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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 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 ...
