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 setX into the power set 2Y ofY. If H,T : X→2Y, then the coincidence problem forH and T is 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 notation T : XY stands for a multimap from a set X into 2Y\ {∅}. For a multimapT : X→2Y, the following notations are used:
(a)T(A)=
x∈AT(x) for A⊆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. Let X and Y be two topological spaces. A multimapT : X→2Yis said to be
(a) upper semicontinuous (u.s.c.) ifT−(B) is closed in X for each closed subset B of Y;
(b) compact ifT(X) is contained in a compact subset of Y;
(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]). LetX and Y be two topological spaces and T : X Y.
(a) IfY is regular and T is u.s.c. with closed values, then T is closed. Conversely, if Y is compact andT is closed, then T is u.s.c. with closed values.
(b) IfT is u.s.c. and compact-valued, then T(A) is compact for any compact subset A of X.
LetX be a subset of a vector space and D a nonempty subset of X. Then (X,D) is called a convex space if the convex hull co(A) of any A∈ Dis contained inX and X has a topology that induces the Euclidean topology on such convex hulls. A subsetC of (X,D) is said to be D-convex if co(A)⊆C for any A∈ DwithA⊆C. If X=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 subset D of X and a map Γ :DX such that for each A∈ Dwith|A| =n + 1, there exists a continuous function ϕA:Δn→Γ(A) such that J∈ AimpliesϕA(ΔJ)⊆Γ(J), where ΔJdenotes the face ofΔncorresponding toJ∈ A. A subsetK of a G-convex space (X,D;Γ) is said to be Γ-convex if for any A∈ K∩D, Γ(A)⊆K.
In what follows we will expressΓ(A) by ΓA, and we just say that (X,Γ) is a G-convex space provided thatD=X.
Thec-space introduced by Horvath [7] is an example ofG-convex space.
For topological spacesX and Y, Ꮿ(X,Y) denote the class of all continuous (single- valued) functions fromX to Y.
Given a classᏸ of multimaps, ᏸ(X,Y) denotes the set of multimaps T : X→2Y be- longing toᏸ, and ᏸcthe set of finite composites of multimaps inᏸ. Park and Kim [12]
introduced the class U to be the one satisfying
(a) U contains the classᏯ of (single-valued) continuous functions;
(b) eachT∈Ucis upper semicontinuous and compact-valued; and (c) for any polytopeP, each T∈Uc(P,P) has a fixed point.
Further, Park defined the following
T∈Ukc(X,Y)⇐⇒ for any compact subsetK of X, there is a
Γ∈Uc(X,Y) such that Γ(x)⊆T(x) for each x∈K. (2.1)
A uniformity for a setX is a nonempty family ᐁ of subsets of X×X such that (a) each member ofᐁ contains the diagonal Δ;
(b) ifU∈ᐁ, then U−1∈ᐁ;
(c) ifU∈ᐁ, then V◦V⊆U for some V in ᐁ;
(d) ifU and V are members of ᐁ, then U∩V∈ᐁ; and (e) ifU∈ᐁ and U⊆V⊆X×X, then V∈ᐁ.
If (X,ᐁ) is a uniform space the topology ᐀ induced by ᐁ is the family of all subsets W of X such that for each x in W there is U in ᐁ such that U[x]⊆W, where U[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. LetX be a nonempty set, (Y,D;Γ) a G-convex space and Z a topological space. IfS : XD, T : YZ and F : XZ are three multimaps satisfying
TΓS(A)
⊆F(A) (3.1)
for anyA∈ X, thenF is called a S-KKM mapping with respect to T. If the multimap T : YZ satisfies that for any S-KKM mapping F with respect to T, the family{F(x) : x∈ X}has the finite intersection property, thenT is said to have the S-KKM property. The classS-KKM(X,Y,Z) is defined to be the set{T : XY : T has the S-KKM property}.
