# Brouwer's Fixed Point Theorem with Isolated Fixed Points and His Fan Theorem

Authors: Yasuhito Tanaka
Journal: ISRN Computational Mathematics
 (2012)

---

## Abstract

We show that Brouwer’s fixed point theorem with isolated fixed points is equivalent to Brouwer’s fan theorem.


---

## Body

## 1. Introduction

It is well known that Brouwer's fixed point theorem cannot be constructively proved.Kellogg et al. [1] provided a constructive proof of Brouwer's fixed point theorem. But it is not constructive from the view point of constructive mathematics á la Bishop. It is sufficient to say that one-dimensional case of Brouwer's fixed point theorem, that is, the intermediate value theorem is nonconstructive (see [2, 3]).Sperner's lemma which is used to prove Brouwer's theorem, however, can be constructively proved. Some authors have presented an approximate version of Brouwer's theorem using Sperner's Lemma (see [3, 4]). Thus, Brouwer's fixed point theorem is constructively, in the sense of constructive mathematics á la Bishop, proved in its approximate version.Recently Berger and Ishihara [5] showed that the following theorem is equivalent to Brouwer's fan theorem.Each uniformly continuous function from a compact metric space into itself with at most one fixed point and approximate fixed points has a fixed point.In this paper we require a more general condition that each uniformly continuous function from a compact metric space into itself may have only isolated fixed points and show that the proposition that such a function has a fixed point is equivalent to Brouwer's fan theorem.In another paper we have shown that if a uniformly continuous function in a compact metric space satisfies stronger condition,sequential local non-constancy, then without the fan theorem we can constructively show that it has an exact fixed point (see [6]).


