FUNCTIONS IN S -KKM CLASS ON GENERALIZED CONVEX SPACES
TIAN-YUAN KUO, YOUNG-YE HUANG, JYH-CHUNG JENG, AND CHEN-YUH SHIH
Received 25 October 2004; Revised 13 July 2005; Accepted 1 September 2005
We establish a coincidence theorem inS-KKM class by means of the basic defining prop- erty for multifunctions inS-KKM. Based on this coincidence theorem, we deduce some useful corollaries and investigate the fixed point problem on uniform spaces.
Copyright © 2006 Tian-Yuan Kuo et al. 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.
1. Introduction
A multimapT:X→2Y is a function from a setXinto the power set 2Y ofY. IfH,T: X→2Y, then the coincidence problem forH andTis concerned with conditions which guarantee thatH(x) ∩T(x)=∅for somex∈X. Park [11] established a very general coincidence theorem in the class Ukcof admissible functions, which extends and improves many results of Browder [1,2], Granas and Liu [6].
On the other hand, Huang together with Chang et al. [3] introduced theS-KKM class which is much larger than the class Ukc. A lot of interesting and generalized results about fixed point theory on locally convex topological vector spaces have been studied in the setting ofS-KKM class in [3]. In this paper, we will at first construct a coincidence theo- rem inS-KKM class on generalized convex spaces by means of the basic defining property for multimaps inS-KKM class. And then based on this coincidence theorem, we deduce some useful corollaries and investigate the fixed point problem on uniform spaces.
2. Preliminaries
Throughout this paper,Ydenotes the class of all nonempty finite subsets of a nonempty setY. The notationT:XY stands for a multimap from a setXinto 2Y\ {∅}. For a multimapT:X→2Y, the following notations are used:
(a)T(A)=
x∈AT(x) forA⊆X;
(b)T−(y)= {x∈X:y∈T(x)}fory∈Y; (c)T−(B)= {x∈X:T(x)∩B=∅}forB⊆Y.
Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2006, Article ID 72184, Pages1–9 DOI10.1155/FPTA/2006/72184
All topological spaces are supposed to be Hausdorff. LetX andY be two topological spaces. A multimapT:X→2Yis said to be
(a) upper semicontinuous (u.s.c.) ifT−(B) is closed inXfor each closed subsetBof Y;
(b) compact ifT(X) is contained in a compact subset ofY;
(c) closed if its graph Gr(T)= {(x,y) :y∈T(x),x∈X}is a closed subset ofX×Y. Lemma 2.1 (Lassonde [9, Lemma 1]). LetXandY be two topological spaces andT:X Y.
(a) IfY is regular andTis u.s.c. with closed values, thenT is closed. Conversely, ifY is compact andTis closed, thenTis u.s.c. with closed values.
(b) IfTis u.s.c. and compact-valued, thenT(A) is compact for any compact subsetAof X.
LetX be a subset of a vector space and Da nonempty subset ofX. Then (X,D) is called a convex space if the convex hull co(A) of anyA∈ Dis contained inXandX has a topology that induces the Euclidean topology on such convex hulls. A subsetCof (X,D) is said to beD-convex if co(A)⊆Cfor anyA∈ DwithA⊆C. IfX=D, then X=(X,X) becomes a convex space in the sense of Lassonde [9]. The concept of convexity is further generalized under an extra condition by Park and Kim [12]. Later, Lin and Park [10] give the following definition by removing the extra condition.
Definition 2.2. A generalized convex space or aG-convex space (X,D;Γ) consists of a topological spaceX, a nonempty subsetDofXand a mapΓ:DXsuch that for each A∈ Dwith|A| =n+ 1, there exists a continuous functionϕA:Δn→Γ(A) such that J∈ AimpliesϕA(ΔJ)⊆Γ(J), whereΔJdenotes the face ofΔncorresponding toJ∈ A. A subsetKof aG-convex space (X,D;Γ) is said to beΓ-convex if for anyA∈ K∩D, Γ(A)⊆K.
