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A fixed point of a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f0405201.png" /> on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f0405202.png" /> is a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f0405203.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f0405204.png" />. Proofs of the existence of fixed points and methods for finding them are important mathematical problems, since the solution of every equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f0405205.png" /> reduces, by transforming it to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f0405206.png" />, to finding a fixed point of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f0405207.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f0405208.png" /> is the identity mapping. Depending on the structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f0405209.png" />, or the properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052010.png" />, there arise various fixed-point principles. Of greatest interest is the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052011.png" /> is a topological space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052012.png" /> is a continuous operator in some sense.
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The simplest among them is the contraction-mapping principle (cf. also [[Contracting-mapping principle|Contracting-mapping principle]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052013.png" /> be a complete metric space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052014.png" /> an operator such that
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
A fixed point of a mapping  $  F $
 +
on a set  $  X $
 +
is a point  $  x \in X $
 +
for which  $  F ( x) = x $.
 +
Proofs of the existence of fixed points and methods for finding them are important mathematical problems, since the solution of every equation  $  f ( x) = 0 $
 +
reduces, by transforming it to  $  x \pm  f ( x) = x $,
 +
to finding a fixed point of the mapping  $  F = I \pm  f $,
 +
where  $  I $
 +
is the identity mapping. Depending on the structure on  $  X $,
 +
or the properties of  $  F $,
 +
there arise various fixed-point principles. Of greatest interest is the case when  $  X $
 +
is a topological space and  $  F $
 +
is a continuous operator in some sense.
  
Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052016.png" /> has precisely one fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052017.png" />, which can be obtained as the limit of successive approximations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052019.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052020.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052021.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052022.png" />. The condition (1) cannot, in general, be replaced by
+
The simplest among them is the contraction-mapping principle (cf. also [[Contracting-mapping principle|Contracting-mapping principle]]). Let  $  X $
 +
be a complete metric space and $  F: X \rightarrow X $
 +
an operator such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{1 }
 +
\rho ( F ( x),\
 +
F ( y))  \leq  \
 +
q \rho ( x, y),\ \
 +
0 < q < 1.
 +
$$
  
however, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052024.png" /> is compact, then (2), as before, ensures that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052025.png" /> has a unique fixed point.
+
Then  $  F $
 +
has precisely one fixed point  $  \overline{x}\; $,  
 +
which can be obtained as the limit of successive approximations  $  x _ {n} = F ( x _ {n - 1 }  ) $,  
 +
$  n = 1, 2 \dots $
 +
where  $  x _ {0} \in X $
 +
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  $  \{ x _ {n} \} $
 +
to  $  \overline{x}\; $.  
 +
The condition (1) cannot, in general, be replaced by
  
More general is the generalized contraction principle. Suppose, as above, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052026.png" /> is a complete metric space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052027.png" /> and
+
$$ \tag{2 }
 +
\rho ( F ( x),\
 +
F ( y))  < \
 +
\rho ( x, y);
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052028.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
however, if  $  X $
 +
is compact, then (2), as before, ensures that  $  F $
 +
has a unique fixed point.
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052030.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052031.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052032.png" /> has a unique fixed point. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052033.png" /> is a Banach space, then (1) is nothing but a Lipschitz condition for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052034.png" /> 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.
+
More general is the generalized contraction principle. Suppose, as above, that  $  X $
 +
is a complete metric space, $  F: X \rightarrow X $
 +
and
  
There are other conditions of a topological nature that guarantee the existence of a fixed point for an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052035.png" />. The best known of them is Schauder's principle. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052036.png" /> be a Banach space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052037.png" /> a [[Completely-continuous operator|completely-continuous operator]] mapping a bounded convex closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052038.png" /> into itself. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052039.png" /> has in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052040.png" /> 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.
+
$$ \tag{3 }
 +
\rho ( F ( x),\
 +
F ( y))  \leq  \
 +
q ( \alpha , \beta )
 +
\rho ( x, y)
 +
$$
  
