International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

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

Yasuhito Tanaka, "A Proof of Constructive Version of Brouwer's Fixed Point Theorem with Uniform Sequential Continuity", International Scholarly Research Notices, vol. 2011, Article ID 276040, 9 pages, 2011. 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 ๐‘›-dimensional simplex for uniformly sequentially continuous functions. We follow the Bishop style constructive mathematics.

1. Introduction

It is often said that Brouwerโ€™s fixed point theorem cannot be constructively proved.

Reference [1] provided a constructive proof of Brouwerโ€™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 Brouwerโ€™s fixed point theorem, that is, the intermediate value theorem is nonconstructive. See [2] or [3]. Brouwerโ€™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, Spernerโ€™s lemma, which is used to prove Brouwerโ€™s theorem, can be constructively proved. Some authors have presented a constructive (or an approximate) version of Brouwerโ€™s fixed point theorem using Spernerโ€™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 Brouwerโ€™s fixed point theorem in an ๐‘›-dimensional simplex for uniformly sequentially continuous functions.

In the next section, we consider Spernerโ€™s lemma. In Section 3, we prove a constructive version of Brouwerโ€™s fixed point theorem for uniformly sequentially continuous functions from an ๐‘›-dimensional simplex to itself using Spernerโ€™s lemma. We follow the Bishop style constructive mathematics according to [2, 8, 9].

2. Spernerโ€™s Lemma

Let ฮ” denote an ๐‘›-dimensional simplex. ๐‘› 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 ฮ” in ๐‘š equal segments and draw the lines parallel to the sides of ฮ”. ๐‘š is a finite natural number. Then, the 2-dimensional simplex is partitioned into ๐‘š2 triangles. We consider partition of ฮ” inductively for cases of higher dimension. In a 3-dimensional case, each face of ฮ” is a 2-dimensional simplex, and so it is partitioned into ๐‘š2 triangles in the above-mentioned way, and draw the planes parallel to the faces of ฮ”. Then, the 3-dimensional simplex is partitioned into ๐‘š3 trigonal pyramids and similarly for cases of higher dimension.

Let ๐พ denote the set of small ๐‘›-dimensional simplices of ฮ” constructed by partition. Vertices of these small simplices of ๐พ are labeled with the numbers 0,1,2,โ€ฆ,๐‘› subject to the following rules. (1)The vertices of ฮ” are, respectively, labeled with 0 to ๐‘›. We label a point (1,0,โ€ฆ,0) with 0, a point (0,1,0,โ€ฆ,0) with 1, a point (0,0,1โ€ฆ,0) with 2,โ€ฆ,and๐‘Ž point (0,โ€ฆ,0,1) with ๐‘›. That is, a vertex whose ๐‘˜th coordinate (๐‘˜=0,1,โ€ฆ,๐‘›) is 1 and all other coordinates are 0 is labeled with ๐‘˜.(2) If a vertex of ๐พ is contained in an ๐‘›โˆ’1-dimensional face of ฮ”, 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 ๐พ is contained in an ๐‘›โˆ’2-dimensional face of ฮ”, 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 ฮ” is labeled with an arbitrary number among 0,1,โ€ฆ,๐‘›.

A small simplex of ๐พ which is labeled with the numbers 0,1,โ€ฆ,๐‘› is called a fully labeled simplex. Spernerโ€™s lemma is stated as follows.

Lemma 2.1 (Spernerโ€™s lemma). If one labels the vertices of ๐พ 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 Spernerโ€™s lemma see [10] or [11].

Since ๐‘› and partition of ฮ” are finite, the number of small simplices constructed by partition is also finite. Thus, we can constructively find a fully labeled ๐‘›-dimensional simplex of ๐พ through finite steps.

3. Constructive Version of Brouwerโ€™s Fixed Point Theorem with Uniform Sequential Continuity

Let us consider a function ๐‘“ from an ๐‘›-dimensional simplex ฮ” to itself. Denote a point in ฮ” by ๐ฉ. Uniform continuity, sequential continuity, and uniform sequential continuity of functions are defined as follows.

