Progress in Nonlinear Differential Equations and Their Applications

Volume 95

Series Editor Haïm Brezis Université Pierre et Marie Curie, Paris, France Technion – Israel Institute of Technology, Haifa, Israel Rutgers University, New Brunswick, NJ, USA

Editorial Board Members Antonio Ambrosetti, Scuola Internationale Superiore di Studi Avanzati, Trieste, Italy Luigi Ambrosio, Scuola Normale Superiore, Pisa, Italy Henri Berestycki, Ecoles des Hautes en Science Sociales, Paris France Luis Caffarelli, The University of Texas, Austin, TX, USA Sun-Yung Chang, Princeton University, NJ, USA Jean-Michel Coron, University Pierre et Marie Curie, Paris, France Manuel del Pino, University of Chile, Santiago, Chile Lawrence Evans, University of California, Berkeley, CA, USA Alessio Figalli, The University of Texas, Austin, TX, USA Rupert Frank, CalTech, CA, USA Nicola Fusco, Univeristy of Naples Federico II, Naples, Italy Sergiu Klainerman, Princeton University, NJ, USA Robert Kohn, New York University, NY, USA Pierre Louis Lions, Collège de France, Paris, France Andrea Malchiodi, Scuola Normale Superiore, Pisa, Italy Jean Mawhin, Université Catholique de Louvain, Louvain-la-Neuve, Belgium Frank Merle, U de Cergy-Ponoise and IHES, Paris France Giuseppe Mingione, Universita degli Studi di Parma, Parma, Italy Louis Nirenberg, New York University, NY, USA Felix Otto, Max Planck Institute, Leipzig, Germany Paul Rabinowitz, University of Wisconsin, Madison, WI, USA John Toland, Isaac Newton Institute, Cambridge, UK Michael Vogelius, Rutgers, NJ, USA

More information about this series at http://www.springer.com/series/4889 George Dinca • Jean Mawhin

Brouwer Degree The Core of Nonlinear Analysis George Dinca Jean Mawhin Faculty of Mathematics & Computer IRMP Science Université Catholique de Louvain University of Bucharest Louvain-la-Neuve, Belgium Bucharest, Romania

ISSN 1421-1750 ISSN 2374-0280 (electronic) Progress in Nonlinear Differential Equations and Their Applications ISBN 978-3-030-63229-8 ISBN 978-3-030-63230-4 (eBook) https://doi.org/10.1007/978-3-030-63230-4

Mathematics Subject Classification: 58E95, 55M25, 47H05, 47H11, 47J15, 39A23, 35J60, 34B15, 34C25

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface

Solving equations and widening the concept of solutions has been a driving force of the progress and evolution of mathematics. After the study of linear scalar algebraic equations led to the concept of rational numbers, the algebraic equations of the second degree forced mathematicians to introduce the irrational numbers. The complex numbers, introduced to give a meaning to a formula for the solutions of algebraic equations of the third order, have shown, through the existing formulas for solutions, that the algebraic equations of order smaller or equal to four with complex coefficients always have exactly four complex solutions (counted with their multiplicity). Before the vain quest of formulas for solutions in the case of degree five or more (justified later by Abel and Galois), it was conjectured that an algebraic equation of any order n with complex coefficients should always have at least one complex solution, and hence n complex solutions (counting multiplicity), a result first proved by Gauss in 1799 after the less fruitful efforts of Laplace, Lagrange and others. In between, the ‘geometrically evident’ fact that an algebraic equation of odd degree with real coefficients always had a real solution, already used by Cardano and Stevin to approximate the roots of some equations of the third degree, received several unsatisfactory proofs by Lagrange and others until Bolzano in 1817 and Cauchy in 1821 showed rigorously that this ‘evident’ fact was a consequence of a deep feature (the Bolzano intermediate value theorem) of the wider class of continuous real functions, a concept that they had defined clearly for the first time. Great efforts had also been made, starting with Descartes, to find conditions on the coefficients of a real algebraic equation to evaluate the number of its real solutions on a given interval, culminating, after the works of Budan and Fourier, to the elegant final solution by Sturm in 1829. In the complex case, the same problem was solved by Cauchy between 1831 and 1837 through the introduction of a suitable contour integral in the complex case in the frame of holomorphic functions, other proofs being given by Sturm and Liouville. Through a more algebraic approach, Hermite obtained in 1853 similar results for systems of algebraic equations. The way was paved for Kronecker to introduce in 1869 his famous index and integral, providing a bold generalization and synthesis of the work of Cauchy,

