IT is well known that Brouwer s fixed point theorem can

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1 IAENG Iteratioal Joural of Applied Mathematics, 4:, IJAM_4 0 Costructive Proof of Brouwer s Fixed Poit Theorem for Sequetially Locally No-costat ad Uiformly Sequetially Cotiuous Fuctios Yasuhito Taaka, Member, IAENG Abstract We preset a costructive proof of Brouwer s fixed poit theorem for sequetially locally o-costat ad uiformly sequetially cotiuous fuctios based o the existece of approximate fixed poits. Ad we will show that our Brouwer s fixed poit theorem implies Sperer s lemma for a simplex. Sice the existece of approximate fixed poits is derived from Sperer s lemma, our Brouwer s fixed poit theorem is equivalet to Sperer s lemma. Idex Terms Brouwer s fixed poit theorem, Sperer s lemma, sequetially locally o-costat fuctios, uiformly sequetially cotiuous fuctios, costructive mathematics. I. INTRODUCTION IT is well kow that Brouwer s fixed poit theorem ca ot be costructively proved. Sperer s lemma which is used to prove Brouwer s theorem, however, ca be costructively proved. Some authors have preseted a approximate versio of Brouwer s theorem usig Sperer s lemma. See [3] ad [4]. Thus, Brouwer s fixed poit theorem is costructively, i the sese of costructive mathematics á la Bishop, proved i its approximate versio. Also Dale i [3] states a cojecture that a uiformly cotiuous fuctio f from a simplex ito itself, with property that each ope set cotais a poit x such that x f(x), which meas x f(x) > 0, ad also at every poit x o the boudaries of the simplex x f(x), has a exact fixed poit. We call such a property of fuctios local o-costacy. I this paper we preset a partial aswer to Dale s cojecture. Recetly [5] showed that the followig theorem is equivalet to Brouwer s fa theorem. Each uiformly cotiuous fuctio f from a compact metric space X ito itself with at most oe fixed poit ad approximate fixed poits has a fixed poit. I [3], [4] ad [5] uiform cotiuity of fuctios is assumed. We cosider a weaker uiform sequetial cotiuity of fuctios accordig to [6]. I classical mathematics uiform cotiuity ad uiform sequetial cotiuity are equivalet. Mauscript received Jue, 0; revised September, 0. This work was supported i part by the Miistry of Educatio, Sciece, Sports ad Culture of Japa, Grat-i-Aid for Scietific Research (C), , ad the Special Costs for Graduate Schools of the Special Expeses for Hitech Promotio by the Miistry of Educatio, Sciece, Sports ad Culture of Japa i 0. Yasuhito Taaka is with the Faculty of Ecoomics, Doshisha Uiversity, Kyoto, Japa. yasuhito@mail.doshisha.ac.jp. [] provided a costructive proof of Brouwer s fixed poit theorem. But it is ot costructive from the view poit of costructive mathematics á la Bishop. It is sufficiet to say that oe dimesioal case of Brouwer s fixed poit theorem, that is, the itermediate value theorem is o-costructive. See [] or [3]. I costructive mathematics á la Bishop, however, uiform sequetial cotiuity is weaker tha uiform cotiuity Ad by referece to the otio of sequetially at most oe maximum i [9] we require a coditio that a fuctio f is sequetially locally o-costat, ad will show the followig result. Each sequetially locally o-costat ad uiformly sequetially cotiuous fuctio f from a -dimesioal simplex ito itself has a fixed poit, without the fa theorem 3. Our sequetial local ocostacy, the coditio i [3] (local o-costacy) ad the coditio that a fuctio has at most oe fixed poit i [5] are mutually differet. [] costructed a computably coded cotiuous fuctio f from the uit square ito itself, which is defied at each computable poit of the square, such that f has