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 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.

Figure 1 

Partition and labeling of 2-dimensional simplex.

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,…,and𝑎 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,𝐩^∈Δ and 𝜀>0 there, exists 𝛿>0 such that If||𝐩−𝐩^||<𝛿,then||𝑓(𝐩)−𝑓𝐩^||<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 (𝐩_𝑛)_𝑛≥1 and (𝑓(𝐩_𝑛))_𝑛≥1 in Δ, 𝑓𝐩_𝑛⟶𝑓(𝐩)whenever𝐩_𝑛⟶𝐩.(3.2)

Definition 3.3 (Uniform sequential continuity). A function 𝑓 is uniformly sequentially continuous in Δ if, for sequences (𝐩_𝑛)_𝑛≥1, (𝐩^_n)_𝑛≥1, (𝑓(𝐩_𝑛))_𝑛≥1, and (𝑓(𝐩^_n))_𝑛≥1 in Δ, ||𝑓𝐩_𝑛−𝑓𝐩^_n||⟶0whenever||𝐩_𝑛−𝐩^_n||⟶0.(3.3)|𝐩_𝑛−𝐩^_n|→0 means ∀𝜀>0∃𝑁∀𝑛≥𝑁||𝐩_𝑛−𝐩^_n||<𝜀,(3.4) where 𝜀 is a real number and 𝑛 and 𝑁 are natural numbers. Similarly, |𝑓(𝐩_𝑛)−𝑓(𝐩^_n)|→0 means ∀𝜀>0∃𝑁′∀𝑛≥𝑁′||𝑓𝐩_𝑛−𝑓𝐩^_n||<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^∗−𝑓𝐩^∗||<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,…,𝑝_𝑛) be a vertex of a simplex of 𝐾, and denote the 𝑖th component of 𝑓(𝐩) by 𝑓_𝑖. Then, we label a vertex 𝐩 according to the following rule: If𝑝_𝑘>𝑓_𝑘or𝑝_𝑘+𝜏>𝑓_𝑘,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 𝑝_𝑖=0 for one 𝑖 among 0,1,2,…,𝑛 (its 𝑖th coordinate is 0). With 𝜏>0, we have 𝑓_𝑖>0 or 𝑓_𝑖<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 𝑓_𝑖>0, from ∑^𝑛𝑗=0𝑝_𝑗=1, ∑^𝑛𝑗=0𝑓_𝑗=1, and 𝑝_𝑖=0, ^𝑛_𝑗_{=0,𝑗≠𝑖}𝑝_𝑗>^𝑛_𝑗=_{0,𝑗≠𝑖}𝑓_𝑗.(3.8) Then, for at least one 𝑗 (denote it by 𝑘), we have 𝑝_𝑘>𝑓_𝑘 and we label 𝐩 with 𝑘, where 𝑘 is one of the numbers which satisfy 𝑝_𝑘>𝑓_𝑘. Since 𝑓_𝑖>𝑝_𝑖=0, 𝑖 does not satisfy this condition. Assume that 𝑓_𝑖<𝜏⋅𝑝_𝑖=0 implies ∑^{𝑛𝑗=0,𝑗≠𝑖}𝑝_𝑗=1. Since ∑^{𝑛𝑗=0,𝑗≠𝑖}𝑓_𝑗≤1, we obtain ^𝑛_𝑗_{=0,𝑗≠𝑖}𝑝_𝑗≥^𝑛_𝑗=_{0,𝑗≠𝑖}𝑓_𝑗.