A Proof of Constructive Version of Brouwer's Fixed Point Theorem with Uniform Sequential Continuity

Abstract

It is often said that Brouwer's fixed point theorem cannot be constructively proved. On the other hand, Sperner's lemma, which is used to prove Brouwer's theorem, can be constructively proved. Some authors have presented a constructive (or an approximate) version of Brouwer's fixed point theorem using Sperner's lemma. They, however, assume uniform continuity of functions. We consider uniform sequential continuity of functions. In classical mathematics, uniform continuity and uniform sequential continuity are equivalent. In constructive mathematics a la Bishop, however, uniform sequential continuity is weaker than uniform continuity. We will prove a constructive version of Brouwer's fixed point theorem in an -dimensional simplex for uniformly sequentially continuous functions. We follow the Bishop style constructive mathematics.

1. Introduction

It is often said that Brouwer’s fixed point theorem cannot be constructively proved.

Reference [1] provided a constructive proof of Brouwer’s fixed point theorem. But it is not constructive from the view point of constructive mathematics a 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] or [3]. Brouwer’s fixed point theorem can be constructively, in the sense of constructive mathematics a la Bishop, proved only approximately. The existence of an exact fixed point of a function which satisfies some property of local non-constancy may be constructively proved.

On the other hand, Sperner’s lemma, which is used to prove Brouwer’s theorem, can be constructively proved. Some authors have presented a constructive (or an approximate) version of Brouwer’s fixed point theorem using Sperner’s lemma. See [3, 4]. They, however, assume uniform continuity of functions. We consider uniform sequential continuity of functions according to [5]. In classical mathematics uniform continuity and uniform sequential continuity are equivalent. In constructive mathematics a la Bishop, however, uniform sequential continuity is weaker than uniform continuity. Also in constructive mathematics, sequential continuity is weaker than continuity, and uniform continuity (resp., uniform sequential continuity) is stronger than continuity (resp., sequential continuity) even in a compact space. See, for example, [6]. As stated in [7], all proofs of the equivalence between continuity and sequential continuity involve the law of excluded middle, and so they are nonconstructive. We will prove a constructive version of Brouwer’s fixed point theorem in an -dimensional simplex for uniformly sequentially continuous functions.

In the next section, we consider Sperner’s lemma. In Section 3, we prove a constructive version of Brouwer’s fixed point theorem for uniformly sequentially continuous functions from an -dimensional simplex to itself using Sperner’s lemma. We follow the Bishop style constructive mathematics according to [2, 8, 9].

2. Sperner’s Lemma

Let denote an -dimensional simplex. is a finite natural number. For example, a 2-dimensional simplex is a triangle. Let partition or triangulate the simplex. Figure 1 is an example of partition (triangulation) of a 2-dimensional simplex. In a 2-dimensional case, we divide each side of in equal segments and draw the lines parallel to the sides of . is a finite natural number. Then, the 2-dimensional simplex is partitioned into triangles. We consider partition of inductively for cases of higher dimension. In a 3-dimensional case, each face of is a 2-dimensional simplex, and so it is partitioned into triangles in the above-mentioned way, and draw the planes parallel to the faces of . Then, the 3-dimensional simplex is partitioned into trigonal pyramids and similarly for cases of higher dimension.

Figure 1 

Partition and labeling of 2-dimensional simplex.

Let denote the set of small -dimensional simplices of constructed by partition. Vertices of these small simplices of are labeled with the numbers subject to the following rules. (1)The vertices of are, respectively, labeled with 0 to . We label a point with 0, a point with 1, a point with point with . That is, a vertex whose th coordinate () is 1 and all other coordinates are 0 is labeled with .(2) If a vertex of is contained in an -dimensional face of , then this vertex is labeled with some number which is the same as the number of one of the vertices of that face.(3)If a vertex of is contained in an -dimensional face of , then this vertex is labeled with some number which is the same as the number of one of the vertices of that face and so on for cases of lower dimension.(4) A vertex contained inside of is labeled with an arbitrary number among .

A small simplex of which is labeled with the numbers is called a fully labeled simplex. Sperner’s lemma is stated as follows.

Lemma 2.1 (Sperner’s lemma). If one labels the vertices of following the rules (1)~(4), then there are an odd number of fully labeled simplices, and so there exists at least one fully labeled simplex.

Proof. About constructive proofs of Sperner’s lemma see [10] or [11].

Since and partition of are finite, the number of small simplices constructed by partition is also finite. Thus, we can constructively find a fully labeled -dimensional simplex of through finite steps.

3. Constructive Version of Brouwer’s Fixed Point Theorem with Uniform Sequential Continuity

Let us consider a function from an -dimensional simplex to itself. Denote a point in by . Uniform continuity, sequential continuity, and uniform sequential continuity of functions are defined as follows.

