Coincidence theorems on nonconvex sets and its applications

Abstract

In this paper, we establish some coincidence theorems, generalized variational inequality theorems and minimax inequality theorems for the family KKM * (X, Y) and the generalized Φ-mapping on a nonconvex set.

Full text

TAIWANESE JOURNAL OF MATHEMATICS
Vol. 13, No. 2A, pp. 501-513, April 2009
This paper is available online at http://www.tjm.nsysu.edu.tw/
COINCIDENCE THEOREMS ON NONCONVEX SETS AND ITS
APPLICATIONS
Chi-Ming Chen, Tong-Huei Chang* and Chiao-Wei Chung
Abstract. In this paper, we establish some coincidence theorems, generalized
variational inequality theorems and minimaxinequality theorems for the family
KKM∗(X, Y )and the generalized Φ-mapping on a nonconvex set.
1. INTRODUCTION AND PRELIMINARIES
In 1929, Knaster, Kurnatoaski and Mazurkiewicz [11] had proved the well-
known KKM theorem on n-simplex. In 1961, Ky Fan [7] 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 the-
orem, coincidence theorem, variational inequalities, minimax inequalities and so on
had been presented a grand occasions. Rcecntly, Chang and Yen [4] introduced the
family KKM(X, Y ), and got some results about fixed point theorems, coincidence
theorems and some applications on this family. In this paper, we establish some
coincidence theorems, generalized variational inequality theorems and minimax in-
equality theorems for the family KKM∗(X, Y )and the generalized Φ-mapping.
Let Xand Ybe two sets, 2Xdenotes the class of all nonempty subsets of X,
and let T:X→2Ybe a set-valued mapping. We shall use the following notations
in the sequel.
(i)T(x)={y∈Y:y∈T(x)},
(ii)T(A)=∪x∈AT(x),
Received August 15, 2007, accepted September 12, 2007.
Communicated by Sen-Yen Shaw.
2000 Mathematics Subject Classification: 47H10, 54C60, 54H25, 55M20.
Key words and phrases: Almost-convex set, KKM∗(X,Y ), Generalized Φ-mapping, Coincidence
theorem, Variational inequality theorem, Minimax theorem.
This Research supported by the NSC.
*Corresponding author.
501

502 Chi-Ming Chen, Tong-Huei Chang and Chiao-Wei Chung
(iii)T−1(y)={x∈X:y∈T(x)},
(iv)T−1(B)={x∈X:T(x)∩B=φ},
(v)T∗(y)={x∈X:y/∈T(x)}, and
(vi)if Dis a nonempty subset of X, then Ddenotes the class of all nonempty
finite subset of D.
For the case that Xand Yare two topological spaces. Then T:X→2Yis
said to be closed if its graph GT={(x, y)∈X×Y:y∈T(x)}is closed. T
is said to be compact if the image T(X)of Xunder Tis contained in a compact
subset of Y.
A convex space Xis a convex set (in a linear space) with any topology that
induces the Euclidean topology on the convex hull of its finite subset.
A nonempty subset Xof a Hausdorff topological vector space Eis said to be
almost-convex [14] if for every finite subset A={x1,x
2, ..., xn}of Xand every
neighborhood Vof the origin 0of E, there is a mapping hA,V :A→Xsuch that
hA,V (xi)∈xi+Vfor each i∈{1,2, ..., n}and co(hA,V (A)) ⊂X. We call hA,V
a convex-inducing mapping.
We now introduce some properties of the almost-convex sets of a Hausdorff
topological vector space E, as follows:
(i)In general, the convex-inducing mapping hA,V is not unique. If U⊂V, then
it is clear that any hA,U can be regarded as an hA,V .
(ii)It is clear that the convex set is almost-convex, but the converse is not true,
for an counterexample,
Let E=l2(R∞). Then the set B(1) = {x∈E:0<x<1}is an
almost-convex subset of E, not a convex set.
Lemma 1. If Eis a Hausdorff topopogical vector space, Xan almost-convex
subset of E, and Yan open convex subset of E, then X∩Yis also almost-convex.
Proof. Let A={x1,x
2, ..., xn}⊂X∩Y. Since Yis open, there exists an open
neighborhood Uof the origin 0of Esuch that A+U⊂Y. For any neighborhood
Vof the origin 0of Ewith V⊂U, since A⊂Xand Xis almost-convex, there
exists a convex-inducing mapping hA,V :A→Xsuch that hA,V (xi)∈xi+Vfor
all i=1,2, ..., n and co(hA,V (A)) ⊂X. Since hA,V (A)⊂co(hA,V (A)) ⊂Xand
hA,V (A)⊂A+V, we get hA,V (A)⊂(A+V)∩X⊂(A+U)∩X⊂X∩Y,
and so co(hA,V (A)) ⊂Y, since Yis convex. Therefore, we conclude that X∩Y
is almost-convex.
Remark 1. Let us note that the open condition of the above Lemma1 is
really needed. For instance, if we consider the Euclidean topology in 2,X=

