principle.**
*(English)*
Zbl 0901.47039

Mathematics and its Applications (Dordrecht). 424. Dordrecht: Kluwer Academic Publishers. x, 220 p. (1997).

This small book deals with some aspects of nonlinear functional analysis concerning fixed points and their “natural” neighborhood. The book consists of 5 chapters.

Chapter 1: “Introductory concepts and fixed point theorems” is a brief summary on classical fixed point theorems (Brouwer, Banach-Caccioppoli, Darbo-Sadovskiĭ, Browder-Göhde-Kirk, Ky Fan, and others), successive approximations and other iteration processes for mappings in Banach and Hilbert spaces; applications to integral equations are also considered.

Chapter 2: “Ky Fan’s best approximation theorem” presents an exhaustive amount of generalizations and modifications of Ky Fan’s best approximation theorem including Prolla’s result, and variants for multifunctions. Applications to the fixed point theory for (essentially, non-self) maps are discussed in details.

Chapter 3: “Principle and applications of KKM-maps” is devoted to the analysis of the classical KKM (Knaster-Kuratowski-Mazurkiewicz) map principle and its infinite-dimensional variant by Ky Fan, and also others’ up-to-date variants and modifications. Here some, equivalent to the KKM map, principle results are presented; deep relations between the KKM map principle and the fixed point theory, theory of variational inequalities are described.

Chapter 4: “Partitions of unity and applications” deals with the partition of unity arguments in fixed point theory. The base of the chapter is the classical Browder theorem about fixed point existence for multivalued mappings with open pre-images of points; the main part of the chapter is devoted to maps defined on compact sets, but the case of paracompact sets is also considered. The applications to coincidence theorems, variational and minimax inequalities are considered.

Chapter 5: “Applications of fixed points to approximation theory” is a collection of applications of results presented in the previous chapters to approximation theory, variational inequalities, and complementarity problems; in particular, equivalent theorems in these fields are presented. The bibliography consists of more than 300 references.

Thus, the book represents sufficiently rich information in an important area of current and growing interest. The book’s format allows the authors to give simple and unified proofs of all results presented here. Undoubtedly, the book will be useful to all specialists in nonlinear functional analysis, operator theory, approximation theory, the game theory and mathematical economics.

Chapter 1: “Introductory concepts and fixed point theorems” is a brief summary on classical fixed point theorems (Brouwer, Banach-Caccioppoli, Darbo-Sadovskiĭ, Browder-Göhde-Kirk, Ky Fan, and others), successive approximations and other iteration processes for mappings in Banach and Hilbert spaces; applications to integral equations are also considered.

Chapter 2: “Ky Fan’s best approximation theorem” presents an exhaustive amount of generalizations and modifications of Ky Fan’s best approximation theorem including Prolla’s result, and variants for multifunctions. Applications to the fixed point theory for (essentially, non-self) maps are discussed in details.

Chapter 3: “Principle and applications of KKM-maps” is devoted to the analysis of the classical KKM (Knaster-Kuratowski-Mazurkiewicz) map principle and its infinite-dimensional variant by Ky Fan, and also others’ up-to-date variants and modifications. Here some, equivalent to the KKM map, principle results are presented; deep relations between the KKM map principle and the fixed point theory, theory of variational inequalities are described.

Chapter 4: “Partitions of unity and applications” deals with the partition of unity arguments in fixed point theory. The base of the chapter is the classical Browder theorem about fixed point existence for multivalued mappings with open pre-images of points; the main part of the chapter is devoted to maps defined on compact sets, but the case of paracompact sets is also considered. The applications to coincidence theorems, variational and minimax inequalities are considered.

Chapter 5: “Applications of fixed points to approximation theory” is a collection of applications of results presented in the previous chapters to approximation theory, variational inequalities, and complementarity problems; in particular, equivalent theorems in these fields are presented. The bibliography consists of more than 300 references.

Thus, the book represents sufficiently rich information in an important area of current and growing interest. The book’s format allows the authors to give simple and unified proofs of all results presented here. Undoubtedly, the book will be useful to all specialists in nonlinear functional analysis, operator theory, approximation theory, the game theory and mathematical economics.

Reviewer: P.Zabreiko (Minsk)

### MSC:

47H10 | Fixed-point theorems |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

41A50 | Best approximation, Chebyshev systems |

41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |

47J20 | Variational and other types of inequalities involving nonlinear operators (general) |