International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 276040 | 9 pages | https://doi.org/10.5402/2011/276040

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

Academic Editor: C. I. Siettos
Received01 Apr 2011
Accepted18 May 2011
Published09 Jul 2011

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 𝑛-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 𝑛-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 𝑛-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 Δ denote an 𝑛-dimensional simplex. 𝑛 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 Δ in 𝑚 equal segments and draw the lines parallel to the sides of Δ. 𝑚 is a finite natural number. Then, the 2-dimensional simplex is partitioned into 𝑚2 triangles. We consider partition of Δ inductively for cases of higher dimension. In a 3-dimensional case, each face of Δ is a 2-dimensional simplex, and so it is partitioned into 𝑚2 triangles in the above-mentioned way, and draw the planes parallel to the faces of Δ. Then, the 3-dimensional simplex is partitioned into 𝑚3 trigonal pyramids and similarly for cases of higher dimension.

Let 𝐾 denote the set of small 𝑛-dimensional simplices of Δ constructed by partition. Vertices of these small simplices of 𝐾 are labeled with the numbers 0,1,2,…,𝑛 subject to the following rules. (1)The vertices of Δ are, respectively, labeled with 0 to 𝑛. We label a point (1,0,…,0) with 0, a point (0,1,0,…,0) with 1, a point (0,0,1…,0) with 2,…,andğ‘Ž point (0,…,0,1) with 𝑛. That is, a vertex whose 𝑘th coordinate (𝑘=0,1,…,𝑛) is 1 and all other coordinates are 0 is labeled with 𝑘.(2) If a vertex of 𝐾 is contained in an 𝑛−1-dimensional face of Δ, 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 𝐾 is contained in an 𝑛−2-dimensional face of Δ, 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 Δ is labeled with an arbitrary number among 0,1,…,𝑛.

A small simplex of 𝐾 which is labeled with the numbers 0,1,…,𝑛 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 𝐾 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 𝑛 and partition of Δ are finite, the number of small simplices constructed by partition is also finite. Thus, we can constructively find a fully labeled 𝑛-dimensional simplex of 𝐾 through finite steps.

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

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

Definition 3.1 (Uniform continuity). A function 𝑓 is uniformly continuous in Δ if, for any 𝐩,ğ©î…žâˆˆÎ” and 𝜀>0 there, exists 𝛿>0 such that ||Ifğ©âˆ’ğ©î…ž||||𝑓𝐩<𝛿,then(𝐩)âˆ’ğ‘“î…žî€¸||<𝜀.(3.1)𝛿 depends on only 𝜀.

Definition 3.2 (Sequential continuity). A function 𝑓 is sequentially continuous at 𝐩∈Δ in Δ if, for sequences (𝐩𝑛)𝑛≥1 and (𝑓(𝐩𝑛))𝑛≥1 in Δ, 𝑓𝐩𝑛⟶𝑓(𝐩)whenever𝐩𝑛⟶𝐩.(3.2)

Definition 3.3 (Uniform sequential continuity). A function 𝑓 is uniformly sequentially continuous in Δ if, for sequences (𝐩𝑛)𝑛≥1, (ğ©î…žn)𝑛≥1, (𝑓(𝐩𝑛))𝑛≥1, and (𝑓(ğ©î…žn))𝑛≥1 in Δ, ||ğ‘“î€·ğ©ğ‘›î€¸î€·ğ©âˆ’ğ‘“î…žn||||𝐩⟶0wheneverğ‘›âˆ’ğ©î…žn||⟶0.(3.3)|ğ©ğ‘›âˆ’ğ©î…žn|→0 means ||𝐩∀𝜀>0âˆƒğ‘âˆ€ğ‘›â‰¥ğ‘ğ‘›âˆ’ğ©î…žn||<𝜀,(3.4) where 𝜀 is a real number and 𝑛 and 𝑁 are natural numbers. Similarly, |𝑓(𝐩𝑛)−𝑓(ğ©î…žn)|→0 means ||𝑓𝐩∀𝜀>0âˆƒğ‘â€²âˆ€ğ‘›â‰¥ğ‘â€²ğ‘›î€¸î€·ğ©âˆ’ğ‘“î…žn||<𝜀.(3.5)𝑁′ 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 𝑓 is defined as follows.

