Metric Spaces: Including Fixed Point Theory and Set-valued Maps

Description

METRIC SPACES is intended for undergraduate students offering a course of metric spaces and post graduate students offering a course of nonlinear analysis or fixed point theory. The first six chapters cover basic concepts of metric spaces, separable spaces, compact spaces, connected spaces and continuity of functions defined on a metric space. Chapter seven is devoted to the metric fixed point theory. Banach contraction theorem and several of its generalizations along with their applications and Caristi s fixed point theorem are also given in this chapter. The introductory set-valued analysis with special emphysis on continuity and fixed point theory of set-valued maps is given in chapter eight. One of the most useful and important results from nonlinear analysis is Ekeland s variational principle. This principle along with several of its equivalent forms, Takahashi s minimization theorem, introduction of theory of equilibrium problems and the equilibrium version of Ekeland s variational principle and several of its equivalent forms are presented in the last chapter. This book will also be useful for researchers working in nonlinear analysis, optimization and theory of equilibrium problems Content: Machine generated contents note: 1.Basic Concepts -- 1.1.Definition and Examples of Metric Spaces -- 1.2.Distance between Sets and Diameter of a Set -- 1.3.Open Sets and Interior Points -- 1.4.Closed Sets and Closure of Sets -- 1.5.Subspaces -- 2.Complete Metric Spaces -- 2.1.Convergent Sequences -- 2.2.Cauchy Sequences -- 2.3.Complete Metric Spaces -- 2.4.Completion -- 3.Separable Spaces -- 3.1.Countability -- 3.2.Dense Sets -- 3.3.Nowhere Dense Sets -- 4.Compact Spaces -- 4.1.Definitions and Basic Concepts -- 4.2.Sequentially Compact Spaces -- 4.3.Totally Bounded Spaces -- 5.Continuous Functions -- 5.1.Definition and Characterizations -- 5.2.Continuous Functions and Compact Spaces -- 5.3.Uniform Continuous Functions -- 5.4.Homeomorphism and Equivalent Metrics -- 5.5.Uniform Convergence of Sequences of Functions -- 6.Connected Spaces -- 6.1.Separated Sets -- 6.2.Connected Sets -- 6.3.Continuous Functions and Connected Sets -- 6.4.Components -- 7.Fixed Point Theorems -- 7.1.Banach Contraction Theorem and its Applications -- 7.1.1.Application to System of Linear Equations -- 7.1.2.Application to Differential Equations -- 7.1.3.Application to Integral Equations -- 7.2.Further Extension of Banach Contraction Theorem -- 7.3.Caristi's Fixed Point Theorem -- 8.Set-Valued Maps -- 8.1.Basic Concepts and Definitions -- 8.2.Continuity of Set-Valued Maps -- 8.3.Nadler's Fixed Point Theorem -- 8.4.Some Fixed Point Theorems for Set-Valued Maps -- 9.Ekeland's Variational Principle and its Applications -- 9.1.Ekeland Variational's Principle in Complete Metric Spaces -- 9.2.Applications to Fixed Point Theorems -- 9.3.Applications to Optimization -- 9.4.Equilibrium Problems and Extended Ekeland's Variational Principle -- 9.4.1.Equilibrium Problems -- 9.4.2.Extended Ekeland's Variational Principle -- A.Some Basic Inequalities -- B.Partial Ordering -- C.Nested Interval Property -- References.