##
**Fixed point theory.**
*(English)*
Zbl 1025.47002

Springer Monographs in Mathematics. New York, NY: Springer. xv, 690 p. (2003).

This book cannot be considered as a new edition of the authors’ book [“Fixed Point Theory. Vol. I” (Monografie Matematyczne 61, PWN, Warszawa) (1982; Zbl 0483.47038)] although it evolved from it. Really, this book is an up-to-date and carefully worked out unified account of (not only) classical results in fixed point theory that lie on the border-line of topology and nonlinear functional analysis. The presentation is self-contained and is accessible to a broad spectrum of readers. The main text is complemented by numerous exercises, detailed comments, and a comprehensive bibliography. On the book cover, one can read three endorsements of well-known specialists in the field and I completely share these high estimates. Felix Browder writes: “Granas-Dugundji’s book is an encyclopedic survey of the classical fixed point theory of continuous mappings (the work of Poincaré, Brouwer, Lefschetz-Hopf, Leray-Schauder) and all its various modern extensions. This is certainly the most learned book ever likely to be published on this subject.” Haim Brézis echoes: “The theory of fixed points is one of the most powerful tools of modern mathematics. Not only is it used on a daily basis in pure and applied mathematics, but it also serves as a bridge between analysis and topology, and provides a very fruitful area of interaction between the two. This book contains a clear, detailed and well-organized presentation of major results, together with an entertaining set of historical notes and an extensive bibliography describing further development and applications.” And Isaac Namioka: “In this monograph, no effort has been spared, even to the smallest detail, be it mathematical, historical or bibliographical. In particular, the necessary background materials are generously provided for non-specialists. In fact, the book could even serve as an introduction to algebraic topology, among others. It is certain that the book will be a standard work on fixed point theory for many years to come.”

The book consists of 21 sections divided into six parts, Appendix, Bibliography, List of Standard Symbols and Indexes. Each section is complemented by two subsections: “Miscellaneous Results and Examples” and “Notes and Comments”. These are wonderful “adornments”: here one can find further applications and extensions of the theory, references, and some other information about the results in the main text of the section. These parts contain very important and rich, and often, unexpected, information on the subject, perhaps more important than the main parts. Section 0, “Introduction”, gives a brief general look at the subject of the book and discusses some simple notions and techniques of fixed point theory. Subsections in this section are: Fixed Point Spaces, Forming New Fixed Point Spaces from Old, Topological Transversality, and Factorization Technique.

Part I, “Elementary Fixed Point Theorems”, consists of Section 1-4. Section 1, “Results Based on Completeness”, deals with the Banach contraction principle, the corresponding domain invariance theorem, the continuation method and the nonlinear alternative for contractive mappings as well as some modifications and extensions of the Banach theorem. Section 2, “Order-Theoretic Results”, is devoted to the Knaster-Tarski theorem, Bishop-Phelps and Caristi theorems, fixed points for set-valued contractive maps, and some geometrical applications of these results, as well as applications to the theory of critical points. Section 3, “Results Based on Convexity”, is concerned with KKM-maps and the KKM-principle, the Neumann theorem about systems of inequalities, the Markov-Kakutani theorem, fixed points for families of mappings, and others. Section 4, “Further Results and Applications”, presents the theory of nonexpansive mappings, some applications of the Banach principle and the elementary domain invariance theorem to differential and integral equations.

Part II, “Theorem of Borsuk and Topological Transversality” consists of Sections 5-7. Section 5, “Theorems of Brouwer and Borsuk”, presents classical results of these authors together with results tightly connected to them, such as the Lyusternik-Schnirelmann and Borsuk-Ulam theorems, the topological KKM-principle, and so on. Section 6, “Fixed Points for Compact Maps in Normed Linear Spaces”, deals with Schauder projections, infinite-dimensional analogues of the Brouwer and Borsuk theorems, the Leray-Schauder principle, the hof-Kellogg theorem, the theory of compact vector fields, essential and non-essential mappings, and so on. Section 6, “Further Results and Applications”, presents applications of the results described in Sections 5-6 to differential equations, Galerkin approximation theory, and also the invariant subspace problem, the theory of absolute retractions, fixed points for set-valued Kakutani maps, the Ryll-Nardzewski theorem, and so on.