Definition 3.4 (Approximate fixed point). For each 𝜀>0,  𝐩∗ is an approximate fixed point of 𝑓 if we have ||𝐩∗𝐩−𝑓∗||<𝜀.(3.6)

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 𝑛-dimensional simplex Δ to itself has an approximate fixed point for each 𝜀>0.

Proof. (1) First, we show that we can partition Δ so that the conditions for Sperner’s lemma are satisfied. We partition Δ according to the method in Sperner’s lemma, and label the vertices of simplices constructed by partition of Δ. It is important how to label the vertices contained in the faces of Δ. Let 𝐾 be the set of small simplices constructed by partition of Δ, let 𝐩=(𝑝0,𝑝1,…,𝑝𝑛) be a vertex of a simplex of 𝐾, and denote the 𝑖th component of 𝑓(𝐩) by 𝑓𝑖. Then, we label a vertex 𝐩 according to the following rule: If𝑝𝑘>𝑓𝑘or𝑝𝑘+𝜏>𝑓𝑘,thenwelabel𝐩with𝑘,(3.7) where 𝜏 is a positive number. If there are multiple 𝑘’s which satisfy this condition, then we label 𝐩 conveniently for the conditions for Sperner’s lemma to be satisfied. We do not randomly label the vertices.
For example, let 𝐩 be a point contained in an 𝑛−1-dimensional face of Δ such that 𝑝𝑖=0 for one 𝑖 among 0,1,2,…,𝑛 (its 𝑖th coordinate is 0). With 𝜏>0, we have 𝑓𝑖>0 or 𝑓𝑖<𝜏.
In constructive mathematics, for any real number 𝑥, we cannot prove that 𝑥≥0 or 𝑥<0, that 𝑥>0 or 𝑥=0 or 𝑥<0. But for any distinct real numbers 𝑥, 𝑦, and 𝑧 such that 𝑥>𝑧, we can prove that 𝑥>𝑦 or 𝑦>𝑧.
When 𝑓𝑖>0, from ∑𝑛𝑗=0𝑝𝑗=1, ∑𝑛𝑗=0𝑓𝑗=1, and 𝑝𝑖=0, 𝑛𝑗=0,𝑗≠𝑖𝑝𝑗>𝑛𝑗=0,𝑗≠𝑖𝑓𝑗.(3.8) Then, for at least one 𝑗 (denote it by 𝑘), we have 𝑝𝑘>𝑓𝑘 and we label 𝐩 with 𝑘, where 𝑘 is one of the numbers which satisfy 𝑝𝑘>𝑓𝑘. Since 𝑓𝑖>𝑝𝑖=0, 𝑖 does not satisfy this condition. Assume that 𝑓𝑖<𝜏⋅𝑝𝑖=0 implies ∑𝑛𝑗=0,𝑗≠𝑖𝑝𝑗=1. Since ∑𝑛𝑗=0,𝑗≠𝑖𝑓𝑗≤1, we obtain 𝑛𝑗=0,𝑗≠𝑖𝑝𝑗≥𝑛𝑗=0,𝑗≠𝑖𝑓𝑗.(3.9) Then, for a positive number 𝜏, we have 𝑛𝑗=0,𝑗≠𝑖𝑝𝑗>+𝜏𝑛𝑗=0,𝑗≠𝑖𝑓𝑗.(3.10) There is at least one 𝑗(≠𝑖) which satisfies 𝑝𝑗+𝜏>𝑓𝑗. Denote it by 𝑘, and we label 𝐩 with 𝑘. 𝑘 is one of the numbers other than 𝑖 such that 𝑝𝑘+𝜏>𝑓𝑘 is satisfied. 