Definition 3.1 (Uniform continuity). A function ๐‘“ is uniformly continuous in ฮ” if, for any ๐ฉ,๐ฉ๎…žโˆˆฮ” and ๐œ€>0 there, exists ๐›ฟ>0 such that ||If๐ฉโˆ’๐ฉ๎…ž||||๐‘“๎€ท๐ฉ<๐›ฟ,then(๐ฉ)โˆ’๐‘“๎…ž๎€ธ||<๐œ€.(3.1)๐›ฟ depends on only ๐œ€.

Definition 3.2 (Sequential continuity). A function ๐‘“ is sequentially continuous at ๐ฉโˆˆฮ” in ฮ” if, for sequences (๐ฉ๐‘›)๐‘›โ‰ฅ1 and (๐‘“(๐ฉ๐‘›))๐‘›โ‰ฅ1 in ฮ”, ๐‘“๎€ท๐ฉ๐‘›๎€ธโŸถ๐‘“(๐ฉ)whenever๐ฉ๐‘›โŸถ๐ฉ.(3.2)

Definition 3.3 (Uniform sequential continuity). A function ๐‘“ is uniformly sequentially continuous in ฮ” if, for sequences (๐ฉ๐‘›)๐‘›โ‰ฅ1, (๐ฉ๎…žn)๐‘›โ‰ฅ1, (๐‘“(๐ฉ๐‘›))๐‘›โ‰ฅ1, and (๐‘“(๐ฉ๎…žn))๐‘›โ‰ฅ1 in ฮ”, ||๐‘“๎€ท๐ฉ๐‘›๎€ธ๎€ท๐ฉโˆ’๐‘“๎…žn๎€ธ||||๐ฉโŸถ0whenever๐‘›โˆ’๐ฉ๎…žn||โŸถ0.(3.3)|๐ฉ๐‘›โˆ’๐ฉ๎…žn|โ†’0 means ๎€ท||๐ฉโˆ€๐œ€>0โˆƒ๐‘โˆ€๐‘›โ‰ฅ๐‘๐‘›โˆ’๐ฉ๎…žn||๎€ธ<๐œ€,(3.4) where ๐œ€ is a real number and ๐‘› and ๐‘ are natural numbers. Similarly, |๐‘“(๐ฉ๐‘›)โˆ’๐‘“(๐ฉ๎…žn)|โ†’0 means ๎€ท||๐‘“๎€ท๐ฉโˆ€๐œ€>0โˆƒ๐‘โ€ฒโˆ€๐‘›โ‰ฅ๐‘โ€ฒ๐‘›๎€ธ๎€ท๐ฉโˆ’๐‘“๎…žn๎€ธ||๎€ธ<๐œ€.(3.5)๐‘โ€ฒ 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 ๐‘“ is defined as follows.

Definition 3.4 (Approximate fixed point). For each ๐œ€>0,โ€‰โ€‰๐ฉโˆ— is an approximate fixed point of ๐‘“ if we have ||๐ฉโˆ—๎€ท๐ฉโˆ’๐‘“โˆ—๎€ธ||<๐œ€.(3.6)

Now, we show the following theorem.

Theorem 3.5 (Constructive version of Brouwerโ€™s fixed point theorem with uniform sequential continuity). Any uniformly sequentially continuous function from an ๐‘›-dimensional simplex ฮ” to itself has an approximate fixed point for each ๐œ€>0.