Part III, “Homology and Fixed Points”, consists of Sections 8-9. Section 8, “Simplicial Homology”, presents an elegant exposition of this theory. Section 9, “The Lefschetz-Hopf Theorem and Brouwer Degree”, deals with the Lefschetz-Hopf fixed point theorem, and the classical Brouwer degree theory of mappings \({\mathbf S}^n\to F{\mathbf S}^n\), the Borsuk-Hirsch theorem, and numerous related results and constructions.

Part IV, “Leray-Schauder Degree and Fixed Point Index”, contains Sections 10-13. Section 10 is devoted to the systematical theory of topological degree in \(\mathbb{R}^n\). Here the authors present a sufficiently simple exposition of this theory (their construction is based on PL-maps of polyhedra), a variant of the axiomatic approach to this theory, and new modifications of basic results, such as the Borsuk antipodal theorem and others. Sections 11-12, “Absolute Neighborhood Retracts” and “Fixed Point Index in ANRs”, deal with this theory and generalizations of Leray-Schauder’s principle and Leray-Schauder’s index for ANRs. Section 13, “Further Results and Applications”, is devoted to bifurcation results for ANRs, Leray-Schauder degree theory extensions of the Borsuk and Borsuk-Ulam theorems, generalizations to locally convex spaces and applications to nonlinear partial differential equations.

Part V, “The Lefschetz-Hopf Theory”, consists of Sections 14-17: “Singular Homology”, “Lefschetz Theory for Maps of ANRs”, “The Hopf Index Theorem”, and “Further Results and Applications”. Here one can find analysis of relations between different homologies and topological degree, the Künneth formula for homologies of products of complexes, theorems about generalized Lefschetz number, the theory of asymptotic fixed point theorems for ANRs, the theory of the periodic index, the Hopf index theorem in arbitrary ANRs, the theory of fixed points in linear topological spaces, and other related themes.

Part VI, “Selected Topics”, consists of Section 18, “Finite-Codimensional Čech Cohomology”, Section 19, “Vietoris Fractions and Coincidence Theory”, and Section 20, “Further Results and Supplements”. Here one can find the description of Čech cohomology groups \(H^{\infty-n}(X)\) and the functor \(H^{\infty-n}(X):({\mathfrak L},\backsim)\to{\mathbf A}{\mathbf b}\), coincidence theorems, fixed points for compact and acyclic set-valued maps, the degree for equivariant maps, infinite-dimensional \(E^+\)-cohomologies, the Lefschetz theorem for \(\mathbb{N}\mathbb{B}\)-maps of compacta, and other things.

A small Appendix, “Preliminaries”, is a summary of basic notions and facts about sets, topological spaces, linear topological spaces, algebraic structures, categories and functors. The rich bibliography comprises almost 50 pages.

The book is useful to all mathematicians who deal with fixed points. It will become an obligatory volume for every serious mathematical library.

The book consists of 21 sections divided into six parts, Appendix, Bibliography, List of Standard Symbols and Indexes. Each section is complemented by two subsections: “Miscellaneous Results and Examples” and “Notes and Comments”. These are wonderful “adornments”: here one can find further applications and extensions of the theory, references, and some other information about the results in the main text of the section. These parts contain very important and rich, and often, unexpected, information on the subject, perhaps more important than the main parts. Section 0, “Introduction”, gives a brief general look at the subject of the book and discusses some simple notions and techniques of fixed point theory. Subsections in this section are: Fixed Point Spaces, Forming New Fixed Point Spaces from Old, Topological Transversality, and Factorization Technique.

Part I, “Elementary Fixed Point Theorems”, consists of Section 1-4. Section 1, “Results Based on Completeness”, deals with the Banach contraction principle, the corresponding domain invariance theorem, the continuation method and the nonlinear alternative for contractive mappings as well as some modifications and extensions of the Banach theorem. Section 2, “Order-Theoretic Results”, is devoted to the Knaster-Tarski theorem, Bishop-Phelps and Caristi theorems, fixed points for set-valued contractive maps, and some geometrical applications of these results, as well as applications to the theory of critical points. Section 3, “Results Based on Convexity”, is concerned with KKM-maps and the KKM-principle, the Neumann theorem about systems of inequalities, the Markov-Kakutani theorem, fixed points for families of mappings, and others. Section 4, “Further Results and Applications”, presents the theory of nonexpansive mappings, some applications of the Banach principle and the elementary domain invariance theorem to differential and integral equations.

