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), 053065, 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. e-mail: 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)
IAENG Iteratioal Joural of Applied Mathematics, 4:, IJAM_4 0 Fig.. 0 0 0 0 0 0 0 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)
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)
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 0 0 + 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 0 0 + 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)
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)
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. 473 483, 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. 340 344, 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. 360 364, 005. [6] D. Bridges, H. Ishihara, P. Schuster, ad L. Vîţă, Strog cotiuity implies uiform sequetial cotiuity, Archive for Mathematical Logic, vol. 44, pp. 887 895, 005. [7] H. Ishihara, Cotiuity properties i costructive mathematics, Joural of Symbolic Logic, vol. 57, pp. 557 565, 99. [8] D. Bridges ad R. Mies, Sequetially cotiuous liear mappigs i costructive aalysis, Joural of Symbolic Logic, vol. 63, pp. 579 583, 998. [9] J. Berger, D. Bridges, ad P. Schuster, The fa theorem ad uique existece of maxima, Joural of Symbolic Logic, vol. 7, pp. 73 70, 006. [0] Y. Taaka, Costructive proof of Brouwer s fixed poit theorem for sequetially locally o-costat fuctios, http://arxiv.org/abs/03. 776, 0. [] V. P. Orevkov, A costructive mappig of a square oto itself displacig every costructive poit, Soviet Math., vol. 4, pp. 53 56, 963. [] J. L. Hirst, Notes o reverse mathematics ad Brouwer s fixed poit theorem, http://www.mathsci.appstate.edu/ 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. 930 94, 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. 33 40, 0. (Advace olie publicatio: 7 February 0)