J. Korean Math. Soc. 32 (1995), No. 2, pp. 251-264
COINCIDENCES OF COMPOSITES OF V.S.C.
MAPS ON H-SPACES AND APPLICATIONS
SEHIE PARK AND HOONJOO KIM
1. Introduction
Applications of the classical Knaster - Kuratowski - Mazurkiewicz (si- mply, KKM) theorem and the fixed point theory of multifunctions defined on convex subsets of topological vector spaces have been greatly improved by adopting the concept of convex spaces due to Lassonde [LIJ. In this direction, the first author [P5J found that certain coinci- dence theorems on a large class of composites of upper semicontinuous multifunctions imply many fundamental results in the KKM theory.
On the other hand, the concept of convex spaces was extended by Horvath [HI-5J to spaces having certain families of contractible subsets orH-spaces. In this direction, a number of authors extended important results on convex spaces to those on H-spaces. See Bardaro and Cep- pitelli [BCI-3J, Ding et al. [Di, DKTI-2, DTrJ, Park [PI-3]' Tarafdar [T], and H. Kim [KJ.
In the present paper, we extend the main coincidence theorem in [P5J to H-spaces and apply it to obtain a far-reaching generalization of the KKM theorem, fixed point or coincidence theorems for H-spaces, and other results. We also obtain open-valued versions of a KKM theorem and a coincidence theorem for H-spaces. Many of the main results in the above-mentioned papers are extended and unified. Especially, the main theorems of [DiJ are improved and corrected.
Received January 25, 1994.
1991 AMS Subject Classifications: Primary 54C65, 54H25, 52A07, 47HI0, 55M20.
Key words and phrases: KKM theorem, multifunction, upper semicontinuous (u.s.c.), contractible, acyclic, convex space, D-convex, polytope, H-space, c-space, H-convex, H-subspace, Kakutani map, acyclic map, open-valued KKM theorem
Supported in part by the Basic Sciences Research Institute Program, Ministry of Education, 1992.
2. Preliminaries
For the terminology and notations, we follow mainly [Pl-5J.
A multi/unction (simply, map) F : X ~ Y is a function from a set X into the power set of a set Y. Let F(A) = U{Fx : x E A} for A C X and F- y ={x EX: y E Fx} for y E Y. As usual, the set Gr(F)={(x,y): y E Fx} is called the graphof F.
For topological spaces X and Y, a map F : X ~ Y is said to be upper semicontinuous (u.s.c.) iffor each closed set B C Y, F-(B) = {x EX: FxnB =F 0} is closed inX; and compactifF( X) is contained in a compact subset ofY.
A subset C of a topological space X is said to be compactly closed [resp. openJ inX if for every compact setK eX,the setCnKis closed [resp. open] in K. A topological space X is said to be contractibleif the identity·map Ix ·ofXis homotopic to a constant map; and acyclic if all of its reduced Cech homology groups over rationals vanish.
Int and - denote the interior and closure, resp.
For a set D, let (D) denote the set of all nonempty finite subsets of
D.
Let X be a set (in a vector space) and D a nonempty subset of X. Then (X,D) is called. a convex space [P5] if for eachN E (D), its convex hull coN is contained. in X and X has a topology that induces the Euclidean topology on such convex hulls. Such convex hull will be called a polytope. A subset A of X is said to be D-convex if, for any N E (D), N c A implies coN C A. If X = D, then X = (X,X) becomes a convex space in the sense of Lassonde [LIJ.
A triple(X, D;r) is called an H-space [Pl,2] if X isa topological space, D a nonempty subset· of X, and f = {fA} a family of con- tractible subsets ofX indexed by A E (D) such that fAC fB when- ever A CBE (D). (The triple is called a c-space in [H5J whenever X =D.) If X = D, we denote (Xjr) instead of (X,Xjr). For an H-space(X;f) and any nonempty subsetY ofX,we have anH-space (X,Yjr).
Any convex space(X,D) is an H-space (X,Djr) by puttingfA = .coA'fo:t;f'~'(D).