Coincidence Theorems on Nonconvex Sets and its Applications 503
int(co({(1,1),(−1,1),(−1,−1),(1,−1)})) ∪{(1,1),(−1,1),(−1,−1),(1,−1)},
and Y=co({(−1,1),(−2,1),(−2,−1),(−1,−1)}), then X∩Y={(−1,1),(−1,
−1)}is not almost-convex.
In [4], Chang and Yen had introducedthe class KKM(X, Y ), we now extended
this class to be the class KKM ∗(X, Y )for the almost-convex set X.
Definition 1. Let Xbe a nonempty almost-convex subset of a topological
vector space E, and Ya topological space. If T,F :X→2Yare two set-valued
mappings such that for each finite subset Aof Xand every neighborhood Vof
the origin 0of E, there exists a convex-inducing mapping hA,V :A→Xsuch
that T(co(hA,V (A))) ⊂F(A), then we call Fa generalized KKM∗mapping with
respect to T.
If the set-valued mapping T:X→2Ysatisfies the requirement that for any
generalized KKM∗mapping F:X→2Ywith respect to T, the family {Fx :x∈
X}has the finite intersection property, then Tis said to have the KKM∗property.
Denote
KKM∗(X, Y )={T:X→2Y|Thas the KKM∗property}.
Definition 2. Let Ybe a topological space and Xbe a convex space. A
set-valued mapping T:Y→2Xis called a Φ-mapping if there exists a set-valued
mapping F:Y→2Xsuch that
(i)for each y∈Y,A∈F(y)implies co(A)⊂T(y), and
(ii)Y=∪x∈XintF−1(x).
Moreover, the mapping Fis called a companion mapping of T.
Definition 3. Let Ybe a topological space, Xa nonempty almost-convex
subset of a topological vector space E, and Vbe a neighborhood of the origin 0of
E. A set-valued mapping T:Y→2Xis called a generalized Φ-mapping if there
exists a set-valued mapping F:Y→2Xsuch that
(i)for each y∈Y,A∈F(y)implies co(hA,V (A)) ⊂T(y), where hA,V is a
convex-inducing mapping , and
(ii)Y=∪x∈XintF−1(x).
Moreover, the mapping Fis called a generalized companion mapping of T.
Remark 2.
(i)AΦ-mapping is also a generalized Φ-mapping, but the converse is not true.