o computable fixed poit. His map cosists of a retract of the computable elemets of the square to its boudary followed by a rotatio of the boudary of the square. As poited out by [], sice there is o retract of the square to its boudary, his map does ot have a total extesio. I the ext sectio we preset Sperer s lemma. Its proof is omitted idicatig refereces. I Sectio 3 we preset our Brouwer s fixed poit theorem ad its proof. The first part of the proof proves the existece of a approximate fixed poit of uiformly sequetially cotiuous fuctios usig Sperer s lemma, ad the secod part proves the existece of a exact fixed poit of sequetially locally o-costat ad uiformly sequetially cotiuous fuctios. I Sectio 4 we will derive Sperer s lemma from Brouwer s fixed poit theorem for uiformly sequetially cotiuous ad sequetially locally o-costat fuctios. II. SPERNER S LEMMA Let deote a -dimesioal simplex. is a fiite atural umber. For example, a -dimesioal simplex is a triagle. Let partitio or triagulate the simplex. Fig. is a example of partitio (triagulatio) of a -dimesioal simplex. I a -dimesioal case we divide each side of i m equal segmets, ad draw the lies parallel to the sides Also i costructive mathematics sequetial cotiuity is weaker tha cotiuity, ad uiform cotiuity (respectively, uiform sequetial cotiuity) is stroger tha cotiuity (respectively, sequetial cotiuity) eve i a compact space. See, for example, [7]. As stated i [8] all proofs of the equivalece betwee cotiuity ad sequetial cotiuity ivolve the law of excluded middle, ad so the equivalece of them is o-costructive. 3 I aother paper [0] we have preseted a partial aswer to Dale s cojecture with uiform cotiuity ad sequetial local o-costacy, that is, a proof of the existece of a fixed poit for a uiformly cotiuous ad sequetially locally o-costat fuctios. (Advace olie publicatio: 7 February 0)

2 IAENG Iteratioal Joural of Applied Mathematics, 4:, IJAM_4 0 Fig Partitio ad labelig of -dimesioal simplex of. m is a fiite atural umber. The, the -dimesioal simplex is partitioed ito m triagles. We cosider partitio of iductively for cases of higher dimesio. I a 3 dimesioal case each face of is a -dimesioal simplex, ad so it is partitioed ito m triagles i the above-metioed way, ad draw the plaes parallel to the faces of. The, the 3-dimesioal simplex is partitioed ito m 3 trigoal pyramids. Ad similarly for cases of higher dimesio. Let K deote the set of small -dimesioal simplices of costructed by partitio. Vertices of these small simplices of K are labeled with the umbers 0,,,..., subject to the followig rules. ) The vertices of are respectively labeled with 0 to. We label a poit (, 0,..., 0) with 0, a poit (0,, 0,..., 0) with, a poit (0, 0,..., 0) with,..., a poit (0,..., 0, ) with. That is, a vertex whose k-th coordiate (k = 0,,..., ) is ad all other coordiates are 0 is labeled with k. ) If a vertex of K is cotaied i a -dimesioal face of, the this vertex is labeled with some umber which is the same as the umber of oe of the vertices of that face. 3) If a vertex of K is cotaied i a -dimesioal face of, the this vertex is labeled with some umber which is the same as the umber of oe of the vertices of that face. Ad so o for cases of lower dimesio. 