In what follows we will expressΓ(A) byΓA, and we just say that (X,Γ) is aG-convex space provided thatD=X.
Thec-space introduced by Horvath [7] is an example ofG-convex space.
For topological spacesX andY,Ꮿ(X,Y) denote the class of all continuous (single- valued) functions fromXtoY.
Given a classᏸof multimaps,ᏸ(X,Y) denotes the set of multimapsT:X→2Y be- longing toᏸ, andᏸcthe set of finite composites of multimaps inᏸ. Park and Kim [12]
introduced the class U to be the one satisfying
(a) U contains the classᏯof (single-valued) continuous functions;
(b) eachT∈Ucis upper semicontinuous and compact-valued; and (c) for any polytopeP, eachT∈Uc(P,P) has a fixed point.
Further, Park defined the following
T∈Ukc(X,Y)⇐⇒ for any compact subsetKofX, there is a
Γ∈Uc(X,Y) such thatΓ(x)⊆T(x) for eachx∈K. (2.1)
A uniformity for a setXis a nonempty familyᐁof subsets ofX×Xsuch that (a) each member ofᐁcontains the diagonalΔ;
(b) ifU∈ᐁ, thenU−1∈ᐁ;
(c) ifU∈ᐁ, thenV◦V⊆Ufor someVinᐁ;
(d) ifUandVare members ofᐁ, thenU∩V∈ᐁ; and (e) ifU∈ᐁandU⊆V⊆X×X, thenV∈ᐁ.
If (X,ᐁ) is a uniform space the topology᐀induced byᐁis the family of all subsets WofX such that for eachxinW there isU inᐁsuch thatU[x]⊆W, whereU[x] is defined as{y∈X: (x,y)∈U}. For details of uniform spaces we refer to [8].
3. The results
The concept ofS-KKM property of [3] can be extented toG-convex spaces.
Definition 3.1. LetXbe a nonempty set, (Y,D;Γ) aG-convex space andZa topological space. IfS:XD,T:YZandF:XZare three multimaps satisfying
TΓS(A)
⊆F(A) (3.1)
for anyA∈ X, thenFis called aS-KKM mapping with respect toT. If the multimapT: YZsatisfies that for anyS-KKM mappingFwith respect toT, the family{F(x) :x∈ X}has the finite intersection property, thenTis said to have theS-KKM property. The classS-KKM(X,Y,Z) is defined to be the set{T:XY:Thas theS-KKM property}.
WhenD=Y is a nonempty convex subset of a linear space withΓB=co(B) forB∈ Y, theS-KKM(X,Y,Z) is just that as in [3]. In the case thatX=DandSis the identity mapping 1D,S-KKM(X,Y,Z) is abbreviated as KKM(Y,Z), and a 1D-KKM mapping with respect toTis called a KKM mapping with respect toT, and 1D-KKM property is called KKM property. Just as [3, Propositions 2.2 and 2.3], forX a nonempty set, (Y,D;Γ) a G-convex space, Z a topological space and anySD, one has T∈KKM(Y,Z)⊆S- KKM(X,Y,Z). By the corollary to [13, Theorem 2], we have Ukc(Y,Z)⊆KKM(Y,Z), and so Ukc(Y,Z)⊆S-KKM(X,Y,Z).
Here we like to give a concrete multimapThaving KKM property on aG-convex space.
LetX=[0, 1]×[0, 1] be endowed with the Euclidean metric. For anyA= {x1,..., xn} ∈ X, defineΓA=n
i=1[0, xi], where [0, xi] denotes the line segment joining 0 and xi. It is easy to see that (X,Γ) is ac-space, and so it is a G-convex space. LetT:XX be defined byT(x)=[(0, 0), (0, 1)]∪[(0, 0), (1, 0)]. IfF:XXis any KKM mapping with respect toT, then for anyA= {x1,..., xn} ∈ X, sinceT(ΓA)⊆F(A) and (0, 0)∈T(0, 0), we infer that (0, 0)∈T(xi)⊆F(xi) for anyi=1,...,n, so (0, 0)∈n
i=1F(xi). This shows thatThas the KKM property.