504 Chi-Ming Chen, Tong-Huei Chang and Chiao-Wei Chung
(ii)If T:Y→2Xis a generalized Φ-mapping (Φ-mapping ), then for each
nonempty subset Y1of Y,T|Y1:Y1→2Xis also a generalized Φ-mapping
(Φ-mapping).
Let Xbe a convex space, and Ya topological space. A real-valued function
f:X×Y→is said to be quasiconvex in the first variable if for each y∈Y
and for each ξ∈, the set {x∈X:f(x, y)≤ξ}is convex, and fis said to be
quasiconcave if −fis quasiconvex.
Definition 4. Let Xbe a nonempty almost-convex subset of a topological
vector space, and Ya topological space. A real-valued function f:X×Y→is
said to be almost quasiconvex in the first variable if for each y∈Yand for each
ξ∈, the set {x∈X:f(x, y)≤ξ}is almost-convex, and fis said to be almost
quasiconcave if −fis almost quasiconvex.
Definition 5. Let Xbe a convex space, Y a nonempty set, and let f, g :
X×Y→be two real-valued functions. For any y∈Y,gis said to be f-
quasiconcave in the first variable if for each A={x1,x
2, ..., xn}∈X, we have
min
1≤i≤nf(xi,y)≤g(x, y),for all x∈co(A).
Definition 6. Let Xbe a convex space, Y a nonempty set, and let f, g :
X×Y→be two real-valued functions. For any y∈Y,gis said to be f-
quasiconvex in the first variable if for each A={x1,x
2, ..., xn}∈X, we have
max
1≤i≤nf(xi,y)≥g(x, y),for all x∈co(A).
Definition 7. Let Xbe a nonempty almost-convex subset of a topological
vector space E, Y a nonempty set, and let f, g :X×Y→be two real-valued
functions. For any y∈Y,gis said to be almost f-quasiconcave in the first variable
if for each A={x1,x
2, ..., xn}∈Xand for every neighborhood Vof the origin
0of E, there exists a convex-inducing mapping hA,V :A→Xsuch that
min
1≤i≤nf(xi,y)≤g(x, y),for all x∈co(hA,V (A)).
Remark 3. It is clear that if f(x, y)≤g(x, y)for each (x, y)∈X×Y, and
if for each y∈Y, the mapping x→ f(x, y)is almost quasiconcave( quasiconcave
), then gis almost f-quasiconcave( f-quasiconcave ) in the first variable.

Coincidence Theorems on Nonconvex Sets and its Applications 505
2. COINCIDENCE THEOREMS
The following lemma will plays an important role for this section, in order to
establish some coincidence theorems.
Lemma 2. Let Xbe a compact set, and Ya nonempty almost-convex subset
of a Hausdorff topological vector space E.IfT:X→2Yis a generalized Φ-
mapping with a companion mapping F:X→2Y, then there exists a continuous
function f:X→Ysuch that for each x∈X,f(x)∈T(x), that is, Thas a
continuous selection.
Proof. Let Vbe a neighborhood of the origin 0of E. Since Xis compact,
there exists A={y1,y
2, ..., yn}⊂Ysuch that X=∪n
i=1intF−1(yi). Since Yis
almost-convex and A∈Y, there exists a convex-inducing mapping hA,V :A→Y
such that co(hA,V (A)) ⊂Y.
Let {λi}n
i=1 be a partition of the unity subordinated to the cover {intF−1(yi)}n
i=1
of X. Define a continuous mapping f:X→co(hA,V (A)) by
f(x)=
n

i=1
λi(x)hA,V (yi)= 
i∈I(x)
λi(x)hA,V (yi),for each x∈X.
where I(x)={i∈{1,2, ..., n}:λi=0}. Noting that i∈I(x)if and only if
x∈F−1(yi); that is, yi∈F(x). Since Tis a generalized Φ-mapping, we conclude
that f(x)=n
i=1 λi(x)hA,V (yi)∈co(hA,V (A)) ⊂T(x), for each x∈X. This
completes the proof.
By above Lemma 2, we immediately get the following corollary.
Corollary 1. Let Xbe a compact set, Ya convex space, and let T:X→2Y
be a Φ-mapping. Then Thas a continuous selection.
A polytope in Xis denoted by ∆=co(A)for each A∈X.
Theorem 1. Let Xbe a convex space, Ya nonempty almost-convex subset
of a Hausdorff topological vector space, and let T:X→2Ybe a generalized
Φ-mapping. Then T∈KKM(X, Y ).
Proof. Since T:X→2Yis a generalized Φ-mapping, T|∆is also a gen-
eralized Φ-mapping. Since ∆is compact, Thas a continuous selection, and so
T|∆∈KKM(∆,Y). Applying Proposition 3(i)[4], we conclude that T∈KKM
(X, Y )
Lemma 3. Let Xbe a nonempty almost-convex subset of a Hausdorff topo-
logical vector space E, and let Y,Zbe two topological spaces. Then