4) A vertex cotaied iside of is labeled with a arbitrary umber amog 0,,...,. A small simplex of K which is labeled with the umbers 0,,..., is called a fully labeled simplex. Sperer s lemma is stated as follows. Lemma (Sperer s lemma): If we label the vertices of K followig the rules ) 4), the there are a odd umber of fully labeled simplices, ad so there exists at least oe fully labeled simplex. Proof: About costructive proofs of Sperer s lemma see [3] or [4]. Sice ad partitio of are fiite, the umber of small simplices costructed by partitio is also fiite. Thus, we ca costructively fid a fully labeled -dimesioal simplex of K through fiite steps. III. BROUWER S FIXED POINT THEOREM FOR SEQUENTIALLY LOCALLY NON-CONSTANT AND UNIFORMLY SEQUENTIALLY CONTINUOUS FUNCTIONS Let x = (x 0, x,..., x ) be a poit i a -dimesioal simplex, ad cosider a fuctio f from to itself. Deote the i-th compoets of x ad f(x) by, respectively, x i ad f i (x) or f i. Uiform cotiuity, sequetial cotiuity ad uiform sequetial cotiuity of fuctios are defied as follows; Defiitio (Uiform cotiuity): A fuctio f is uiformly cotiuous i if for ay x, x ad ε > 0 there exists δ > 0 such that If x x < δ, the f(x) f(x ) < ε. δ depeds o oly ε. Defiitio (Sequetial cotiuity): A fuctio f is sequetially cotiuous at x i if for sequeces (x ) ad (f(x )) i f(x ) f(x) wheever x x. Defiitio 3 (Uiform sequetial cotiuity): A fuctio f is uiformly sequetially cotiuous i if for sequeces (x ), (x ), (f(x )) ad (f(x )) i f(x ) f(x ) 0 wheever x x 0. x x 0 meas ε > 0 N N ( x x < ε), where ε is a real umber, ad ad N are atural umbers. Similarly, f(x ) f(x ) 0 meas ε > 0 N N ( f(x ) f(x ) < ε). N is a atural umber. I classical mathematics uiform cotiuity ad uiform sequetial cotiuity of fuctios are equivalet. But i costructive mathematics á la Bishop uiform sequetial cotiuity is weaker tha uiform cotiuity. O the other had, the defiitio of local o-costacy of fuctios is as follows; Defiitio 4: (Local o-costacy of fuctios) ) At a poit x o a face (boudary) of a simplex f(x) x. This meas f i (x) > x i or f i (x) < x i for at least oe i. ) I ay ope set of there exists a poit x such that f(x) x. The otio that φ has at most oe fixed poit i [5] is defied as follows; Defiitio 5 (At most oe fixed poit): For all x, y, if x y, the f(x) x or f(y) y. Next, by referece to the otio of sequetially at most oe maximum i [9], we defie the property of sequetial local o-costacy. First we recapitulate the compactess (total boudedess with completeess) of a set i costructive mathematics. is totally bouded i the sese that for each ε > 0 there exists a fiitely eumerable ε-approximatio to 4. A ε- approximatio to is a subset of such that for each x there exists y i that ε-approximatio with x y < ε. 4 A set S is fiitely eumerable if there exist a atural umber N ad a mappig of the set {,,..., N} oto S. (Advace olie publicatio: 7 February 0)

3 IAENG Iteratioal Joural of Applied Mathematics, 4:, IJAM_4 0 Accordig to Corollary.. of [5] we have the followig result. Lemma : For each ε > 0 there exist totally bouded sets H, H,..., H, each of diameter less tha or equal to ε, such that = i= H i. The defiitio of sequetial local o-costacy is as follows; Defiitio 6: (Sequetial local o-costacy of fuctios) There exists ε > 0 with the followig property. For each ε > 0 less tha or equal to ε there exist totally bouded sets H, H,..., H m, each of diameter less tha or equal to ε, such that = m i= H i, ad if for all sequeces (x ), (y ) i each H i, f(x ) x 0 ad f(y ) y 0, the x y 0. Now we show the followig lemma, which is based o Lemma of [9]. Lemma 3: Let f be a uiformly cotiuous fuctio from ito itself. Assume if x Hi f(x) x = 0 for some H i defied above. If the followig property holds: For each δ > 0 there exists ε > 0 such that if x, y H i, f(x) x < ε ad f(y) y < ε, the x y δ. The, there exists a poit z H i such that f(z) = z, that is, f has a fixed poit. Proof: Choose a sequece (x ) i H i such that f(x ) x 0. Compute N such that f(x ) x < ε for all N. The, for m, N we have x m x δ. Sice δ > 0 is arbitrary, (x ) is a Cauchy