Example (Peano's theorem). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052041.png" /> be continuous in both variables in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052043.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052044.png" /> in this domain. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052045.png" />, then on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052046.png" /> there is at least one solution of the equation
+
for  $  \alpha \leq  \rho ( x, y) \leq  \beta $,
 +
where  $  q ( \alpha , \beta ) < 1 $
 +
for  $  0 < \alpha \leq  \beta < \infty $.  
 +
Then  $  F $
 +
has a unique fixed point. If $  X $
 +
is a Banach space, then (1) is nothing but a Lipschitz condition for  $  F $
 +
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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052047.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
There are other conditions of a topological nature that guarantee the existence of a fixed point for an operator  $  F $.
 +
The best known of them is Schauder's principle. Let  $  X $
 +
be a Banach space and  $  F $
 +
a [[Completely-continuous operator|completely-continuous operator]] mapping a bounded convex closed set  $  C \subset  X $
 +
into itself. Then  $  F $
 +
has in  $  C $
 +
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  $  f ( t, x) $
 +
be continuous in both variables in a domain  $  | t - t _ {0} | \leq  a $,
 +
$  | x - x _ {0} | \leq  b $,
 +
and let  $  \beta = \max  | f ( t, x) | $
 +
in this domain. If  $  h = \min \{ a, b/ \beta \} $,
 +
then on the interval  $  [ t _ {0} - h, t _ {0} + h] $
 +
there is at least one solution of the equation
 +
 
 +
$$ \tag{4 }
 +
x  ^  \prime  ( t)  = \
 +
f ( t, x)
 +
$$
  
 
such that
 
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052048.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
$$ \tag{5 }
 +
x ( t _ {0} )  = x _ {0} .
 +
$$
  
 
Equation (4) together with (5) is equivalent to the integral equation
 
Equation (4) together with (5) is equivalent to the integral equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052049.png" /></td> </tr></table>
+
$$
 +
x ( t)  = \
 +
x _ {0} +
 +
\int\limits _ {t _ {0} } ^ { t }
 +
f ( \tau , x ( \tau ))  d \tau .
 +
$$
  
 
The operator
 
The operator
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052050.png" /></td> </tr></table>
+
$$
 +
F ( x)  = \
 +
x _ {0} +
 +
\int\limits _ {t _ {0} } ^ { t }
 +
f ( \tau , x ( \tau ))  d \tau
 +
$$
  
maps, under the conditions of the theorem, the ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052051.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052052.png" /> into itself and it is completely continuous on this ball. Therefore, by Schauder's principle, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052053.png" /> has a fixed point, which is also a solution of the Cauchy problem (see [[#References|[4]]], [[#References|[5]]]). A generalization of Schauder's principle is Tikhonov's principle. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052054.png" /> be a separable locally convex space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052055.png" /> a continuous operator mapping a convex compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052056.png" /> into itself. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052057.png" /> has in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052058.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052059.png" /> is convex, for without this Schauder's theorem and its generalizations become false. One can combine Schauder's principle and the contraction principle. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052060.png" /> be an operator that transforms a bounded closed convex set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052061.png" /> of a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052062.png" /> into itself and that can be represented in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052063.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052064.png" /> is completely continuous and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052065.png" /> contracting. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052066.png" /> has at least one fixed point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052067.png" />.
+
maps, under the conditions of the theorem, the ball $  \| x - x _ {0} \| \leq  b $
 +
of the space $  C [ t _ {0} - h, t _ {0} + h] $
 +
into itself and it is completely continuous on this ball. Therefore, by Schauder's principle, $  F $
 +
has a fixed point, which is also a solution of the Cauchy problem (see [[#References|[4]]], [[#References|[5]]]). A generalization of Schauder's principle is Tikhonov's principle. Let $  X $
 +
be a separable locally convex space and $  F $
 +
a continuous operator mapping a convex compact set $  C \subset  X $
 +
into itself. Then $  F $
 +
has in $  C $
 +
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 $  C $
 +
is convex, for without this Schauder's theorem and its generalizations become false. One can combine Schauder's principle and the contraction principle. Let $  F $
 +
be an operator that transforms a bounded closed convex set $  C $
 +
of a Banach space $  X $
 +
into itself and that can be represented in the form $  F = F _ {1} + F _ {2} $,  
 +
where $  F _ {1} $
 +
is completely continuous and $  F _ {2} $
 +
contracting. Then $  F $
 +
has at least one fixed point in $  C $.
  