506 Chi-Ming Chen, Tong-Huei Chang and Chiao-Wei Chung
(i)if T∈KKM∗(X, Y )and f∈C(Y, Z ), then fT ∈KKM∗(X, Z);
(ii)if T∈KKM∗(X, Y )and Dis a nonempty almost-convex subset of X, then
T|D∈KKM∗(D, Y ).
Proof. The proof is analyogous to the proof of Lemma 2 of Chang and Yen
[4].
The following theorem and corollary are well-known, cf [4] and [8].
Theorem 2. Let Xbe a nonempty almost-convex subset of a locally convex
space E.IfT∈KKM∗(X, X )is compact and closed, then Thas a fixed point
in X.
Corollary 2. Let Xbe a nonempty convex subset of a locally convex space
E.IfT∈KKM(X, X)is compact and closed, then Thas a fixed point in X.
By Lemma 2, we have the following coincidence theorem.
Theorem 3. Let Xbe a nonempty almost-convex subset of a locally convex
space E, and let Ybe a topological space. Assume that
(i)T∈KKM∗(X, Y )is compact and closed, and
(ii)F:Y→2Xis a generalized Φ-mapping.
Then there exists (x, y)∈X×Ysuch that y∈T(x)and x∈F(y).
Proof. Since Tis compact, we have K=T(X)is compact in Y.By(ii),
F|Kis also a generalized Φ-mapping. By Lemma 2, F|Khas a continuous selection
f:K→X. So, by Lemma 3, we have fT ∈KKM∗(X, X), and so, by Theorem
2, there exists x∈Xsuch that x∈fT(x)⊂FT(x); that is, there exists y∈T(x)
such that x∈F(y).
Corollary 3. [1] Let Xbe a nonempty convex subset of a locally convex
space E, and let Ybe a topological space. Assume that
(i)T∈KKM(X, Y )is compact and closed, and
(ii)F:Y→2Xis a Φ-mapping.
Then there exists (x, y)∈X×Ysuch that y∈T(x)and x∈F(y).
Applying Theorem 1 and Corollary 3, we also have the following theorem.
Theorem 4. Let Xbe a nonempty convex subset of a locally convex space E1,
and Ya nonempty almost-convex subset of a Hausdorff topological vector space

Coincidence Theorems on Nonconvex Sets and its Applications 507
E2.IfT:X→2Yis a generalized Φ-mapping, F:Y→2XisaΦ-mapping,
and if Tis compact and closed, then exists (x, y)∈X×Ysuch that y∈T(x)
and x∈F(y).
We next establish the another coincidence theorem, as follows:
Theorem 5. Let Xbe a nonempty almost-convex subset of a topological vector
space E, and let Ybe a topological space. Suppose that T, F :X→2Yare two
mutifunctions satisfying
(i)T∈KKM∗(X, Y )is compact, and
(ii)F−1:Y→2Xis a generalized Φ-mapping.
Then there exists x0∈Xsuch that T(x0)∩F(x0)=φ.
Proof. Since Tis compact, T(X)is compact. By (ii), there exists a companion
mapping G:Y→2Xsuch that Y=∪x∈XintG−1(x). Hence, there exists
A∈Xsuch that T(X)⊂∪
x∈AintG−1(x).
Case 1. If T(X)⊂intG−1(x0)for some x0∈X, then T(x0)⊂intG−1(x0).
Take y0∈T(x0). Then y0∈intG−1(x0), which implies x0∈G(y0). And, by
(ii), we have x0∈F−1(y0),y0∈F(x0). This shows T(x0)∩F(x0)=φ.
Case 2. If T(X)intG−1(x)for all x∈X, then T(X)\intG−1(x)=φ
for all x∈X. Define S:X→2Yby
S(x)=T(x)\intG−1(x)for x∈X.
Then S(x)is nonempty and closed for all x∈X. Let A={x1,x
2, ..., xn}∈X.
We claim that Sis not a generalized KKM∗mapping with respect to T. Suppose,
on the contrary, Sis a generalized KKM∗mapping with respect to T. Since
T∈KKM∗(X, Y ),{S(x):x∈X}has finite intersection property. Thus,
∩x∈NS(x)=φfor each N∈X, which implies T(X)∪x∈NintG−1(x)for
each N∈X, and so we get a contradiction. Therefore, there exists a neighborhood
Vof the origin 0of Esuch that for any convex-inducing mapping hA,V :A→X
one has T(co(hA,V (A))) S(A). Choose x0∈co(hA,V (A)) and y0∈T(x0)⊂Y
such that y0/∈S(A). By the definition of S,y0∈intG−1(xi)for all i=1,2, .., n.
This implies xi∈G(y0)for all i=1,2, .., n.By(ii), we have co(hA,V (A)) ⊂
F−1(y0), and so y0∈F(x0). This shows T(x0)∩F(x0)=φ. We complete the
proof.
Corollary 4. Let Xbe a convex space, and let Ybe a topological space.
Suppose that T,F :X→2Yare two mutifunctions satisfying