Proof. (1) First, we show that we can partition ฮ” so that the conditions for Spernerโ€™s lemma are satisfied. We partition ฮ” according to the method in Spernerโ€™s lemma, and label the vertices of simplices constructed by partition of ฮ”. It is important how to label the vertices contained in the faces of ฮ”. Let ๐พ be the set of small simplices constructed by partition of ฮ”, let ๐ฉ=(๐‘0,๐‘1,โ€ฆ,๐‘๐‘›) be a vertex of a simplex of ๐พ, and denote the ๐‘–th component of ๐‘“(๐ฉ) by ๐‘“๐‘–. Then, we label a vertex ๐ฉ according to the following rule: If๐‘๐‘˜>๐‘“๐‘˜or๐‘๐‘˜+๐œ>๐‘“๐‘˜,thenwelabel๐ฉwith๐‘˜,(3.7) where ๐œ is a positive number. If there are multiple ๐‘˜โ€™s which satisfy this condition, then we label ๐ฉ conveniently for the conditions for Spernerโ€™s lemma to be satisfied. We do not randomly label the vertices.
For example, let ๐ฉ be a point contained in an ๐‘›โˆ’1-dimensional face of ฮ” such that ๐‘๐‘–=0 for one ๐‘– among 0,1,2,โ€ฆ,๐‘› (its ๐‘–th coordinate is 0). With ๐œ>0, we have ๐‘“๐‘–>0 or ๐‘“๐‘–<๐œ.
In constructive mathematics, for any real number ๐‘ฅ, we cannot prove that ๐‘ฅโ‰ฅ0 or ๐‘ฅ<0, that ๐‘ฅ>0 or ๐‘ฅ=0 or ๐‘ฅ<0. But for any distinct real numbers ๐‘ฅ, ๐‘ฆ, and ๐‘ง such that ๐‘ฅ>๐‘ง, we can prove that ๐‘ฅ>๐‘ฆ or ๐‘ฆ>๐‘ง.
When ๐‘“๐‘–>0, from โˆ‘๐‘›๐‘—=0๐‘๐‘—=1, โˆ‘๐‘›๐‘—=0๐‘“๐‘—=1, and ๐‘๐‘–=0, ๐‘›๎“๐‘—=0,๐‘—โ‰ ๐‘–๐‘๐‘—>๐‘›๎“๐‘—=0,๐‘—โ‰ ๐‘–๐‘“๐‘—.(3.8) Then, for at least one ๐‘— (denote it by ๐‘˜), we have ๐‘๐‘˜>๐‘“๐‘˜ and we label ๐ฉ with ๐‘˜, where ๐‘˜ is one of the numbers which satisfy ๐‘๐‘˜>๐‘“๐‘˜. Since ๐‘“๐‘–>๐‘๐‘–=0, ๐‘– does not satisfy this condition. Assume that ๐‘“๐‘–<๐œโ‹…๐‘๐‘–=0 implies โˆ‘๐‘›๐‘—=0,๐‘—โ‰ ๐‘–๐‘๐‘—=1. Since โˆ‘๐‘›๐‘—=0,๐‘—โ‰ ๐‘–๐‘“๐‘—โ‰ค1, we obtain ๐‘›๎“๐‘—=0,๐‘—โ‰ ๐‘–๐‘๐‘—โ‰ฅ๐‘›๎“๐‘—=0,๐‘—โ‰ ๐‘–๐‘“๐‘—.(3.9) Then, for a positive number ๐œ, we have ๐‘›๎“๐‘—=0,๐‘—โ‰ ๐‘–๎€ท๐‘๐‘—๎€ธ>+๐œ๐‘›๎“๐‘—=0,๐‘—โ‰ ๐‘–๐‘“๐‘—.