sequece i H i, ad coverges to a limit z H i. The cotiuity of f yields f(z) z = 0, that is, f(z) = z. Next we show Lemma 4: If X is a totally bouded space, ad φ is a uiformly sequetially cotiuous fuctio of X ito a metric space, the φ(x) is totally bouded. Proof: Cosider a sequece of positive real umbers (ε m ) m such that ε > ε > > ε m ad ε m 0, ad a sequece of ε m -approximatio L m = {x m, x m,..., x m } to X. For each x X ad each ε m, there exists a poit x i m L m such that x x i m < ε m. Thus, we ca costruct a sequece (x im ) m such that x x im 0. The uiform sequetial cotiuity implies φ(x) φ(x i m ) 0. x x i m 0 meas ε m > 0 M m M ( x x im < ε m ). M is a atural umber. Similarly, φ(x) φ(x im ) 0 meas ε m > 0 M m M ( φ(x) φ(x i m ) < ε m ). M is a atural umber. Let m max(m, M ). The, correspodig to a ε m -approximatio to X there exists a ε m -approximatio to φ(x). Therefore, φ(x) is totally bouded. This is a modified versio of Propositio..6 of [5]. From this lemma we see that φ has the ifimum i X by Corollary..7 of [5]. The, f(x) x has the ifimum i. With these prelimiaries we show the followig theorem. Theorem : (Brouwer s fixed poit theorem for sequetially locally o-costat ad uiformly sequetially cotiuous fuctios) Ay sequetially locally o-costat ad uiformly sequetially cotiuous fuctio from a - dimesioal simplex to itself has a fixed poit. Proof: ) First we show that we ca partitio so that the coditios for Sperer s lemma are satisfied. We partitio accordig to the method i Sperer s lemma, ad label the vertices of simplices costructed by partitio of. It is importat how to label the vertices cotaied i the faces of. Let K be the set of small simplices costructed by partitio of, x = (x 0, x,..., x ) be a vertex of a simplex of K, ad deote the i-th compoet of f(x) by f i. The, we label a vertex x accordig to the followig rule, If x k > f k or x k + τ > f k, we label x with k, where τ is a positive umber. If there are multiple k s which satisfy this coditio, we label x coveietly for the coditios for Sperer s lemma to be satisfied. We do ot radomly label the vertices. For example, let x be a poit cotaied i a -dimesioal face of such that x i = 0 for oe i amog 0,,,..., (its i-th coordiate is 0). With τ > 0, we have f i > 0 or f i < τ. I costructive mathematics for ay real umber x we ca ot prove that x 0 or x < 0, that x > 0 or x = 0 or x < 0. But for ay distict real umbers x, y ad z such that x > z we ca prove that x > y or y > z. Whe f i > 0, from x j =, f j = ad x i = 0, x j > f j.,j i,j i The, for at least oe j (deote it by k) we have x k > f k, ad we label x with k, where k is oe of the umbers which satisfy x k > f k. Sice f i > x i = 0, i does ot satisfy this coditio. Assume f i < τ. x i = 0 implies,j i x j =. Sice,j i f j, we obtai x j f j.,j i,j i The, for a positive umber τ we have (x j + τ) > f j.,j i,j i There is at least oe j( i) which satisfies x j +τ > f j. Deote it by k, ad we label x with k. k is oe of the umbers other tha i such that x k + τ > f k is satisfied. i itself satisfies this coditio (x i + τ > f i ). But, sice there is a umber other tha i which satisfies this coditio, we ca select a umber other tha i. We have proved that we ca label the vertices cotaied i a -dimesioal face of such that x i = 0 for oe i amog 0,,,..., with the umbers other tha i. By similar procedures we ca show that we ca label the vertices cotaied i a -dimesioal face of such that x i = 0 for two i s amog 0,,,..., with the umbers other tha those i s, ad so o. Therefore, the coditios for Sperer s lemma are satisfied, ad there exists a odd umber of fully labeled simplices i K. (Advace olie publicatio: 7 February 0)

