Elementary fixed point theorems
This monograph is written by a well-known expert in fixed point theory and presents his choice of results from this wide area of research. \par Besides the introductory chapter containing analytic and topological preliminaries, the book contains 11 chapters treating some of the most useful fixed point results. Chapter 2 describes the basic properties of iterates of real and complex functions, and includes theorems of Thron, Cohen, Shield, Sharkovsky and Bergweiler. Chapter 3 explores the existence of fixed points of mappings acting on partially ordered sets, including Tarski's theorem and several of its applications. Most of Chapter 4 deals with Ward's theory of partially ordered topological spaces and includes Schweigerts's fixed point theorem and Klee's counterexample. Several variants of the contraction principle for single- and multivalued mappings, including some generalizations of Nadler's theorem are presented in Chapter 5. Chapter 6 is dedicated to various applications, amongst others, to differential and functional equations, the Cauchy-Kowalevskaya theorem and the central limit theorem. Caristi's fixed point result is the theme of Chapter 7, together with its connection with Ekeland's variational principle, Takahashi's minimization theorem and metrical completeness. \par Contractive and non-expansive mappings are treated in Chapter 8, which includes Edelstein's result, Browder-Gohde-Kirk's fixed point theorem, Pasicki's theory and related problems. Chapter 9 uses the concepts of the geometry of Banach spaces to establish fixed point results. Brouwer's theorem, with two of its possible proofs, is treated in Chapter 10, and the existence of Walrasian and Nash equilibria is deduced as an application. Schauder's theorem and its extensions and several applications constitute the major part of Chapter 11. Chapter 12 describes the finite-dimensional degree theory. At the end, three appendices are given summarizing some of the classical counterexamples and presenting a connection of fractals and fixed points. \par The monograph can serve as a very useful introduction into the fixed point topic, which is one of the most applicable parts, both of Topology and Nonlinear Analysis.
scientific article; zbMATH DE number 6982490