Principles of Schauder type can be extended in the following way to non-compact operators. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052068.png" /> be a bounded set in a complete metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052069.png" />. The measure of non-compactness <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052070.png" /> of this set is defined as the greatest lower bound of those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052071.png" /> for which there is a finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052072.png" />-net for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052073.png" /> (cf. [[Net (of sets in a topological space)|Net (of sets in a topological space)]]). For compact sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052074.png" />. An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052075.png" /> is said to be compressing if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052076.png" /> for any non-compact bounded set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052077.png" />. Suppose that a compressing operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052078.png" /> transforms a bounded convex closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052079.png" /> into itself. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052080.png" /> has at least one fixed point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052081.png" />. 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.
+
Principles of Schauder type can be extended in the following way to non-compact operators. Let $  M $
 +
be a bounded set in a complete metric space $  X $.  
 +
The measure of non-compactness $  \chi ( M) $
 +
of this set is defined as the greatest lower bound of those $  \epsilon > 0 $
 +
for which there is a finite $  \epsilon $-
 +
net for $  M $(
 +
cf. [[Net (of sets in a topological space)|Net (of sets in a topological space)]]). For compact sets $  \chi ( M) = 0 $.  
 +
An operator $  F: X \rightarrow X $
 +
is said to be compressing if $  \chi ( F ( M)) < \chi ( M) $
 +
for any non-compact bounded set $  M \subset  X $.  
 +
Suppose that a compressing operator $  F $
 +
transforms a bounded convex closed set $  C \subset  X $
 +
into itself. Then $  F $
 +
has at least one fixed point in $  C $.  
 +
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052082.png" /> of a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052083.png" /> in a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052084.png" /> a non-degenerate vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052085.png" /> is given, that is, every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052086.png" /> is put in correspondence with a non-zero vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052087.png" />. To such a field one can assign under certain conditions an integer, the so-called index (rotation) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052088.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052089.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052090.png" />. Suppose, to begin with, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052091.png" /> is finite dimensional and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052092.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052093.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052094.png" /> is defined as the topological degree of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052095.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052096.png" /> onto the unit sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052097.png" /> (cf. [[Degree of a mapping|Degree of a mapping]]). Now let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052098.png" /> be an infinite-dimensional Banach space and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f04052099.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520100.png" /> is a completely-continuous operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520101.png" />. Such fields are called completely continuous.
+
An appeal to more subtle topological concepts leads to stronger criteria for the existence of fixed points. Suppose that on the boundary $  \partial  M $
 +
of a bounded domain $  M $
 +
in a Banach space $  X $
 +
a non-degenerate vector field $  \Phi $
 +
is given, that is, every point $  x \in \partial  M $
 +
is put in correspondence with a non-zero vector $  \Phi ( x) $.  
 +
To such a field one can assign under certain conditions an integer, the so-called index (rotation) $  \gamma ( \Phi , \partial  M) $
 +
of $  \Phi $
 +
on $  \partial  M $.  
 +
Suppose, to begin with, that $  X $
 +
is finite dimensional and that $  \Phi $
 +
is continuous on $  \partial  M $.  
 +
Then $  \gamma ( \Phi , \partial  M) $
 +
is defined as the topological degree of the mapping $  \Psi ( x) = \Phi ( x)/ \| \Phi ( x) \| $
 +
of $  \partial  M $
 +
onto the unit sphere $  \| x \| = 1 $(
 +
cf. [[Degree of a mapping|Degree of a mapping]]). Now let $  X $
 +
be an infinite-dimensional Banach space and let $  \Phi ( x) = x - F ( x) $,  
 +
where $  F $
 +
is a completely-continuous operator on $  \overline{M}\; $.  
 +
Such fields are called completely continuous.
  
Suppose that a finite-dimensional subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520102.png" /> gives a fairly good approximation to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520103.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520104.png" /> is the projection operator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520105.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520106.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520107.png" /> is sufficiently small for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520108.png" />, then the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520109.png" /> is also continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520110.png" /> and its index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520111.png" /> does not depend on the choice of the approximating subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520112.png" /> nor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520113.png" />. This number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520114.png" /> is called the index of the completely-continuous vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520115.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520116.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520117.png" />. An important property of the rotation is the fact that it does not change under homotopy transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520118.png" />.
+
Suppose that a finite-dimensional subspace $  X _ {n} $
 +
gives a fairly good approximation to $  F ( \overline{M}\; ) $
 +
and that $  P _ {n} $
 +
is the projection operator of $  F ( \overline{M}\; ) $
 +
onto $  X _ {n} $.  
 +
If $  \| P _ {n} F ( x) - F ( x) \| $
 +
is sufficiently small for $  x \in \partial  M $,  
 +
then the field $  \Phi _ {n} = I - P _ {n} F $
 +
is also continuous on $  \partial  M \cap X _ {n} $
 +
and its index $  \gamma _ {n} $
 +
does not depend on the choice of the approximating subspace $  X _ {n} $
 +
nor on $  P _ {n} $.  
 +
This number $  \gamma _ {n} $
 +
is called the index of the completely-continuous vector field $  \Phi $
 +
on $  \partial  M $
 +
and is denoted by $  \gamma ( \Phi , \partial  M) $.  
 +
An important property of the rotation is the fact that it does not change under homotopy transformations of $  \Phi $.
  