4 IAENG Iteratioal Joural of Applied Mathematics, 4:, IJAM_4 0 Cosider a sequece ( m ) m of partitios of, ad a sequece of fully labeled simplices (δ m ) m. The larger m, the fier partitio. The larger m, the smaller the diameter of a fully labeled simplex. Let x 0 m, x m,... ad x m be the vertices of a fully labeled simplex δ m. We ame these vertices so that x 0 m, x m,..., x m are labeled, respectively, with 0,,...,. The values of f at theses vertices are f(x 0 m), f(x m),... ad f(x m). We ca cosider sequeces of vertices of fully labeled simplices. Deote them by (x 0 m) m, (x m) m,..., ad (x m) m. Ad cosider sequeces of the values of f at vertices of fully labeled simplices. Deote them by (f(x 0 m)) m, (f(x m)) m,..., ad (f(x m)) m. By the uiform sequetial cotiuity of f (f(x i m)) (f(x j m)) m 0 wheever (x i m) (x j m) m 0, for i j. (x i m) m (x j m) m 0 meas ε > 0 M m M ( x i m x j m < ε) i j, ad (f(x i m)) m (f(x j m)) m 0 meas ε > 0 M m M ( f(x i m) f(x j m) < ε) i j. Cosider a fully labeled simplex δ l i partitio of such that l max(m, M ). Deote vertices of δ l by x 0, x,..., x. We ame these vertices so that x 0, x,..., x are labeled, respectively, with 0,,...,. The, x i x j < ε ad f(x i ) f(x j ) < ε. About x 0, from the labelig rules we have x τ > f(x 0 ) 0. About x, also from the labelig rules we have x + τ > f(x ) which implies x > f(x ) τ. f(x 0 ) f(x ) < ε meas f(x ) > f(x 0 ) ε. O the other had, x 0 x < ε meas x 0 > x ε. Thus, from x 0 > x ε, x > f(x ) τ, f(x ) > f(x 0 ) ε we obtai x 0 > f(x 0 ) ε τ By similar argumets, for each i other tha 0, x 0 i > f(x 0 ) i ε τ. () For i = 0 we have x τ > f(x 0 ) 0. The, x 0 0 > f(x 0 ) 0 τ () Addig () ad () side by side except for some i (deote it by k) other tha 0, x 0 j > f(x 0 ) j ( )ε τ.,j k,j k From x0 j =, f(x0 ) j = we have x 0 k > f(x0 ) k ( )ε τ, which is rewritte as x 0 k < f(x 0 ) k + ( )ε + τ. Sice () implies x 0 k > f(x0 ) k ε τ, we have f(x 0 ) k ε τ < x 0 k < f(x 0 ) k + ( )ε + τ. Thus, x 0 k f(x 0 ) k < ( )ε + τ (3) is derived. O the other had, addig () from to yields x 0 j > j= f(x 0 ) j ε τ. j= From x0 j =, f(x0 ) j = we have x 0 0 > f(x 0 ) 0 ε τ. (4) The, from () ad (4) we get x 0 0 f(x 0 ) 0 < ε + τ. (5) From (3) ad (5) we obtai the followig result, Thus, x 0 i f(x 0 ) i < ε + τ for all i. x 0 f(x 0 ) < ( + )(ε + τ). (6) Sice is fiite, x 0 is a approximate fixed poit of f 5. Ad sice ε > 0 ad τ are arbitrary, if x f(x) = 0. x By Lemma we have if x Hi x f(x) = 0 for some H i defied i that lemma. ) Choose a sequece (ξ m ) m i H i such that f(ξ m ) ξ m 0. I view of Lemma 3 it is eough to prove that the followig coditio holds. For each δ > 0 there exists ε > 0 such that if x, y H i, f(x) x < ε ad f(y) y < ε, the x y δ. Assume that the set K = {(x, y) H i H i : x y δ} is oempty ad compact 6. Sice the mappig (x, y) max( f(x) x, f(y) y ) is uiformly sequetially cotiuous, by Lemma 4 we ca costruct a icreasig biary sequece (λ m ) m such that λ m = 0 λ m = if max( f(x) x, f(y) y ) < (x,y) K m, if (x,y) K max( f(x) x, f(y) y ) > m. It suffices to fid m such that λ m =. I that case, if f(x) x < m ad f(y) y < m, we have (x, y) / K ad x y δ. Assume λ = 0. If λ m = 0, choose (x m, y m ) K such that max( f(x m ) x m, f(y m ) y m ) < m, ad if λ m =, set x m = y m = ξ m. The, f(x m ) x m 0 ad f(y m ) y m 0, so x m y m 0. Computig M such that x M y M < δ, we must have λ M =. We have completed the proof. 5 I aother paper [6] we have show the existece of a approximate fixed poit of each uiformly cotiuous fuctio i a locally-covex space. 6 See Theorem..3 of [5]. (Advace olie publicatio: 7 February 0)

