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Fixed Point Theorems
John Hillas
University of Auckland
Contents
Chapter 1. Fixed Point Theorems
1. Definitions
2. The Contraction Mapping Theorem
3. Brouwer’s Theorem
4. Kakutani’s Theorem
5. Schauder’s Theorem
6. The Fan-Glicksberg Theorem
Bibliography
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CHAPTER 1
Fixed Point Theorems
These notes were written as a reference for the Intensive Course on the Theory
of Strategic Equilibrium, given at SUNY at Stony Brook in the summer of 1990.
They are not complete and are not intended, nor are they suitable, as a substitute
for a good book on the topic.
I have found the part of Franklin (1980) on fixed point theorems to be very
good for the intuition behind many of the results covered in these notes. It also
contains three different proofs of Brouwer’s Theorem, which is not proved here.
I have borrowed heavily from a number of sources. Most of the proof of Schauder’s Theorem comes from Franklin (1980). The proof of Kakutani’s Theorem is a
mixture of Franklin (1980) and Kakutani (1941). The proof of the Fan-Glicksberg
Theorem is from Glicksberg (1952). Many of the definitions in the following section
come from Franklin (1980), Kelley (1955), and Zeidler (1986).
1. Definitions
Before starting to discuss the fixed point theorems that are the subject of these
notes we state for reference a number of relevant definitions. These will, at best,
serve as a revision. If you are not familiar with these concepts you should consult a
good text. A standard text for much of the material is Kelley (1955). The concept
of a topological vector space (also called a topological linear space) is covered in
most books on functional analysis, for example Rudin (1973). The material on
correspondences or multivalued functions is less standard in the mathematics literature, though it is of central importance in economics and game theory. Some of
this material is covered in Berge (1963). The theory of correspondences is covered
in great detail in Klein and Thompson (1984).
Definition 1. A topological space is a pair (X, τ ) where X is a set and τ is a
collection of subsets of X satisfying
(i) both X and ∅ belong to τ , and
(ii) the intersection of a finite number and the union of an arbitrary number of
sets in τ belong to τ .
The sets in τ are called open sets and τ is called the topology. An open neighbourhood
of a point x ∈ X is an open set containing x. The complement of an open set is
called a closed set.
Definition 2. Let (X, τ ) and (X 0 , τ 0 ) be two topological spaces. The function
f : X → X 0 is said to be continuous if for all sets A ∈ τ 0 the set f −1 (A) = { x ∈
X | f (x) ∈ A } ∈ τ . We express this by saying the inverse images of open sets are
open.
Definition 3. A collection of open sets {Oα }α∈A is called an open cover of
the set M if M ⊂ ∪α∈A Oα . A set M is compact if every open cover of M has a
finite subcover, i.e., a finite subcollection of {Oα }α∈A say {O1 , . . . , Ok } such that
M ⊂ ∪ki=1 Oi .
1
2
1. FIXED POINT THEOREMS
Definition 4. A topological space (X, τ ) is called a Hausdorff space if for any
two points x, y ∈ X with x 6= y there are open neighbourhoods of x and y say U (x)
and U (y) such that U (x) ∩ U (y) = ∅.
Just as we can speak of convergent sequences in a metric space (see below) we
can define convergent sequences in a topological space. It happens that this is not
the most useful notion of convergence for a topological space. A somewhat more
general notion is the concept of Moore-Smith convergence defined below.
Definition 5. Sequential Convergence.
Let (X, τ ) be a topological
space. A sequence (xn ) in X is a function from N the natural numbers to X.
We say that almost all xn lie in M if and only if there is N such that xn ∈ M for
all n ≥ N . Also (xn ) is frequently in M if and only if for every m there is n ≥ m
such that xn ∈ M .
• A sequence (xn ) converges to x if and only if every neighbourhood of x
contains almost all xn . We write xn → x.
• A point x is a cluster point of (xn ) if and only if this sequence is frequently
in every neighbourhood of x.
• A function f : X → Y is sequentially continuous at x if f (xn ) → f (x)
whenever xn → x.
