Some Results for the Family KKM(X,Y) and Φ-Mapping in...

International Scholarly Research NetworkISRN Applied MathematicsVolume 2011, Article ID 696917, 13 pagesdoi:10.5402/2011/696917

Research ArticleSome Results for the Family KKM(X,Y) andthe Φ-Mapping in FC-Spaces

Rong-Hua He

Department of Mathematics, Chengdu University of Information Technology,Chengdu, Sichuan 610103, China

Correspondence should be addressed to Rong-Hua He, [email protected]

Received 15 July 2011; Accepted 24 August 2011

Academic Editor: A. Bellouquid

Copyright q 2011 Rong-Hua He. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

We first establish a fixed point theorem for a k-set contraction map on the family KKM(X,X),which does not need to be a compact map. Next, we present the KKM type theorems, matchingtheorems, coincidence theorems, and minimax theorems on the family KKM(X,Y ) and the Φ-mapping in FC-spaces. Our results improve and generalize some recent results.

1. IntroductionIn 1929, Knaster et al. [1] first established the well-known KKM theorem in finite-dimensional spaces. In 1961, Fan [2] generalized the KKM theorem to infinite dimensionaltopological vector spaces and gave some applications in several directions. Later, Changand Yen [3] introduced the family KKM(X,Y ) and got some results about fixed pointtheorems, coincidence theorems, and its applications on this family. Recently, Lin and Chen[4] studied the coincidence theorems for two families of multivalued functions, Chen andChang [5] obtained some results for the familyKKM(X,Y ) and theΦ-mapping in Hausdorfftopological vector spaces. For the related results, the reader may consult [4–10]. In this paper,we first establish a fixed point theorem for a k-set contractionmap on the familyKKM(X,X),which does not need to be a compact map. Next, we present the KKM type theorems,matching theorems, coincidence theorems, andminimax theorems on the familyKKM(X,Y )and the Φ-mapping in FC-spaces. Our results improve and generalize the correspondingresults in [5, 8, 9].

2. PreliminariesLet Y be a nonempty set. We denote by 2Y and 〈Y〉 the family of all subsets of Y and thefamily of all nonempty finite subsets of Y , respectively. For each A ∈ 〈X〉, we denote by

2 ISRN Applied Mathematics

|A| the cardinality of A. Let Δn denote the standard n-dimensional simplex with the vertices{e0, . . . , en}. If J is a nonempty subset of {0, 1, . . . , n}, we will denote by ΔJ the convex hull ofthe vertices {ej : j ∈ J}.

Let X and Y be two sets, and T : X → 2Y be a set-valued mapping. We will use thefollowing notations in the following material:

(i) T(x) = {y ∈ Y : y ∈ T(x)},(ii) T(A) =

⋃x∈A T(x),

(iii) T−1(y) = {x ∈ X : y ∈ T(x)},(iv) T−1(B) = {x ∈ X : T(x)

⋂B /= ∅},

(v) C(X,Y ) denote the family of single-valued continuous mappings from X to Y .

For topological spaces X and Y , T : X → 2Y is said to be closed if its graphGT = {(x, y) ∈ X × Y : y ∈ T(x)} is closed. T is said to be compact if the image T(X) ofX under T is contained in a compact subset of Y . A subset D of X is said to be compactlyopen (resp., compactly closed) if for each nonempty compact subset K of X, D

⋂K is open

(resp., closed) in K. The compact closure of A and the compact interior of A (see [5]) aredefined, respectively, by

cclA =⋂{

B ⊂ X : A ⊂ B and B is compactly closed in X},

cintA =⋃{

B ⊂ X : B ⊂ A and B is compactly open in X}.

(2.1)

It is easy to see that ccl(X \ A) = X \ cint A, int A ⊂ cint A ⊂ A,A ⊂ ccl A ⊂ cl A,A is compactly open (resp., compactly closed) in X if and only if cint A = A (resp., ccl A =A). For each nonempty compact subset K of X, cclA

⋂K = clK(A

⋂K) and cint A

⋂K =

intK(A⋂K), where clK(A

⋂K) (resp., intK(A

⋂K)) denotes the closure (resp., interior) of

A⋂K in K.

A set-valued mapping T : X → 2Y is said to be transfer compactly closed valued onX(see [5]) if for each x ∈ X and y /∈ T(x), there exists x′ ∈ X such that y /∈ ccl T(x′). T is saidto be transfer compactly open valued on X if for each x ∈ X and y ∈ T(x), there exists x′ ∈ Xsuch that y ∈ cint T(x′). T is said to have the compactly local intersection property onX if foreach nonempty compact subsetK of X and for each x ∈ X with T(x)/= ∅, there exists an openneighborhoodN(x) of x in X such

⋂z∈N(x)

⋂K T(z)/= ∅.

LetAi (i = 1, . . . , m) is transfer compactly open valued, then⋂mi=1 cint Ai = cint

⋂mi=1Ai.

It is clear that each transfer open valued correspondence is transfer compactly open valued.The inverse is not true in general.

Throughout this paper, all topological spaces are assumed to be Hausdorff. Thefollowing notion of a finitely continuous topological space (in short, FC-space) wasintroduced by Ding in [11].