For an (X, Djr), a subset C of X is said to be H -convex if for each A E (D), A C C implies fA C C. A subset L of X is called an H-subspace of (X, D;r) ifL nD =F 0and for every A E (LnD),
Coincidences of composites of U.s.c. maps 253
fAn L is contractible. This is equivalent to saying that the triple (L, L n D; {fAn L}) is an H-space.
Given a class X of maps, X(X, Y) denotes the set of all maps F :
X -<l Y belonging toX, andXc the set of all finite composites of maps
in X.
For topological spaces X and Y, we define
1E qx,Y) ~ 1is a (single-valued) continuous function.
T E lK(X,Y) ~ T is a Kakutani map; that is, Y is a convex space and T is U.S.c. with nonempty compact convex values.
T E V(X,Y) ~ T is an acyclic map; that is, T is U.S.c. with compact acyclic values.
A class Q( of maps is one satisfying:
(i) Q( contains the class C of (single-valued) continuous functions;
(ii) each FE Q(c is u.s.c. and compact-valued; and
(iii) for any polytope P, each F EQ(c(P, P) has a fixed point.
Examples ofQ( are C, lK, V, the Aronszajn mapsM (with R6 values) [Gr], the O'Neill maps N (with values consisting of one or m acyclic components, where m is fixed) [Gr], the approachable maps in topo- logical vector spaces [BDl,2], the admissible maps of G6miewicz [Go], the permissible maps of Dzedzej [Dz], and others.
Further, we define the following:
T E 2(~(X,Y) ~ for any a-compact subset 1\ of X, there is a
T EQ(c(1\,Y) such that Tx c Tx for each xE K.
T E Q(~(X,Y) ~ for any compact subset K of X, there is a
TEQ(c(I{,Y) as above.
The classlKt due to Lassonde [L3] andvt due to Park, Singh, and Watson [PSW] belong to Q(~. Note that Q( C Q(c C 2t~ C Q(~. For details, see Park [P5J or [PKJ.
3. Coincidence theorems for H-spaces
In this section, we give a basic coincidence theorem.
Let 6n denote an n-simplex. We need the following:
LEMMA. Let(X, D;f) be an H -space where D = {IQ, Xl, . . . ,In} E (X). Then there exists an 1 E q6n,X) such that 1(1::.J) C fJ for each J E (D), where 6J is the face of 6 n corresponding to J.
Lemma is given implicitly by Horvath [H2, Theoreme I), [H3, The- orem 1] and explicitly by Ding and Tan [DT, Lemma I) and Horvath [H4, Theoreme 1], [H5, Theorem 1.1].
Our main result is the following general coincidence theorem related to the class~~:
THEOREM 1. Let (X, D;r) be an H -space, Y a Hausdorff space, F E 2{~(X,V), and K a nonempty compact subset ofY. Let S: D-o Y ai1dT :X - 0Y satisfy the following:
(1.1) for each x E D, Sx is compactly openin Y;
(1.2) for each yE F(X), ME (S-y) impliesrM C T-y;
(1.3) F(X)nKC SeD); and (1.4) either
(i) Y\K c S(M) for some M E (D); or
(ii) for each N E (D), there exists a compact H-subspace LN of X containingN such that F(LN)\Kc S(LNnD).
Then T and F has a coincidence point x EX; that is, Tx nFx =1= 0.
Proof. Since F(X)nK is compact and covered by compactly open sets Sx by (1.1) and (1.3), there exists an N E (D) such that F(X)n
Kc S(N).