• The set M is sequentially closed if whenever xn ∈ M for all n and xn → x
then x ∈ M
Unfortunately the concepts of “sequentially continuous” and “sequentially closed”
do not coincide with the concepts of “continuous” and “closed.” We can however
characterize these concepts in terms of Moore-Smith sequences.
Definition 6. A set A is directed if and only if there is a relation defined
on certain pairs (α, β) with α, β ∈ A such that for all elements of A
(i) α α,
(ii) if α β and β γ then α γ, and
(iii) for α, β ∈ A there exists δ ∈ A such that α δ and β δ.
Definition 7. Let X be a topological space and A a directed set. A MooreSmith sequence (xα ) is a function from A to X.
We say that almost all xα lie in M if and only if there is γ such that xα ∈ M
for all γ α. Also (xα ) is frequently in M if and only if for every γ there is α such
that γ α such that xα ∈ M . (Recall the definition for sequences above.)
A section of the Moore-Smith sequence (xα ) is a set {xα | α γ} for some
fixed γ.
Let (xα ) and (yβ ) be Moore-Smith sequences with index sets A and B respectively. Then (yβ ) is a Moore-Smith subsequence of (xα ) if and only if every section
of (xα ) contains almost all yβ .
Convergence and cluster points are defined as for sequences.
Remark 1. Moore-Smith sequences are sometimes called nets. (For example,
in Kelley (1955).)
Proposition 1. A topological space is Hausdorff if and only if each MooreSmith sequence converges to at most one point.
Proposition 2. Let X be a topological space. The set M ⊂ X is compact if
and only if every Moore-Smith sequence in M has a cluster point in M .
1. DEFINITIONS
3
Definition 8. A metric space is a pair (X, d) where X is a set and d : X ×X →
[0, ∞) is a function satisfying
(i) d(x, y) = d(y, x),
(ii) d(x, z) ≤ d(x, y) + d(y, z), and
(iii) d(x, y) = 0 if and only if x = y.
The function d is called a metric. Property (ii) is called the triangle inequality.
Definition 9. The open -ball about the point x in the metric space (X, d)
denoted B (x) is the set { y ∈ X | d(x, y) < }. A set A ⊂ X is open if for any
x ∈ A there is some > 0 such that B (x) ⊂ A. A set is closed if its complement
is open.
A sequence of points x1 , x2 , . . . in X is said to converge to x ∈ X if for any
> 0 there is some integer N such that for all n > N xn ∈ B (x). The point x is
called the limit of the sequence. Any sequence with a limit is called a convergent
sequence. This allows an alternate definition of a closed set. A set is closed if every
convergent sequence contained in the set converges to a point in the set.
Definition 10. A sequence x1 , x2 , . . . in a metric space (X, d) is called a
Cauchy sequence if limp,q→∞ d(xp , xq ) = 0.
Definition 11. A metric space (X, d) is called a complete metric space if every
Cauchy sequence in (X, d) converges (to a limit in X).
Definition 12. Let (X, d) be a metric space. A function f : X → X is called
a contraction if there is some 0 ≤ θ < 1 such that for all x, y ∈ X d(f (x), f (y)) ≤
θd(x, y).
Definition 13. A (real) linear space or vector space is a set X together with
two operations, addition and scalar multiplication such that for all x, y ∈ X and
all α ∈ R both x + y and αx are in X, and for all x, y, z ∈ X and all α, β ∈ R the
following properties are satisfied
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
x + y = y + x,
(x + y) + z = x + (y + z),
(α + β)x = αx + βx,
α(x + y) = αx + αy,
α(βx) = (αβ)x,
there exists 0 ∈ X such that for all x0 ∈ X x0 + 0 = x0 , and
there exists w ∈ X such that x + 0 = x.
This linear structure allows us to define the notion of a convex set.
Definition 14. A subset S of a vector space X is said to be convex if for all
x, y ∈ S and all λ ∈ [0, 1] we have λx + (1 − λ)y ∈ S.