𝑖 itself satisfies this condition (𝑝𝑖+𝜏>𝑓𝑖). But, since there is a number other than 𝑖 which satisfies this condition, we can select a number other than 𝑖. We have proved that we can label the vertices contained in an 𝑛−1-dimensional face of Δ such that 𝑝𝑖=0 for one 𝑖 among 0,1,2,…,𝑛 with the numbers other than 𝑖. By similar procedures, we can show that we can label the vertices contained in an 𝑛−2-dimensional face of Δ such that 𝑝𝑖=0 for two 𝑖’s among 0,1,2,…,𝑛 with the numbers other than those 𝑖’s, and so on.
Consider the case where 𝑝𝑖=𝑝𝑖+1=0. We see that, when 𝑓𝑖>0 or 𝑓𝑖+1>0, 𝑛𝑗=0,𝑗≠𝑖,𝑖+1𝑝𝑗>𝑛𝑗=0,𝑗≠𝑖,𝑖+1𝑓𝑗,(3.11) and so for at least one 𝑗 (denote it by 𝑘), we have 𝑝𝑘>𝑓𝑘, and we label 𝐩 with 𝑘. On the other hand, when 𝑓𝑖<𝜏 and 𝑓𝑖+1<𝜏, we have 𝑛𝑗=0,𝑗≠𝑖,𝑖+1𝑝𝑗≥𝑛𝑗=0,𝑗≠𝑖,𝑖+1𝑓𝑗.(3.12) Then, for a positive number 𝜏, we have 𝑛𝑗=0,𝑗≠𝑖,𝑖+1𝑝𝑗>+𝜏𝑛𝑗=0,𝑗≠𝑖,𝑖+1𝑓𝑗.(3.13) Thus, there is at least one 𝑗(≠𝑖,𝑖+1) which satisfies 𝑝𝑗+𝜏>𝑓𝑗. Denote it by 𝑘, and we label 𝐩 with 𝑘.
Next, consider the case where 𝑝𝑖=0 for all 𝑖 other than 𝑛. If, for some 𝑖,  𝑓𝑖>0, then we have 𝑝𝑛>𝑓𝑛 and label 𝐩 with 𝑛. On the other hand, if 𝑓𝑗<𝜏 for all 𝑗≠𝑛, then we obtain 𝑝𝑛≥𝑓𝑛. It implies 𝑝𝑛+𝜏>𝑓𝑛. Thus, we can label 𝐩 with 𝑛.
Therefore, the conditions for Sperner’s lemma are satisfied and there exists an odd number of fully labeled simplices in 𝐾.
(2) Consider a sequence (Δ𝑚)𝑚≥1 of partitions of Δ and a sequence of fully labeled simplices (𝛿𝑚)𝑚≥1. The larger 𝑚, the finer partition. The larger 𝑚, the smaller the diameter of a fully labeled simplex. Let 𝐩0𝑚,𝐩1𝑚,… and 𝐩𝑛𝑚 be the vertices of a fully labeled simplex 𝛿𝑚. We name these vertices so that 𝐩0𝑚,𝐩1𝑚,…,𝐩𝑛𝑚 are labeled, respectively, with 0,1,…,𝑛. The values of 𝑓 at theses vertices are 𝑓(𝐩0𝑚),𝑓(𝐩1𝑚),… and 𝑓(𝐩𝑛𝑚). We can consider sequences of vertices of fully labeled simplices. Denote them by (𝐩0𝑚)𝑚≥1,(𝐩1𝑚)𝑚≥1,…, and (𝐩𝑛𝑚)𝑚≥1. And consider sequences of the values of 𝑓 at vertices of fully labeled simplices. Denote them by (𝑓(𝐩0𝑚))𝑚≥1,(𝑓(𝐩1𝑚))𝑚≥1,…, and (𝑓(𝐩𝑛𝑚))𝑚≥1. By the uniform sequential continuity of 𝑓, ||𝑓𝐩𝑖𝑚𝑚≥1−𝑓𝐩𝑗𝑚𝑚≥1||||𝐩⟶0whenever𝑖𝑚𝑚≥1−𝐩𝑗𝑚𝑚≥1||⟶0,(3.14) for 𝑖≠𝑗. |(𝐩𝑖𝑚)𝑚≥1−(𝐩𝑗𝑚)𝑚≥1|→0 means ||𝐩∀𝜀>0∃𝑀∀𝑚≥𝑀𝑖𝑚−𝐩𝑗𝑚||<𝜀𝑖≠𝑗,(3.15) and |(𝑓(𝐩𝑖𝑚))𝑚≥1−(𝑓(𝐩𝑗𝑚))𝑚≥1|→0 means ∀𝜀>0âˆƒğ‘€î…žâˆ€ğ‘šâ‰¥ğ‘€î…žî€·||𝑓𝐩𝑖𝑚𝐩−𝑓𝑗𝑚||<𝜀𝑖≠𝑗.