Fixed Point Theorems And Their Applications

Ioannis Farmakis, Martin Moskowitz

• Publisher: WSPC
• Year: 2013
• eBook ISBN: 9789814458931
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

This is the only book that deals comprehensively with fixed point theorems throughout mathematics. Their importance is due, as the book demonstrates, to their wide applicability. Beyond the first chapter, each of the other seven can be read independently of the others so the reader has much flexibility to follow his/her own interests. The book is written for graduate students and professional mathematicians and could be of interest to physicists, economists and engineers.

Contents:

• Early Fixed Point Theorems
• Fixed Point Theorems in Analysis
• The Lefschetz Fixed Point Theorem
• Fixed Point Theorems in Geometry
• Fixed Points of Volume Preserving Maps
• Borel's Fixed Point Theorem in Algebraic Groups
• Miscellaneous Fixed Point Theorems
• A Fixed Point Theorem in Set Theory

Readership: Graduate students and professionals in analysis, approximation theory, algebra and geometry.
Key Features:

• Our book gives a complete treatment of the diverse and manifold use of fixed point theorems and their many applications throughout Mathematics and is another example of the unity within Mathematics
• As such it serves as a valuable resource for researchers in diverse fields of Mathematics
• It also serves as solid introduction for students to several subjects in modern Mathematics such as Functional Analysis, Topology, Differential Geometry, Dynamical Systems and Algebraic Groups

Chapter 1

Early Fixed Point
Theorems

1.1The Picard-Banach Theorem

One of the earliest and best known fixed point theorems is that of Picard-Banach 1.1.1. Either explicitly or implicitly this theorem is the usual way one proves local existence and uniqueness theorems for systems of ordinary differential equations (see for example [79], sections 7.3 and 7.5). This theorem also can be used to prove the inverse function theorem (see [79], pp. 179-181).

Theorem 1.1.1.Let (X, d) be a complete metric space and f : X → X be a contraction mapping, that is, one in which there is a 1 > b > 0 so that for all x, y ∈ X, d(f(x), f(y)) ≤ bd(x, y). Then f has a unique fixed point.

Proof. Choose a point x₁ ∈ X (in an arbitrary manner) and construct the sequence x_n ∈ X by x_n+1 = fⁿ(x₁), n ≥ 2. Then x_n is a Cauchy sequence. For n ≥ m,

But

The latter term is ≤ (b^n−m−1 + ... + b + 1)d(f(x₁), x₁) which is itself

. Since 0 < b < 1 this geometric series converges to

. Since b^m → 0 we see for n and m sufficiently large, given є > 0,

Hence x_n is Cauchy. Because X is complete x_n → x for some x ∈ X. As f is a contraction map it is (uniformly) continuous. Hence f(x_n) → f(x). But as a subsequence f(x_n) → x so the uniqueness of limits tells us x = f(x).

Now suppose there was another fixed point y ∈ X. Then d(f(x), f(y)) = d(x, y) ≤ bd(x, y) so that if d(x, y) ≠ 0 we conclude b ≥ 1, a contradiction. Therefore d(x, y) = 0 and x = y.

We remark that the reader may wish to consult Bessaga ([11]), or Jachymski, ([60]) where the following converse to the Picard-Banach theorem has been proved.

Theorem 1.1.2. Let f: X → X be a self map of a set X and 0 < b < 1. If fⁿ has at most 1 fixed point for every integer n, then there exists a metric d on X for which d(f(x), f(y)) ≤ bd(x, y), for all x and y ∈ X. If, in addition, some fⁿ has a fixed point, then d can be chosen to be complete.

As a corollary to the Picard-Banach theorem we have the following precursor to the Brouwer theorem.

Corollary 1.1.3. Let X be the closed unit ball in

and f: X → X be a nonexpanding map; that is one that satisfies d(f(x), f(y)) ≤ d(x, y) for all x, y ∈ X . Then f has a fixed point.

Proof. For positive integers, n, define

. Then each f_n is a contraction mapping of X. Since X is compact, it is complete. By the contraction mapping principle each f_n has a fixed point, x_n ∈ X and since X is compact x_n has a subsequence converging to say x. Taking limits as n → ∞ shows x is fixed by f.

We now state the Brouwer fixed point theorem.

Theorem 1.1.4. Any continuous map f of the closed unit ball Bⁿ in

to itself has a fixed point.

In other words, if one stirs a mug of coffee then at any given time there is at least one particle of coffee that is in exactly the position it started in.

Of course, since the Brouwer theorem is stated in ...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Contents
• Preface and Acknowledgments
• Introduction
• 1 Early Fixed Point Theorems
• 2 Fixed Point Theorems in Analysis
• 3 The Lefschetz Fixed Point Theorem
• 4 Fixed Point Theorems in Geometry
• 5 Fixed Points of Volume Preserving Maps
• 6 Borel’s Fixed Point Theorem in Algebraic Groups
• 7 Miscellaneous Fixed Point Theorems
• 8 A Fixed Point Theorem in Set Theory
• Afterword
• Bibliography
• Index