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Author: Kim C. Border
Publisher: Cambridge University Press
Published: 1985
Total Pages: 144
ISBN-13: 9780521388085
DOWNLOAD EBOOKThis book explores fixed point theorems and its uses in economics, co-operative and noncooperative games.
Author: Ken Urai
Publisher: World Scientific
Published: 2010
Total Pages: 311
ISBN-13: 9812837191
DOWNLOAD EBOOK1. Introduction. 1.1. Mathematics is language. 1.2. Notes on some mathematical tools in this book. 1.3. Basic mathematical concepts and definitions -- 2. Fixed-point theorems. 2.1. Classical results and basic extensions. 2.2. Convexity and duality for general spaces. 2.3. Extension of classical results to general spaces -- 3. Nash equilibrium and abstract economy. 3.1. Multi-agent product settings for games. 3.2. Nash equilibrium. 3.3. Abstract economy -- 4. Gale-Nikaido-Debreu's theorem. 4.1. Gale-Nikaido-Debreu's theorem. 4.2. Market equilibria in general vector spaces. 4.3. Demand-supply coincidence in general spaces -- 5. General economic equilibrium. 5.1. General preferences and basic existence theorems. 5.2. Pareto optimal allocations. 5.3. Existence of general equilibrium -- 6. The C̮ech type homology theory and fixed points. 6.1. Basic concepts in algebraic topology. 6.2. Vietoris-Begle mapping and local connectedness. 6.3. Nikaido's analogue of Sperner's lemma. 6.4. Eilenberg-Montgomery's theorem -- 7. Convex structure and fixed-point index. 7.1. Lefschetz's fixed-point theorem and its extensions. 7.2. Cohomology theory for general spaces. 7.3. Dual-system structure and differentiability. 7.4. Linear Approximation for Isolated Fixed Points. 7.5. Indices for compact set of fixed points -- 8. Applications to related topics. 8.1. KKM, KKMS, and core existence. 8.2. Eaves' theorem. 8.3. Fan-Browder's coincidence theorem. 8.4. L-majorized mappings. 8.5. Variational inequality problem. 8.6. Equilibrium with cooperative concepts. 8.7. System of inequalities and affine transformations -- 9. Mathematics and social science. 9.1. Basic concepts in axiomatic set theory. 9.2. Individuals and rationality. 9.3. Society and values -- 10. Concluding discussions. 10.1. Fixed points and economic equilibria. 10.2. Rationality and fixed-point views of the world
Author: Andrew McLennan
Publisher: Springer
Published: 2018-07-03
Total Pages: 441
ISBN-13: 9811307105
DOWNLOAD EBOOKThis book develops the central aspect of fixed point theory – the topological fixed point index – to maximal generality, emphasizing correspondences and other aspects of the theory that are of special interest to economics. Numerous topological consequences are presented, along with important implications for dynamical systems. The book assumes the reader has no mathematical knowledge beyond that which is familiar to all theoretical economists. In addition to making the material available to a broad audience, avoiding algebraic topology results in more geometric and intuitive proofs. Graduate students and researchers in economics, and related fields in mathematics and computer science, will benefit from this book, both as a useful reference and as a well-written rigorous exposition of foundational mathematics. Numerous problems sketch key results from a wide variety of topics in theoretical economics, making the book an outstanding text for advanced graduate courses in economics and related disciplines.
Author: Vittorino Pata
Publisher: Springer Nature
Published: 2019-09-22
Total Pages: 171
ISBN-13: 3030196704
DOWNLOAD EBOOKThis book addresses fixed point theory, a fascinating and far-reaching field with applications in several areas of mathematics. The content is divided into two main parts. The first, which is more theoretical, develops the main abstract theorems on the existence and uniqueness of fixed points of maps. In turn, the second part focuses on applications, covering a large variety of significant results ranging from ordinary differential equations in Banach spaces, to partial differential equations, operator theory, functional analysis, measure theory, and game theory. A final section containing 50 problems, many of which include helpful hints, rounds out the coverage. Intended for Master’s and PhD students in Mathematics or, more generally, mathematically oriented subjects, the book is designed to be largely self-contained, although some mathematical background is needed: readers should be familiar with measure theory, Banach and Hilbert spaces, locally convex topological vector spaces and, in general, with linear functional analysis.