5 IAENG Iteratioal Joural of Applied Mathematics, 4:, IJAM_4 0 IV. FROM BROUWER S FIXED POINT THEOREM FOR SEQUENTIALLY LOCALLY NON-CONSTANT AND UNIFORMLY SEQUENTIALLY CONTINUOUS FUNCTIONS TO SPERNER S LEMMA I this sectio we will derive Sperer s lemma from Brouwer s fixed poit theorem for sequetially locally ocostat ad uiformly sequetially cotiuous fuctios. Let be a -dimesioal simplex. Deote a poit o by x. Cosider a fuctio f from to itself. Partitio i the way depicted i Fig.. Let K deote the set of small - dimesioal simplices of costructed by partitio. Vertices of these small simplices of K are labeled with the umbers 0,,,..., subject to the same rules as those i Lemma. Now we derive Sperer s lemma expressed i Lemma from Brouwer s fixed poit theorem for sequetially locally ocostat ad uiformly sequetially cotiuous fuctios. Deote the vertices of a -dimesioal simplex of K by x 0, x,..., x, the j-th coordiate of x i by x i j, ad deote the label of x i by l(x i ). Let τ be a positive umber which is smaller tha x i l(x i ) for all xi, ad defie a fuctio f(x i ) as follows 7 : ad f(x i ) = (f 0 (x i ), f (x i ),..., f (x i )), f j (x i ) = { x i j τ for j = l(x i ), x i j + τ for j l(x i ). f j deotes the j-th compoet of f. From the labelig rules we have x i l(x i ) > 0 for all xi, ad so τ > 0 is well defied. Sice f j(x i ) = xi j =, we have f(x i ). We exted f to all poits i the simplex by covex combiatios o the vertices of the simplex. Let z be a poit i the - dimesioal simplex of K whose vertices are x 0, x,..., x. The, z ad f(z) are expressed as follows; z = λ i x i, ad f(z) = λ i f(x i ), λ i 0, λ i =. We verify that f is uiformly sequetially cotiuous. Cosider sequeces (z()), (z ()), (f(z())) ad (f(z ())) such that z() z () 0. Deote each compoet of z() by z() j ad so o. Whe z() z () 0, z() j z () j 0 for each j. The, sice τ > 0 is fiite, we have f(z()) f(z ()) 0, ad so f is uiformly sequetially cotiuous. Next we verify that f is sequetially locally o-costat. ) Assume that a poit z is cotaied i a - dimesioal small simplex δ costructed by partitio of a -dimesioal face of such that its i-th coordiate is z i = 0. Deote the vertices of δ by x j, j = 0,,..., ad their i-th coordiate by x j i. The, we have f i (z) = λ j f i (x j ), λ j 0, λ j =. (7) Sice all vertices of δ are ot labeled with i, (7) meas f i (x j ) > x j i for all j = {0,,..., }. 7 We refer to [7] about the defiitio of this fuctio. The, there exists o sequece (z(m)) m such that f(z(m)) z(m) 0 i a -dimesioal face of. ) Let z be a poit i a -dimesioal simplex δ. Assume that o vertex of δ is labeled with i. The f i (z) = λ j f i (x j ) = z i + ( + ) τ. (8) The, there exists o sequece (z(m)) m such that f(z(m)) z(m) 0 i δ. 3) Assume that z is cotaied i a fully labeled - dimesioal simplex δ, ad reame vertices of δ so that a vertex x i is labeled with i for each i. The, f i (z) = λ j f i (x j ) = λ j x j i + τ λ j λ iτ j i = z i + λ j λ i τ for each i. j i Cosider sequeces (z(m)) m = (z(), z(),... ), (z (m)) m = (z (), z (),... ) such that f(z(m)) z(m) 0 ad f(z (m)) z (m) 0. Let z(m) = λ(m) ix i ad z (m) = λ (m) i x i. The, we have λ(m) j λ(m) i 0, ad j i λ (m) j λ (m) i 0 for all i. j i Therefore, we obtai λ(m) i +, ad λ (m) i +. These mea z(m) z (m) 0. Thus, f is sequetially locally o-costat, ad it has a fixed poit. Let z be a fixed poit of f. We have z i = f i (z ) for all i. (9) Suppose that z is cotaied i a small -dimesioal simplex δ. Let z 0, z,..., z be the vertices of δ. The, z ad f(z ) are expressed as z = λ i z i ad f(z ) = λ i f(z i ), λ i 0, λ i =. (7) implies that if oly oe z k amog z 0, z,..., z is labeled with i, we have f i (z ) = λ j f i (z j ) = λ j z j i + τ λ j λ kτ This meas j k = z i (z i is the i th coordiate of z ). λ j λ k = 0. j k The, (9) is satisfied with λ k = + for all k. If o zj is labeled with i, we have (8) with z = z ad the (9) ca ot (Advace olie publicatio: 7 February 0)