v vi Preface

Sturm, Liouville and Hermite mentioned above. In this epoch-making but not widely read memoir, Kronecker showed among many other things that, in order to obtain a count of the number of solutions of a system of equations insensitive to small perturbations, it was necessary to replace the naive count by an ‘algebraic’ one, attributing in some way a sign to each solution in counting it. For example, the = R = 2 − equation fε(x) 0, in with fε(x) x √ ε, has no solution for ε<0, one solution for√ ε = 0 and two√ solutions ± ε for ε>0. In this elementary example, f (− ε) < 0 and f ( ε) > 0, so that the difference between the number of solutions with positive derivative and the number of solutions with negative derivative is zero. Hence, this algebraic count of the number of zeros of fε remains thesameforε<0 and ε>0. One can even say that it is zero for ε = 0, as f (0) = 0 for the solution 0. In the case of a system f(x) = 0ofn equations of class C1 with n unknowns, Kronecker replaced the sign of the derivative by that of the Jacobian and showed that the corresponding algebraic count of solutions in a given bounded domain D of Rn with smooth boundary could be expressed by an integral on ∂D of some expression depending upon f and its partial derivatives. For the two-dimensional system associated to the complex equation f(z)= 0inC,the Kronecker integral reduces to the Cauchy integral mentioned above. Interestingly, Kronecker borrowed most of its technical tools to potential theory in constructing his theory. The memoir of Kronecker did not have many immediate followers, but had illustrious ones. In 1883, Poincaré used it to obtain an n-dimensional generalization of the Bolzano intermediate value theorem and applied it in various questions of celestial mechanics and automorphic functions. Between 1891 and 1894, Picard used it to the study of the number of solutions of systems of two or three equations, including his results in his famous Traité d’analyse. From 1888 till 1898, Dyck developed interesting geometrical and topological consequences of the Kronecker index in a series of papers. In 1904, Bohl applied Kronecker’s integral to some questions of mechanics and topology. An important step forward in the topological development of the Kronecker index was made in 1910 by Hadamard, in extending Kronecker’s definition from smooth to continuous mappings and including in his paper an unpublished fixed point theorem of Brouwer, known to Hadamard before its publication through a letter of the Dutch mathematician. The contribution of Hadamard makes the link between the Kronecker integral and a closely related concept introduced implicitly in 1911 by Brouwer in his proof of the invariance of dimension and made explicit in 1912 in a paper ending by the fixed point theorem sent to Hadamard. This concept is the degree of a continuous mapping between two oriented n-dimensional boundaryless topological manifolds. Brouwer’s construction, completely different from Kronecker-Hadamard’s one, was based upon approximations of the manifolds by polyhedra and of the mappings by simplicial ones. Other important topological consequences of his degree were given by Brouwer between 1912 and 1921. From those considerations, a Brouwer degree for a continuous mapping f on the closure of an open bounded set D ⊂ Rn into Rn such that 0 ∈ f(∂D) was constructed in 1935 by Alexandroff and Hopf in their book Topologie, generalizing the algebraic count of the number of solutions of Preface vii equation f(x)= 0inD to the situations where the solutions are not isolated and/or the Jacobians are not defined. This degree is insensitive to small perturbations of f . The richness of the concept of Brouwer degree is already revealed by its history and the quality of its protagonists. It is linked to fundamental problems coming from many different parts of mathematics, like algebra, analysis, topology and geometry and is associated to the names of the greatest mathematicians. Its richness will be illustrated in its subsequent history by the number and variety of its definitions, which can come from combinatorics, analysis, algebra, differential topology, real algebraic topology or, of course, can be homological or cohomological like in most treatises of algebraic topology. In this monograph, we present the analytical approach proposed by Heinz in 1959 in the somewhat simpler, shorter and better motivated way developed by Mawhin in 2004, linking it to the older concept of Kronecker index and using the language of differential forms, at the level of an advanced calculus book. The requested versions of the Stokes formula integrate the exterior differential of a C2 (n−1)-form in Rn over an open bounded subset D with smooth oriented boundary ∂D (to relate the definitions of Kronecker and Heinz) and integrate the exterior differential of a C2 differential (n−1)-form with compact support in an arbitrary open subset D (to justify the definition of Heinz and prove the homotopy invariance of the degree). That property, which is the globalization of the invariance with respect to small perturbations of f mentioned above, is inspired by an unpublished result of Tartar of 1977, quoted in the Introduction without being adopted in the recent interesting monograph on degree theory of Fonseca and Gangbo. When written in the language of exterior calculus, this result immediately follows from an elementary straightforward computation. The richness of the concept of Brouwer degree is also illustrated by the number and variety of its applications. Although we do not refrain to give the simplest topological consequences of the concept, we concentrate the displayed applications, following our expertise, to the use of the degree in discussing the existence, multiplicity, bifurcation and stability of solutions of some nonlinear difference, differential and partial differential equations, to the obtention of various fixed point theorems and some applications to game theory and economics, and to the use of degree in some questions of mechanics and elasticity. The reader interested in applications to differential geometry and topology can consult with profit the excellent monograph of Outerelo and Ruiz. We describe now briefly the contents and the structure of this book. Each chapter, divided in sections and subsections, ends with a subsection containing historical information and another one containing bibliographical suggestions. Chapter 1 first introduces the Kronecker index, its properties and some applications to the Gauss linking number between two curves in R3 and to some work of Béthuel, Brezis and Helein on the Ginzburgh-Landau theory of liquid crystals. Then the Brouwer degree is defined and developed, first for mappings of Rn into itself and then for mappings between vector spaces of the same finite dimension, including the uniqueness of an axiomatic characterization and its localization named the Brouwer index of a zero. Some useful reduction formulas, relating the degree of some mappings in some dimension to some degree in a lower one, are proved and applied to the computation viii Preface of the Brouwer index. Chapter 1 ends with the important concept of retraction due to Borsuk, its application to proving the Brouwer fixed point theorem, and to constructing a fixed point index for mappings in a retract of Rn, with applications to fixed point theorems in cones of Rn. The role of the dimension in topology is illustrated by the hairy ball theorem on spheres, and the importance of various linking results in some fundamental theorems of critical point theory in Banach spaces is emphasized. Chapter 2 starts with an extension of the homotopy invariance property of the Brouwer degree to a one-parameter