Author: D. R. Smart
Publisher: CUP Archive
Published: 1980-02-14
Total Pages: 108
ISBN-13: 9780521298339
DOWNLOAD EBOOKAuthor: M. J. Todd
Publisher: Springer Science & Business Media
Published: 2013-03-09
Total Pages: 138
ISBN-13: 3642503276
DOWNLOAD EBOOKFixed-point algorithms have diverse applications in economics, optimization, game theory and the numerical solution of boundary-value problems. Since Scarf's pioneering work [56,57] on obtaining approximate fixed points of continuous mappings, a great deal of research has been done in extending the applicability and improving the efficiency of fixed-point methods. Much of this work is available only in research papers, although Scarf's book [58] gives a remarkably clear exposition of the power of fixed-point methods. However, the algorithms described by Scarf have been super~eded by the more sophisticated restart and homotopy techniques of Merrill [~8,~9] and Eaves and Saigal [1~,16]. To understand the more efficient algorithms one must become familiar with the notions of triangulation and simplicial approxi- tion, whereas Scarf stresses the concept of primitive set. These notes are intended to introduce to a wider audience the most recent fixed-point methods and their applications. Our approach is therefore via triangu- tions. For this reason, Scarf is cited less in this manuscript than his contri- tions would otherwise warrant. We have also confined our treatment of applications to the computation of economic equilibria and the solution of optimization problems. Hansen and Koopmans [28] apply fixed-point methods to the computation of an invariant optimal capital stock in an economic growth model. Applications to game theory are discussed in Scarf [56,58], Shapley [59], and Garcia, Lemke and Luethi [24]. Allgower [1] and Jeppson [31] use fixed-point algorithms to find many solutions to boundary-value problems.
Author: Jean-Pierre Aubin
Publisher: Courier Corporation
Published: 2007-01-01
Total Pages: 658
ISBN-13: 048646265X
DOWNLOAD EBOOKMathematical economics and game theory approached with the fundamental mathematical toolbox of nonlinear functional analysis are the central themes of this text. Both optimization and equilibrium theories are covered in full detail. The book's central application is the fundamental economic problem of allocating scarce resources among competing agents, which leads to considerations of the interrelated applications in game theory and the theory of optimization. Mathematicians, mathematical economists, and operations research specialists will find that it provides a solid foundation in nonlinear functional analysis. This text begins by developing linear and convex analysis in the context of optimization theory. The treatment includes results on the existence and stability of solutions to optimization problems as well as an introduction to duality theory. The second part explores a number of topics in game theory and mathematical economics, including two-person games, which provide the framework to study theorems of nonlinear analysis. The text concludes with an introduction to non-linear analysis and optimal control theory, including an array of fixed point and subjectivity theorems that offer powerful tools in proving existence theorems.
Author: S.P. Singh
Publisher: Springer Science & Business Media
Published: 2013-04-17
Total Pages: 231
ISBN-13: 9401588228
DOWNLOAD EBOOKThe aim of this volume is to make available to a large audience recent material in nonlinear functional analysis that has not been covered in book format before. Here, several topics of current and growing interest are systematically presented, such as fixed point theory, best approximation, the KKM-map principle, and results related to optimization theory, variational inequalities and complementarity problems. Illustrations of suitable applications are given, the links between results in various fields of research are highlighted, and an up-to-date bibliography is included to assist readers in further studies. Audience: This book will be of interest to graduate students, researchers and applied mathematicians working in nonlinear functional analysis, operator theory, approximations and expansions, convex sets and related geometric topics and game theory.
Author: Zaifu Yang
Publisher: Springer Science & Business Media
Published: 2013-04-17
Total Pages: 349
ISBN-13: 1475748396
DOWNLOAD EBOOKComputing Equilibria and Fixed Points is devoted to the computation of equilibria, fixed points and stationary points. This volume is written with three goals in mind: (i) To give a comprehensive introduction to fixed point methods and to the definition and construction of Gröbner bases; (ii) To discuss several interesting applications of these methods in the fields of general equilibrium theory, game theory, mathematical programming, algebra and symbolic computation; (iii) To introduce several advanced fixed point and stationary point theorems. These methods and topics should be of interest not only to economists and game theorists concerned with the computation and existence of equilibrium outcomes in economic models and cooperative and non-cooperative games, but also to applied mathematicians, computer scientists and engineers dealing with models of highly nonlinear systems of equations (or polynomial equations).
Author: Ravi P. Agarwal
Publisher: Cambridge University Press
Published: 2001-03-22
Total Pages: 182
ISBN-13: 1139433792
DOWNLOAD EBOOKThis book provides a clear exposition of the flourishing field of fixed point theory. Starting from the basics of Banach's contraction theorem, most of the main results and techniques are developed: fixed point results are established for several classes of maps and the three main approaches to establishing continuation principles are presented. The theory is applied to many areas of interest in analysis. Topological considerations play a crucial role, including a final chapter on the relationship with degree theory. Researchers and graduate students in applicable analysis will find this to be a useful survey of the fundamental principles of the subject. The very extensive bibliography and close to 100 exercises mean that it can be used both as a text and as a comprehensive reference work, currently the only one of its type.
