R. Brown – Handbook of Topological Fixed Point Theory

Автор: R. Brown
Название книги: Handbook of Topological Fixed Point Theory
Формат: PDF
Жанр: Математика
Страницы: 966
Качество: Изначально компьютерное, E-book

Fixed point theory concerns itself with a very simple, and basic, mathematical
setting. For a function f that has a set X as both domain and range, a fixed
point of f is a point x of X for which f(x) = x. Two fundamental theorems
concerning fixed points are those of Banach and of Brouwer. In Banach’s theorem,
X is a complete metric space with metric d and f:X → X is required to be
a contraction, that is, there must exist L < 1 such that d(f(x), f(y)) ≤ Ld(x, y) for all x, y ∈ X. The conclusion is that f has a fixed point, in fact exactly one of them. Brouwer’s theorem requires X to be the closed unit ball in a Euclidean space and f:X → X to be a map, that is, a continuous function. Again we can conclude that f has a fixed point. But in this case the set of fixed points need not be a single point, in fact every closed nonempty subset of the unit ball is the fixed point set for some map. The metric on X in Banach’s theorem is used in the crucial hypothesis about the function, that it is a contraction. The unit ball in Euclidean space is also metric, and the metric topology determines the continuity of the function, but the focus of Brouwer’s theorem is on topological characteristics of the unit ball, in particular that it is a contractible finite polyhedron. The theorems of Banach and Brouwer illustrate the difference between the two principal branches of fixed point theory: metric fixed point theory and topological fixed point theory. The Handbook of Metric Fixed Point Theory, edited by Art Kirk and Brailey Sims and published by Kluwer in 2001, presented that portion of the subject and, in this companion volume, we take up the other part of the fixed point story. The classification of mathematical content is seldom easy. For instance, the distinction between the metric and topological fixed point theories is far from precise and it can be difficult to determine to which a specific topic belongs. In the same way, although fixed point theory is generally considered a branch of topology, the influence of nonlinear analysis, and the related subject of dynamics, is so profound that much of fixed point theory could just as well be considered a part of analysis. The papers in this Handbook reflect the varied, and not easily classified, nature of the mathematics that makes up topological fixed point theory. To impose some structure on its contents, the papers have been divided into four “chapters”, each of which consists of papers that have something important in common with each other. The title of Chapter I, homological methods in fixed point theory, points out the importance of algebraic topology, specifically homology theory, as the source of many of the mathematical tools used in fixed point theory. This chapter also illustrates the fact that topological fixed point theory is not just about the equation f(x) = x. For instance, if the function f is multi-valued, taking points of X to subsets of the same space, a fixed point is a point such that x ∈ f(x). The pioneering work of Lefschetz was in the context of coincidence theory. For two maps f, g:X → Y between closed orientable manifolds of the same dimension, a nonzero value of the homotopy invariant that Lefschetz introduced implies the existence of a coincidence, that is, a point x ∈ X such that f(x) = g(x). Another modification of the fixed point equation is fn(x) = x where fn denotes the n-times iteration of a map f:X → X. A point x such that fn(x) = x is called a periodic point. The iterates of f constitute a discrete dynamical system on X and the periodic points can furnish important dynamical information. The influence of dynamics can also be observed in the fact that such homogenous spaces as nilmanifolds and solvmanifolds are the setting for several of the papers in this chapter. Since homogeneous spaces are spaces of cosets, algebra plays an important role in the study of maps on such spaces. Another way in which algebra impacts fixed point theory is through the study of equivariant maps. If a space is acted on by a group, then an equivariant map is one that respects the action. Chapter II is devoted to the fixed point theory of equivariant maps and its application to analysis. Topological fixed point theory is often referred to as “Nielsen theory”. This terminology reflects the importance of the concepts introduced by Jacob Nielsen that furnish a homotopy invariant lower bound for the number of solutions to an equation. All the papers in Chapter III contain the name of Nielsen, or of Wecken who expanded Nielsen’s ideas, in their title. Again the objects of study are not just fixed points. Coincidences, periodic points and fixed points of multivalued maps all make their appearance in this chapter, as do roots, the solutions to the equation f(x) = a where f:X → Y is a map and a ∈ Y . Nielsen theory is particularly interesting, and challenging, when the spaces are compact surfaces, as some of the papers in Chapter III demonstrate. The substantial size, and content, of Chapter IV indicates the importance of the applications of topological fixed point theory to nonlinear analysis and dynamics. Problems are formulated in terms or fixed or periodic points, coincidences and roots. The tools of fixed point theory are those of the previous chapters: the Lefschetz number, fixed point index, Nielsen number and, for root problems, the topological degree. Again the fuctions considered may be multivalued as well as single valued. However, a notable difference between the papers in this chapter and many of those earlier in the Handbook is the extention of the tools to more general settings than those of the purely topological investigations. These powerful tools are then employed to obtain results about periodic solutions and solutions satisfying boundary conditions and other constraints, for differential equations and differential inclusions. We have not attempted a definition of the “topological fixed point theory” that is the subject of this Handbook; neither will we try to define precisely what a “handbook” is. A handbook contains information that will furnish the mathematician reader, whether an established researcher or a graduate student, with a foundation in its subject and a guide to further study. An up-to-date handbook also gives its readers a sense of the present state of the art and, ideally, offers some clues as to where the subject will go in the future. But a handbook is not a textbook in which the reader starts on the first page expecting to find a complete and detailed exposition following a clearly indicated line of development that extends to the very last page. Instead, the reader of a handbook is invited to view its table of contents as a buffet from which to taste some items while perhaps consuming others that seem particularly attractive. Each of the 28 authors who contributed to this handbook was asked to do so because the editors consider that person an expert on the topic that he or she was invited to write about. The style of presentation and level of mathematical detail was determined by the authors, based on their own mathematical taste and their judgment of the best way to present their specialty. This handbook is the sum of the contributions of its authors. It exists because these busy people were willing to expend a considerable amount of time and effort and we are grateful to them for doing it. We very much appreciate the help of the Juliusz Schauder Center of the Nicolaus Copernicus University in Toruń. Mariusz Czerniak managed the collection of the papers and Jolanta Szelatyńska converted them into a uniform style. We are grateful for the support of our editors at Kluwer: Liesbeth Mol who initiated the project and Lynn Brandon who saw if through to completion and Marlies Vlot who supervised the producion of the handbook