(3.16) Consider a fully labeled simplex 𝛿𝑙 in partition of Δ such that 𝑙≥max(𝑀,𝑀′). Denote vertices of 𝛿𝑙 by 𝐩0,𝐩1,…,𝐩𝑛. We name these vertices so that 𝐩0,𝐩1,…,𝐩𝑛 are labeled, respectively, with 0,1,…,𝑛. Then, |𝐩𝑖−𝐩𝑗|<𝜀 and |𝑓(𝐩𝑖)−𝑓(𝐩𝑗)|<𝜀.
About 𝐩0, from the labeling rules, we have 𝐩00+𝜏>𝑓(𝐩0)0. About 𝐩1, also from the labeling rules, we have 𝐩11+𝜏>𝑓(𝐩1)1 which implies 𝐩11>𝑓(𝐩1)1−𝜏. |𝑓(𝐩0)−𝑓(𝐩1)|<𝜀 means 𝑓(𝐩1)1>𝑓(𝐩0)1−𝜀. On the other hand, |𝐩0−𝐩1|<𝜀 means 𝐩01>𝐩11−𝜀. Thus, from 𝐩01>𝐩11−𝜀,𝐩11𝐩>𝑓11𝐩−𝜏,𝑓11𝐩>𝑓01−𝜀,(3.17) we obtain 𝐩01𝐩>𝑓01−2𝜀−𝜏.(3.18) By similar arguments, for each 𝑖 other than 0, 𝐩0𝑖𝐩>𝑓0𝑖−2𝜀−𝜏.(3.19) For 𝑖=0, we have 𝐩00+𝜏>𝑓(𝐩0)0. Then, 𝐩00𝐩>𝑓00−𝜏.(3.20) Adding (3.19) and (3.20) side by side except for some 𝑖 (denote it by 𝑘) other than 0, 𝑛𝑗=0,𝑗≠𝑘𝐩0𝑗>𝑛𝑗=0,𝑗≠𝑘𝑓𝐩0𝑗−2(𝑛−1)𝜀−𝑛𝜏.(3.21) From ∑𝑛𝑗=0𝐩0𝑗=1, ∑𝑛𝑗=0𝑓(𝐩0)𝑗=1, we have 1−𝐩0𝑘>1−𝑓(𝐩0)𝑘−2(𝑛−1)𝜀−𝑛𝜏, which is rewritten as 𝐩0𝑘𝐩<𝑓0𝑘+2(𝑛−1)𝜀+𝑛𝜏.(3.22) Since (3.19) implies 𝐩0𝑘>𝑓(𝐩0)𝑘−2𝜀−𝜏, we have 𝑓𝐩0𝑘−2𝜀−𝜏<𝐩0𝑘𝐩<𝑓0𝑘+2(𝑛−1)𝜀+𝑛𝜏.(3.23) Thus, ||𝐩0𝑘𝐩−𝑓0𝑘||<2(𝑛−1)𝜀+𝑛𝜏(3.24) is derived. On the other hand, adding (3.19) from 1 to 𝑛 yields 𝑛𝑗=1𝐩0𝑗>𝑛𝑗=1𝑓𝐩0𝑗−2𝑛𝜀−𝑛𝜏.(3.25) From ∑𝑛𝑗=0𝐩0𝑗=1, ∑𝑛𝑗=0𝑓(𝐩0)𝑗=1, we have 1−𝐩00𝐩>1−𝑓00−2𝑛𝜀−𝑛𝜏.(3.26) Then, from (3.20) and (3.26), we get ||𝐩00𝐩−𝑓00||<2𝑛𝜀+𝑛𝜏.(3.27) From (3.24) and (3.27), we obtain the following result: ||𝐩0𝑖𝐩−𝑓0𝑖||<2𝑛𝜀+𝑛𝜏∀𝑖.(3.28) Thus, ||𝐩0𝐩−𝑓0||<𝑛(𝑛+1)(2𝜀+𝜏).(3.29) Since 𝑛 is finite, 𝐩0 is an approximate fixed point of 𝑓. Similarly, we can prove that every other vertex, 𝐩1,𝐩2,…,𝐩𝑛, and all points in a fully-labeled simplex of 𝐾 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 © 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.

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