(3.10) There is at least one ๐‘—(โ‰ ๐‘–) which satisfies ๐‘๐‘—+๐œ>๐‘“๐‘—. Denote it by ๐‘˜, and we label ๐ฉ with ๐‘˜. ๐‘˜ is one of the numbers other than ๐‘– such that ๐‘๐‘˜+๐œ>๐‘“๐‘˜ is satisfied. ๐‘– itself satisfies this condition (๐‘๐‘–+๐œ>๐‘“๐‘–). But, since there is a number other than ๐‘– which satisfies this condition, we can select a number other than ๐‘–. We have proved that we can label the vertices contained in an ๐‘›โˆ’1-dimensional face of ฮ” such that ๐‘๐‘–=0 for one ๐‘– among 0,1,2,โ€ฆ,๐‘› with the numbers other than ๐‘–. By similar procedures, we can show that we can label the vertices contained in an ๐‘›โˆ’2-dimensional face of ฮ” such that ๐‘๐‘–=0 for two ๐‘–โ€™s among 0,1,2,โ€ฆ,๐‘› with the numbers other than those ๐‘–โ€™s, and so on.
Consider the case where ๐‘๐‘–=๐‘๐‘–+1=0. We see that, when ๐‘“๐‘–>0 or ๐‘“๐‘–+1>0, ๐‘›๎“๐‘—=0,๐‘—โ‰ ๐‘–,๐‘–+1๐‘๐‘—>๐‘›๎“๐‘—=0,๐‘—โ‰ ๐‘–,๐‘–+1๐‘“๐‘—,(3.11) and so for at least one ๐‘— (denote it by ๐‘˜), we have ๐‘๐‘˜>๐‘“๐‘˜, and we label ๐ฉ with ๐‘˜. On the other hand, when ๐‘“๐‘–<๐œ and ๐‘“๐‘–+1<๐œ, we have ๐‘›๎“๐‘—=0,๐‘—โ‰ ๐‘–,๐‘–+1๐‘๐‘—โ‰ฅ๐‘›๎“๐‘—=0,๐‘—โ‰ ๐‘–,๐‘–+1๐‘“๐‘—.(3.12) Then, for a positive number ๐œ, we have ๐‘›๎“๐‘—=0,๐‘—โ‰ ๐‘–,๐‘–+1๎€ท๐‘๐‘—๎€ธ>+๐œ๐‘›๎“๐‘—=0,๐‘—โ‰ ๐‘–,๐‘–+1๐‘“๐‘—.(3.13) Thus, there is at least one ๐‘—(โ‰ ๐‘–,๐‘–+1) which satisfies ๐‘๐‘—+๐œ>๐‘“๐‘—. Denote it by ๐‘˜, and we label ๐ฉ with ๐‘˜.
Next, consider the case where ๐‘๐‘–=0 for all ๐‘– other than ๐‘›. If, for some ๐‘–,โ€‰โ€‰๐‘“๐‘–>0, then we have ๐‘๐‘›>๐‘“๐‘› and label ๐ฉ with ๐‘›. On the other hand, if ๐‘“๐‘—<๐œ for all ๐‘—โ‰ ๐‘›, then we obtain ๐‘๐‘›โ‰ฅ๐‘“๐‘›. It implies ๐‘๐‘›+๐œ>๐‘“๐‘›. Thus, we can label ๐ฉ with ๐‘›.
Therefore, the conditions for Spernerโ€™s lemma are satisfied and there exists an odd number of fully labeled simplices in ๐พ.