Case (i). Since Y\K C S(M) for some M E (D) by (i), we have F(X) C SeA) where A =MU N = {xo,XI,'" ,xn } E (D)~ Then, by Lemma, there e:ll..;sts an f E C(~n,X) such that f(~n)C r.4 and
f(~J)erJfor each J E (A), where ~n= co{eo,el'''' ,en} and ~J is the face of~ncorresponding toJ. Sincef(~n)is compact inX and F E 2{~(X,V), there e.xists a F E ~cU(~n),Y) such that Fx C Fx for each x E f(~n)' Then Ff(~n)is compact inF(X), since it is the image of the compact set f(~n) under the compact valued u.s.c. map
F. Let Pd7=0 be the partition of unity subordinated to the cover {SXinFf(~n)}7=0 of Pf(~n)'
. Define a continuous map p: Ff(~n) ~ ~n by
n
py=E Ai{y)ei= LAi(y}e; .fory E Ff(i1n),
i=O iENN
Coincidences of composites of U.S.c. maps 255
we have jp(y) E j(6.N.) C fN. C T-y for each yE Fj(6.n)j that is, yE (Tjp)y.
Since pFj E 2tc(6.n,6.n ), pFj has a fixed point z E 6.nj that is, z E (pFJ)z. Put x = jz. Since p-zn(FJ)z =p-znFx =I 0,for any yE p-znFx, we haveyE Fj(6.n ),(fp)y = jz = x,andyE (Tjp)y=
Tx. Therefore,p-znFx c Tx and hence TxnFx c TxnFx =I 0.
Case (ii). For an N E (D) such that F(X)nK c S(N), consider the set LN in (1.4).
We claim that F(LN) c S(LN nD) for F E 2lc(LN,Y) satisfying Fx C Fx for each x E LN. In fact, note that
F(LN)nKc F(X) nKc S(N) C S(LN nD).
On the other hand, F(LN)\K C F(LN)\K C S(LN nD) by (1.4).
Therefore, we have F(LN) C S(LN nD).
Note that F(LN) is compact since it is the image of the com- pact set LN under F. Therefore, F(LN) c S( A) for some A = {xo,Xl, . . . ,X n } E (LN nD). Then, by Lemma, there exists an j E
C(6.n,X) such that j(6.n ) C fAn LN = fA and j(6.J) C fJ for
each J E (A), where .6.n = co{eo, el, ... ,en} and 6.J is the face of6.n
corresponding to J. Let {Ai}~=obe the partition of unity subordinated to the cover {SXi nFj(6.nnr:o of Fj(6.n ) C F(LN)'
For the remainder of the proof, we can just follow that of Case (i).
This completes our proof.
REMARKS. 1. Theorem 1 for Case (ii) is a correct and generalized form of Ding [Di, Theorem 3.3]. As in [Di], Theorem 1 can be applied to intersection theorems concerning sets with H-conveX sections and the von Neumann type minimax theorems.
2. Note that, if F is single-valued, the Hausdorft'ness assumption is not necessary. See [P1, Theorem 6], [P2, Theorem 4].
PARTICULAR FORMS. 1. IT (X,Djr) is a convex space with fN = coN for each N E (D), then Theorem 1 for Case (ii) includes Park (P5, Theorem 5], which was shown to be equivalent to a number of fundamental theorems in the KKM theory. For details, see [P5].
2. For H-spaces, Theorem 1 includes Horvath [H5, Theorem 4.2], Ding, Kim, and Tan [DKTl, Corollaries 3-5]' Ding and Tan [DT, The-
orems 10-12 and Corollaries 2-4], Tarafdar [T, Theorem 2}, Chen [Ch, Theorem 2], and Park [PI, Theorem 6], [P2, Theorem 4].
3. IfF is compact in Theorem 1, by putting F(X) = K, the coer- civity condition (1.4) (ii) holds automatically. Therefore, we have the following:
COROLLARY 1. Let(X,Djr) bean H-space, Y a Hausdorffspace, F E2!~(X,Y) a compact map, andS : D - 0 Y, T: X - 0 Y. Suppose that
(1) for each x E D, Sx is compactly open;
(2) for each yE F(X), ME (S-y) impliesrM C T-y; and (3) F(X) c SeD).
Then F andT havea coincidence point x EX; that is, Fx nTx i-0.
REMARK. Condition (2) in. Corollary 1can be replaced by the fol- lowing:
(2)' for each x E D, Sx C Tx and, for each y E F(X), T-y is H-convex.