A subsetBof a topological spaceZ is said to be compactly open if for any compact subsetKofZ,K∩Bis open inK. We begin with the following coincidence theorem.
Theorem 3.2. LetXbe any nonempty set, (Y,D;Γ) aG-convex space andZa topological space. Supposes:X→D,W:D→2Z,H:Y→2Z andT ∈s-KKM(X,Y,Z) satisfy the
following conditions:
(3.2.1)Tis compact;
(3.2.2) for anyy∈D,W(y)⊆H(y) andW(y) is compactly open inZ;
(3.2.3) for anyz∈T(Y),M∈ W−(z)implies thatΓM⊆H−(z);
(3.2.4)T(Y)⊆
x∈XW(s(x)).
ThenTandHhave a coincidence point.
Proof. We prove the theorem by contradiction. Assume that T(y)∩H(y)=∅for any y∈Y. PutK=T(Y). By (3.2.1),Kis a compact subset ofZ. DefineF:X→2Zby
F(x)=K\Ws(x) (3.2)
forx∈X. SinceW(s(x)) is compactly open,F(x) is closed for eachx∈X. The assump- tion thatT(y)∩H(y)=∅for any y∈Y implies that T(s(x))∩H(s(x))=∅for any x∈X, so
∅=T(s(x))⊆K\Hs(x)
⊆K\Ws(x)
=F(x).
(3.3)
HenceFis a nonempty and compact-valued multimap. Since
x∈X
F(x)=
x∈X
K\Ws(x)
=K\
x∈X
Ws(x)
⊆K\K by (3.2.4)
=∅,
(3.4)
Fis not as-KKM mapping with respect toT. Hence there isA= {x1,...,xn} ∈ Xsuch that
TΓ{s(x1),...,s(xn)} n
i=1
Fxi. (3.5)
Choosey∈Γ{s(x1),...,s(xn)}andz∈T(y) such thatz /∈n
i=1F(xi). It follows from
z∈K\n
i=1
Fxi
= n i=1
K\Fxi
⊆n
i=1
Wsxi
⊆n
i=1
Hsxi
(3.6)
that s(xi)∈W−(z) ⊆H−(z) for anyi∈ {1,...,n}. Therefore by (3.2.3), Γ{s(x1),...,s(xn)}⊆ H−(z).In particular,y∈H−(z), and soz∈H(y)∩T(y), a contradiction. This completes
the proof.
Corollary 3.3. Let (Y,D) be a convex space andZa topological space. SupposeH:Y→2Z andT∈KKM(Y,Z) satisfy the following conditions:
(3.3.1)Tis compact;
(3.3.2) for anyz∈T(Y),H−(z) isD-convex;
(3.3.3)T(Y)⊆
y∈DInt(H(y)).
ThenTandHhave a coincidence point.
Proof. PuttingX=D,s:X→Dbe the identity mapping 1DandW:D→2Z be defined byW(y)=Int(H(y)) in the above theorem, the result follows immediately.
Here we like to mention thatCorollary 3.3is an improvement for Theorem 4 of Chang and Yen [4], where except the conditions (3.3.1)∼(3.3.3), they requireT be closed. For Ukc(Y,Z) instead of KKM(Y,Z),Corollary 3.3is due to Park [11]. We now give a concrete example showing thatCorollary 3.3extends both of [4, Theorem 4] and [11, Theorem 2]
properly. LetX=[0, 1] andV be any convex open subset of 0 inR. DefineT:XX byT(x)= {1}forx∈[0, 1); and [0, 1) forx=1, andH:XXbyH(x)=(x+V)∩X.
Then we have
(a)Tbelongs to KKM(X,X) and is compact;
(b)H−(y) is convex for eachy∈X, and (c) eachH(x) is open andT(X)⊆
x∈XH(x).