family of mappings defined on a parameter- dependent set and, to its consequence, the important continuation theorem obtained in 1934 by Leray and Schauder in a more general frame. The application of this result to various types of mappings provides several existence theorems including the Rothe fixed point theorem, various surjectivity theorems, the Poincaré-Miranda, the Vishik and the Hadamard theorems for the existence of zeros, and some fixed point theorems for expansive-compressive-type and Poincaré-Miranda-type mappings. Those last fixed point theorems have applications in the theory of topological chaos. The concept of characteristic values and their multiplicity for couples of linear mappings in Rn is then introduced, together with the computation of their Brouwer index. Application is made to the bifurcation of solutions of mappings depending upon a real parameter, for which both local results of the Krasnosel’skii type and global results of the Rabinowitz type are obtained, including bifurcation from infinity. The chapter ends with the Kakutani fixed point theorem, natural extension of the Brouwer fixed point theorem to set-valued mappings, and its applications to fundamental results of game theory and mathematical economy, like the von Neumann minimax theorem and the existence of Nash equilibria for non-cooperative games. Although the Brouwer degree is a finite-dimensional tool, it can be combined with basic techniques of linear functional analysis to prove a number of fixed point and existence theorems in infinite-dimensional spaces. This is the object of Chap. 3, starting with the Ky Fan fixed point theorem, extending the Kakutani theorem to compact convex subsets of a locally convex topological vector space, and its single-valued important special cases in normed and locally convex vector spaces associated to the names of the Schauder and Tychonov. The variational inequalities are extensions to non-variational mappings of the characterization of a minimum of a functional on a closed convex set through some inequalities. Using the techniques of the KKM mappings, the main results, obtained by Hartman and Stampacchia in 1966, are first proved on a compact convex set of a locally convex topological vector space. In the case of monotone mappings from a reflexive Banach space to its dual, the results hold on a closed convex subset. Existence theorems are then obtained for hemivariational inequalities in a reflexive Banach space, a generalization of the variational inequalities useful in various problems of mechanics. Chapter 3 continues with the Browder-Minty surjectivity theorem for coercive monotone applications from a reflexive Banach space into its dual, with applications to some monotone and nonmonotone quasilinear Dirichlet problems. The variational formulation of the stationary Navier-Stokes equations is then Preface ix introduced and conditions for the existence and for the uniqueness of a weak solution are obtained, following Fujita. Chapter 3 ends with an introduction to the extension of the Brouwer degree to compact perturbations of the identity defined on the closure of a bounded open subset of a normed space, introduced by Leray and Schauder in their epoch-making paper of 1934. A very natural field of applications of the Brouwer degree is given by the semilinear difference equations submitted to some boundary conditions. They are considered in Chap. 4, starting with first order difference equations with periodic boundary conditions. Special emphasis is made upon the method of lower and upper solutions and its application to the obtention of multiplicity results of the Ambrosetti-Prodi type for parameter-dependent equations and to discrete models of the Lotka-Volterra type in population dynamics. An Ambrosetti-Prodi-type result, initiated in 1972 by Ambrosetti and Prodi semilinear elliptic Dirichlet problems, is essentially the generalization to this situation of the fact that the scalar equation x2 = s depending upon the parameter s has no solution for s<0, one solution for s = 0 and two solutions for s>0. The chapter continues with the study of semilinear second order difference equations with Dirichlet boundary conditions. After some applications of bifurcation theory from the trivial solution, a combination of bifurcation from infinity and of the Brouwer degree is used to prove the existence of multiple solutions for some nonlinear equations when its linear part crosses an eigenvalue. Sharp existence results are then obtained when the linear part is at resonance and the nonlinear perturbation is bounded from below or from above. The chapter ends with the method of lower and upper solutions and its application to multiplicity results of the Ambrosetti-Prodi type for second order equations. For ordinary differential systems with corresponding Cauchy problem uniquely solvable, the search of periodic solutions of period T can be reduced to the search of fixed points of the Poincaré operator mapping any initial condition x0 to the value for t = T of the solution starting from x0 at t = 0. This methodology, which reduces the infinite-dimensional problem to a finite-dimensional one, is developed and applied in Chap. 5. A little known variant of the approach introduced by Stampacchia in 1947 and avoiding the uniqueness condition for the Cauchy problem is described. An existence theorem proved by Krasnosel’skii and Perov in 1958 is applied to the computation of the Brouwer degree of gradient systems under various hypotheses. The method of guiding functions initiated by Krasnosel’skii and Perov in 1963 is developed, which reduces the existence of a T-periodic to finding a suitable real function whose inner product with the vector field of the differential systems is positive near infinity and satisfies some further conditions. The chapter ends with a version of the method of guiding functions to evolution complementarity systems, where solutions take values in a closed convex cone and satisfy, instead of a differential system, a corresponding variational inequality. Chapter 6 collects a number of results which depend upon the computation of degree of two-dimensional mappings. It starts with developing a method of lower and upper solutions for second order differential equations with periodic boundary conditions initiated by Knobloch in 1963. It is followed by a study of the relation between the stability of the periodic solutions of a second order differential equation x Preface of Duffing type and the Brouwer index of some associated Poincaré operators, obtained by Ortega in 1989. Applications are given to the case of convex and of periodic nonlinearities. Planar differential systems are then considered, and existence conditions proved by Fabry and Fonda in 1998 are described and related to the Fucikˇ spectrum introduced independently in 1976 by Dancer and Fucikˇ for second order piecewise linear equations. A series of results and methods for computing the Brouwer degree or Kronecker index in two dimensions are then given, including the use of Sturm sequences, and applications to planar mappings associated to holomorphic functions. The chapter ends with some results related to the classical Routh-Hurwitz and the Schur-Cohn conditions occurring in the study of the stability of linear autonomous differential and difference systems. Chapter 7 is devoted to the computation of the Brouwer degree for various classes of mappings and to corresponding applications. First we describe results, essentially obtained by Cronin in 1954, for homogeneous polynomial mappings, before considering the orientation-preserving mappings, introduced by Browder in 1970, and containing as special cases the monotone mappings, the holomorphic