PARTICULAR FORMS. For H-spaces, Corollary 1 includes Horvath [Hl, Theoreme 4.1], [H2,Theoreme 2 et Lemma 1], and [H3, Theorem
2'J.
Moreover, from Corollary 1, we have
COROLLARY 2. Let (X, Djr) bean H-space, Y a Hausdorffspace, F E 2!~(X,Y) compact, andG: X - 0Y. Suppose that
(1) for each y EF(X); G-yis H-convex; and (2) {IntGx : x E DJ covers F(X).
Then F and G havea coincidencepoint.
Proof. Put G =T and let S: D - 0 Y be defined by Sx =IntGx
for x E D in Corollary l.
PARTICULAR FORMS. IfeX, Djr)is a convex space, then Corollary 2,reduces to Park (P5,,'!J)heol'em 2]: 'Por H-ffPaces;Borollary 2 includes Horvath {H4, Corollaire 61 and Dingand Tarafdar [DTr, Theorem 3.11.
Coincidences of composites of u.s.c. maps 257
4. The KKM theorems
In this section, we show that Theorem 1 is equivalent to the following KKM type theorem:
THEOREM 2. Let (X, Djr) be an H -space, Y a Hausdorff space, and F E 21~(X,Y). Let G :D - 0 Y be a map such that
(2.1) for each x E D, Gx is compactly closedin Y;
(2.2) for each N E (D), F(f N) c G( N); and
(2.3) there exists a nonempty compact subset K of Y such that either
(i) n{Gx: x EM} c K for some M E (D); or
(ii) for each N E (D), there exists a compact H -subspace LN of X containing N such that
F(LN)nn{Gx: x ELNnD} c E.
Then we have
Proof. Let S : D - 0 Y, H : Y - 0 X, and T : X - 0 Y be defined by Sx = Y\Gx for x E D, Hy = U{fM : M E (S-y)} for y E Y, and Tx =H-x for x E X. Then (1.1) and (1.2) hold by (2.1) and the definitions of S and T. Suppose that F(X) n E n n{Gx : x E
D} = 0; that is, F(X) nK c S(D), which is just (1.3). Note that (2.3) is equivalent to (1.4). Therefore, by Theorem 1, T and F have a coincidence point i EX; that is, Fi nTi :I 0. For y E Fi nTi, we have i E T-y and hence, there exists an N E (S-y) C (D) such that i E fN. Sincey E Fi C F(fN) C G(N) = y\n{Sx: x EN}, Y ~ Sx for some x E N; that is, x ~ S-y and x EN, which is a contradiction.
This completes our proof.
Theorems 1 and 2 are equivalent:
Proof of Theorem 1 using Theorem 2. Let Gx = Y\Sx for x E D.
Then (2.1) and (2.3) follow from (1.1) and (1.4), respectively. More- over, from (1.3), we have
F(X)nKC S(D) = U(Y\Gx) = Y\ nGx,
xED xED
contrary to the conclusion of Theorem 2. Therefore, F and G does not satisfy (2.2) and hence, there exist an NE (D) and ayE F(rN)\G(N)j that is, y f/. Gx or y E Sx for all x E N. Since N E (S-y) c (D), we have rN C T-y by (1.2). Since y E F(rN), there exists an i E rN
such that y E Fi: Note that i E rN C T-Yi that is, yETi. This completes our proof.
PARTICULAR FORMS. 1. IT(X,Djr) is a convex space, then The- orem 2 reduces to Park [P5, Theorem 7] which includes many known generalizations of the KKMtheorem.
2. For H-spaces, Theorem 2 generalizes Horvath [H1, Theoreme 3.1 et Corollaire 3], [B3, I,Theorem 1 and Corollary 11, Bardaro and Cep- pitelli [BC1, Theorem 1], Ding and Tan [DT, Corollary 1 and Theorem
SI, Ding, Kim, and Tan [DKT1, Lemma 11, and Park [P1, Theorems 1 and 4], [P2, Theorem1].