(3.9) Then, for a positive number 𝜏, we have ^𝑛_𝑗_{=0,𝑗≠𝑖}𝑝_𝑗+𝜏>^𝑛_𝑗=_{0,𝑗≠𝑖}𝑓_𝑗.(3.10) There is at least one 𝑗(≠𝑖) which satisfies 𝑝_𝑗+𝜏>𝑓_𝑗. Denote it by 𝑘, and we label 𝐩 with 𝑘. 𝑘 is one of the numbers other than 𝑖 such that 𝑝_𝑘+𝜏>𝑓_𝑘 is satisfied. 𝑖 itself satisfies this condition (𝑝_𝑖+𝜏>𝑓_𝑖). 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 𝑝_𝑖=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 𝑝_𝑖=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 𝑝_𝑖=𝑝_𝑖+1=0. We see that, when 𝑓_𝑖>0 or 𝑓_𝑖+1>0, ^𝑛_𝑗=0_{,𝑗≠𝑖,𝑖+1}𝑝_𝑗>^𝑛_𝑗=0,_{𝑗≠𝑖,𝑖+1}𝑓_𝑗,(3.11) and so for at least one 𝑗 (denote it by 𝑘), we have 𝑝_𝑘>𝑓_𝑘, and we label 𝐩 with 𝑘. On the other hand, when 𝑓_𝑖<𝜏 and 𝑓_𝑖+1<𝜏, we have ^𝑛_𝑗=0_{,𝑗≠𝑖,𝑖+1}𝑝_𝑗≥^𝑛_𝑗=0,_{𝑗≠𝑖,𝑖+1}𝑓_𝑗.(3.12) Then, for a positive number 𝜏, we have ^𝑛_𝑗=0_{,𝑗≠𝑖,𝑖+1}𝑝_𝑗+𝜏>^𝑛_𝑗=0,_{𝑗≠𝑖,𝑖+1}𝑓_𝑗.(3.13) Thus, there is at least one 𝑗(≠𝑖,𝑖+1) which satisfies 𝑝_𝑗+𝜏>𝑓_𝑗. Denote it by 𝑘, and we label 𝐩 with 𝑘.
Next, consider the case where 𝑝_𝑖=0 for all 𝑖 other than 𝑛. If, for some 𝑖,  𝑓_𝑖>0, then we have 𝑝_𝑛>𝑓_𝑛 and label 𝐩 with 𝑛. On the other hand, if 𝑓_𝑗<𝜏 for all 𝑗≠𝑛, then we obtain 𝑝_𝑛≥𝑓_𝑛. It implies 𝑝_𝑛+𝜏>𝑓_𝑛. 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 (Δ_𝑚)_𝑚≥1 of partitions of Δ and a sequence of fully labeled simplices (𝛿_𝑚)_𝑚≥1. The larger 𝑚, the finer partition. The larger 𝑚, the smaller the diameter of a fully labeled simplex. Let 𝐩^0𝑚,𝐩^1𝑚,… and 𝐩^𝑛𝑚 be the vertices of a fully labeled simplex 𝛿_𝑚. We name these vertices so that 𝐩^0𝑚,𝐩^1𝑚,…,𝐩^𝑛𝑚 are labeled, respectively, with 0,1,…,𝑛. The values of 𝑓 at theses vertices are 𝑓(𝐩^0𝑚),𝑓(𝐩^1𝑚),… and 𝑓(𝐩^𝑛𝑚). We can consider sequences of vertices of fully labeled simplices. Denote them