WhenD=Y is a nonempty convex subset of a linear space with ΓB=co(B) for B∈
Y, theS-KKM(X,Y,Z) is just that as in [3]. In the case thatX=D and S is the identity mapping 1D,S-KKM(X,Y,Z) is abbreviated as KKM(Y,Z), and a 1D-KKM mapping with respect toT is called a KKM mapping with respect to T, 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 any SD, 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 multimapT having KKM property on a G-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 a c-space, and so it is a G-convex space. Let T : XX be defined byT(x)=[(0, 0), (0, 1)]∪[(0, 0), (1, 0)]. IfF : XX is any KKM mapping with respect toT, then for any A= {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 thatT has the KKM property.
A subsetB of a topological space Z is said to be compactly open if for any compact subsetK of Z, K∩B is open in K. We begin with the following coincidence theorem.
Theorem 3.2. LetX be any nonempty set, (Y,D;Γ) a G-convex space and Z a topological space. Supposes : X→D, W : D→2Z,H : Y→2Z andT ∈s-KKM(X,Y,Z) satisfy the
following conditions:
(3.2.1)T is compact;
(3.2.2) for anyy∈D, W(y)⊆H(y) and W(y) is compactly open in Z;
(3.2.3) for anyz∈T(Y), M∈ W−(z)implies thatΓM⊆H−(z);
(3.2.4)T(Y)⊆
x∈XW(s(x)).
ThenT and H have a coincidence point.
Proof. We prove the theorem by contradiction. Assume that T(y)∩H(y)=∅for any y∈Y. Put K=T(Y). By (3.2.1), K is a compact subset of Z. Define F : X→2Zby
F(x)=K\Ws(x) (3.2)
forx∈X. Since W(s(x)) is compactly open, F(x) is closed for each x∈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)
HenceF is 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)
F is not a s-KKM mapping with respect to T. Hence there is A= {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 any i ∈ {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 and Z a topological space. Suppose H : Y→2Z andT∈KKM(Y,Z) satisfy the following conditions:
(3.3.1)T is compact;
(3.3.2) for anyz∈T(Y), H−(z) is D-convex;
(3.3.3)T(Y)⊆
y∈DInt(H(y)).
ThenT and H have a coincidence point.
Proof. PuttingX=D, s : X→D be 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 require T 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 : XX by H(x)=(x + V)∩X.
Then we have
(a)T belongs to KKM(X,X) and is compact;
(b)H−(y) is convex for each y∈X, and (c) eachH(x) is open and T(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 exist Γ∈Uc(X,X) such that Γ(x)⊆T(x) for eachx∈[0, 1]. SinceX is a polytope, Γ must 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 thatT and H have a coincidence point.
Corollary 3.4. LetX be any nonempty set, (Y,D) a convex space and Z a topological space.
Supposes : X→D, H : Y→2ZandT∈s-KKM(X,Y,Z) satisfy the following conditions:
(3.4.1)T is compact;
(3.4.2) for anyz∈T(Y), H−(z) is D-convex;
(3.4.3)T(Y)⊆
x∈XInt(H(s(x))).
ThenT and H have 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→E the inclusion map. Then for each closed subsetB of Y, (T−i)(B) is closed in E.
Corollary 3.6. LetX be any nonempty set and Y, C be two nonempty convex subsets of a locally convex topological vector spaceE. Suppose s : X→Y and T∈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)T is compact and closed.
(3.6.2)T(Y)⊆s(X) + C.
(3.6.3)Y is closed and C is compact.
(3.6.3)Y is compact and C is closed.
(3.6.3)C= {0}.
Then there isy∈Y such (y + C)∩T(y)=∅.
Proof. LetV be any convex open neighborhood of 0∈E and K=T(Y). Define H : Y→ 2Y+Cto beH(y)=(y + C + V)∩K for each y∈Y. Each H(y) is open in K and H−(z)= (z−C−V)∩Y is convex for any z∈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∈Y and zV∈K such that zV∈ 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→E is 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 ofC and the regularity of E implies that (T−i)(Y)∩C=∅, that is, there exists ay∈Y such that T(y)∩(y + C)=∅. In case that (3.6.3)holds, since (T−i)(Y) is compact byLemma 2.1and sinceC is closed, the conclusion follows as the previous case. Finally, assume that (3.6.3)holds. By (∗), for every convex open neigh- borhoodV of 0, there are yV andzV inY such that zV∈T(yV) andzV−yV∈V. Since T(Y) is compact, we may assume that zV→ y for somey∈T(Y). Then we also have that yV→ y. The closedness of T implies 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 a G-convex space whose topology is induced by a uniformityᐁ. Suppose s : X→D and T∈s-KKM(X,Y,Y) satisfy that
(3.7.1)T is compact; and (3.7.2)T(Y)⊆s(X).
IfV∈ᐁ is symmetric and satisfies that V[y] is Γ-convex for any y∈Y, then there is yV∈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 of V it is easy to see that H−(z)=V[z] for any z∈Y, and so H−(z) is Γ-convex. Also, it fol- lows from condition (3.6.2) that for anyz∈T(Y), there is x0∈s(X) such that z=s(x0).
Then in view of (s(x0),s(x0))∈V we see that z=s(x0)∈V[s(x0)]=H(s(x0)), and hence z∈
x∈XH(s(x)), that is T(Y)⊆
x∈XH(s(x)). Finally, noting H is open-valued and puttingW : D→2Yto beW(y)=H(y) for any y∈D, we see that all the requirements ofTheorem 3.2are satisfied. Thus there isyV∈Y such that H(yV)∩T(yV)=∅, that is
V[yV]∩T(yV)=∅.
Definition 3.8 [14]. AG-convex space (X,D;Γ) is said to be a locally G-convex uniform space if the topology ofX is induced by a uniformity ᐁ which has a base ᏺ consisting of symmetric entourages such that for anyV∈ᏺ and x∈X, V[x] is Γ-convex.
Recall that the concepts ofl.c. space and l.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 locally G-convex space. Suppose s : X→D and T∈s-KKM(X,Y,Y) satisfy that
(3.9.1)T is compact and closed;
(3.9.2)T(Y)⊆s(X).
ThenT has 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). Since T is compact, we may as- sume that{zV}V∈ᏺconverges toz0. For anyW∈ᏺ, choose U∈ᏺ such that U◦U⊆W.
Since{zV}V∈ᏺconverges toz0, there isV0∈ᏺ such that V0⊆U and zV∈U z0
, ∀V∈ᏺ with V⊆V0, (3.9)
that is,
zV,z0
∈U, ∀V∈ᏺ with V⊆V0. (3.10)
Thus, forV∈ᏺ with V⊆V0, it follows from
yV,zV
∈V⊆U, zV,z0
∈U (3.11)
that (yV,z0)∈U◦U⊆W. Hence yV∈W[z0]. This shows that{yV}V∈ᏺconverges toz0. SinceT is closed, we conclude that z0∈T(z0), completing the proof.
For a topological spaceX and locally G-convex uniform space (Y,Γ), define T∈(X,Y)⇐⇒T : X−→Y is a Kakutani map, that is,
T is u.s.c. with nonempty compact Γ-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 some n.
Lemma 3.10 (Watson [14]). Let (X,Γ) be a compact locally G-convex uniform space. Then any u.s.c.T : XX with closed Γ-convex values has a fixed point.
By the above lemma, we see that, in the setting of locallyG-convex uniform spaces, the class is an example of the Park’s class U. Therefore, for any locally G-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 locally G-convex uniform space. If T∈c(X,X) is com- pact, then it has a fixed point.
Proof. SinceX is 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). Since T is compact and closed, it follows from Theorem 3.9
thatT has 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, then T has 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:sc038@mail.fy.edu.tw
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:yueh@mail.stut.edu.tw
Jyh-Chung Jeng: Nan-Jeon Institute of Technology, Yen-Shui, Tainan Hsien 737, Taiwan E-mail address:jhychung@pchome.com.tw
Chen-Yuh Shih: Department of Mathmatics, Cheng Kung University, Tainan 701, Taiwan E-mail address:cyshih@math.ncku.edu.tw
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Shouyang Wang, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; sywang@amss.ac.cn
K. K. Lai, Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong; mskklai@cityu.edu.hk
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