The Leray–Schauder principle. Suppose that on the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520119.png" /> of a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520120.png" /> in a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520121.png" /> one is given a completely-continuous vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520122.png" /> that is non-degenerate on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520123.png" /> and suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520124.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520125.png" /> vanishes at at least one point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520126.png" />, that is, the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520127.png" /> has in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520128.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520129.png" /> one constructs a family of fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520130.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520131.png" />, such that they are all homotopic to each other and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520132.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520133.png" />. If for another <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520134.png" /> the index of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520135.png" /> is easy to compute, and is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520136.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520137.png" /> 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.
+
The Leray–Schauder principle. Suppose that on the closure $  \overline{M}\; $
 +
of a bounded domain $  M $
 +
in a Banach space $  X $
 +
one is given a completely-continuous vector field $  \Phi $
 +
that is non-degenerate on $  \partial  M $
 +
and suppose that $  \gamma ( \Phi , \partial  M) \neq 0 $.  
 +
Then $  \Phi $
 +
vanishes at at least one point $  x \in M $,  
 +
that is, the operator $  F $
 +
has in $  M $
 +
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 $  \Phi ( x) $
 +
one constructs a family of fields $  \Phi ( x, \lambda ) $,  
 +
$  \alpha \leq  \lambda \leq  \beta $,  
 +
such that they are all homotopic to each other and $  \Phi ( x, \lambda _ {0} ) = \Phi ( x) $
 +
for some $  \lambda _ {0} \in [ \alpha , \beta ] $.  
 +
If for another $  \lambda \in [ \alpha , \beta ] $
 +
the index of $  \Phi ( x, \lambda ) $
 +
is easy to compute, and is $  k $,  
 +
then $  \gamma ( \Phi , \partial  M) = k $
 +
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520138.png" /> is called non-expansive if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520139.png" />. Suppose that the Banach space is uniformly convex (for example, a Hilbert space, cf. [[Banach space|Banach space]]) and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520140.png" /> is a non-expansive operator taking a bounded closed convex set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520141.png" /> into itself. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520142.png" /> has in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520143.png" /> at least one fixed point.
+
By strengthening the conditions on the space one can weaken the restrictions on the operator. For example, an operator $  F $
 +
is called non-expansive if $  \rho ( F ( x), F ( y)) \leq  \rho ( x, y) $.  
 +
Suppose that the Banach space is uniformly convex (for example, a Hilbert space, cf. [[Banach space|Banach space]]) and that $  F $
 +
is a non-expansive operator taking a bounded closed convex set $  C \subset  X $
 +
into itself. Then $  F $
 +
has in $  C $
 +
at least one fixed point.
  
All preceding fixed-point principles assume the continuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520144.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520145.png" /> is endowed with the structure of a partially ordered set, then in some cases the requirement of continuity can be dropped.
+
All preceding fixed-point principles assume the continuity of $  F $.  
 +
If $  X $
 +
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520146.png" /> be a [[Complete lattice|complete lattice]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520147.png" /> an isotone operator (cf. [[Isotone mapping|Isotone mapping]]) from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520148.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520149.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520150.png" /> has at least one fixed point. There is another version of this principle. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520151.png" /> be a conditionally-complete lattice, that is, every bounded subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520152.png" /> has in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520153.png" /> a least upper and a greatest lower bound. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520154.png" /> is isotone and maps the ordered interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520155.png" /> into itself, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520156.png" /> has in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520157.png" /> at least one fixed point.
+
The Birkhoff–Tarski principle. Let $  X $
 +
be a [[Complete lattice|complete lattice]] and $  F $
 +
an isotone operator (cf. [[Isotone mapping|Isotone mapping]]) from $  X $
 +
to $  X $.  
 +
Then $  F $
 +
has at least one fixed point. There is another version of this principle. Let $  X $
 +
be a conditionally-complete lattice, that is, every bounded subset in $  X $
 +
has in $  X $
 +
a least upper and a greatest lower bound. If $  F $
 +
is isotone and maps the ordered interval $  [ a, b] = \{ {x } : {a \leq  x \leq  b } \} \subset  X $
 +
into itself, then $  F $
 +
has in $  [ a, b] $
 +
at least one fixed point.
  
