Brouwer's Fixed Point Theorem with Isolated Fixed Points and His Fan Theorem
We show that Brouwer’s fixed point theorem with isolated fixed points is equivalent to Brouwer’s fan theorem.
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.
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.
Theorem 7. Every positive-valued uniformly continuous function on a compact metric space has positive infimum.
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
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.
This research was partially supported by the Ministry of Education, Science, Sports and Culture of Japan, Grant-in-Aid for Scientific Research (C), 20530165.
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