Definition 15. Let X be a vector space. A function k · k : X → [0, ∞) is
called a norm if and only if for all x, y ∈ X and all α ∈ R
(i) kαxk = |α|kxk,
(ii) kx + yk ≤ kxk + kyk, and
(iii) kxk = 0 if and only if x = 0.
The pair (X, k · k) is then called a normed vector space. Property (ii) is again called
the triangle inequality.
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1. FIXED POINT THEOREMS
A normed vector space (X, k · k) defines a metric space (X, d) with d defined by
d(x, y) = kx − yk. Thus we already have all the objects we defined for metric spaces
automatically defined for normed vector spaces. The following definition identifies
an important class of normed vector spaces.
Definition 16. A Banach space is a complete normed vector space.
We see that a Banach space puts all of the topological structure, all of the
metric structure, and indeed more, on a vector space. It is possible to put the
topological structure on a vector space without going as far as putting on all of the
structure of a Banach space. The following definition does exactly this. Of course
we want the topological structure to “respect” the linear structure of the vector
space.
Definition 17. A topological vector space is a linear space X together with
a topology τ under which both addition and scalar multiplication are continuous
functions.
Definition 18. A topological vector space is said to be locally convex if every
neighbourhood of zero contains a convex neighbourhood of zero.
Definition 19. A correspondence F between the sets X and Y , written F :
X Y , is a function from the set X to the set of all subsets of Y (written P) such
that for all x ∈ X F (x) 6= ∅.
The notion of continuity that we shall be using for correspondences is called
upper hemi-continuity. It is most satisfactorily defined by defining a topology on
the space of all subsets of the space Y . This is done in great detail in Klein and
Thompson [1984]. When we confine our attention to the case that the space Y is
compact and that F takes on as values only closed sets there is an easier way to
make the definition.
Definition 20. The graph of the correspondence F : X Y denoted Gf(F )
is the set { (x, y) ∈ X × Y | y ∈ F (x) }.
Definition 21. Let X be a topological space. Let Y be a compact topological
space. Let F : X Y be a closed valued correspondence. Then F is upper
hemi-continuous if and only if Gf(F ) is closed in X × Y .
Finally we give a number of definitions having to do with simplices which are
used in the proof of Kakutani’s Theorem
Definition 22. An n-dimensional simplex defined by the n+1 points v0 , v1 , . . . , vn
in Rp , p ≥ n, is denoted hv0 , v1 , . . . , vn i and is defined to be the set
n
n
X
X
{ x ∈ Rp | x =
θj vj
θj = 1, θj ≥ 0 }.
j=0
j=0
The simplex is said to be nondegenerate
if the n vectors v1 − v0 , . . . , vn − v0
Pn
are linearly independent. If x = j=0 θj vj the numbers θ0 , . . . , θn are called the
barycentric coordinates of x.
The barycenter of the simplex hv0 , v1 , . . . , vn i is the point having barycentric
coordinates θ0 = θ1 = · · · = θn = 1/(n + 1).
Definition 23. Barycentric Subdivision of a Simplex. A 0-dimensional simplex—
which is a single point—is itself its subdivided simplex. A 1-dimensional simplex
hx0 , x1 i is subdivided into two simplices of the same dimension hx0 , yi and hy, x1 i
where y is the barycenter of hx0 , x1 i.
2. THE CONTRACTION MAPPING THEOREM
5
Now suppose that we have defined the subdivision of all simplices of dimension
less than k and that for dimension ` < k an `-dimensional simplex is subdivided
into (` + 1)! simplices of the same dimension. (This is clearly true for ` = 0 and
` = 1 defined above.) Thus all (k − 1)-dimensional faces of a k-dimensional simplex
hv0 , v1 , . . . , vk i are assumed to have been subdivided to k! simplices of dimension k−
1. Let y be the barycenter of hv0 , v1 , . . . , vk i and let hy0 , y1 , . . . , yk−1 i be a simplex
obtained by subdividing a (k − 1)-dimensional face of hv0 , v1 , . . . , vk i. Since there
are k + 1 faces of dimension k − 1, and each face is subdivided to k! simplices, there
are (k + 1)! simplices such as hy0 , y1 , . . . , yk−1 , yi. Now hv0 , v1 , . . . , vk i is divided
into these (k + 1)! simplices for the following reason. Any point of hv0 , v1 , . . . , vk i
which neither is the barycenter nor lies on any proper face is an a segment joining
the barycenter to a point of some simplex such as hy0 , y1 , . . . , yk−1 i and so belongs
to some hy0 , y1 , . . . , yk−1 , yi.
