Fixed Point Theory and Applications

Welcome to the future Book on Fixed Point Theory and Applications on the Web. I would like to ask anyone who is interested to be part of this wonderful project. The idea is very simple: Choose one of the following topics and write down something in Latex, mail it to me and I will be glad to put it here under the author's name. This way this project will not be a burden on one person.

Chapter 0. Introduction

Chapter 1. The Contraction Principle

1.

Historical Notes

2.

The Contraction Principle in Complete Metric Spaces

3.

Applications

3.1

Differential Equations

3.2

Implicit Theorem

3.3

Holomorphic Mappings

3.4

Perron-Frobenius Theorem

3.5

Fractal sets: The Cantor set

4.

Menger Spaces

Chapter 2. Nonexpansive Mappings in Hilbert Spaces

1.

Some Examples

2.

Classical Theorems

2.1

Demi-Closeness Principle

2.2

Existence of Fixed Point

3.

Semi-group of Nonexpansive Mappings

3.1

Basic Definitions

3.2

Generator

3.3

Accretivity, Monotone Operators

3.4

Resolvent

3.5

Theorems

3.6

Example:

4.

Commuting Families

5.

Uniformly Lipschitzian Maps

6.

Applications

6.1

Periodic Solutions

6.2

Forcing

Chapter 3. Nonexpansive Mappings in Banach Spaces

1.

Classical Counter Examples

1.1

,

1.2

1.3

Alspach's counter example

2.

Uniformly Convex and Uniformly Smooth Banach Spaces

2.1

Basic Definitions

2.2

Examples

2.3

Properties

2.4

Scaling of the Unit Ball

3.

Fixed Point Theorems

3.1

Demi-Closeness Principle

3.2

Browder-Gohde Theorem

3.3

Contraction of Type-( )

3.4

Acretive Operators

3.5

Asymptotic Center Technique

3.6

Minimal points and Optimal Points

4.

Normal Structure Property

4.1

Basic Definitions and Examples

4.2

Characterizations: Brodskii-Milman-Landes

4.3

Some Counter Examples

4.4

Spaces with Normal Structure Property

4.5

Normal Structure Property in Spaces with Bases

5.

Ultraproduct Techniques

5.1

Basic Definitions

5.2

Basic Results in Ultraproduct Language

5.3

Fixed Point Theorems

5.3.1

Uniformly Convex Spaces

5.3.2

Lin's results and its Extensions

5.3.3

Maurey's Fixed Point Theorems

6.

Lattice Banach Spaces

6.1

Remarks on Maurey's -Fixed point Theorem

6.2

Borwein-Sim's Fixed Point Theorem

7.

-Fixed Point Property

7.1

Examples and Counter Examples

7.2

-Fixed Point Property

7.3

-Normal Structure Property

7.4

in Lattice Banach Spaces

Chapter 4. Orbit, Omega-set

1.

Basic Definitions

2.

Compact Orbit

3.

Classical Techniques in Hilbert space

4.

Periodic Points

5.

Iterative Processes

6.

Applications

6.1

Atractors

6.2

Chaos

Chapter 5. Ergodic Theorems

1.

Classical Linear results

2.

Baillon's Theorem

3.

Extensions: Amenable Semi-Group

4.

Asymptotic Behavior

5.

Examples

Chapter 6. Approximation Techniques

Chapter 7. Non-classical Banach Spaces

1.

Orlicz spaces

2.

James' spaces

3.

Tsirelson' spaces

Chapter 8. Metric Spaces

1.

Basic Definitions

2.

Examples

2.1

Unit Ball of

2.2

Unit Ball of the Hilbert space

3.

Nonexpansive Mappings in Metric Spaces

3.1

Convexity Structures in Metric Spaces

3.2

Normal Structure Property in Metric Spaces

3.3

Kirk's analogue in Metric Spaces

3.4

Holomorphic Maps on the Unit Ball of the Hilbert space

4.

Hyperconvex Metric Spaces

4.1

Basic Definitions

4.2

Basic Properties

4.3

Fixed Point Theorems

Chapter 9. Measure of Non-compactness

1.

Basic Definitions

2.

Examples

3.

Condensing Mappings

4.

Approximation

5.

Applications

Chapter 10. Caristi's Fixed Point Theorem

1.

Caristi's Fixed Point Theorem

2.

Ekeland's Principle

3.

Equivalent Theorems

4.

Normal Solvability

5.

Examples and Applications

Chapter 11. Bifurcation Theory

Chapter 12. Multivalued Mappings

Chapter 13. Generalized Structures

1.

Definition of an Ordered Set

2.

Tarski's Fixed point Theorem

3.

Applications

3.1

Godel's Theory

3.2

Graph Theory

3.3

Logic Programming

4.

Generalized Metric Spaces

4.1

Basic Definitions

4.2

Examples

4.3

Hyperconvex Generalized Metric Spaces

4.4

Fixed Point Theorems in Generalized Metric Spaces

4.5

Fixed Point Theorems for Families

4.6

Product Spaces

4.7

Applications

5.

Modular Spaces

Chapter 14. Topological Fixed Point Theory

1.

Finite Dimension

1.1

Brower's Theorem

1.2

Minimax Theorems

1.3

KKM-Maps

1.4

Homology and Kohomology

1.5

Degree Theory

1.6

Sperner's Lemma

1.7

Computation Techniques

1.8

Applications

1.8.1 Game Theory

1.8.2 Economics

1.8.3 Logic Programming

1.9

Hoft Construction

1.10

discrete Brower's Theorem

2.

Infinite Dimension

2.1

Leray-Schauder's Fixed Point Theorem

2.2

Degree Theory

2.3

ANR' Sets

2.4

Nielson Theorems

2.5

Lefschetz Fixed Point Theorems

2.6

Bifurcation Theory

2.7

Applications

3.

Miscellanous

3.1

Complementarity Problems

3.2

Renorming Techniques

3.3

Most Recent Fixed Point Theorems