## 2. Brouwer's Fixed Point Theorem with Isolated Fixed Points and His Fan Theorem

LetX be a compact (totally bounded and complete) metric space, x be a point in X, and consider a uniformly continuous function f from X into itself.According to [3, 4] f has an approximate fixed point. It means the following,
(1)For eachɛ>0there existsx∈Xsuchthat|x-f(x)|<ɛ.Sinceɛ>0 is arbitrary,
(2)inf⁡x∈X|x-f(x)|=0.The notion thatf has at most one fixed point in [5] is defined as follows.Definition 1 (at most one fixed point).
For allx,y∈X, if x≠y, then f(x)≠x or f(y)≠y.Now we consider a condition thatf may have only isolated fixed points. First we recapitulate the compactness of a set in constructive mathematics. We say that X is totally bounded if for each ɛ>0 there exists a finitely enumerable ɛ-approximation to X. (A set S is finitely enumerable if there exist a natural number N and a mapping of the set {1,2,…,N} onto S.) An ɛ-approximation to X is a subset of X such that for each x∈X there exists y in that ɛ-approximation with |x-y|<ɛ. According to Corollary 2.2.12 of [7], about totally bounded set we have the following result.Lemma 2.
IfX is totally bounded, for each ɛ>0 there exist totally bounded sets H1,…,Hn, each of diameter less than or equal to ɛ, such that X=⋃i=1nHi.Sinceinfx∈X|x-f(x)|=0, we have infx∈Hi|x-f(x)|=0 for some Hi⊂X such that X=⋃i=1nHi.The definition that a function may have only isolated fixed points is as follows.Definition 3 (isolated fixed points).
There existsɛ¯>0 with the following property. For each ɛ>0 less than or equal to ɛ¯, there exist totally bounded sets H1,…,Hn, each of diameter less than or equal to ɛ, such that X=⋃i=1nHi, and in each Hi if x≠y, then f(x)≠x or f(y)≠y.In eachHi, f has at most one fixed point. Now we show the following lemma, which is based on Lemma 2 of [8].Lemma 4.
Letf be a uniformly continuous function from X into itself. Assume infx∈Hif(x)=0 for some Hi⊂X defined above. If the following property holds: for eachɛ>0 there exists δ>0 such that if x,y∈Hi, |f(x)-x|≤δ and |f(y)-y|≤δ, then |x-y|≤ɛ.Then, there exists a point z∈Hi such that f(z)=z, that is, f has a fixed point.Proof.
Choose a sequence(xn)n≥1 in Hi such that |f(xn)-xn|→0. Compute N such that |f(xn)-xn|<δ for all n≥N. Then, for m,n≥N we have |xm-xn|≤ɛ. Since ɛ>0 is arbitrary, (xn)n≥1 is a Cauchy sequence in Hi and converges to a limit z∈Hi. The continuity of f yields |f(z)-z|=0, that is, f(z)=z.LetY={0,1}ℕ, the set of all binary sequences, {0,1}n with a finite natural number n be the set of finite binary sequences with length n+1. We write x,y,…, for the elements (xn)n≥0,(yn)n≥0,… of Y. Also for each x∈Y and each natural number n we write
(3)x¯(n)=(x0,x1,…,xn-1).Y is compact under the metric defined by (see [2, 8])
(4)|x-y|=inf⁡{2-n:x¯(n)=y¯(n)}.LetB be a set of finite binary sequences. B is(i)
detachable if
(5)∀x∈Y∀n(x¯(n)∈B∨x¯(n)∉B);(ii)
a bar  if
(6)∀x∈Y∃n(x¯(n)∈B);(iii)
a uniform bar  if
(7)∃N∀x∈Y∃n≤N(x¯(n)∈B).In [8] the following lemma has been proved (their Lemma  4).Lemma 5.
LetY={0,1}ℕ, and B a detachable bar for Y. Then, for each x∈Y,
(8)σ(x)=inf⁡{n:x¯(n)∈B}
exists, and the mapping x→4-σ(x) is uniformly continuous in Y.Brouwer's fan theorem is as follows.Theorem 6.
Every detachable bar for{0,1}ℕ is a uniform bar.It has been shown in [2, 5] that this theorem is equivalent to the following theorem.Theorem 7.
Every positive-valued uniformly continuous function on a compact metric space has positive infimum.Now, according to the Proof of Theorem  5 in [8] and the Proof of Proposition in [5], we show the following result.Theorem 8.
Brouwer's fixed point theorem with isolated fixed points in a compact metric space is equivalent to Brouwer's fan theorem.Proof.
(1)  Assume that each uniformly continuous function from a compact metric space into itself with isolated fixed points has a fixed point. It implies that each uniformly continuous function from a compact metric space into itself with at most one fixed point has a fixed point. ConsiderY={0,1}ℕ and a function φ:Y→Y. Let x∈Y, and T be an infinite tree with at most one infinite path (A tree is a detachable set in {0,1}ℕ which is closed under restriction.) and define
(9)φ(x)n={xnifx¯(n)∈T,1-xnifx¯(n)∉T.
Sincex¯(n)=y¯(n) implies φ(x)¯(n)=φ(y)¯(n), φ is uniformly continuous. Thus, φ has a fixed point. From the definition of φ its fixed print is an infinite branch. Thus, T has an infinite branch. Let B be a detachable bar and set(10)B′={x:∃n(x¯(n)∈B)}.
Then, B′ is also a detachable bar. For x=(x0,…,xn-1)  and   y=(y0,…,ym-1)  set
(11)x*y=(x0,…,xn-1,y0,…,ym-1).
If x∈B, then x*y∈B′. Consider a tree {0,1}ℕ∖B′. Define for each n a un∈{0,1}n by the following procedure. If {0,1}n⊄B′, let un be any element of {0,1}n∖B′. If{0,1}n⊂B′, let k be the largest number such that {0,1}k⊄B′ and define
(12)un=uk*(0,…,0︸)n-ktimes.
Set
(13)T=({0,1}N∖B′)∪{un:{0,1}n⊂B′}.
Then, T is an infinite tree since it contains each un. For all x with length n we have
(14)x∈B′∩T⟹x=un.
Let x,y∈{0,1}ℕ and suppose x≠y. Then, there is n such that x¯(n)∈B′, y¯(n)∈B′, and x¯(n)≠y¯(n). Thus, x¯(n)≠un or y¯(n)≠un, and so x¯(n)∉T or y¯(n)∉T. Therefore, T has at most one infinite branch. From the argument above it has an infinite branch x. Since B′ is a bar, there is m such that x¯(m)∈B′. Thus, x¯(m)∈B′∩T, and so x¯(m)=um. Therefore, {0,1}m⊂B′, and B′ is a uniform bar. It means that B is also a uniform bar.
(2)  Assume Brouwer's Fan theorem. Consider a compact metric spaceX and a uniformly continuous function f from X into itself with isolated fixed points. Then, |x-f(x)| is uniformly continuous. Let x∈X, and Hi,  i=1,…,n be totally bounded subsets of X, each of diameter less than or equal to ɛ¯ in Definition 3, such that X=⋃i=1nHi. Given ɛ>0 assume that the set
(15)K={(x,y)∈Hi×Hi:|x-y|≥ɛ},
is nonempty and compact (see Theorem  2.2.13 of [7]). For x,y∈Hi let
(16)F(x,y)=|x-f(x)|+|y-f(y)|.
Then, F is uniformly continuous and positive-valued on K. So, by Theorem 7(17)0<δ=13inf⁡{F(x,y):(x,y)∈K}.
For each (x,y)∈K we have
(18)|x-f(x)|+|y-f(y)|=F(x,y)>2δ.
Thus, either |x-f(x)|>δ or |y-f(y)|>δ. It follows that if x,y∈Hi, |x-f(x)|≤δ and |y-f(y)|≤δ, then (x,y)∉K and so |x-y|≤ɛ. Then, from Lemma 4 there exists a fixed point of f in Hi*⊂X such that infx∈Hi*|x-f(x)|=0. Thus, Brouwer's fan theorem implies his fixed point theorem for uniformly continuous functions with isolated fixed points.



---

*Source: 843256-2012-01-26.xml*