An m times iterative application of barycentric subdivision to a given simplex
hv0 , v1 , . . . , vk i gives rise to the mth barycentric subdivision by which the simplex
decomposes to derived simplices of order m.
The following Proposition confirms the intuition that the larger m becomes the
smaller the derived simplices become.
Proposition 3. Let T (m) be any derived simplex of order m in the mth barycentric subdivision of S = hv0 , v1 , . . . , vk i. Then their diameters δ(T (m) ) and δ(S)
satisfy
m
k
δ(S).
δ(T (m) ) ≤
k+1
Remark 2. The previous Definition and Proposition are taken from Nikaido
(1968).
2. The Contraction Mapping Theorem
Theorem 1 (Banach 1922). Let (M, d) be a complete metric space. Let f :
M → M be a contraction. That is, for any x, y ∈ M we have d(f (x), f (y)) ≤
θd(x, y) where 0 ≤ θ < 1. Then f has a unique fixed point in M , i.e., there is a
unique point x∗ ∈ M such that x∗ = f (x∗ ).
Proof. Uniqueness. Suppose that there were two fixed points x and y. Then
d(x, y) = d(f (x), f (y)) ≤ θd(x, y)
or
(1 − θ)d(x, y) ≤ 0.
Thus, since (1 − θ) > 0, we have d(x, y) = 0 or x = y. Any two fixed points are
identical.
Existence. Choose any x0 ∈ M . Define the sequence x1 , x2 , . . . by
xn+1 = f (xn ).
We first show that this sequence is a Cauchy sequence. Note that
d(xn , xn+1 ) = d(f (xn−1 ), f (xn )) ≤ θd(xn−1 , xn ) ≤ · · · ≤ θn d(x0 , x1 ).
Now, for any p and q such that p < q the triangle inequality implies that
d(xp , xq )
≤
d(xp , xp+1 ) + · · · + d(xq−1 , xq )
≤
d(x0 , x − 1)(θp + · · · + θq−1 )
≤
d(x0 , x − 1)θp (1 + · · · + θq−p−1 )
< d(x0 , x − 1)θp (1 − θ)−1 .
6
1. FIXED POINT THEOREMS
Now this last term goes to zero as p goes to infinity. Thus the sequence (xn ) is a
Cauchy sequence and, since M is a complete metric space this sequence converges,
say xn → x. We now show that such an x is a fixed point. Consider d(x, f (x)). By
the triangle inequality
d(x, f (x))
≤ d(x, xn+1 ) + d(xn+1 , f (x))
= d(x, xn+1 ) + d(f (xn ), f (x))
≤ d(x, xn+1 ) + θd(xn , x).
and both these last two terms go to zero as n goes to infinity. Thus d(x, f (x)) = 0.
(It is less than any strictly positive number.) And so x = f (x), as required.
3. Brouwer’s Theorem
Theorem 2 (Brouwer 1910). Let f : B n → B n be a continuous function from
the n-ball to itself. Then there is some x∗ ∈ B n such that x∗ = f (x∗ ), i.e., there is
a fixed point.
The following generality is costless.
Theorem 3. Let M ⊂ Rn be a compact convex set. Let f : M → M be a
continuous function. Then there is some x∗ ∈ M such that x∗ = f (x∗ ).
4. Kakutani’s Theorem
Theorem 4 (Kakutani 1941). Let M ⊂ Rn be a compact convex set. Let F :
M M be an upper hemi-continuous convex valued correspondence. Then there is
some x∗ ∈ M such that x∗ ∈ F (x∗ ).