by (𝐩^0𝑚)_𝑚≥1,(𝐩^1𝑚)_𝑚≥1,…, and (𝐩^𝑛𝑚)_𝑚≥1. And consider sequences of the values of 𝑓 at vertices of fully labeled simplices. Denote them by (𝑓(𝐩^0𝑚))_𝑚≥1,(𝑓(𝐩^1𝑚))_𝑚≥1,…, and (𝑓(𝐩^𝑛𝑚))_𝑚≥1. By the uniform sequential continuity of 𝑓, ||𝑓𝐩^𝑖𝑚_𝑚≥1−𝑓𝐩^𝑗𝑚_𝑚≥1||⟶0whenever||𝐩^𝑖𝑚_𝑚≥1−𝐩^𝑗𝑚_𝑚≥1||⟶0,(3.14) for 𝑖≠𝑗. |(𝐩^𝑖𝑚)_𝑚≥1−(𝐩^𝑗𝑚)_𝑚≥1|→0 means ∀𝜀>0∃𝑀∀𝑚≥𝑀||𝐩^𝑖𝑚−𝐩^𝑗𝑚||<𝜀𝑖≠𝑗,(3.15) and |(𝑓(𝐩^𝑖𝑚))_𝑚≥1−(𝑓(𝐩^𝑗𝑚))_𝑚≥1|→0 means ∀𝜀>0∃𝑀^∀𝑚≥𝑀^||𝑓𝐩^𝑖𝑚−𝑓𝐩^𝑗𝑚||<𝜀𝑖≠𝑗.(3.16) Consider a fully labeled simplex 𝛿_𝑙 in partition of Δ such that 𝑙≥max(𝑀,𝑀′). Denote vertices of 𝛿_𝑙 by 𝐩⁰,𝐩¹,…,𝐩^𝑛. We name these vertices so that 𝐩⁰,𝐩¹,…,𝐩^𝑛 are labeled, respectively, with 0,1,…,𝑛. Then, |𝐩^𝑖−𝐩^𝑗|<𝜀 and |𝑓(𝐩^𝑖)−𝑓(𝐩^𝑗)|<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>𝑓𝐩¹₁−𝜏,𝑓𝐩¹₁>𝑓𝐩⁰₁−𝜀,(3.17) we obtain 𝐩⁰¹>𝑓𝐩⁰₁−2𝜀−𝜏.(3.18) By similar arguments, for each 𝑖 other than 0, 𝐩^0𝑖>𝑓𝐩⁰_𝑖−2𝜀−𝜏.(3.19) For 𝑖=0, we have 𝐩⁰⁰+𝜏>𝑓(𝐩⁰)₀. Then, 𝐩⁰⁰>𝑓𝐩⁰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≠𝑘}𝐩^0𝑗>^𝑛_𝑗=0_,𝑗≠𝑘𝑓𝐩⁰_𝑗−2(𝑛−1)𝜀−𝑛𝜏.(3.21) From ∑^𝑛𝑗=0𝐩^0𝑗=1, ∑^𝑛𝑗=0𝑓(𝐩⁰)_𝑗=1, we have 1−𝐩^0𝑘>1−𝑓(𝐩⁰)_𝑘−2(𝑛−1)𝜀−𝑛𝜏, which is rewritten as 𝐩^0𝑘<𝑓𝐩⁰_𝑘+2(𝑛−1)𝜀+𝑛𝜏.(3.22) Since (3.19) implies 𝐩^0𝑘>𝑓(𝐩⁰)_𝑘−2𝜀−𝜏, we have 𝑓𝐩⁰_𝑘−2𝜀−𝜏<𝐩^0𝑘<𝑓𝐩⁰_𝑘+2(𝑛−1)𝜀+𝑛𝜏.(3.23) Thus, ||𝐩^0𝑘−𝑓𝐩⁰_𝑘||<2(𝑛−1)𝜀+𝑛𝜏(3.24) is derived. On the other hand, adding (3.19) from 1 to 𝑛 yields ^𝑛_𝑗=1𝐩^0𝑗>^𝑛_𝑗=1𝑓𝐩⁰_𝑗−2𝑛𝜀−𝑛𝜏.(3.25) From ∑^𝑛𝑗=0𝐩^0𝑗=1, ∑^𝑛𝑗=0𝑓(𝐩⁰)_𝑗=1, we have 1−𝐩⁰⁰>1−𝑓𝐩⁰₀−2𝑛𝜀−𝑛𝜏.(3.26) Then, from (3.20) and (3.26), we get ||𝐩⁰⁰−𝑓𝐩⁰₀||<2𝑛𝜀+𝑛𝜏.(3.27) From (3.24) and (3.27), we obtain the following result: ||𝐩^0𝑖−𝑓𝐩⁰_𝑖||<2𝑛𝜀+𝑛𝜏∀𝑖.(3.28) Thus, ||𝐩⁰−𝑓𝐩⁰||<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,…,𝐩^𝑛, 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.