A Study On Fixed Point Theory - Suranjoy Singh - Bog

Fixed Point Theory is an attractive and interesting subject with a large number of applications in various fields of mathematics and other branches of science. This book offers encouragement and empowers readers to embrace the basic history, the basic types of fixed point theories, important theorems and some common selected applications of fixed point theory. Above all, only lucid language is used so that this book will also be beginners friendly.A brief description of the contents is given below: Chapter 1: Introduction: 2 pagesConcept of fixed point, a brief history, basic types of the theory.Chapter 2: Topological Fixed Point Theory: 68 pagesIntroduction and Background: Topological Fixed Point Property, Background on Simplexes and Triangulations, Background in Analysis And Topology, Upper Semi-continuous Multifunctions.The Brower's Fixed point Theorem: Proof by Non Analytic Methods, Proof by Analytic Methods. Sperner's Lemma and Brower's Fixed Point Theorem, Barycentric Coordinates. Generalization: Extension in Infinite Dimensions, Fixed Point Properties for Closed Boundary ConvexSets, Theorems with boundary ConditionsCondition on Compactness. Multifunctions and Kakutani's Theorem, Theory with Boundary Conditions.Applications.Chapter 3: Metric Fixed Point Theory: 45 pagesIntroduction: Banach's Contraction Principle, Extensions of the Contraction Principle, Generalization.Converse of Banach's Contraction Principle.Metric Fixed Point Theory in Banach Spaces.Metric Fixed Point Theory in Metric Spaces.Applications.Chapter 4: Order-Theoretic Fixed Point Theory: 17 pages Introduction, Completeness Condition for Posets, Conditionally Complete Posets, Countably Chain-Complete Posets, Chain-Complete Posets.Iterative Fixed Point Theorems, The Tarski-Kantorovitch Fixed Point Theorem, Applications toFunctional Equations, The Contraction Mapping Theorem. The Tarski's Fixed Point Theorems, The Knaster-Tarski's Theorem, Application.The Abian-Brown Theorem, Application.An extensive bibliography.