Thus,Corollary 3.3guarantees thatT(x)∩H(x) =∅for somex∈[0, 1]. But, Theorem 4 of Chang and Yen [4] is not applicable in this case because T is not closed. On the other hand, ifT∈Ukc(X,X), then there would existΓ∈Uc(X,X) such thatΓ(x)⊆T(x) for eachx∈[0, 1]. SinceXis a polytope,Γmust have a fixed a point which is impossible by noting thatT has no fixed point. Consequently,T /∈Ukc(X,X), and hence we can not apply Theorem 2 of Park [11] to conclude thatTandHhave a coincidence point.
Corollary 3.4. LetXbe any nonempty set, (Y,D) a convex space andZa topological space.
Supposes:X→D,H:Y→2ZandT∈s- KKM(X,Y,Z) satisfy the following conditions:
(3.4.1)Tis compact;
(3.4.2) for anyz∈T(Y),H−(z) isD-convex;
(3.4.3)T(Y)⊆
x∈XInt(H(s(x))).
ThenTandHhave a coincidence point.
Proof. InTheorem 3.2, putting W:D→2Z be W(y)=Int(H(y)) for each y∈Y, the
result follows immediately.
Lemma 3.5 (Lassonde [9, Lemma 2]). LetY be a nonempty subset of a topological vector spaceE,T:Y→2Ea compact and closed multimap andi:Y→Ethe inclusion map. Then for each closed subsetBofY, (T−i)(B) is closed inE.
Corollary 3.6. LetXbe any nonempty set andY,Cbe two nonempty convex subsets of a locally convex topological vector spaceE. Supposes:X→Y andT∈s- KKM(X,Y,Y+C) satisfy the following conditions (3.6.1), (3.6.2) and any one of (3.6.3), (3.6.3)and (3.6.3).
(3.6.1)Tis compact and closed.
(3.6.2)T(Y)⊆s(X) +C.
(3.6.3)Y is closed andCis compact.
(3.6.3)Yis compact andCis closed.
(3.6.3)C= {0}.
Then there isy∈Ysuch (y+C)∩T(y)=∅.
Proof. LetVbe any convex open neighborhood of 0∈EandK=T(Y). DefineH:Y→ 2Y+Cto beH(y)=(y+C+V)∩Kfor eachy∈Y. EachH(y) is open inKandH−(z)= (z−C−V)∩Yis convex for anyz∈K. Moreover,
x∈X
H(s(x))=
x∈X
s(x) +C+V∩K
=
s(X) +C+V∩K
=T(Y) by (3.6.2).
(3.7)
Therefore, it follows fromCorollary 3.4that there areyV∈YandzV∈Ksuch thatzV∈ T(yV)∩H(yV). Then in view of the definition ofH,zV−yV∈C+V. Up to now, we have proved the assertion.
(∗) For each convex open neighborhood V of 0 in E, (T−i)(Y)∩(C+V)=∅, wherei:Y→Eis the inclusion map.
Now take into account of conditions (3.6.3), (3.6.3)and (3.6.3). Suppose (3.6.3) holds.
SinceY is closed, so is (T−i)(Y) byLemma 3.5, and then the assertion (∗) in conjunc- tion with the compactness ofCand the regularity ofEimplies that (T−i)(Y)∩C=∅, that is, there exists ay∈Ysuch thatT(y)∩(y+C)=∅. In case that (3.6.3)holds, since (T−i)(Y) is compact byLemma 2.1and sinceCis closed, the conclusion follows as the previous case. Finally, assume that (3.6.3)holds. By (∗), for every convex open neigh- borhoodVof 0, there areyV andzV inY such thatzV∈T(yV) andzV−yV∈V. Since T(Y) is compact, we may assume thatzV→yfor somey∈T(Y). Then we also have that yV→y. The closedness ofTimplies thaty∈T(y).This completes the proof.
The above corollary extends Park [11, Theorem 3], which in turn is a generalization to Lassonde [9, Theorem 1.6 and Corollary 1.18].
We now turn to investigate the fixed point problem on uniform spaces. At first we applyTheorem 3.2to establish a useful lemma.