A combination of topological and order conditions leads to new fixed-point principles. For example, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520158.png" /> be a partially ordered Banach space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520159.png" /> a continuous isotone operator mapping the ordered interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520160.png" /> into itself. If the semi-order on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520161.png" /> is regular, that is, if every monotone increasing order-bounded sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520162.png" /> converges in the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520163.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520164.png" /> has in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520165.png" /> at least one fixed point. Here the conditions of the theorem do not require a lattice order on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520166.png" />, that is, not for every pair of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520167.png" /> their sup and inf need exist in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520168.png" />.
+
A combination of topological and order conditions leads to new fixed-point principles. For example, let $  X $
 +
be a partially ordered Banach space and $  F $
 +
a continuous isotone operator mapping the ordered interval $  [ a, b] $
 +
into itself. If the semi-order on $  X $
 +
is regular, that is, if every monotone increasing order-bounded sequence $  \{ x _ {n} \} \subset  X $
 +
converges in the norm of $  X $,  
 +
then $  F $
 +
has in $  [ a, b] $
 +
at least one fixed point. Here the conditions of the theorem do not require a lattice order on $  X $,  
 +
that is, not for every pair of elements $  x, y \in X $
 +
their sup and inf need exist in $  X $.
  
Finally, a fixed point of a linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520169.png" /> is an eigen element of it corresponding to the eigen value 1.
+
Finally, a fixed point of a linear operator $  F $
 +
is an eigen element of it corresponding to the eigen value 1.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.A. Lyusternik,  V.I. Sobolev,  "Elemente der Funktionalanalysis" , Akademie Verlag  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.A. Krasnosel'skii,  "Topological methods in the theory of nonlinear integral equations" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.A. Krasnosel'skii,  P.P. Zabreiko,  "Geometric methods of non-linear analysis" , Springer  (1983)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.V. Nemytskii,  ''Uspekhi Mat. Nauk'' , '''1''' :  1  (1946)  pp. 141–174</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J. Leray,  J. Schauder,  "Topology and functional equations"  ''Uspekhi Mat. Nauk'' , '''1''' :  3–4  (1946)  pp. 71–95  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B.N. Sadovskii,  "Limit-compact and condensing operators"  ''Russian Math. Surveys'' , '''27''' :  1  (1972)  pp. 85–155  ''Uspekhi Mat. Nauk'' , '''27''' :  1  (1972)  pp. 81–146</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.A. Lyusternik,  V.I. Sobolev,  "Elemente der Funktionalanalysis" , Akademie Verlag  (1968)  (Translated from Russian) {{MR|0539144}} {{ZBL|0044.32501}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.A. Krasnosel'skii,  "Topological methods in the theory of nonlinear integral equations" , Pergamon  (1964)  (Translated from Russian) {{MR|}} {{ZBL|0111.30303}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.A. Krasnosel'skii,  P.P. Zabreiko,  "Geometric methods of non-linear analysis" , Springer  (1983)  (Translated from Russian) {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.V. Nemytskii,  ''Uspekhi Mat. Nauk'' , '''1''' :  1  (1946)  pp. 141–174 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J. Leray,  J. Schauder,  "Topology and functional equations"  ''Uspekhi Mat. Nauk'' , '''1''' :  3–4  (1946)  pp. 71–95  (In Russian) {{MR|}} {{ZBL|0060.27704}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B.N. Sadovskii,  "Limit-compact and condensing operators"  ''Russian Math. Surveys'' , '''27''' :  1  (1972)  pp. 85–155  ''Uspekhi Mat. Nauk'' , '''27''' :  1  (1972)  pp. 81–146 {{MR|0428132}} {{ZBL|}} </TD></TR></table>
 
 
 
 
  
 
====Comments====
 
====Comments====
Line 64: Line 262:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Dugundji,  A. Granas,  "Fixed point theory" , PWN  (1982)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Dugundji,  A. Granas,  "Fixed point theory" , PWN  (1982) {{MR|0660439}} {{ZBL|0483.47038}} </TD></TR></table>
  
A fixed point of a fractional-linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520170.png" /> of the closed complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520171.png" /> is a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520172.png" /> for which
+
A fixed point of a fractional-linear transformation $  A $
 +
of the closed complex plane $  \overline{\mathbf C}\; $
 +
is a point $  \rho \in \overline{\mathbf C}\; $
 +
for which
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520173.png" /></td> </tr></table>
+
$$
 +
\rho  = \
 +
 
 +
\frac{a \rho + b }{c \rho + d }
 +
,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520174.png" /></td> </tr></table>
+
$$
 +
A: \overline{\mathbf C}\; \rightarrow  \overline{\mathbf C}\; ,\ \
 +
z  \rightarrow  w  = A ( z)  = \
 +
 