508 Chi-Ming Chen, Tong-Huei Chang and Chiao-Wei Chung
(i)T∈KKM(X, Y )is compact, and
(ii)F−1:Y→2Xis a Φ-mapping.
Then there exists x0∈Xsuch that T(x0)∩F(x0)=φ.
Corollary 5. Let Xand Ybe two convex spaces. Suppose that T,F :X→2Y
are two mutifunctions satisfying
(i)Tis a Φ-mapping and compact, and
(ii)F−1:Y→2Xis a Φ-mapping.
Then there exists x0∈Xsuch that T(x0)∩F(x0)=φ.
A subset Xof a topological vector space Eis said to be admissible (in the
sense of Klee [10]) provided that, for any nonempty compact subset Aof Xand
every neighborhood Vof 0of E, there exists a continuous mapping h
A,V :A→X
such thta h(x)∈x+Vfor all x∈Aand h(A)is contained in a finite-dimensional
subspace Lof E.
In [3], Chang et al. had introduced the class S−KKM(D, X, Y)on an
admissible onvex set X, we now apply the Corollary 1 and Theorem 3.1[3], we
also have the following coincidence theorem for the Φ-mapping and the class S−
KKM(D, X, Y ).
Theorem 6. Let Xbe an admissible convex subset of a topological vector
space, Dbe a nonempty subset of X, and let Ybe a topological space. Suppose
that
(i)s:D→Xis surjective,
(ii)T∈s−KKM(D, X, Y )is comapct and closed, and
(iii)F:Y→2Xis a Φ-mapping.
Then there exists (x0,y
0)∈X×Ysuch that y0∈T(x0)and x0∈F(y0).
Proof. Since the proof is analyogous to the proof of Theorem 3, we omit it.
Applying Theorem 1 and Theorem 6, we also have the following coincidence
theorem.
Theorem 7. Let Xbe an admissible convex subset of a Hausdorff topological
vector space E1, and Yan almost-convex subset of a Hausdorff topological vector
space E2. Suppose that T:X→2Yand F:Y→2Xsatisfy
(i)Tis comapct and closed,
(ii)Tis a generalized Φ-mapping, and