(2) Consider a sequence (ฮ”๐‘š)๐‘šโ‰ฅ1 of partitions of ฮ” and a sequence of fully labeled simplices (๐›ฟ๐‘š)๐‘šโ‰ฅ1. The larger ๐‘š, the finer partition. The larger ๐‘š, the smaller the diameter of a fully labeled simplex. Let ๐ฉ0๐‘š,๐ฉ1๐‘š,โ€ฆ and ๐ฉ๐‘›๐‘š be the vertices of a fully labeled simplex ๐›ฟ๐‘š. We name these vertices so that ๐ฉ0๐‘š,๐ฉ1๐‘š,โ€ฆ,๐ฉ๐‘›๐‘š are labeled, respectively, with 0,1,โ€ฆ,๐‘›. The values of ๐‘“ at theses vertices are ๐‘“(๐ฉ0๐‘š),๐‘“(๐ฉ1๐‘š),โ€ฆ and ๐‘“(๐ฉ๐‘›๐‘š). We can consider sequences of vertices of fully labeled simplices. Denote them by (๐ฉ0๐‘š)๐‘šโ‰ฅ1,(๐ฉ1๐‘š)๐‘šโ‰ฅ1,โ€ฆ, and (๐ฉ๐‘›๐‘š)๐‘šโ‰ฅ1. And consider sequences of the values of ๐‘“ at vertices of fully labeled simplices. Denote them by (๐‘“(๐ฉ0๐‘š))๐‘šโ‰ฅ1,(๐‘“(๐ฉ1๐‘š))๐‘šโ‰ฅ1,โ€ฆ, and (๐‘“(๐ฉ๐‘›๐‘š))๐‘šโ‰ฅ1. By the uniform sequential continuity of ๐‘“, ||๎€ท๐‘“๎€ท๐ฉ๐‘–๐‘š๎€ธ๎€ธ๐‘šโ‰ฅ1โˆ’๎€ท๐‘“๎€ท๐ฉ๐‘—๐‘š๎€ธ๎€ธ๐‘šโ‰ฅ1||||๎€ท๐ฉโŸถ0whenever๐‘–๐‘š๎€ธ๐‘šโ‰ฅ1โˆ’๎€ท๐ฉ๐‘—๐‘š๎€ธ๐‘šโ‰ฅ1||โŸถ0,(3.14) for ๐‘–โ‰ ๐‘—. |(๐ฉ๐‘–๐‘š)๐‘šโ‰ฅ1โˆ’(๐ฉ๐‘—๐‘š)๐‘šโ‰ฅ1|โ†’0 means ๎€ท||๐ฉโˆ€๐œ€>0โˆƒ๐‘€โˆ€๐‘šโ‰ฅ๐‘€๐‘–๐‘šโˆ’๐ฉ๐‘—๐‘š||๎€ธ<๐œ€๐‘–โ‰ ๐‘—,(3.15) and |(๐‘“(๐ฉ๐‘–๐‘š))๐‘šโ‰ฅ1โˆ’(๐‘“(๐ฉ๐‘—๐‘š))๐‘šโ‰ฅ1|โ†’0 means โˆ€๐œ€>0โˆƒ๐‘€๎…žโˆ€๐‘šโ‰ฅ๐‘€๎…ž๎€ท||๐‘“๎€ท๐ฉ๐‘–๐‘š๎€ธ๎€ท๐ฉโˆ’๐‘“๐‘—๐‘š๎€ธ||๎€ธ<๐œ€๐‘–โ‰ ๐‘—.(3.16) Consider a fully labeled simplex ๐›ฟ๐‘™ in partition of ฮ” such that ๐‘™โ‰ฅmax(๐‘€,๐‘€โ€ฒ). Denote vertices of ๐›ฟ๐‘™ by ๐ฉ0,๐ฉ1,โ€ฆ,๐ฉ๐‘›. We name these vertices so that ๐ฉ0,๐ฉ1,โ€ฆ,๐ฉ๐‘› are labeled, respectively, with 0,1,โ€ฆ,๐‘›. Then, |๐ฉ๐‘–โˆ’๐ฉ๐‘—|<๐œ€ and |๐‘“(๐ฉ๐‘–)โˆ’๐‘“(๐ฉ๐‘—)|<๐œ€.
