# Metric Spaces: Including Fixed Point Theory and Set-valued Maps 1842656554, 978-1-84265-655-6

METRIC SPACES is intended for undergraduate students offering a course of metric spaces and post graduate students offer

289 12 1MB

English Pages 208 [209] Year 2010

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 --
8.4.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.