Definition 3.1 (Uniform continuity). A function is uniformly continuous in if, for any and there, exists such that depends on only .

Definition 3.2 (Sequential continuity). A function is sequentially continuous at in if, for sequences and in ,

Definition 3.3 (Uniform sequential continuity). A function is uniformly sequentially continuous in if, for sequences , , , and in , means where is a real number and and are natural numbers. Similarly, means is a natural number.

In classical mathematics, uniform continuity and uniform sequential continuity of functions are equivalent. But in constructive mathematics a ala Bishop, uniform sequential continuity is weaker than uniform continuity and uniform sequential continuity is stronger than sequential continuity.

An approximate fixed point of is defined as follows.

Definition 3.4 (Approximate fixed point). For each ,   is an approximate fixed point of if we have

Now, we show the following theorem.

Theorem 3.5 (Constructive version of Brouwer’s fixed point theorem with uniform sequential continuity). Any uniformly sequentially continuous function from an -dimensional simplex to itself has an approximate fixed point for each .

Proof. (1) First, we show that we can partition so that the conditions for Sperner’s lemma are satisfied. We partition according to the method in Sperner’s lemma, and label the vertices of simplices constructed by partition of . It is important how to label the vertices contained in the faces of . Let be the set of small simplices constructed by partition of , let be a vertex of a simplex of , and denote the th component of by . Then, we label a vertex according to the following rule: where is a positive number. If there are multiple ’s which satisfy this condition, then we label conveniently for the conditions for Sperner’s lemma to be satisfied. We do not randomly label the vertices.
For example, let be a point contained in an -dimensional face of such that for one among (its th coordinate is 0). With , we have or .
In constructive mathematics, for any real number , we cannot prove that or , that or or . But for any distinct real numbers , , and such that , we can prove that or .
When , from , , and , Then, for at least one (denote it by ), we have and we label with , where is one of the numbers which satisfy . Since , does not satisfy this condition. Assume that implies . Since , we obtain Then, for a positive number , we have There is at least one which satisfies . Denote it by , and we label with . is one of the numbers other than such that is satisfied. itself satisfies this condition (). But, since there is a number other than which satisfies this condition, we can select a number other than . We have proved that we can label the vertices contained in an -dimensional face of such that for one among with the numbers other than . By similar procedures, we can show that we can label the vertices contained in an -dimensional face of such that for two ’s among with the numbers other than those ’s, and so on.
Consider the case where . We see that, when or , and so for at least one (denote it by ), we have , and we label with . On the other hand, when and , we have Then, for a positive number , we have Thus, there is at least one which satisfies . Denote it by , and we label with .
Next, consider the case where for all other than . If, for some ,  , then we have and label with . On the other hand, if for all , then we obtain . It implies . Thus, we can label with .
Therefore, the conditions for Sperner’s lemma are satisfied and there exists an odd number of fully labeled simplices in .
(2) Consider a sequence of partitions of and a sequence of fully labeled simplices . The larger , the finer partition. The larger , the smaller the diameter of a fully labeled simplex. Let and be the vertices of a fully labeled simplex . We name these vertices so that are labeled, respectively, with . The values of at theses vertices are and . We can consider sequences of vertices of fully labeled simplices. Denote them by , and . And consider sequences of the values of at vertices of fully labeled simplices. Denote them by , and . By the uniform sequential continuity of , for . means and means Consider a fully labeled simplex in partition of such that . Denote vertices of by . We name these vertices so that are labeled, respectively, with . Then, and .
About , from the labeling rules, we have . About , also from the labeling rules, we have which implies . means . On the other hand, means . Thus, from we obtain By similar arguments, for each other than 0, For , we have . Then, Adding (3.19) and (3.20) side by side except for some (denote it by ) other than 0, From , , we have , which is rewritten as Since (3.19) implies , we have Thus, is derived. On the other hand, adding (3.19) from 1 to yields From , , we have Then, from (3.20) and (3.26), we get From (3.24) and (3.27), we obtain the following result: Thus, Since is finite, is an approximate fixed point of . Similarly, we can prove that every other vertex, , and all points in a fully-labeled simplex of are approximate fixed points.

4. Concluding Remarks

There are some themes to which we can apply the result of this paper. In [11], we studied a proof of the existence of an approximate equilibrium in a competitive economy with uniformly continuous excess demand functions by Sperner’s lemma. Using the result of this paper, we can prove the existence of an approximate equilibrium with uniformly sequentially continuous excess demand functions.

Acknowledgments

This research was partially supported by the Ministry of Education, Science, Sports and Culture of Japan, Grant-in-Aid for Scientific Research (C), 20530165, and the Special Costs for Graduate Schools of the Special Expenses for Hitech Promotion by the Ministry of Education, Science, Sports and Culture of Japan in 2011.

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Copyright

Copyright © 2011 Yasuhito Tanaka. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.