About ๐ฉ0, from the labeling rules, we have ๐ฉ00+๐œ>๐‘“(๐ฉ0)0. About ๐ฉ1, also from the labeling rules, we have ๐ฉ11+๐œ>๐‘“(๐ฉ1)1 which implies ๐ฉ11>๐‘“(๐ฉ1)1โˆ’๐œ. |๐‘“(๐ฉ0)โˆ’๐‘“(๐ฉ1)|<๐œ€ means ๐‘“(๐ฉ1)1>๐‘“(๐ฉ0)1โˆ’๐œ€. On the other hand, |๐ฉ0โˆ’๐ฉ1|<๐œ€ means ๐ฉ01>๐ฉ11โˆ’๐œ€. Thus, from ๐ฉ01>๐ฉ11โˆ’๐œ€,๐ฉ11๎€ท๐ฉ>๐‘“1๎€ธ1๎€ท๐ฉโˆ’๐œ,๐‘“1๎€ธ1๎€ท๐ฉ>๐‘“0๎€ธ1โˆ’๐œ€,(3.17) we obtain ๐ฉ01๎€ท๐ฉ>๐‘“0๎€ธ1โˆ’2๐œ€โˆ’๐œ.(3.18) By similar arguments, for each ๐‘– other than 0, ๐ฉ0๐‘–๎€ท๐ฉ>๐‘“0๎€ธ๐‘–โˆ’2๐œ€โˆ’๐œ.(3.19) For ๐‘–=0, we have ๐ฉ00+๐œ>๐‘“(๐ฉ0)0. Then, ๐ฉ00๎€ท๐ฉ>๐‘“0๎€ธ0โˆ’๐œ.(3.20) Adding (3.19) and (3.20) side by side except for some ๐‘– (denote it by ๐‘˜) other than 0, ๐‘›๎“๐‘—=0,๐‘—โ‰ ๐‘˜๐ฉ0๐‘—>๐‘›๎“๐‘—=0,๐‘—โ‰ ๐‘˜๐‘“๎€ท๐ฉ0๎€ธ๐‘—โˆ’2(๐‘›โˆ’1)๐œ€โˆ’๐‘›๐œ.(3.21) From โˆ‘๐‘›๐‘—=0๐ฉ0๐‘—=1, โˆ‘๐‘›๐‘—=0๐‘“(๐ฉ0)๐‘—=1, we have 1โˆ’๐ฉ0๐‘˜>1โˆ’๐‘“(๐ฉ0)๐‘˜โˆ’2(๐‘›โˆ’1)๐œ€โˆ’๐‘›๐œ, which is rewritten as ๐ฉ0๐‘˜๎€ท๐ฉ<๐‘“0๎€ธ๐‘˜+2(๐‘›โˆ’1)๐œ€+๐‘›๐œ.(3.22) Since (3.19) implies ๐ฉ0๐‘˜>๐‘“(๐ฉ0)๐‘˜โˆ’2๐œ€โˆ’๐œ, we have ๐‘“๎€ท๐ฉ0๎€ธ๐‘˜โˆ’2๐œ€โˆ’๐œ<๐ฉ0๐‘˜๎€ท๐ฉ<๐‘“0๎€ธ๐‘˜+2(๐‘›โˆ’1)๐œ€+๐‘›๐œ.(3.23) Thus, ||๐ฉ0๐‘˜๎€ท๐ฉโˆ’๐‘“0๎€ธ๐‘˜||<2(๐‘›โˆ’1)๐œ€+๐‘›๐œ(3.24) is derived. On the other hand, adding (3.19) from 1 to ๐‘› yields ๐‘›๎“๐‘—=1๐ฉ0๐‘—>๐‘›๎“๐‘—=1๐‘“๎€ท๐ฉ0๎€ธ๐‘—โˆ’2๐‘›๐œ€โˆ’๐‘›๐œ.(3.25) From โˆ‘๐‘›๐‘—=0๐ฉ0๐‘—=1, โˆ‘๐‘›๐‘—=0๐‘“(๐ฉ0)๐‘—=1, we have 1โˆ’๐ฉ00๎€ท๐ฉ>1โˆ’๐‘“0๎€ธ0โˆ’2๐‘›๐œ€โˆ’๐‘›๐œ.(3.26) Then, from (3.20) and (3.26), we get ||๐ฉ00๎€ท๐ฉโˆ’๐‘“0๎€ธ0||<2๐‘›๐œ€+๐‘›๐œ.(3.27) From (3.24) and (3.27), we obtain the following result: ||๐ฉ0๐‘–๎€ท๐ฉโˆ’๐‘“0๎€ธ๐‘–||<2๐‘›๐œ€+๐‘›๐œโˆ€๐‘–.(3.28) Thus, ||๐ฉ0๎€ท๐ฉโˆ’๐‘“0๎€ธ||<๐‘›(๐‘›+1)(2๐œ€+๐œ).(3.29) Since ๐‘› is finite, ๐ฉ0 is an approximate fixed point of ๐‘“. Similarly, we can prove that every other vertex, ๐ฉ1,๐ฉ2,โ€ฆ,๐ฉ๐‘›, and all points in a fully-labeled simplex of ๐พ 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 Spernerโ€™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.

References

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Copyright ยฉ 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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