Coincidence Theorems on Nonconvex Sets and its Applications 509
(iii)Fis a Φ-mapping.
Then there exists (x0,y
0)∈X×Ysuch that y0∈T(x0)and x0∈F(y0).
Proof. Since Tis a generalized Φ-mapping, by Theorem 1, we get T∈
KKM(X, Y ). Let S=iX, then T∈s−KKM(X, X, Y ). Thus all of the
assumptions of Theorem 6 are satisfied. So, there exists (x0,y
0)∈X×Ysuch
that y0∈T(x0)and x0∈F(y0).
3. GENERALIZED VARIATIONAL THEOREMS AND MINIMAX INEQUALITY THEOREMS
Definition 8. Let Xand Ybe two topological spaces, and let F:X→2Y.
(i)Fis said to be tranfer open if for any x∈Xand y∈F(x), there exists an
x∈Xsuch that y∈intF(x), and
(ii)Fis said to be tranfer closed if for any x∈Xand y/∈F(x), there exists an
x∈Xsuch that y/∈intF(x).
Definition 9. Let Xand Ybe two topologicalspaces. A function f:X×Y→
is said to be transfer upper semicontinuous(resp. transfer lower semicontinuous)
in the first variable if for each λ∈and all x∈X,y∈Ywith f(x, y)<λ
(resp. f(x, y)>λ), there exists a y∈Yand a neighborhood Nxof xsuch that
f(u, y)<λ(resp. f(u, y)>λ) for all u∈Nx.
Remark 4. It is easy to prove (see [12], Lemma 2.2) that fis transfer upper
semicontinuous(resp. transfer lower semicontinuous) in the first variable if and only
if the set-valued mapping F:Y→2X,F(y)={x∈X:f(x, y)<λ}(resp.
F(y)={x∈X:f(x, y)>λ}) is transfer open valued.
Lemma 4. [13]. Let Xand Ybe two topological spaces, and let F:X→2Y
be a set-valued mapping. Then the following conditions are equivalent:
(i)F−1is tranfer open valued on Y, and
(ii)X=∪y∈YintF−1(y).
Lemma 5. [2]. Let Xand Ybe two topological spaces, and let F:X→2Y
be a set-valued mapping. Then Fis transfer closed if and only if ∩x∈XF(x)=
∩x∈XF(x)
Applying the above Lemma 5, we immediately obtain the following variational
inequalities and minimax inequalities.
Theorem 8. Let Xbe a nonempty almost-convex subset of a Hausdorff topo-
logical vector space E,Ya topological space, and let F∈KKM ∗(X, Y )be
compact. If f, g :X×Y→are two real-valued functions satisfying the follow-
ing conditions:

510 Chi-Ming Chen, Tong-Huei Chang and Chiao-Wei Chung
(i)for each x∈X, the mapping y→ f(x, y)is transfer lower sem-continuous
on Y, and
(ii)for each y∈Y,gis almost f-quasiconcave,
then for each ξ∈, one of the following properties holds:
(1) there exists (x, y)∈G
Fsuch that
g(x, y)>ξ, or
(2) there exists y∈Ysuch that
f(x, y)≤ξ, for all x∈X.
Proof. Let ξ∈. Since Fis compact, F(X)is compact in Y. Define
T,S :X→2Yby
T(x)={y∈F(X):g(x, y)≤ξ},for all x∈X, and
S(x)={y∈F(X):f(x, y)≤ξ},for all x∈X.
Suppose the conclusion (1) is false. Then for each (x, y)∈G
F,g(x, y)≤ξ. This
implies that GF⊂G
T.
Let A={x1,x
2, ...xn}∈X. By the condition (ii), we claim that Sis
a generalized KKM∗mapping with respect to T. If the above statement is not
true, then there exists a neighborhood Vof the orgin 0of Esuch that for any
convex-inducing mapping hA,V :A→Xone has T(co(hA,V (A))) S(A).So
there exist x0∈co(hA,V (A)) and y0∈T(x0)such that y0/∈S(A). From the
definitions of Tand S, it follows that g(x0,y
0)≤ξand f(xi,y
0)>ξfor all
i=1,2, ..., n. This contradicts the condition (ii). Therefore, Sis a generalized
KKM∗mapping with respect to T, and so we get Sis a generalized KKM∗
mapping with respect to F. Since F∈KKM∗(X, Y ), the family {S(x):x∈X}
has the finite intersection property, and since S(x)is compact for each x∈X,
so we have ∩x∈XS(x)=φ. From Lemma 5 and the condition (i), we have that
∩x∈XS(x)=φ. Take y0∈∩
x∈XS(x), then f(x, y0)≤ξfor all x∈X
Theorem 9. If all of the assumptions of Theorem 8 hold, then we immediately
conclude the following inequality.
inf
y∈Ysup
x∈X
f(x, y)≤sup
(x,y)∈GF
g(x, y).
Proof. Let ξ=sup
(x,y)∈GFg(x, y). Then the conclusion (1) of Theorem
8 is false. So there exist y0∈Ysuch that f(x, y0)≤ξfor all x∈X.

