#### Abstract

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

#### 1. Introduction

It is well known that Brouwer's fixed point theorem cannot be constructively proved.

Kellogg et al.  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  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 ).

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

Let be a compact (totally bounded and complete) metric space, be a point in , and consider a uniformly continuous function from into itself.

According to [3, 4] has an approximate fixed point. It means the following,

Since is arbitrary,

The notion that has at most one fixed point in  is defined as follows.

Definition 1 (at most one fixed point). For all , if , then or .

Now we consider a condition that may have only isolated fixed points. First we recapitulate the compactness of a set in constructive mathematics. We say that is totally bounded if for each there exists a finitely enumerable -approximation to . (A set is finitely enumerable if there exist a natural number and a mapping of the set onto .) An -approximation to is a subset of such that for each there exists in that -approximation with . According to Corollary 2.2.12 of , about totally bounded set we have the following result.

Lemma 2. If is totally bounded, for each there exist totally bounded sets , each of diameter less than or equal to , such that .

Since , we have for some such that .

The definition that a function may have only isolated fixed points is as follows.

Definition 3 (isolated fixed points). There exists with the following property. For each less than or equal to , there exist totally bounded sets , each of diameter less than or equal to , such that , and in each if , then or .

In each , has at most one fixed point. Now we show the following lemma, which is based on Lemma 2 of .

Lemma 4. Let be a uniformly continuous function from into itself. Assume for some defined above. If the following property holds: for each there exists such that if , and , then . Then, there exists a point such that , that is, has a fixed point.

Proof. Choose a sequence in such that . Compute such that for all . Then, for we have . Since is arbitrary, is a Cauchy sequence in and converges to a limit . The continuity of yields , that is, .

Let , the set of all binary sequences, with a finite natural number be the set of finite binary sequences with length . We write , for the elements of . Also for each and each natural number we write is compact under the metric defined by (see [2, 8])

Let be a set of finite binary sequences. is(i)detachable if (ii)a bar if (iii)a uniform bar if

In  the following lemma has been proved (their Lemma 4).

Lemma 5. Let , and a detachable bar for . Then, for each , exists, and the mapping is uniformly continuous in .

Brouwer's fan theorem is as follows.

Theorem 6. Every detachable bar for 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  and the Proof of Proposition in , 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. Consider and a function . Let , and be an infinite tree with at most one infinite path (A tree is a detachable set in which is closed under restriction.) and define Since implies , is uniformly continuous. Thus, has a fixed point. From the definition of its fixed print is an infinite branch. Thus, has an infinite branch. Let be a detachable bar and set Then, is also a detachable bar. For and set If , then . Consider a tree . Define for each a by the following procedure. If , let be any element of . If, let be the largest number such that and define Set  Then, is an infinite tree since it contains each . For all with length we have Let and suppose . Then, there is such that , , and . Thus, or , and so or . Therefore, has at most one infinite branch. From the argument above it has an infinite branch . Since is a bar, there is such that . Thus, , and so . Therefore, , and is a uniform bar. It means that is also a uniform bar.
(2) Assume Brouwer's Fan theorem. Consider a compact metric space and a uniformly continuous function from into itself with isolated fixed points. Then, is uniformly continuous. Let , and , be totally bounded subsets of , each of diameter less than or equal to in Definition 3, such that . Given assume that the set is nonempty and compact (see Theorem 2.2.13 of ). For let Then, is uniformly continuous and positive-valued on . So, by Theorem 7 For each we have Thus, either or . It follows that if , and , then and so . Then, from Lemma 4 there exists a fixed point of in such that . Thus, Brouwer's fan theorem implies his fixed point theorem for uniformly continuous functions with isolated fixed points.

#### Acknowledgment

This research was partially supported by the Ministry of Education, Science, Sports and Culture of Japan, Grant-in-Aid for Scientific Research (C), 20530165.