mappings, with the interesting results of Rabinowitz in 1973, and the quaternionic monomials. The important case of symmetrical mappings is then considered, starting with the odd ones, the results of Borsuk and Ulam, the Lusternik-Schnirelmann covering theorem and the Krasnosel’skii genus, important in the critical point theory for even functionals. We then give Nirenberg’s version and proof of an important reduction formula in the case of equivariant mappings for the S1-symmetry, also widely used in critical point theory. The product formula for the degree of composed mappings, introduced by Leray in 1935, is then stated, proved and applied to the obtention of important topological properties like the Jordan-Brouwer separation theorem, the Brouwer theorem of invariance of domain, and to the computation of the degree of one-to-one mappings. Various surjectivity and homeomorphism conditions, like the Banach-Mazur theorem, are also obtained. Interesting applications to elasticity theory, due to Ciarlet, end the chapter. Even if it was stated and proved without any comment by Brouwer at the very end of his 1912 paper on topological degree, and even if Brouwer never came back to it in the sequel, the Brouwer fixed point theorem has rapidly became the most spectacular and widely applied consequence of the Brouwer degree theory. Its history is far from straightforward, particularly because of the large number of equivalent forms it can take. Its history also especially emphasizes the human aspects of mathematical activity, with its exemplary and less glorious aspects. Hence we thought that it was appropriate to end this volume in Chap. 8 by a short essay on this history, which of course crosses in many points the history of the Brouwer degree. It is not surprising that the sections of this chapter are centered on mathematicians rather than on mathematics. Despite the variety and the number of the applications proposed in those chapters, a fact illustrated by the (selected) biography of more than 600 items, the book is far from covering all aspects of the subject. As told earlier, we only discussed the basic applications to topology, left outside the consequences to differential geometry, and skipped the numerical aspects of degree theory. But above all, we were forced to Preface xi ignore completely the surprising development linked to the construction, started by Brezis and Coron in 1985, of a Brouwer degree for mappings between manifolds of the same finite dimension belonging to some Sobolev space, instead of belonging to space of continuous mapping. This research was motivated by some important questions on nonlinear partial differential equations, geometry and mathematical physics and has led to the astonishing construction of a Brouwer degree for some classes of discontinuous mappings. Its fast and important development would require to add not only a second volume to this book, but also a second life to its authors, which is far from granted. Hence, we must content ourselves with sending the reader to the interesting papers [65–67, 71, 73, 426], their bibliographies and the more recent publications, and to a forthcoming book of Brezis and Mironescu [74]. Summarizing, the Brouwer degree has been for more than a century and still remains an unavoidable tool in nonlinear analysis and in its applications. It remains the core of most of the existence theorems for nonlinear abstract equations and for the construction of the many generalized degree or index theories in more and more general spaces and for more and more classes of mappings. It remains the best tool for proving general and elegant existence, multiplicity and bifurcation theorems for various types of nonlinear equations. In critical point theorem, if one excepts the direct method of the calculus of variations or the special minimax theorems based upon convexity properties, one should not forget that the Brouwer degree is requested for proving the linking properties underlying the most useful theorems of mountain pass or saddle point type. In Lusternik-Schnirelmann-type theories, the properties of the Brouwer degree for symmetrical mappings are important intermediate tools in their construction. This book can be read in different ways, depending on the interests of the reader. Chapter 1 contains the basic information about the notions of Kronecker index, Brouwer degree, fixed point index and their relations. The uniqueness of the Brouwer degree can be skipped in a first reading, as well as the material on linking which is not used in the sequel. The four first sections of Chap. 2 are fundamental for all applications of Brouwer degree to the existence, multiplicity and bifurcation of solutions of nonlinear equations. Section 2.4 can be reserved to those interested in the use of degree to game theory and mathematical economics. Chapter 3 introduces the interested reader to several important parts and prob- lems of nonlinear functional analysis using a topological approach and some of its striking applications to nonlinear partial differential equations. Its reading is not necessary for understanding Chap. 4, an introduction to the theory of nonlinear boundary value for difference equations, and Chap. 5, which illustrates the use of Brouwer degree in the study of the existence of periodic solutions of nonlinear differential systems or arbitrary dimension. Chapter 6 concentrates a number of results in dimension two: periodic solutions of planar systems, computation of the Brouwer degree and questions related to the stability and control of linear systems. Chapter 7 collects general results about the Brouwer degree of various classes of mappings in Rn: homogeneous polyno- mials, orientation-preserving applications, including monotone, holomorphic and 1 quaternionic ones, mappings having the symmetry Z2 or S , including applications xii Preface to Lusternik-Schnirelmann covering and genus, and finally composed mappings, with their topological consequences (Jordan-Brouwer separation theorem and the Brouwer theorem of invariance of domain) and their application to elasticity. It is independent from Chaps. 3–6, so that the reader only interested on the Brouwer degree and its properties can restrict his or her reading to Chaps. 1, 2 and 7. The last chapter, devoted to the history of Brouwer fixed point theorem and its various formulations presented in the preceding chapters, should be of interest to the mathematicians having some historical fiber. It exemplifies how mathematics is, before all, a wonderful human adventure. For the reader’s convenience, the statements of the results already mentioned in previous chapters are repeated. Acknowledgments This book is a fruit of many years of collaboration and friend- ship of the authors. It would not have existed without the constant encouragements of their friend Haïm Brezis. The elaboration of the book has greatly profited from a two weeks stay at the Oberwolfach Mathematische Forschungs Institut, as a Research in Pairs, from February 24 till March 8, 2008. Several stays of George Dinca as a visiting professor at the University of Louvains and several visits of Jean Mawhin at the University of Bucharest have been most useful as well. All those institutions are warmly thanked for their help. Special thanks also due to Dr. Michael Ban, from RWTH Aachen University, for his help in dealing with the many technical problems linked to the publication of the book. We are also indebted to the Publishing House Birkhäuser for its excellent work, and in particular to Mr. Christopher Tominich (Editing department) and Mr. Mario Gabriele (Production Department). Last but not least, the patience and support of our respective wives have been without limits.