Proof. We first prove Kakutani’s Theorem for the case that M is a nondegenerate simplex in Rn . We then generalize to the case of compact convex subsets
of Rn .
Let M be the simplex defined by the n + 1 points v0 , v1 , . . . , vn . That is
M = hv0 , v1 , . . . , vn i. Now form the mth barycentric subdivision of M . We define
the continuous function f m : M → M as follows: If x is the vertex of any cell of
the subdivision let f m (x) = y for some y ∈ F (x). For any other x we define f m (x)
by extending the function in aP
linear manner inside
Pn each cell. That is, if x is in the
n
cell hx0 , x1 , . . . , xn i and x = j=0 θj xj with j=0 θj = 1, θj ≥ 0 then f m (x) is
Pn
defined to be j=0 θj f m (xj ).
Now we apply Brouwer’s Theorem to obtain a fixed point of the map f m , say
m
x . If xm is a vertex of one of the cells in the subdivision then we are done since
xm = f m (xm ) ∈ F (xm ). If xm is not a vertex of one of the cells then let the cell in
m
m
m m
m
which it does lie be hxm
0 , x1 , . . . , xn i, and let θ0 , θ1 , . . . , θn be the barycentric
m
coordinates of x relative to that cell. Thus
xm =
n
X
θjm xm
j ,
j=0
and
(1)
m
x
m
m
= f (x ) =
n
X
θjm yjm
j=0
m
where yjm = f m (xm
j ) ∈ F (xj ) for all j = 0, 1, . . . , n. Now we choose a subsequence
of m → ∞ say mk → ∞ such that xmk → x∗ , θjmk → θj , and yjmk → yj . Also, since
5. SCHAUDER’S THEOREM
7
the cells shrink to points as mk → ∞ each of the vertices of the cell containing xmk
k
also converges to x∗ , i.e., xm
→ x∗ . Thus from (1)
j
x∗ =
n
X
θj yj .
j=0
Also since F is upper hemi-continuous yj ∈ F (x∗ ). Since F (x∗ ) is convex and x∗
is a convex combination of the yj ’s this implies that x∗ ∈ F (x∗ ) as required.
Now what happens if M is not a simplex? We take some simplex M 0 containing
M and a retraction ψ : M 0 → M (i.e., a continuous function taking M 0 to M that
leaves all points of M fixed.) Then F 0 : M 0 M 0 defined by F 0 (x) = F (ψ(x))
is clearly an upper hemi-continuous correspondence and a fixed point of F 0 clearly
lies in M and so is also a fixed point of F .
Remark: This proof is essentially the one originally given by Kakutani. Another way of presenting the proof is by showing that for any convex valued upper
hemi-continuous correspondence there is a continuous function whose graph is close
to the graph of the correspondence. In the proof we have given we have essentially
constructed such a function.
5. Schauder’s Theorem
Theorem 5 (Schauder 1930). Let M be a nonempty convex subset of a Banach
space X. Let N be a compact subset of M . Let f : M → N be a continuous function.
Then there is some x∗ ∈ M such that x∗ = f (x∗ ).
Proof. For any fixed > 0 cover N with a collection of -balls. Since N is
compact we may take a finite subset that also covers N , say the balls centred at
y1 , y2 , . . . , yn . Thus every point in N is within of at least some yj . Now let
n
X
M = {
θi yi |
i=1
n
X
θi = 1, θi ≥ 0}.
i=1
Clearly M ⊂ M since M is convex. Also M lies in the finite dimensional
linear subspace of X spanned by y1 , y2 , . . . , yn . There is a natural identification of
this space with the Euclidean space of the same dimension and M then becomes
a compact convex subset of this space to which we may apply Brouwer’s Theorem.
We now define a continuous function p : N → M that for any y ∈ N chooses
a point in M that approximates y to within , i.e., kp (y) − yk < for all y in N .
We do so in the following way. First let
0
if kyi − yk ≥ ϕi (y) =
− kyi − yk otherwise.
Clearly, each ϕi is continuous and for all y ∈ N there is at least one i such that
ϕi (y) > 0. Now let
ϕi (y)
θi (y) = Pn
j=1 ϕj (y)
and
p (y) =
n
X
i=1
θi (y)yi .