Coincidence Theorems on Nonconvex Sets and its Applications 511
This implies supx∈Xf(x, y0)≤ξ, and so we have infy∈Ysupx∈Xf(x, y)≤
sup(x,y)∈GFg(x, y).
Proposition 1. Let Xbe a nonempty almost-convex subset of a Hausdorff
topological vector space E,Ya topological space, Va neighborhood of the origin
oof E, and let T, F :X→2Ybe two set-valued mappings. Then the following
two statements are equivalent.
(i)for each y∈Y,A∈T∗(y)implies co(hA,V (A)) ⊂F∗(y), where hA,V :
A→Xis a convex-inducing mapping.
(ii)Tis a generalized KKM ∗mapping with respect to F.
Theorem 10. Let Xbe a nonempty almost-convex subset of a Hausdorff
topological vector space E,Ya compact topological space, and let S, T :X→2Y
be two set-valued mappings satisfying the following conditions:
(i)T∈KKM∗(X, Y ),
(ii)Sis transfer closed valued on X,
(iii)for each y∈Y,T∗(y)is almost-convex, and
(iv)for each x∈X,T(x)⊂S(x).
Then there exists y∈Ysuch that S∗(y)=φ.
Proof. Let Vbe a neighborhood of the origin 0of E. By the conditions (iii)
and (iv), we have that for each y∈Yand any A∈S∗(y),co(hA,V (A)) ⊂T∗(y),
where hA,V :A→Xis a convex-inducing mapping. So, by Proposition 1, Sis a
generalized KKM∗mapping with respect to T. Therefore, the family {S(x):x∈
X}has the finite intersection property. Since Yis compact, ∩x∈XS(x)=φ.By
Lemma 5, we have ∩x∈XS(x)=φ. Let y∈∩
x∈XS(x). Then S∗(y)=φ.
Corollary 6. Let Xbe a convex space, Ya compact topological space, and
let S, T :X→2Ybe two set-valued mappings satisfying the following conditions:
(i)T∈KKM(X, Y ),
(ii)Sis transfer closed valued on X,
(iii)for each y∈Y,T∗(y)is convex, and
(iv)for each x∈X,T(x)⊂S(x).
Then there exists y∈Ysuch that S∗(y)=φ.
Theorem 11. Let Xbe a convex space, Ya nonempty set, and Za compact
convex space. Let s, g, t :X×Z→and f:X×Y→be four functions such
that