Lemma 3.7. LetX be any nonempty set, (Y,D;Γ) be aG-convex space whose topology is induced by a uniformityᐁ. Supposes:X→DandT∈s- KKM(X,Y,Y) satisfy that
(3.7.1)Tis compact; and (3.7.2)T(Y)⊆s(X).
IfV∈ᐁis symmetric and satisfies thatV[y] isΓ-convex for anyy∈Y, then there isyV∈Y such that
V[yV]∩T(yV)=∅. (3.8)
Proof. Define H:Y →2Y to beH(y)=V[y] for any y∈Y. By symmetry ofV it is easy to see that H−(z)=V[z] for anyz∈Y, and soH−(z) isΓ-convex. Also, it fol- lows from condition (3.6.2) that for anyz∈T(Y), there isx0∈s(X) such thatz=s(x0).
Then in view of (s(x0),s(x0))∈Vwe see thatz=s(x0)∈V[s(x0)]=H(s(x0)), and hence z∈
x∈XH(s(x)), that isT(Y)⊆
x∈XH(s(x)). Finally, noting H is open-valued and puttingW:D→2Yto beW(y)=H(y) for anyy∈D, we see that all the requirements ofTheorem 3.2are satisfied. Thus there isyV∈Y such thatH(yV)∩T(yV)=∅, that is
V[yV]∩T(yV)=∅.
Definition 3.8 [14]. AG-convex space (X,D;Γ) is said to be a locallyG-convex uniform space if the topology ofXis induced by a uniformityᐁwhich has a baseᏺconsisting of symmetric entourages such that for anyV∈ᏺandx∈X,V[x] isΓ-convex.
Recall that the concepts ofl.c.space andl.c.metric space in Horvath [7]. IfD=X andΓx= {x}for anyx∈X, then it is obvious that both of them are examples of locally G-convex uniform space.
Theorem 3.9. LetX be any nonempty set, (Y,D;Γ) a locallyG-convex space. Supposes: X→DandT∈s- KKM(X,Y,Y) satisfy that
(3.9.1)Tis compact and closed;
(3.9.2)T(Y)⊆s(X).
ThenThas a fixed point.
Proof. By Lemma 3.7, for any V ∈ᏺ there is yV ∈Y such that V[yV]∩T(yV)=∅. ChoosezV∈V[yV]∩T(yV). Then (yV,zV)∈V∩Gr(T). SinceTis compact, we may as- sume that{zV}V∈ᏺconverges toz0. For anyW∈ᏺ, chooseU∈ᏺsuch thatU◦U⊆W.
Since{zV}V∈ᏺconverges toz0, there isV0∈ᏺsuch thatV0⊆Uand zV∈U z0
, ∀V∈ᏺwithV⊆V0, (3.9)
that is,
zV,z0
∈U, ∀V∈ᏺwithV⊆V0. (3.10)
Thus, forV∈ᏺwithV⊆V0, it follows from yV,zV
∈V⊆U, zV,z0
∈U (3.11)
that (yV,z0)∈U◦U⊆W. HenceyV∈W[z0]. This shows that{yV}V∈ᏺconverges toz0. SinceTis closed, we conclude thatz0∈T(z0), completing the proof.
For a topological spaceXand locallyG-convex uniform space (Y,Γ), define T∈(X,Y)⇐⇒T:X−→Y is a Kakutani map, that is,
Tis u.s.c. with nonempty compactΓ-convex values. (3.12)
c(X,Y) denotes the set of finite composites of multimaps in of which ranges are contained in locallyG-convex uniform spaces (Yi,Γi) (i=0,...,n) for somen.
Lemma 3.10 (Watson [14]). Let (X,Γ) be a compact locallyG-convex uniform space. Then any u.s.c.T:XXwith closedΓ-convex values has a fixed point.
By the above lemma, we see that, in the setting of locallyG-convex uniform spaces, the classis an example of the Park’s class U. Therefore, for any locallyG-convex uniform space (X,Γ),c(X,X)⊆KKM(X,X), and so we have the following theorem.
