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
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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 xn ∈ X by xn+1 = fn(x1), n ≥ 2. Then xn is a Cauchy sequence. For n ≥ m,
But
The latter term is ≤ (bn−m−1 + ... + b + 1)d(f(x1), x1) which is itself
. Since 0 < b < 1 this geometric series converges to
. Since bm → 0 we see for n and m sufficiently large, given є > 0,
Hence xn is Cauchy. Because X is complete xn → x for some x ∈ X. As f is a contraction map it is (uniformly) continuous. Hence f(xn) → f(x). But as a subsequence f(xn) → 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 fn is a contraction mapping of X. Since X is compact, it is complete. By the contraction mapping principle each fn has a fixed point, xn ∈ X and since X is compact xn 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 ...