 +
\frac{az + b }{cz + d }
 +
,
 +
$$
  
is the fractional-linear transformation, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520175.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520176.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520177.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520178.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520179.png" /> is the identity transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520180.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520181.png" /> has one or two fixed points. By means of the fixed points one can classify the fractional-linear mappings (cf. [[Fractional-linear mapping|Fractional-linear mapping]]) (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040520/f040520182.png" /> is excluded from the discussion).
+
is the fractional-linear transformation, $  a, b, c, d \in \mathbf C $
 +
and $  ad - bc \neq 0 $,  
 +
$  \overline{\mathbf C}\; = \mathbf C \cup \{ \infty \} $.  
 +
If $  A \neq I $(
 +
where $  I $
 +
is the identity transformation $  w = z $),  
 +
then $  A $
 +
has one or two fixed points. By means of the fixed points one can classify the fractional-linear mappings (cf. [[Fractional-linear mapping|Fractional-linear mapping]]) ( $  I $
 +
is excluded from the discussion).
  
 
''O.M. Fomenko''
 
''O.M. Fomenko''
Line 82: Line 302:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Schwerdtfeger,  "Geometry of complex numbers" , Dover, reprint  (1979)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Schwerdtfeger,  "Geometry of complex numbers" , Dover, reprint  (1979) {{MR|0620163}} {{ZBL|0484.51007}} </TD></TR></table>
  
 
For a fixed point of a system of ordinary differential equations or of a dynamical system see [[Equilibrium position|Equilibrium position]].
 
For a fixed point of a system of ordinary differential equations or of a dynamical system see [[Equilibrium position|Equilibrium position]].

Latest revision as of 19:39, 5 June 2020


A fixed point of a mapping $ F $ on a set $ X $ is a point $ x \in X $ for which $ F ( x) = x $. Proofs of the existence of fixed points and methods for finding them are important mathematical problems, since the solution of every equation $ f ( x) = 0 $ reduces, by transforming it to $ x \pm f ( x) = x $, to finding a fixed point of the mapping $ F = I \pm f $, where $ I $ is the identity mapping. Depending on the structure on $ X $, or the properties of $ F $, there arise various fixed-point principles. Of greatest interest is the case when $ X $ is a topological space and $ F $ is a continuous operator in some sense.

The simplest among them is the contraction-mapping principle (cf. also Contracting-mapping principle). Let $ X $ be a complete metric space and $ F: X \rightarrow X $ an operator such that

$$ \tag{1 } \rho ( F ( x),\ F ( y)) \leq \ q \rho ( x, y),\ \ 0 < q < 1. $$

Then $ F $ has precisely one fixed point $ \overline{x}\; $, which can be obtained as the limit of successive approximations $ x _ {n} = F ( x _ {n - 1 } ) $, $ n = 1, 2 \dots $ where $ x _ {0} \in X $ 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 $ \{ x _ {n} \} $ to $ \overline{x}\; $. The condition (1) cannot, in general, be replaced by

$$ \tag{2 } \rho ( F ( x),\ F ( y)) < \ \rho ( x, y); $$

however, if $ X $ is compact, then (2), as before, ensures that $ F $ has a unique fixed point.

More general is the generalized contraction principle. Suppose, as above, that $ X $ is a complete metric space, $ F: X \rightarrow X $ and

$$ \tag{3 } \rho ( F ( x),\ F ( y)) \leq \ q ( \alpha , \beta ) \rho ( x, y) $$

for $ \alpha \leq \rho ( x, y) \leq \beta $, where $ q ( \alpha , \beta ) < 1 $ for $ 0 < \alpha \leq \beta < \infty $. Then $ F $ has a unique fixed point. If $ X $ is a Banach space, then (1) is nothing but a Lipschitz condition for $ F $ 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 $ F $. The best known of them is Schauder's principle. Let $ X $ be a Banach space and $ F $ a completely-continuous operator mapping a bounded convex closed set $ C \subset X $ into itself. Then $ F $ has in $ C $ 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 $ f ( t, x) $ be continuous in both variables in a domain $ | t - t _ {0} | \leq a $, $ | x - x _ {0} | \leq b $, and let $ \beta = \max | f ( t, x) | $ in this domain. If $ h = \min \{ a, b/ \beta \} $, then on the interval $ [ t _ {0} - h, t _ {0} + h] $ there is at least one solution of the equation

$$ \tag{4 } x ^ \prime ( t) = \ f ( t, x) $$

such that

$$ \tag{5 } x ( t _ {0} ) = x _ {0} . $$

Equation (4) together with (5) is equivalent to the integral equation

$$ x ( t) = \ x _ {0} + \int\limits _ {t _ {0} } ^ { t } f ( \tau , x ( \tau )) d \tau . $$