512 Chi-Ming Chen, Tong-Huei Chang and Chiao-Wei Chung
(i)tis s-quasiconvex in the second variable,
(ii)tis g-quasiconcave in the first variable,
(iii)fis transfer upper semicontinuous in the first variable,
(iv)gis upper semi-continuous on X×Y, and
(v)for each y∈Y, there exists z∈Zsuch that s(·,z)≤f(·,y).
Then
inf
z∈Zsup
x∈X
g(x, z)≤sup
(x,z)∈X×Z
t(x, z).
Proof. Let λ=sup(x,z)∈X×Zt(x, z)and define the set-valued mappings
T, S, G :X→2Zand F:X→2Yby
T(x)={u∈Z:t(x, u)≤λ},
G(x)={v∈Z:g(x, v)≤λ},
S(x)={z∈Z:s(x, z)<λ},and
F(x)={y∈Y:f(x, y)<λ}.
By the condition (iii),F−1is transfer open valued on Y, and then by Lemma 4,
we have X=∪y∈YintF−1(y). By the condition (v), for each y∈Y, there exists
z∈Zsuch that F−1(y)⊂S−1(z). So we conclude that X=∪z∈ZintS−1(z).
We claim that for each x∈X,N∈S(x)implies co(N)⊂T(x). Let x∈X,
N={z1,z
2, ..., zn}∈S(x)and u∈co{z1,z
2, ..., zn}. Since zi∈S(x)and tis
s-quasiconvex in the second variable, we have
t(x, u)≤max1≤i≤ns(x, zi)<λ,
and hence u∈T(x). These imply Tis a Φ-mapping with a companion mapping
S. Thus, by Theorem 1, T∈KKM(X, Z).
By the condition (iv),G(x)is closed for each x∈X, and by the condition
(ii), we have T(co(N)) ⊂G(N)for each N∈X.SoGis a generalized
KKM mapping with respect to T. Since T∈KKM(X, Z), hence the family
{G(x):x∈X}has the finite intersection property. Since Zis compact, we
get ∩x∈XG(x)=φ. Let z∈∩
x∈XG(x). Then g(x, z)≤λfor all x∈X,
which implies supx∈Xg(x, z)≤λ. So we have infz∈Zsupx∈Xg(x, z)≤λ=
sup(x,z)∈X×Zt(x, z).
REFERENCES
1. C. M. Chen T. H. Chang, Some results for the family KKM(X,Y) and the Φ-mapping,
J. Math. Anal. Appl.,329 (2007), 92-101.

Coincidence Theorems on Nonconvex Sets and its Applications 513
2. S. S. Chang, B. S. Lee, X. Wu, Y. J. Cho, G. M. Lee, On the generalized quasivari-
ational inequality problems, J. Math. Anal. Appl.,203 (1996), 686-711.
3. T. H. Chang, Y. Y. Huang, J. C. Jeng, K. W. Kuo, On S-KKM property and related
topics, J. Math. Anal. Appl.,229 (1999), 212-227.
4. T. H. Chang, C. L. Yen, KKM property and fixed point theorems, J. Math. Anal.
Appl.,203 (1996), 224-235.
5. X. P. Ding, Coincidence theorems in topologicalspaces and their applications, Ap-
plied. Math. Lett.,12 (1999), 99-105.
6. P. M. Fitzpatrick, W. V. Petryshyn, Fixed point theorems for multivalued noncompact
inward mappings, J. Math. Anal. Appl.,46 (1974), 756-767.
7. Ky Fan, A generalization of Tychonoff’s fixed point theorem, Math. Ann.,142
(1961), 305-310.
8. H. C. Hsu, Fixed point theorems on almost convex sets, Thesis, Dep. of Math, Cheng
Kung University, (2003).
9. J. C. Jeng, H. C. Hsu, Y. Y. Huang, Fixed point theorems for multifunctions having
KKM property on almost convex sets. J. Math. Ann. Appl.,319 (2006), 187-198.
10. V. Klee, Leray-Schauder theory without locally convexity, Math. Ann.,141 (1960),
286-297.
11. B. Knaster, C. Kurnatoaski, S. Mazurkiewicz, Ein Beweis des Fixpunksatzes fur
n-dimensionale simplexe, Fund. Math.,14 (1929), 132-137.
12. L. J. Lin, Applications of a fixed point theorems in G-convex spaces, Nonlinear
Anal.,46 (2001), 601-608.
13. L. J. Lin, System of coincidence theorems with applications, J. Math. Anal. Appl.,
285 (2003), 408-418.
14. G. J. Minty, On the maximal domain of a monotone function, Michigan Math. J.,8
(1961), 179-182.
15. G. Q. Tian, Generalizations of KKM theorem and Ky Fan minimax inequality with
applications to maximal elements, price equalibrium and complementarity, J. Math.
Anal. Appl.,170 (1992), 457-471.
Chi-Ming Chen, Tong-Huei Chang and Chiao-Wei Chung
Department of Applied Mathematics,
National Hsin Chu University of Education,
Taiwan 300, R.O.C.
E-mail: thchang@mail.nhcue.edu.tw