8
1. FIXED POINT THEOREMS
This function maps N into M . Recall that θi (y) = 0 unless kyi − yk < . Thus
kp (y) − yk
=
≤
<
k
n
X
θi (y)(yi − y)k
i=1
n
X
θi (y)k(yi − y)k
i=1
n
X
θi (y) = .
i=1
We can now define the continuous function f : M → M by
f (x) = p (f (x)) for x ∈ M .
As we said earlier we can apply Brouwer’s Theorem to obtain a fixed point of this
function, say x = f (x ). Let y = f (x ). Since y ∈ N and N is compact we may
take a sequence k → 0 such that yk converges to a limit, say y ∗ , in N . Now
x = f (x ) = p (f (x )) = p (y ).
And so
kx − y k = kp (y ) − y k < .
∗
Therefore xk → y , and, since f is continuous f (y ∗ ) = y ∗ .
The following is an immediate corollary to Schauder’s Theorem.
Theorem 6 (Tychonoff 1935). Let M be a nonempty compact convex subset
of a Banach space. Let f : M → M be a continuous function. Then there is some
x∗ ∈ M such that x∗ = f (x∗ ).
6. The Fan-Glicksberg Theorem
Theorem 7 ((Fan, 1952; Glicksberg, 1952)). Let M be a nonempty compact
convex subset of a convex Hausdorff topological vector space. Let F : M M
be an upper hemi-continuous convex valued correspondence. Then there is some
x∗ ∈ M such that x∗ ∈ F (x∗ ).
Proof. First fix V a closed neighbourhood of 0. Now since M is compact we
can find a finite set {y1 , y2 , . . . , yn } ⊂ M such that
M⊂
n
[
({yi } + V ).
i=1
Now let
n
n
X
X
MV = {
θi yi |
θi = 1, θi ≥ 0}.
i=1
i=1
Let
FV (x) = (F (x) + V ) ∩ MV .
Now FV is clearly a convex valued correspondence from MV to MV . Moreover
FV is upper hemi-continuous since if xδ → x, yδ ∈ FV (xδ ), and yδ → y then
yδ ∈ (F (xδ ) + V ) ∩ MV .
Thus there exist zδ ∈ F (xδ ) and vδ ∈ V such that yδ = zδ + vδ ∈ MV . Since
zδ ∈ F (xδ ) ⊂ M and M is compact {zδ } has a cluster point z ∈ M . Also z ∈ F (x)
since xδ → x and F is upper hemi-continuous. Since vδ = yδ − zδ and yδ → y, {vδ }
must have v = y − z as a cluster point and , since vδ ∈ V and V closed v ∈ V .
6. THE FAN-GLICKSBERG THEOREM
9
Thus y = z + v ∈ F (x) + V . Also y ∈ MV since yδ ∈ FV (xδ ) ⊂ MV , yδ → y, and
MV is closed. Thus
y ∈ (F (x) + V ) ∩ MV = FV (x).
Now, as we did in proving Schauder’s Theorem, we can identify MV with a
subset of Euclidean space and this allows us to apply Kakutani’s Theorem to find
a fixed point xV ∈ MV such that xV ∈ FV (xV ).
Now the closed neighbourhoods of 0 are naturally ordered by set inclusion so
that {xV } is a directed system and so has a cluster point x. Now form
∆0 = {(V, U ) | xV ∈ U, U a neighbourhood of x}
which is a directed set, and let
xV,U = xV for (V, U ) ∈ ∆0
so that
xV,U −→
x
0
(2)
∆
(3)
0
∀V0 ∃(V, U ) ∈ ∆ such that V ⊂ V0 .
Then we have zV,U such that
xV,U − zV,U ∈ V and zV,U ∈ F (xV,U )
from xV ∈ F (xV ) + V , so that zV,U →∆0 x by (3). But then x ∈ F (x) since F is
upper hemi-continuous.
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1. FIXED POINT THEOREMS
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1
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