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**KKM theory and applications in nonlinear analysis.**
*(English)*
Zbl 0936.47034

Pure and Applied Mathematics, Marcel Dekker. 218. New York, NY: Marcel Dekker. xii, 621 p. (1999).

A very simple and elegant intersection principle presented in 1929 by three Polish mathematicians B. Knaster, K. Kuratowski and S. Mazurkiewicz [Fundam. Math. 14, 132-137 (1929; JFM 55.0972.01)] has generated interesting and productive directions in nonlinear analysis, topology, optimization theory and other fields of the mathematics of the XX century. The book under review is the first attempt to gather together all these directions and to give a complete and “closed into itself” exposition of what is called the KKM theory.

After the introduction where an outline of the theory and some preliminaries are given, the author comes to the chapter “The KKM Theory and Related Topics” which is the fundamental and the most important part of the book. Here the author establishes general theory giving some versions of the KKM principle in topological and topological vector spaces. Then the relations with the minimax inequality of Ky Fan [“A minimax inequality and its applications”, Inequalities III, 103-113 (1972; Zbl 0302.49019)] are described. Further, several fixed point and coincidence theorems for multimaps are derived, in particular, different versions of well-known Browder-Ky Fan and Ky Fan-Glicksberg fixed point theorems are given. Among other topics considered in this chapter are: continuous dependence of sets of solutions of Ky Fan minimax inequalities, Ky Fan’s best approximation and section theorems, matching theorems, KKM and fixed point theory in spaces with convex structures (H-spaces, hyperconvex metric spaces, abstract convex spaces).

In the next chapter the author studies the topological intersection property which may be formulated as the existence of a constant selection for a given multimap and applications to topological minimax inequalities. In the fourth chapter the author applies the KKM theory to the existence of maximal elements and equilibria for abstract economies (generalized games) under various hypotheses. In the following two chapters the methods developed in the previous ones are applied to the systematic study of variational and quasi-variational inequalities in topological vector spaces. Paying great attention to non-compact situations the author obtains some existence and stability results. As applications, some fixed point, minimization and general complementarity problems are considered. The next two chapters are devoted to applications to multi-objective optimization and existence and generic stability of Nash equilibria in multi-objective and non-cooperative \(n\)-person games. In the last, \(9^{\text{th}}\) chapter the author first gives generalizations of the well-known Debreu-Gale-Nikaido theorem which is crucial in proving the existence of a market equilibrium of an economy. Further the existence, uniqueness and algorithms for endpoints for multivalued dynamical systems are considered. These results are then applied to derive the existence and stability of Pareto optima for maps taking values in ordered Banach spaces. The list of references contains 624 items and is quite comprehensive. The book may be recommended as a good introduction into the sphere of contemporary topological methods of nonlinear analysis and their applications.

After the introduction where an outline of the theory and some preliminaries are given, the author comes to the chapter “The KKM Theory and Related Topics” which is the fundamental and the most important part of the book. Here the author establishes general theory giving some versions of the KKM principle in topological and topological vector spaces. Then the relations with the minimax inequality of Ky Fan [“A minimax inequality and its applications”, Inequalities III, 103-113 (1972; Zbl 0302.49019)] are described. Further, several fixed point and coincidence theorems for multimaps are derived, in particular, different versions of well-known Browder-Ky Fan and Ky Fan-Glicksberg fixed point theorems are given. Among other topics considered in this chapter are: continuous dependence of sets of solutions of Ky Fan minimax inequalities, Ky Fan’s best approximation and section theorems, matching theorems, KKM and fixed point theory in spaces with convex structures (H-spaces, hyperconvex metric spaces, abstract convex spaces).

In the next chapter the author studies the topological intersection property which may be formulated as the existence of a constant selection for a given multimap and applications to topological minimax inequalities. In the fourth chapter the author applies the KKM theory to the existence of maximal elements and equilibria for abstract economies (generalized games) under various hypotheses. In the following two chapters the methods developed in the previous ones are applied to the systematic study of variational and quasi-variational inequalities in topological vector spaces. Paying great attention to non-compact situations the author obtains some existence and stability results. As applications, some fixed point, minimization and general complementarity problems are considered. The next two chapters are devoted to applications to multi-objective optimization and existence and generic stability of Nash equilibria in multi-objective and non-cooperative \(n\)-person games. In the last, \(9^{\text{th}}\) chapter the author first gives generalizations of the well-known Debreu-Gale-Nikaido theorem which is crucial in proving the existence of a market equilibrium of an economy. Further the existence, uniqueness and algorithms for endpoints for multivalued dynamical systems are considered. These results are then applied to derive the existence and stability of Pareto optima for maps taking values in ordered Banach spaces. The list of references contains 624 items and is quite comprehensive. The book may be recommended as a good introduction into the sphere of contemporary topological methods of nonlinear analysis and their applications.

Reviewer: Valerii V.Obukhovskij (Voronezh)

### MSC:

47H10 | Fixed-point theorems |

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

47H04 | Set-valued operators |

47N10 | Applications of operator theory in optimization, convex analysis, mathematical programming, economics |

49J35 | Existence of solutions for minimax problems |

49J40 | Variational inequalities |

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

91B50 | General equilibrium theory |

91A44 | Games involving topology, set theory, or logic |