REMARKS. 1. If (X,Djr) is a convex space with rA = coA for A E(D), then (i) implies (ii). In fact, we can chooseLN ~co(MUN).
2. Ding [Di, Theorem 3.2] claimed that Theorem 2 for Case (ii) holds for a compact-valued u.s.c. map F and
(2.2)' for each N E (D), rN C F-G(N)
instead of (2.2). However, this is false as the following example shows:
EXAMPLE. Let X =Y =K = [0,1], D ={0,1}, and rN =coN
for NE (D). Let Gx= {x} for x E D and Fx = [0,1] for x EX. Then n{Gx: x E D} =0and for each NE (D), coN c F-G(N) =[0,1].
5. Fixed point theorems
Our main coincidence theorem can be applied to some fixed point problems for the class~c.
THEOREM 3. Let (X, Djr) be an H -space such that X has a Haus- dorff uniform structure and F E ~c(X,X) a compact map. Suppose that, for each entourage V of X, there exist maps S : D ---Q X and T : X -oXsatis~mg(1)-(3) m Corollary 1and Gr(F) n Gr(T) c V.
Then F h~ a fixed point. . ... . .
Proof. For each entourage V of the Hausdorfl' uniformity, by Corol- lary 1, there exist a map T :X ---QX and a point (x v , yv) such that
Coincidences of composites of u.s.c. maps 259 (XV,yv) E Gr(F) nGr(T) C V. Therefore, F has a V-fixed point.
Since F(X) is compact and F is u.s.c., F must have a fixed point.
PARfICULAR FORMS. See Horvath [H5, Theorem 4.4] and Park [P3, Theorem 5].
For metric spaces, we have the following from Theorem 3:
THEOREM 4. Let (X, Djr) be ametric H -space with the metric d, andF E2l.c(X,X) a compact map. Supposethat
(4.1) every open ballin X is H-convex; and (4.2) D is densein F(X).
Then F hasa fixed point.
Proof. Note that X is a Hausdorff uniform space. For any c >0 and an entourage Ve = {(x, y) E X x X : d(x, y) < c}, define S : D -<l X and T : X -<lX by
Sx = B(x,c) = {y EX: d(x,y) < c} for x ED, and
Tx =B(x,c) for x EX.
Note that Sx = Txfor x E D andSxis open in X. Moreover, for each yEX,
T-y = {x EX: yE B(x,c)} = B(y,c) is H-convex. Furthermore, F(X) C S(D) by (4.2). Note that
Gr(T) = ((x,y) E X x X : d(x,y) < c} = Ye.
Therefore, by Theorem 3, F has a fixed point.
PARTICULAR FORMS. 1. For 21 = C, Theorem 4 extends Brouwer [Br], Schauder [Se], Rassias[R], and Park [P3; Theorems 3 and 6]. For details, see [P3].
2. For 21=K, Theorem 4 extends Kakutani [Ka] and Bohnenblust and Karlin [BK].
6. Open-valued KKM theorems and coincide,nce theorems In order to obtain the open version of Theorem 2 for the class ~c,
we need the following:
THEOREM 5. Let(X,D;r) bean H-space, D = {XO,Xl,'" ,xn } E
(X), Y a regular space, G :D - 0Y a map, andF :X - 0Y a compact valued u.s.c. map. If G : D - 0 Y isa. open valuedmapsuch that
(5.1) foreach J E{D}, F(rJ) c G(J).
Then thereis a closed valuedmap H : D - 0Y such that H x C Gx
forallx E D and
(5.2) Ff(~J) c H(J) for each J E {D} and f : ~n ~ rD in Lemma, where ~Jis the face of~n.corresponding toJ.
Proof. For any y EG(D), let
H y=
n{
Gx :y E Gx};then Hy is an open set inY containingy. By the regurality ofY, there exists an open neighborhoodUy of y inY such that
y EUy C Uy C Hy •
Clearly for any J E {D}, we have
G(,J) = {Uy : y E G(J)}
and so by (5.1), U{Uy : y EG(J)} is an open cover ofFf(~J), since
Ff(~J)C F(rJ).