6 IAENG Iteratioal Joural of Applied Mathematics, 4:, IJAM_4 0 be satisfied. Thus, oe ad oly oe z j must be labeled with i for each i. Therefore, δ must be a fully labeled simplex, ad so the existece of a fixed poit of f implies the existece of a fully labeled simplex. We have completely proved Sperer s lemma. [7] M. Yoseloff, Topological proofs of some combiatorial theorems, Joural of Combiatorial Theory (A), vol. 7, pp. 95, 974. V. CONCLUDING REMARKS As a future research program we are studyig the followig themes. ) A applicatio of Brouwer s fixed poit theorem for sequetially locally o-costat fuctios to ecoomic theory ad game theory, i particular, the problem of the existece of a equilibrium i a competitive ecoomy with excess demad fuctios which have the property that is similar to sequetial local ocostacy, ad the existece of Nash equilibrium i a strategic game with payoff fuctios which satisfy sequetial local o-costacy. ) A geeralizatio of the result of this paper to Kakutai s fixed poit theorem for multi-valued fuctios with property of sequetial local o-costacy ad its applicatio to ecoomic theory. REFERENCES [] R. B. Kellogg, T. Y. Li, ad J. Yorke, A costructive proof of Brouwer fixed-poit theorem ad computatioal results, SIAM Joural o Numerical Aalysis, vol. 3, pp , 976. [] D. Bridges ad F. Richma, Varieties of Costructive Mathematics. Cambridge Uiversity Press, 987. [3] D. va Dale, Brouwer s ε-fixed poit from Sperer s lemma, Theoretical Computer Sciece, vol. 4, o. 8, pp , Jue 0. [4] W. Veldma, Brouwer s approximate fixed poit theorem is equivalet to Brouwer s fa theorem, i Logicism, Ituitioism ad Formalism, S. Lidström, E. Palmgre, K. Segerberg, ad V. Stolteberg-Hase, Eds. Spriger, 009. [5] J. Berger ad H. Ishihara, Brouwer s fa theorem ad uique existece i costructive aalysis, Mathematical Logic Quarterly, vol. 5, o. 4, pp , 005. [6] D. Bridges, H. Ishihara, P. Schuster, ad L. Vîţă, Strog cotiuity implies uiform sequetial cotiuity, Archive for Mathematical Logic, vol. 44, pp , 005. [7] H. Ishihara, Cotiuity properties i costructive mathematics, Joural of Symbolic Logic, vol. 57, pp , 99. [8] D. Bridges ad R. Mies, Sequetially cotiuous liear mappigs i costructive aalysis, Joural of Symbolic Logic, vol. 63, pp , 998. [9] J. Berger, D. Bridges, ad P. Schuster, The fa theorem ad uique existece of maxima, Joural of Symbolic Logic, vol. 7, pp , 006. [0] Y. Taaka, Costructive proof of Brouwer s fixed poit theorem for sequetially locally o-costat fuctios, 776, 0. [] V. P. Orevkov, A costructive mappig of a square oto itself displacig every costructive poit, Soviet Math., vol. 4, pp , 963. [] J. L. Hirst, Notes o reverse mathematics ad Brouwer s fixed poit theorem, jlh/sp/pdfslides/bfp.pdf, pp. 6, 000. [3] F. E. Su, Retal harmoy: Sperer s lemma for fair devisio, America Mathematical Mothly, vol. 06, pp , 999. [4] Y. Taaka, Equivalece betwee the existece of a approximate equilibrium i a competitive ecoomy ad Sperer s lemma: A costructive aalysis, ISRN Applied Mathematics, vol. Article ID 38465, pp. 5, 0. [5] D. Bridges ad L. Vîţă, Techiques of Costructive Mathematics. Spriger, 006. [6] Y. Taaka, Proof of costructive versio of the Fa-Glicksberg fixed poit theorem directly by Sperer s lemma ad approximate Nash equilibrium with cotiuous strategies: A costructive aalysis, IAENG Iteratioal Joural of Applied Mathematics, vol. 4, o., pp , 0. (Advace olie publicatio: 7 February 0)