A
fixed point of a mapping
on a set
is a point
for which
.
Proofs of the existence of fixed points and
methods for finding them are important mathematical
problems, since the solution of every equation
reduces, by transforming it to
,
to finding a fixed point of the mapping
,
where
is the identity mapping. Depending on the structure on
,
or the properties of
,
there arise various
fixed-point principles.
Of greatest interest is the case when
is a topological space and
is a continuous operator in some sense.
The simplest among them is the
contraction-mapping principle
(cf. also
Contracting-mapping principle).
Let
be a complete metric space and
an operator such that
Then
has precisely one fixed point
,
which can be obtained as the limit of successive approximations
,
where
is arbitrary. This principle not only establishes the existence
of a fixed point, but also indicates a method for finding
it, and it is fairly easy to estimate the rate of convergence of the sequence
to
.
The condition
(1)
cannot, in general, be replaced by
however, if
is compact, then
(2),
as before, ensures that
has a unique fixed point.
More general is the
generalized contraction principle.
Suppose, as above, that
is a complete metric space,
and
for
,
where
for
.
Then
has a unique fixed point. If
is a Banach space, then
(1)
is nothing but a Lipschitz condition for
with a constant less than 1. The contraction principle is
extensively used to prove the existence and uniqueness of
solutions of algebraic, differential, integral, and other equations
and to find approximate solutions of them.
There are other conditions of a topological nature that guarantee
the existence of a fixed point for an operator
.
The best known of them is
Schauder's principle.
Let
be a Banach space and
a
completely-continuous operator
mapping a bounded convex closed set
into itself. Then
has in
at least one fixed point. However, in this case the
question of the number of fixed points remains open and
there is no indication of a method for finding them.
Example
(Peano's theorem).
Let
be continuous in both variables in a domain
,
,
and let
in this domain. If
,
then on the interval
there is at least one solution of the equation
such that
Equation
(4)
together with
(5)
is equivalent to the integral equation
The operator
maps, under the conditions of the theorem, the ball
of the space
into itself and it is completely continuous
on this ball. Therefore, by Schauder's principle,
has a fixed point, which is also a solution of the Cauchy problem (see
[4],
[5]).
A generalization of Schauder's principle is
Tikhonov's principle.
Let
be a separable locally convex space and
a continuous operator mapping a convex compact set
into itself. Then
has in
at least one fixed point. There are also other generalizations of
Schauder's principle, among them to many-valued mappings, but
in all cases one has to assume that
is convex, for without this Schauder's theorem and
its generalizations become false. One can combine
Schauder's principle and the contraction principle. Let
be an operator that transforms a bounded closed convex set
of a Banach space
into itself and that can be represented in the form
,
where
is completely continuous and
contracting. Then
has at least one fixed point in
.
Principles of Schauder type can be extended in
the following way to non-compact operators. Let
be a bounded set in a complete metric space
.
The
measure of non-compactness
of this set is defined as the greatest lower bound of those
for which there is a finite
-net
for
(cf.
Net (of sets in a topological space)).
For compact sets
.
An operator
is said to be
compressing
if
for any non-compact bounded set
.
Suppose that a compressing operator
transforms a bounded convex closed set
into itself. Then
has at least one fixed point in
.
In Banach spaces one can introduce other measures
of non-compactness, and by varying them one can obtain various
versions of the theorem, which make it possible to
prove the existence of solutions of various differential, integral and
other equations with operators that need not be completely continuous.
An appeal to more subtle topological concepts leads to stronger criteria
for the existence of fixed points. Suppose that on the boundary
of a bounded domain
in a Banach space
a non-degenerate vector field
is given, that is, every point
is put in correspondence with a non-zero vector
.
To such a field one can assign under certain conditions an integer, the so-called
index
(rotation)
of
on
.
Suppose, to begin with, that
is finite dimensional and that
is continuous on
.
Then
is defined as the topological degree of the mapping
of
onto the unit sphere
(cf.
Degree of a mapping).
Now let
be an infinite-dimensional Banach space and let
,
where
is a completely-continuous operator on
.
Such fields are called
completely continuous.
Suppose that a finite-dimensional subspace
gives a fairly good approximation to
and that
is the projection operator of
onto
.
If
is sufficiently small for
,
then the field
is also continuous on
and its index
does not depend on the choice of the approximating subspace
nor on
.
This number
is called the
index of the completely-continuous vector field
on
and is denoted by
.
An important property of the rotation is the fact
that it does not change under homotopy transformations of
.
The
Leray–Schauder principle.
Suppose that on the closure
of a bounded domain
in a Banach space
one is given a completely-continuous vector field
that is non-degenerate on
and suppose that
.
Then
vanishes at at least one point
,
that is, the operator
has in
at least one fixed point. The invariance of the
index under homotopy transformations makes it possible to compute the
index in the following way. From the given field
one constructs a family of fields
,
,
such that they are all homotopic to each other and
for some
.
If for another
the index of
is easy to compute, and is
,
then
too. By this device, using the degree of
a mapping to establish that completely-continuous operators have a
fixed point, one can prove that some fairly complicated
partial differential equations of high order have solutions.
By strengthening the conditions on the space one can weaken
the restrictions on the operator. For example, an operator
is called
non-expansive
if
.
Suppose that the Banach space is uniformly convex (for example, a Hilbert space, cf.
Banach space)
and that
is a non-expansive operator taking a bounded closed convex set
into itself. Then
has in
at least one fixed point.
All preceding fixed-point principles assume the continuity of
.
If
is endowed with the structure of a partially ordered set, then
in some cases the requirement of continuity can be dropped.
The
Birkhoff–Tarski principle.
Let
be a
complete lattice
and
an isotone operator (cf.
Isotone mapping)
from
to
.
Then
has at least one fixed point. There is another version of this principle. Let
be a
conditionally-complete lattice,
that is, every bounded subset in
has in
a least upper and a greatest lower bound. If
is isotone and maps the ordered interval
into itself, then
has in
at least one fixed point.
A combination of topological and order conditions leads
to new fixed-point principles. For example, let
be a partially ordered Banach space and
a continuous isotone operator mapping the ordered interval
into itself. If the semi-order on
is regular, that is, if every monotone increasing order-bounded sequence
converges in the norm of
,
then
has in
at least one fixed point. Here the conditions of
the theorem do not require a lattice order on
,
that is, not for every pair of elements
their sup and inf need exist in
.
Finally, a
fixed point of a linear operator
is an eigen element of it corresponding to the eigen value 1.