The operator

$$ F ( x) = \ x _ {0} + \int\limits _ {t _ {0} } ^ { t } f ( \tau , x ( \tau )) d \tau $$

maps, under the conditions of the theorem, the ball $ \| x - x _ {0} \| \leq b $ of the space $ C [ t _ {0} - h, t _ {0} + h] $ into itself and it is completely continuous on this ball. Therefore, by Schauder's principle, $ F $ 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 $ X $ be a separable locally convex space and $ F $ a continuous operator mapping a convex compact set $ C \subset X $ into itself. Then $ F $ has in $ C $ 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 $ C $ is convex, for without this Schauder's theorem and its generalizations become false. One can combine Schauder's principle and the contraction principle. Let $ F $ be an operator that transforms a bounded closed convex set $ C $ of a Banach space $ X $ into itself and that can be represented in the form $ F = F _ {1} + F _ {2} $, where $ F _ {1} $ is completely continuous and $ F _ {2} $ contracting. Then $ F $ has at least one fixed point in $ C $.

Principles of Schauder type can be extended in the following way to non-compact operators. Let $ M $ be a bounded set in a complete metric space $ X $. The measure of non-compactness $ \chi ( M) $ of this set is defined as the greatest lower bound of those $ \epsilon > 0 $ for which there is a finite $ \epsilon $- net for $ M $( cf. Net (of sets in a topological space)). For compact sets $ \chi ( M) = 0 $. An operator $ F: X \rightarrow X $ is said to be compressing if $ \chi ( F ( M)) < \chi ( M) $ for any non-compact bounded set $ M \subset X $. Suppose that a compressing operator $ F $ transforms a bounded convex closed set $ C \subset X $ into itself. Then $ F $ has at least one fixed point in $ C $. 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 $ \partial M $ of a bounded domain $ M $ in a Banach space $ X $ a non-degenerate vector field $ \Phi $ is given, that is, every point $ x \in \partial M $ is put in correspondence with a non-zero vector $ \Phi ( x) $. To such a field one can assign under certain conditions an integer, the so-called index (rotation) $ \gamma ( \Phi , \partial M) $ of $ \Phi $ on $ \partial M $. Suppose, to begin with, that $ X $ is finite dimensional and that $ \Phi $ is continuous on $ \partial M $. Then $ \gamma ( \Phi , \partial M) $ is defined as the topological degree of the mapping $ \Psi ( x) = \Phi ( x)/ \| \Phi ( x) \| $ of $ \partial M $ onto the unit sphere $ \| x \| = 1 $( cf. Degree of a mapping). Now let $ X $ be an infinite-dimensional Banach space and let $ \Phi ( x) = x - F ( x) $, where $ F $ is a completely-continuous operator on $ \overline{M}\; $. Such fields are called completely continuous.

Suppose that a finite-dimensional subspace $ X _ {n} $ gives a fairly good approximation to $ F ( \overline{M}\; ) $ and that $ P _ {n} $ is the projection operator of $ F ( \overline{M}\; ) $ onto $ X _ {n} $. If $ \| P _ {n} F ( x) - F ( x) \| $ is sufficiently small for $ x \in \partial M $, then the field $ \Phi _ {n} = I - P _ {n} F $ is also continuous on $ \partial M \cap X _ {n} $ and its index $ \gamma _ {n} $ does not depend on the choice of the approximating subspace $ X _ {n} $ nor on $ P _ {n} $. This number $ \gamma _ {n} $ is called the index of the completely-continuous vector field $ \Phi $ on $ \partial M $ and is denoted by $ \gamma ( \Phi , \partial M) $. An important property of the rotation is the fact that it does not change under homotopy transformations of $ \Phi $.

The Leray–Schauder principle. Suppose that on the closure $ \overline{M}\; $ of a bounded domain $ M $ in a Banach space $ X $ one is given a completely-continuous vector field $ \Phi $ that is non-degenerate on $ \partial M $ and suppose that $ \gamma ( \Phi , \partial M) \neq 0 $. Then $ \Phi $ vanishes at at least one point $ x \in M $, that is, the operator $ F $ has in $ M $ 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 $ \Phi ( x) $ one constructs a family of fields $ \Phi ( x, \lambda ) $, $ \alpha \leq \lambda \leq \beta $, such that they are all homotopic to each other and $ \Phi ( x, \lambda _ {0} ) = \Phi ( x) $ for some $ \lambda _ {0} \in [ \alpha , \beta ] $. If for another $ \lambda \in [ \alpha , \beta ] $ the index of $ \Phi ( x, \lambda ) $ is easy to compute, and is $ k $, then $ \gamma ( \Phi , \partial M) = k $ 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 $ F $ is called non-expansive if $ \rho ( F ( x), F ( y)) \leq \rho ( x, y) $. Suppose that the Banach space is uniformly convex (for example, a Hilbert space, cf. Banach space) and that $ F $ is a non-expansive operator taking a bounded closed convex set $ C \subset X $ into itself. Then $ F $ has in $ C $ at least one fixed point.