Theorem 3.11. Suppose (X,Γ) is a locallyG-convex uniform space. IfT∈c(X,X) is com- pact, then it has a fixed point.
Proof. SinceXis regular by Kelley [8, Corollary 6.17 on page 188] andT∈c(X,X), it is u.s.c. and compact-valued, and so it is closed. Now due to thatc(X,X)⊆KKM(X,X), we haveT∈KKM(X,X). SinceT is compact and closed, it follows fromTheorem 3.9
thatThas a fixed point.
Since any metric space is regular, we infer that for anyl.c.metric space (X,d) satisfying thatΓx= {x}, ifT∈c(X,X) is compact, thenThas a fixed point. This generalizes the famous Fan-Glicksberg fixed point theorem [5].
References
[1] F. E. Browder, The fixed point theory of multi-valued mappings in topological vector spaces, Math- ematische Annalen 177 (1968), 283–301.
[2] , Coincidence theorems, minimax theorems, and variational inequalities, Conference in Modern Analysis and Probability (New Haven, Conn, 1982), Contemp. Math., vol. 26, American Mathematical Society, Rhode Island, 1984, pp. 67–80.
[3] T.-H. Chang, Y.-Y. Huang, J.-C. Jeng, and K.-H. Kuo, OnS-KKM property and related topics, Journal of Mathematical Analysis and Applications 229 (1999), no. 1, 212–227.
[4] T.-H. Chang and C.-L. Yen, KKM property and fixed point theorems, Journal of Mathematical Analysis and Applications 203 (1996), no. 1, 224–235.
[5] K. Fan, A generalization of Tychonoff’s fixed point theorem, Mathematische Annalen 142 (1960/1961), 305–310.
[6] A. Granas and F. C. Liu, Coincidences for set-valued maps and minimax inequalities, Journal de Math´ematiques Pures et Appliqu´ees. Neuvi`eme S´erie(9) 65 (1986), no. 2, 119–148.
[7] C. D. Horvath, Contractibility and generalized convexity, Journal of Mathematical Analysis and Applications 156 (1991), no. 2, 341–357.
[8] J. L. Kelley, General Topology, D. Van Nostrand, Toronto, 1955.
[9] M. Lassonde, On the use of KKM multifunctions in fixed point theory and related topics, Journal of Mathematical Analysis and Applications 97 (1983), no. 1, 151–201.
[10] L.-J. Lin and S. Park, On some generalized quasi-equilibrium problems, Journal of Mathematical Analysis and Applications 224 (1998), no. 2, 167–181.
[11] S. Park, Foundations of the KKM theory via coincidences of composites of upper semicontinuous maps, Journal of the Korean Mathematical Society 31 (1994), no. 3, 493–519.
[12] S. Park and H. Kim, Coincidence theorems for admissible multifunctions on generalized convex spaces, Journal of Mathematical Analysis and Applications 197 (1996), no. 1, 173–187.
[13] S. Park and H. Kim, Foundations of the KKM theory on generalized convex spaces, Journal of Mathematical Analysis and Applications 209 (1997), no. 2, 551–571.
[14] P. J. Watson, Coincidences and fixed points in locallyG-convex spaces, Bulletin of the Australian Mathematical Society 59 (1999), no. 2, 297–304.
Tian-Yuan Kuo: Fooyin University, 151 Chin-Hsueh Rd., Ta-Liao Hsiang, Kaohsiung Hsien 831, Taiwan
E-mail address:[email protected]
Young-Ye Huang: Center for General Education, Southern Taiwan University of Technology, 1 Nan-Tai St. Yung-Kang City, Tainan Hsien 710, Taiwan
E-mail address:[email protected]
Jyh-Chung Jeng: Nan-Jeon Institute of Technology, Yen-Shui, Tainan Hsien 737, Taiwan E-mail address:[email protected]
Chen-Yuh Shih: Department of Mathmatics, Cheng Kung University, Tainan 701, Taiwan E-mail address:[email protected]