Bucharest, Romania George Dinca Louvain-la-Neuve, Belgium Jean Mawhin September 2020 Contents

1 The Kronecker Index and the Brouwer Degree ...... 1 1.1 The Kronecker Index...... 1 1.1.1 Definition...... 1 1.1.2 The Winding Number in R2 ...... 3 1.1.3 The Gauss Linking Number of Two Curves in R3 ...... 5 1.1.4 The Case Where f : ∂D ⊂ Rn → Sn−1 ...... 6 1.1.5 A Minimum Problem from Ginzburg-Landau’s Theory ...... 9 1.1.6 The Kronecker Integral as a Volume Integral ...... 12 1.1.7 HistoricalNotes...... 14 1.1.8 Bibliographical Notes ...... 15 1.2 TheBrouwerDegree...... 16 1.2.1 Smooth Mappings ...... 16 1.2.2 Continuous Mappings ...... 22 1.2.3 The Case Where f : ∂D → Rn \{0} ...... 26 1.2.4 The Case of an Unbounded Set D ...... 29 1.2.5 Uniqueness of the Degree ...... 30 1.2.6 TheBrouwerIndex...... 37 1.2.7 Generalized Implicit Function Theorem ...... 41 1.2.8 HistoricalNotes...... 44 1.2.9 Bibliographical Notes ...... 46 1.3 Degree in Finite-Dimensional Vector Spaces ...... 46 1.3.1 Composition with Linear Isomorphisms...... 46 1.3.2 Definitions and Basic Properties ...... 47 1.3.3 TheBrouwerIndex...... 49 1.3.4 Reduction Formulas ...... 50 1.3.5 Computation of the Index in Degenerate Case ...... 52 1.3.6 HistoricalNotes...... 56 1.3.7 Bibliographical Notes ...... 56 1.4 Retracts,FixedPoints,VectorFields,Linking...... 57 1.4.1 RetractsandRetractions...... 57 1.4.2 Fixed Point Theorems ...... 60