All preceding fixed-point principles assume the continuity of $ F $. If $ X $ 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 $ X $ be a complete lattice and $ F $ an isotone operator (cf. Isotone mapping) from $ X $ to $ X $. Then $ F $ has at least one fixed point. There is another version of this principle. Let $ X $ be a conditionally-complete lattice, that is, every bounded subset in $ X $ has in $ X $ a least upper and a greatest lower bound. If $ F $ is isotone and maps the ordered interval $ [ a, b] = \{ {x } : {a \leq x \leq b } \} \subset X $ into itself, then $ F $ has in $ [ a, b] $ at least one fixed point.

A combination of topological and order conditions leads to new fixed-point principles. For example, let $ X $ be a partially ordered Banach space and $ F $ a continuous isotone operator mapping the ordered interval $ [ a, b] $ into itself. If the semi-order on $ X $ is regular, that is, if every monotone increasing order-bounded sequence $ \{ x _ {n} \} \subset X $ converges in the norm of $ X $, then $ F $ has in $ [ a, b] $ at least one fixed point. Here the conditions of the theorem do not require a lattice order on $ X $, that is, not for every pair of elements $ x, y \in X $ their sup and inf need exist in $ X $.

Finally, a fixed point of a linear operator $ F $ is an eigen element of it corresponding to the eigen value 1.

References

[1] L.A. Lyusternik, V.I. Sobolev, "Elemente der Funktionalanalysis" , Akademie Verlag (1968) (Translated from Russian) MR0539144 Zbl 0044.32501
[2] M.A. Krasnosel'skii, "Topological methods in the theory of nonlinear integral equations" , Pergamon (1964) (Translated from Russian) Zbl 0111.30303
[3] M.A. Krasnosel'skii, P.P. Zabreiko, "Geometric methods of non-linear analysis" , Springer (1983) (Translated from Russian)
[4] V.V. Nemytskii, Uspekhi Mat. Nauk , 1 : 1 (1946) pp. 141–174
[5] J. Leray, J. Schauder, "Topology and functional equations" Uspekhi Mat. Nauk , 1 : 3–4 (1946) pp. 71–95 (In Russian) Zbl 0060.27704
[6] B.N. Sadovskii, "Limit-compact and condensing operators" Russian Math. Surveys , 27 : 1 (1972) pp. 85–155 Uspekhi Mat. Nauk , 27 : 1 (1972) pp. 81–146 MR0428132

Comments

See also Brouwer theorem; Lefschetz theorem.

References

[a1] J. Dugundji, A. Granas, "Fixed point theory" , PWN (1982) MR0660439 Zbl 0483.47038

A fixed point of a fractional-linear transformation $ A $ of the closed complex plane $ \overline{\mathbf C}\; $ is a point $ \rho \in \overline{\mathbf C}\; $ for which

$$ \rho = \ \frac{a \rho + b }{c \rho + d } , $$

where

$$ A: \overline{\mathbf C}\; \rightarrow \overline{\mathbf C}\; ,\ \ z \rightarrow w = A ( z) = \ \frac{az + b }{cz + d } , $$

is the fractional-linear transformation, $ a, b, c, d \in \mathbf C $ and $ ad - bc \neq 0 $, $ \overline{\mathbf C}\; = \mathbf C \cup \{ \infty \} $. If $ A \neq I $( where $ I $ is the identity transformation $ w = z $), then $ A $ has one or two fixed points. By means of the fixed points one can classify the fractional-linear mappings (cf. Fractional-linear mapping) ( $ I $ is excluded from the discussion).

O.M. Fomenko

Comments

Fractional-linear transformations are also called Möbius transformations. For their classification according to fixed points see also [a1].

References

[a1] H. Schwerdtfeger, "Geometry of complex numbers" , Dover, reprint (1979) MR0620163 Zbl 0484.51007

For a fixed point of a system of ordinary differential equations or of a dynamical system see Equilibrium position.

How to Cite This Entry:
Fixed point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fixed_point&oldid=17902
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article