SinceFf(~J) is compact, there exists a BJ E (G(J)} such that Ff(AJ}C U{Uy :yE BJ}.
Let B =U{BJ : J E {D}}. DefineH : D - 0Y by Hx =U{Uy :y E BnGx}
for each xE D. ThenH x is, closed inY for each x ED andH x C Gx, sinceUy C Hy c Gxify EGx. And for eachJ E {D} andz EF f(~J),
we haveZ EUy for some y E BJ c G(J)nB; that is y E GxnB for
"some'xE'J, hence
Ff(~J)c H(J).
This completes our proof.
Coincidences of composit.es of u.s.c. maps 261
REMARK. The proof was motivated by Shih [Sh, Theorem IJ.
THEOREM 6. Let (X,D;r) be an H-space, D = {XO,Xl,'" ,xn } E (X), Y a Tl regular space, G : D - 0 Y a map, and F E 2lc(X, Y).
Suppose that
(6.1) for each x ED, Gx is open in Y; and (6.2) for each N E (D), F(rN) c G(N).
Then F(rD) nn{Gx :x E D} 1:0.
Proof. Suppose that F(rD) nn{Gx : x E D} = 0. By Theorem 5, there exists a map H : D - 0 Y such that Hx C Gx for each
x E D and (5.2) holds, and so F(rD) nn{Hx : x E D} = 0. Then Fj(f1n ) CT(D) =Y, whereTx =Y\Hx for each x ED. Let {.Adi=o
be the partition of unity subordinate to this cover {Tzilf=o of compact subset Fj(f1n ) ofY. Define p : Ff(6.n ) - 6.n by
n
PY = L Ai(y)ei = L Ai(y)ei
i=O iEN,
for y E Fj(f1n ) where i ENy -<==> Ai(y) 1: 0=>Y E TXi.
By Lemma, pFj E 2{c(6. n, f1n ) has a fixed point Zo E f1 n; that is Zo E pF j Zo. So there is a Yo E F j Zo such that PYo = zo and yo E Fxo where Xo = jzo. Ifi ENyo ' then Yo E TXi, and
Yo E Fxo nn{Txi :i E Nyo } 1: 0.
Put M = {Xi: i ENyo }, then Ff(6.M) nn{Tx: x EM} =1= 0; that is,
F j(f1M) et. H(M), where f1M is the face of 6.n corresponding to M.
This contradicts (5.2). This completes our proof.
REMARKS. 1. In the proof of Theorem 6, we actually showed the following:
Let X be a topological space, j : 6.n - X a continuous function, and Y, D, F and G be the same as in Theorem 6. Suppose that (6.1) and
(6.2)' for eachN C D, Fj(!}.N) C G(N).
ThenFj(!}.n)nn{Gx : x ED} =1= 0.
2. If F is a continuous single-valued map, then the Tl regularity of Y is not necessary. See Park [PI, Theorem 14J.
PARTICULAR FORMS. For convex spaces, Theorem 6 extends W.K.
Kim [Kil-2], Lassonde [L2, Theorem 1], and Park [P4, Theorem 10].
By the same argument to that of Theorems 1and 2, we can show that Theorem6is equivalent to the following:
THEOREM 7. Let (X,D;r) be an H-space, D E(X), Y a Tl regular space, andFE2(c(X, V). Let S: D - 0 Y andT: X - 0 Y satisfy the following:
(7.1) for each x E D, Sx is closed in Yi
(7.2) for each y E f(X), ME (S-y) impliesrM C T-Yi and (7.3) F(fD) CSeD).
Then T and F havea coincidence point i EXi that is, Ti nPi =f: 0.
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Sehie Park
Department of Mathematics Seoul National University Seoul 151-742, Korea Hoonjoo Kim
Department of Mathematics
Daebul Institute of Science and Technology YoungArm 526-890, Korea