xiii xiv Contents

1.4.3 Toward a Fixed Point Index in Retracts of Rn ...... 63 1.4.4 Fixed Point Theorems in Cones ...... 67 1.4.5 The Hairy Ball Theorem...... 68 1.4.6 LinkingandCriticalPoints...... 70 1.4.7 HistoricalNotes...... 74 1.4.8 Bibliographical Notes ...... 76 2 Continuation, Existence and Bifurcation...... 77 2.1 Continuation Theorems ...... 77 2.1.1 Extended Homotopy Invariance Property ...... 77 2.1.2 The Leray–Schauder Continuation Theorem ...... 79 2.1.3 The Method of a Priori Bounds ...... 80 2.1.4 The Leray–Schauder Alternative...... 81 2.1.5 The Case of an Unbounded Set ...... 82 2.1.6 HistoricalNotes...... 83 2.1.7 Bibliographical Notes ...... 83 2.2 Zeros, Surjectivity and Fixed Points of Mappings...... 84 2.2.1 Perturbations of Linear Mappings ...... 84 2.2.2 Surjectivity of Some Coercive Mappings...... 89 2.2.3 Monotone Mappings...... 93 2.2.4 Boundary Conditions for the Existence of a Zero ...... 95 2.2.5 Fixed Points of Expansive-Compressive-Type Mappings .... 101 2.2.6 Fixed Points of Poincaré–Miranda-Type Mappings...... 104 2.2.7 Uniqueness of Zeros and of Fixed Points...... 107 2.2.8 HistoricalNotes...... 108 2.2.9 Bibliographical Notes ...... 109 2.3 CharacteristicValues,IndexandBifurcation...... 110 2.3.1 Characteristic Values of Couples of Linear Mappings ...... 110 2.3.2 Multiplicity of a Characteristic Value...... 110 2.3.3 The Leray–Schauder Formula for the Brouwer Index ...... 113 2.3.4 BifurcationPoints...... 115 2.3.5 Bifurcation Through Linearization...... 117 2.3.6 Global Structure of Bifurcation Branches ...... 120 2.3.7 BifurcationfromInfinity ...... 123 2.3.8 HistoricalNotes...... 126 2.3.9 Bibliographical Notes ...... 126 2.4 Set-Valued Mappings ...... 127 2.4.1 The Kakutani Fixed Point Theorem ...... 127 2.4.2 The von Neumann Minimax Theorem ...... 129 2.4.3 The Nash Equilibrium of a Non-cooperative Game...... 130 2.4.4 HistoricalNotes...... 133 2.4.5 Bibliographical Notes ...... 133 Contents xv

3 Infinite-Dimensional Problems ...... 135 3.1 Fixed Point Theorems...... 135 3.1.1 The Ky Fan Fixed Point Theorem...... 135 3.1.2 The Tychonov and Schauder Fixed Point Theorems ...... 138 3.1.3 HistoricalNotes...... 138 3.1.4 Bibliographical Notes ...... 139 3.2 The KKM Theorem and Variational Inequalities ...... 139 3.2.1 The KKM Theorem and Its Generalizations ...... 139 3.2.2 Variational Inequalities ...... 144 3.2.3 Variational Inequalities for Monotone Operators ...... 147 3.2.4 Hemivariational Inequalities ...... 149 3.2.5 HistoricalNotes...... 157 3.2.6 Bibliographical Notes ...... 159 3.3 Equations in Reflexive Banach Spaces...... 159 3.3.1 The Surjectivity of Monotone Coercive Operators...... 159 3.3.2 A Monotone Quasilinear Dirichlet Problem...... 164 3.3.3 A Non Monotone Quasilinear Dirichlet Problem ...... 166 3.3.4 HistoricalNotes...... 169 3.3.5 BibliograhicalNotes...... 170 3.4 Stationary Navier-Stokes Equations...... 170 3.4.1 TheProblem ...... 170 3.4.2 Variational Formulation ...... 172 3.4.3 Existence of a Solution ...... 175 3.4.4 Uniqueness of the Solution ...... 177 3.4.5 HistoricalNotes...... 178 3.4.6 Bibliographical Notes ...... 178 3.5 Toward the Leray-Schauder Degree ...... 178 3.5.1 Extending the Brouwer Degree to Normed Spaces ...... 178 3.5.2 The Schauder Projection and Approximation Theorems ..... 179 3.5.3 Definition of the Leray-Schauder Degree ...... 180 3.5.4 Justification of the Definition ...... 181 3.5.5 HistoricalNotes...... 182 3.5.6 Bibliographical Notes ...... 183 4 Difference Equations ...... 185 4.1 PeriodicSolutionsofFirstOrderEquations...... 185 4.1.1 PeriodicSolutions ...... 185 4.1.2 Bounded Nonlinearities ...... 186 4.1.3 Lower and Upper Solutions ...... 187 4.1.4 Multiplicity Results of the Ambrosetti-Prodi Type ...... 193 4.1.5 HistoricalNotes...... 196 4.1.6 Bibliographical Notes ...... 197 4.2 Applications to Population Dynamics ...... 197 4.2.1 Nonlinearities Bounded from Below or Above...... 197 4.2.2 Lotka-Volterra-TypeSystems ...... 200 xvi Contents

4.2.3 HistoricalNotes...... 203 4.2.4 Bibliographical Notes ...... 203 4.3 The Dirichlet Problem for Second Order Equations...... 203 4.3.1 Formulation and Spectrum of the Linear Part ...... 203 4.3.2 BifurcationfromtheTrivialSolution...... 205 4.3.3 Multiple Solutions Near Resonance ...... 206 4.3.4 Nonlinearities Bounded from Below or Above...... 210 4.3.5 Lower and Upper Solutions ...... 214 4.3.6 Multiplicity Results of the Ambrosetti-Prodi Type ...... 217 4.3.7 HistoricalNotes...... 221 4.3.8 Bibliographical Notes ...... 222 5 Periodic Solutions of Differential Systems ...... 223 5.1 The Poincaré Operator ...... 223 5.1.1 Initial Value and Periodic Problems ...... 223 5.1.2 Bounded Perturbations of Some Linear Systems...... 224 5.1.3 The Stampacchia Method ...... 226 5.1.4 The Krasnosel’skii-Perov Existence Theorem ...... 229 5.1.5 HistoricalNotes...... 234 5.1.6 Bibliographical Notes ...... 234 5.2 Guiding Functions ...... 235 5.2.1 Definition and Preliminaries ...... 235 5.2.2 Systems with Continuable Solutions...... 237 5.2.3 Systems with Non-continuable Solutions ...... 241 5.2.4 Generalized Guiding Functions ...... 245 5.2.5 HistoricalNotes...... 250 5.2.6 Bibliographical Notes ...... 250 5.3 EvolutionComplementaritySystems ...... 250 5.3.1 EquivalentFormulations...... 250 5.3.2 The Cauchy Problem ...... 251 5.3.3 The Poincaré Operator...... 253 5.3.4 Guiding Functions ...... 254 5.3.5 HistoricalNotes...... 261 5.3.6 Bibliographical Notes ...... 261 6 Two-Dimensional Problems ...... 263 6.1 Lower and Upper Solutions for Second Order Equations ...... 263 6.1.1 Definitions ...... 263 6.1.2 Existence Theorem ...... 264 6.1.3 The Brouwer Degree of the Poincaré Operator ...... 266 6.1.4 HistoricalNotes...... 269 6.1.5 Bibliographical Notes ...... 269 6.2 Stability and Index of Periodic Solutions ...... 269 6.2.1 Planar Periodic Systems ...... 269 6.2.2 Second Order Differential Equations ...... 272 6.2.3 Second Order Equations with Convex Nonlinearity ...... 275 Contents xvii