Part II, “Theorem of Borsuk and Topological Transversality” consists of Sections 5-7. Section 5, “Theorems of Brouwer and Borsuk”, presents classical results of these authors together with results tightly connected to them, such as the Lyusternik-Schnirelmann and Borsuk-Ulam theorems, the topological KKM-principle, and so on. Section 6, “Fixed Points for Compact Maps in Normed Linear Spaces”, deals with Schauder projections, infinite-dimensional analogues of the Brouwer and Borsuk theorems, the Leray-Schauder principle, the hof-Kellogg theorem, the theory of compact vector fields, essential and non-essential mappings, and so on. Section 6, “Further Results and Applications”, presents applications of the results described in Sections 5-6 to differential equations, Galerkin approximation theory, and also the invariant subspace problem, the theory of absolute retractions, fixed points for set-valued Kakutani maps, the Ryll-Nardzewski theorem, and so on.

Part III, “Homology and Fixed Points”, consists of Sections 8-9. Section 8, “Simplicial Homology”, presents an elegant exposition of this theory. Section 9, “The Lefschetz-Hopf Theorem and Brouwer Degree”, deals with the Lefschetz-Hopf fixed point theorem, and the classical Brouwer degree theory of mappings \({\mathbf S}^n\to F{\mathbf S}^n\), the Borsuk-Hirsch theorem, and numerous related results and constructions.

Part IV, “Leray-Schauder Degree and Fixed Point Index”, contains Sections 10-13. Section 10 is devoted to the systematical theory of topological degree in \(\mathbb{R}^n\). Here the authors present a sufficiently simple exposition of this theory (their construction is based on PL-maps of polyhedra), a variant of the axiomatic approach to this theory, and new modifications of basic results, such as the Borsuk antipodal theorem and others. Sections 11-12, “Absolute Neighborhood Retracts” and “Fixed Point Index in ANRs”, deal with this theory and generalizations of Leray-Schauder’s principle and Leray-Schauder’s index for ANRs. Section 13, “Further Results and Applications”, is devoted to bifurcation results for ANRs, Leray-Schauder degree theory extensions of the Borsuk and Borsuk-Ulam theorems, generalizations to locally convex spaces and applications to nonlinear partial differential equations.

Part V, “The Lefschetz-Hopf Theory”, consists of Sections 14-17: “Singular Homology”, “Lefschetz Theory for Maps of ANRs”, “The Hopf Index Theorem”, and “Further Results and Applications”. Here one can find analysis of relations between different homologies and topological degree, the Künneth formula for homologies of products of complexes, theorems about generalized Lefschetz number, the theory of asymptotic fixed point theorems for ANRs, the theory of the periodic index, the Hopf index theorem in arbitrary ANRs, the theory of fixed points in linear topological spaces, and other related themes.

Part VI, “Selected Topics”, consists of Section 18, “Finite-Codimensional Čech Cohomology”, Section 19, “Vietoris Fractions and Coincidence Theory”, and Section 20, “Further Results and Supplements”. Here one can find the description of Čech cohomology groups \(H^{\infty-n}(X)\) and the functor \(H^{\infty-n}(X):({\mathfrak L},\backsim)\to{\mathbf A}{\mathbf b}\), coincidence theorems, fixed points for compact and acyclic set-valued maps, the degree for equivariant maps, infinite-dimensional \(E^+\)-cohomologies, the Lefschetz theorem for \(\mathbb{N}\mathbb{B}\)-maps of compacta, and other things.

A small Appendix, “Preliminaries”, is a summary of basic notions and facts about sets, topological spaces, linear topological spaces, algebraic structures, categories and functors. The rich bibliography comprises almost 50 pages.

The book is useful to all mathematicians who deal with fixed points. It will become an obligatory volume for every serious mathematical library.

Reviewer: Peter Zabreiko (Minsk)

### MSC:

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

47H10 | Fixed-point theorems |

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

54-02 | Research exposition (monographs, survey articles) pertaining to general topology |

37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |