Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2007, Article ID 78696, 13 pages
doi:10.1155/2007/78696
Research Article
Coincidence Theorems, Generalized Variational Inequality
Theorems, and Minimax Inequality Theorems for the Φ-Mapping
on G-Convex Spaces
Chi-Ming Chen, Tong-Huei Chang, and Ya-Pei Liao
Received 14 December 2006; Revised 27 February 2007; Accepted 5 March 2007
Recommended by Simeon Reich
We establish some coincidence theorems, generalized variational inequality theorems,
and minimax inequality theorems for the family G-KKM(X,Y) and the Φ-mapping on
G-convex spaces.
Copyright © 2007 Chi-Ming Chen 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 and preliminaries
In 1929, Knaster et al. [1] had proved the well-known KKM theorem on n-simplex. In
1961, Fan [2] had generalized the KKM theorem in the infinite-dimensional topological
vector space. Later, the KKM theorem and related topics, for example, matching theorem,
fixed point theorem, coincidence theorem, variational inequalities, minimax inequalities,
and so on had been presented in grand occasions. Recently, Chang and Yen [3]intro-
duced the family, KKM(X,Y), and got some results about fixed point theorems, coinci-
dence theorems, and some applications on this family. Later, Ansari et al. [4] and Lin and
Chen [5] studied the coincidence theorems for two families of set-valued mappings, and
they also gave some applications of the existence of minimax inequality and equilibrium
problems. In this paper, we establish some coincidence theorems, generalized variational
inequality theorems, and minimax inequality theorems for the family G-KKM(X,Y)and
the Φ-mapping on G-convex spaces.
Let Xand Ybe two sets, and let T:X→2Ybe a set-valued mapping. We will use the
following notations in the sequel:
(i) T(x)={y∈Y:y∈T(x)},
(ii) T(A)=x∈AT(x),
(iii) T−1(y)={x∈X:y∈T(x)},
(iv) T−1(B)={x∈X:T(x)∩B= φ},
2 Fixed Point Theory and Applications
(v) T∗(y)={x∈X:y/∈T(x)},
(vi) if Dis a nonempty subset of X,thenDdenotes the class of all nonempty finite
subsets of D.
For the case that Xand Yare two topological spaces, then T:X→2Yis said to be
closed if its graph ᏳT={(x,y)∈X×Y:y∈T(x)}is closed. Tis said to be compact if
the image T(X)ofXunder Tis contained in a compact subset of Y.
Let Xbe a topological space. A subset Dof Xis said to be compactly closed (resp.,
compactly open) in Xif for any compact subset Kof X, the set D∩Kis closed (resp.,
closed) in K.Obviously,Dis compactly open in Xif and only if its complement Dcis
compactly closed in X.
The compact closure of Dis defined by
ccl(D)=∩
B⊂X:D⊂B,Bis compactly closed in X, (1.1)
and the compact interior of Dis defined by
cint(D)=∪
B⊂X:B⊂D,Bis compactly open in X.(1.2)
Remark 1.1. It is easy to see that ccl(X\D)=X\cint(D), Dis compactly open in Xif and
only if D=cint(D), and for each nonempty compact subset Kof X,wehavecint(D)∩
K=intK(D∩K), where intK(D∩K) denotes the interior of D∩Kin K.
Definition 1.2 [6,7]. Let Xand Ybe two topological spaces, and let T:X→2Y.
(i) Tis said to be transfer compactly closed (resp., transfer closed) on Xif for any x∈
Xand any y/∈T(x), there exists x∈Xsuch that y/∈cclT(x)(resp.,y/∈clT(x)).
(ii) Tis said to be transfer compactly open (resp., transfer open) on Xif for any x∈X
and any y∈T(x), there exists x∈Xsuch that y∈cintT(x)(resp.,y∈intT(x)).
(iii) Tis said to have the compactly local intersection property on Xif for each
nonempty compact subset Kof Xand for each x∈Xwith T(x)= φ,thereex-
istsanopenneighborhoodN(x)ofxin Xsuch that ∩z∈N(x)∩KT(z)= φ.
Remark 1.3. If T:X→2Yis transfer compactly open (resp., transfer compactly closed)
and Yis compact, then Tis transfer open (resp., transfer closed).
We denote by ∆nthe standard n-simplex with vectors e0,e1,...,en,whereeiis the (i+
1)th unit vector in ᏾n+1.
A generalized convex space [8]oraG-convex space (X,D;Γ) consists of a topological
space X, a nonempty subset Dof X, and a function Γ:D→2Xwith nonempty values
(inthesequal,wewriteΓ(A)byΓAfor each A∈D)suchthat
(i) for each A,B∈D,A⊂Bimplies that ΓA⊂ΓB,
(ii) for each A∈Dwith |A|=n+ 1, there exists a continuous function φA:∆n→
ΓAsuch that J∈Aimplies that φA(∆|J|−1)⊂ΓJ,where∆|J|−1denotes the faces
of ∆ncorresponding to J∈A.
Particular forms of G-convex spaces can be found in [8] and references therein. For a
G-convex space (X,D;Γ)andK⊂X,
(i) Kis G-convex if for each A∈D,A⊂Kimplies ΓA⊂K,
Chi-Ming Chen et al. 3
(ii) the G-convex hull of K, denoted by G-Co(K), is the set ∩{B⊂X|Bis a G-
convex subset of Xcontaining K}.
Definition 1.4 [9]. A G-convex space Xis said to be a locally G-convex space if Xis a
uniform topological space with uniformity ᐁwhich has an open base ᏺ={Vi|i∈I}of
symmetric encourages such that for each V∈ᏺ, the set V[x]={y∈X|(x,y)∈V}is a
G-convex set, for each x∈X.
Let (X,D;Γ)beaG-convexspacewhichhasauniformityᐁand ᐁhasanopensym-
metric base family ᏺ. Then a nonempty subset Kof Xis said to be almost G-convex if
for any finite subset Bof Kand for any V∈ᏺ, there is a mapping hB,V:B→Xsuch that
x∈V[hB,V(x)] for all x∈Band G-Co(hB,V(B)) ⊂K.subsetofK. We call the mapping
hB,V:B→XaG-convex-inducing mapping.
Remark 1.5. (i) In general, the G-convex-inducing mapping hB,Vis not unique. If U⊂V,
then it is clear that any hB,Ucan be regarded as an hB,V.
(ii) It is clear that the G-convex set is almost G-convex, but the inverse is not true, for
a counterexample.
Let E=ℜ
2be the Euclidean topological space. Then the set B={x=(x1,x2)∈E:
x2/3
1+x2/3
2<1}is a G-convex set, but the set B′={x=(x1,x2)∈E:0<x
2/3
1+x2/3
2<1}is
an almost G-convex set, not a G-convex set.
Applying Ding [10, Proposition 1] and Lin [11, Lemma 2.2], we have the following
lemma.
Lemma 1.6. Let Xand Ybe two topological spaces, and let F:X→2Ybe a set-valued
mapping. Then the following conditions are equivalent:
(i) Fhas the compactly local intersection property,
(ii) for each compact subset Kof Xand for each y∈Y,thereexistsanopensubsetOyof
Xsuch that Oy∩K⊂F−1(y)and K=y∈Y(Oy∩K),
(iii) for any compact subset Kof X, there exists a set-valued mapping P:X→2Ysuch
that P(x)⊂F(x)for each x∈X,P−1(y)is open in Xand P−1(y)∩K⊂F−1(y)for
each y∈Yand K=y∈Y(P−1(y)∩K),
(iv) for each compact subset Kof Xand for each x∈K,thereexistsy∈Ysuch that
x∈cintF−1(y)∩Kand K=y∈Y(cintF−1(y)∩K),
(v) F−1is transfer compactly open valued on Y,
(vi) X=y∈YcintF−1(y).
Definition 1.7. Let Ybe a topological space and let Xbe a G-convex space. A set-valued
mapping T:Y→2Xis called a Φ-mapping if there exists a set-valued mapping F:Y→2X
such that
(i) for each y∈Y,A∈F(y)implies that G-Co(A)⊂T(y),
(ii) Fsatisfies one of the conditions (i)–(vi) in Lemma 1.6.
Moreover, the mapping Fis called a companion mapping of T.
Remark 1.8. If T:Y→2Xis a Φ-mapping, then for each nonempty subset Y1of Y,T|Y1:
Y1→2Xis also a Φ-mapping.
4 Fixed Point Theory and Applications
Let Xbe a G-convex space. A real-valued function f:X→ℜ is said to be G-
quasiconvex if for each ξ∈ℜ, the set {x∈X:f(x)≤ξ}is G-convex, and fis said to
be G-quasiconcave if −fis G-quasiconvex.
Definition 1.9. Let Xbe a nonempty almost G-convex subset of a G-convex space. A real-
valued function f:X→ℜis said to be almost G-quasiconvex if for each ξ∈ℜ, the set
{x∈X:f(x)≤ξ}is almost G-convex, and fis said to be almost G-quasiconcave if −f
is almost G-quasiconvex.
Definition 1.10. Let Xbe a G-convex space, Yanonemptyset,andlet f,g:X×Y→ℜ
be two real-valued functions. For any y∈Y,gis said to be f-G-quasiconcave in xif for
each A={x1,x2,...,xn}∈X′,
min
1≤i≤nfxi,y≤g(x,y), ∀x∈G-Co(A).(1.3)
Definition 1.11. Let Xbe a nonempty almost G-convex subset of a G-convex space E
which has a uniformity ᐁand ᐁhas an open symmetric base family ᏺ,Yanonempty
set, and let f,g:X×Y→ℜbe two real-valued functions. For any y∈Y,gis said to be
almost f-G-quasiconcave in xif for each A={x1,x2,...,xn}∈Xand for every V∈ᏺ,
there exists a G-convex-inducing mapping hA,V:A→Xsuch that
min
1≤i≤nfxi,y≤g(x,y), ∀x∈G-CohA,V(A).(1.4)
Remark 1.12. It is clear that if f(x,y)≤g(x,y)foreach(x,y)∈X×Y,andifforeach
y∈Y, the mapping x→f(x,y)isalmostG-quasiconcave (G-quasiconcave), then gis
almost f-G-quasiconcave in x(f-G-quasiconcave).
Definition 1.13. Let Xbe a G-convex space, Ya topological space, and let T,F:X→2Y
be two set-valued functions satisfying
TG-Co(A)⊂F(A)foranyA∈X.(1.5)
Then Fis called a generalized G-KKM mapping with respect to T. If the set-valued func-
tion T:X→2Ysatisfies the requirement that for any generalized G-KKM mapping F
with respect to Tthe family {F(x)|x∈X}has the finite intersection property, then T
is said to have the G-KKM property. The class G-KKM(X,Y)isdefinedtobetheset
{T:X→2Y|Thas the G-KKM property}.
We now generalize the G-KKM property on a G-convex space to the G-KKM∗prop-
erty on an almost G-convex subset of a G-convex space.
Definition 1.14. Let Xbe a nonempty almost G-convex subset of a G-convex space E
which has a uniformity ᐁand ᐁhas an open symmetric base family ᏺ,andYatopo-
logical space. Let T,F:X→2Ybe two set-valued functions satisfying that for each finite
subset Aof Xand for any V∈ᏺ, there exists a G-convex-inducing mapping hA,V:A→X
such that
TG-CohA,V(A) ⊂F(A).(1.6)
Chi-Ming Chen et al. 5
Then Fis called a generalized G-KKM∗mapping with respect to T. If the set-valued
function T:X→2Ysatisfies the requirement that for any generalized G-KKM∗mapping
Fwith respect to Tthe family {F(x)|x∈X}has the finite intersection property, then T
is said to have the G-KKM∗property. The class G-KKM∗(X,Y) is defined to be the set
{T:X→2Y|Thas the G-KKM∗property}.
2. Coincidence theorems for the Φ-mapping and the G-KKM family
Throughout this paper, we assume that the set G-Co(A)iscompactwheneverAis a com-
pact set.
The following lemma will play important roles for this paper.
Lemma 2.1. Let Ybeacompactset,XaG-convex space. Let T:Y→2Xbe a Φ-mapping.
Then there exists a continuous function f:Y→Xsuch that for each y∈Y,f(y)∈T(y),
that is, Thas a continuous selection.
Proof. Since Yis compact, there exists A={x0,x1,...,xn}⊂Xsuch that Y=n
i=0intF−1(xi).
Since Xis a G-convex space and A∈X, there exists a continuous mapping φA:∆n→
Γ(A)suchthatφA(∆|J|−1)⊂ΓJfor each J∈A.
Let {λi}n
i=0be the partition of the unity subordinated to the cover {intF−1(xi)}n
i=0of Y.
Define a continuous mapping g:Y→∆nby
g(y)=
n
i=0
λi(y)ei=
i∈I(y)
λi(y)ei,foreachy∈Y, (2.1)
where I(y)={i∈{0,1,2,...,n}:λi= 0}.Notethati∈I(y)ifandonlyify∈F−1(xi), that
is, xi∈F(y). Since Tis a Φ-mapping, we conclude that φA◦g(y)∈φA(∆I(y))⊂G-Co{xi:
i∈I(y)}⊂T(y), for each y∈Y. This completes the proof.
Let Xbe a G-convex space. A polytope in Xis denoted by ∆=G-Co(A)foreachA∈
X. By the conception of the G-KKM(X,Y) family we immediately have the following
proposition.
Proposition 2.2 [12]. Let Xbe a G-convex space, and let Yand Zbe two topological
spaces. Then
(i) T∈G-KKM(X,Y)if and only if T∈G-KKM(∆,Y)for every polytopy ∆in X,
(ii) if Yis a normal space, ∆apolytopeinX,andifT:X→2Ysatisfies the requirement
that fThas a fixed point in ∆for all f∈Ꮿ(Y,∆), then T∈G-KKM(∆,Y).
Following Lemma 2.1 and Proposition 2.2, we prove the following important lemma
for this paper.
Lemma 2.3. Let Xbe a G-convex space and let Ybe a compact G-convex space. If T:X→2Y
is a Φ-mapping, then T∈G-KKM(X,Y).
Proof. Since Tis a Φ-mapping, we have that for any A∈X,let∆=G-Co(A), T|∆:
∆→Yis also a Φ-mapping. Since ∆is compact and by Lemma 2.1,T|∆has a continuous
selection function, that is, there is a continuous function f:∆→Ysuch that for each