6.2.4 Second Order Equations with Periodic Nonlinearity...... 280 6.2.5 HistoricalNotes...... 285 6.2.6 Bibliographical Notes ...... 286 6.3 Planar Differential Systems...... 286 6.3.1 Autonomous Systems...... 286 6.3.2 Nonresonant Forced Systems ...... 288 6.3.3 Polar-TypeCoordinates...... 290 6.3.4 Resonant Forced Systems ...... 293 6.3.5 Asymmetric Piecewise-Linear Oscillators...... 297 6.3.6 HistoricalNotes...... 300 6.3.7 Bibliographical Notes ...... 300 6.4 ComputingtheDegreeinDimensionTwo...... 301 6.4.1 The Kronecker Index on a Closed Simple Curve ...... 301 6.4.2 Factorized Multilinear Mappings ...... 304 6.4.3 Using Sturm Sequences ...... 310 6.4.4 Holomorphic Functions ...... 313 6.4.5 HistoricalNotes...... 316 6.4.6 Bibliographical Notes ...... 316 6.5 Application to Stability and Control ...... 317 6.5.1 Stability Conditions and a Zero Exclusion Principle...... 317 6.5.2 The First Schur Transform of a Polynomial ...... 319 6.5.3 Routh–Hurwitz Stable Family of Polynomials ...... 320 6.5.4 The Second Schur Transform of a Polynomial ...... 321 6.5.5 Schur–Cohn Stable Family of Polynomials ...... 322 6.5.6 HistoricalNotes...... 323 6.5.7 Bibliographical Notes ...... 324 7 The Degree of Some Classes of Mappings ...... 325 7.1 Homogeneous Polynomial Mappings ...... 325 7.1.1 Cartesian Products of Mappings ...... 325 7.1.2 Homogeneous Polynomial Mappings...... 327 7.1.3 HistoricalNotes...... 330 7.1.4 Bibliographical Notes ...... 330 7.2 Orientation-Preserving Mappings ...... 331 7.2.1 Definitions and Main Properties ...... 331 7.2.2 Monotone Mappings...... 333 7.2.3 Holomorphic Mappings ...... 334 7.2.4 Quaternionic Monomials ...... 344 7.2.5 HistoricalNotes...... 350 7.2.6 Bibliographical Notes ...... 350 7.3 Symmetrical Mappings ...... 351 7.3.1 Odd Mappings ...... 351 7.3.2 Elliptic Differential Operators ...... 355 7.3.3 The Lusternik-Schnirelmann Covering Theorem ...... 356 7.3.4 Measure of Non-compactness of the Unit Sphere...... 358 xviii Contents

7.3.5 The Krasnosel’skii Genus ...... 359 7.3.6 S1-Equivariant Mappings ...... 363 7.3.7 A Reduction Formula for S1-Equivariant Mappings ...... 365 7.3.8 HistoricalNotes...... 369 7.3.9 Bibliographical Notes ...... 369 7.4 One-to-One and Composed Mappings ...... 370 7.4.1 Invariance of Domain and of Dimension ...... 370 7.4.2 The Banach-Mazur Theorem...... 372 7.4.3 The Leray Product Formula ...... 376 7.4.4 The Jordan-Brouwer Separation Theorem...... 378 7.4.5 Degree of One-to-One Mappings ...... 381 7.4.6 DeformationofanElasticBody ...... 382 7.4.7 A Class of Mappings with Orientation-Preserving Character 384 7.4.8 HistoricalNotes...... 387 7.4.9 Bibliographical Notes ...... 389 8 History of the Brouwer Fixed Point Theorem ...... 391 8.1 Discovery,PublicationandAnticipations...... 391 8.1.1 Brouwer...... 391 8.1.2 Hadamard ...... 393 8.1.3 Poincaré ...... 394 8.1.4 Bohl ...... 396 8.1.5 Bibliographical Notes ...... 398 8.2 New Proofs and Infinite-Dimensional Extensions ...... 398 8.2.1 Alexander, Birkhoff and Kellogg ...... 398 8.2.2 Schauder and Tychonoff ...... 399 8.2.3 Sperner ...... 400 8.2.4 Knaster, Kuratowski and Mazurkiewicz ...... 401 8.2.5 Bibliographical Notes ...... 402 8.3 Game Theory, Economics and Computation ...... 402 8.3.1 von Neumann and Kakutani ...... 402 8.3.2 Nash...... 404 8.3.3 Scarf ...... 404 8.3.4 Kellogg, Li and Yorke ...... 405 8.3.5 Bibliographical Notes ...... 405 8.4 Topology or Not Topology ...... 406 8.4.1 HammersteinandGolomb ...... 406 8.4.2 Cinquini, Miranda and Scorza-Dragoni ...... 406 8.4.3 Bibliographical Notes ...... 410 8.5 Variational Inequalities, Simple Proofs, Applications...... 411 8.5.1 Hartman, Stampacchia and Karamardian ...... 411 8.5.2 TheQuestforElementaryProofs...... 411 8.5.3 All-OutApplications ...... 412 8.5.4 Bibliographical Notes ...... 412 Contents xix

Bibliography ...... 413 Index of Names ...... 437 Index ...... 443