Definition 2.1. (Y, ϕN) is said to be an FC-space if Y is a topological space and for eachN = {y0, . . . , yn} ∈ 〈Y〉 where some elements in N may be same, there exists a continuousmapping ϕN : Δn → Y . If A and B are two subsets of Y , B is said to be an FC-subspace of Yrelative toA if for eachN = {y0, . . . , yn} ∈ 〈Y〉 and for any {yi0 , . . . , yik} ⊂N∩A, ϕN(Δk) ⊂ B,where Δk = co({eij : j = 0, . . . , k}).

If A = B, then B is called an FC-subspace of Y . Clearly, each FC-subspace of Y is alsoan FC-space.

ISRN Applied Mathematics 3

For a subset A of X, we can define the FC-hull of A (see [12]) as follows: FC(A) =⋂{B ⊂ X : A ⊂ B and B is FC-subspace of X}.

Definition 2.2 (see [8]). An FC-space (X,ϕN) is said to be a locally FC-space, denoted by(X,U, ϕN), if X is a uniform topological space with a uniform structure U having an openbase β of symmetric entourages such that for each x ∈ X and for each U ∈ U, U[x] = {y ∈X : (y, x) ∈ U} is an FC-subspace of X.

Lemma 2.3 (see [5]). Let X and Y be two topological spaces, and let F : X → 2Y be a set-valuedmapping. Then the following conditions are equivalent:

(1) F has the compactly local intersection property,

(2) for each compact subset K of X and for each y ∈ Y , there exists an open subset Oy of Xsuch that Oy

⋂K ⊂ F−1(y) and K =

⋃y∈Y (Oy

⋂K),

(3) for any compact subset K of X, there exists a set-valued mapping P : X → 2Y such thatP(x) ⊂ F(x) for each x ∈ X, P−1(y) is open in X and P−1(y)

⋂K ⊂ F−1(y) for each

y ∈ Y , and K =⋃y∈Y (P

−1(y)⋂K),

(4) for each compact subset K of X and for each x ∈ K, there exists y ∈ Y such that x ∈cint F−1(y)

⋂K and K =

⋃y∈Y (cint F

−1(y)⋂K),

(5) F−1 is transfer compactly open valued on Y ,

(6) X =⋃y∈Y cint F

−1(y).

Now, we introduce the following Definitions 2.4 and 2.6.

Definition 2.4. Let Y be a topological space and (X,ϕN) be an FC-space. A set-valuedmappingT : Y → 2X is called a Φ-mapping if there exists a set-valued mapping T : Y → 2X such that

(i) for each y ∈ Y , T(y) is a nonempty FC-subspace of X relative to F(y),

(ii) F satisfies one of the conditions (1)–(6) in Lemma 2.3.

The mapping F is said to be a companion mapping of T .

Remark 2.5. If T : Y → 2X be a Φ-mapping, then for each nonempty subset Y1 of Y , T |Y1 :Y1 → 2X is also a Φ-mapping.

The class KKM was introduced by Ding [8]. Let (X,ϕN) be an FC-space and let Ybe a topological space. If F, T : X → 2Y are two set-valued mappings such that for eachN ∈ 〈X〉 and for each {ei0 , . . . , eik} ⊂ {e0, . . . , en}, T(ϕN(Δk)) ⊂

⋃kj=0 F(xij ), then F is said to be

a generalized F-KKM mapping with respect to T . Let T : X → 2Y be a set-valued mappingsuch that if F : X → 2Y is a generalized F-KKM mapping with respect to T , then the family{Fx : x ∈ X} has the finite intersection property, where Fx denotes the closure of Fx, then Tis said to have the KKM property. Write

KKM(X,Y ) ={T : X −→ 2Y : T has the KKM property

}. (2.2)

Let (X,ϕN) be an FC-space and B(X) be the family of nonempty bouned subsets. LetP = {P : P is a family of seminorms which determines the topology on X. Let R+ be the


set of all nonnegative real numbers. If {p ∈ P : P ∈ P} is a family of seminorms whichdetermines the topology on X}, then for each P ∈ P and Ω ∈ X, we define the set-measure ofnoncompactness αp : 2X → R+ by

αp(Ω) = inf{ε > 0 : Ω can be covered by a finite number of sets

and each p-diameter of the sets is less than ε},

(2.3)

where p-diameter of a set D = sup{p(x − y) : x, y ∈ D}.

Definition 2.6. Let (X,ϕN) be an FC-space, and a mapping T : X → 2X is said to be a k-setcontraction map, if there exists P ∈ P such that for each p ∈ P , αp(T(Ω)) ≤ kαp(Ω) with0 < k < 1 for each bounded subset Ω of X and T(X) is bounded.

3. Main Results

In order to prove our main results, we need the following Lemmas. The following results are[8, Lemma 3.1(i) and Theorem 3.1].

Lemma 3.1. Let (X,ϕN) be an FC-space and let Y be a topological space. Then we have T ∈KKM(X,Y ) if and only if T |D ∈ KKM(D,Y ) for each nonempty subset D of X.

Lemma 3.2. Let (X,U, ϕN) be a locally FC-space. If T ∈ KKM(X,Y ) is a compact mapping, thenfor each open entourageUi ∈ U there exists xi ∈ X such that T(xi)

⋂Ui[xi]/= ∅.

Lemma 3.3. Let Y be a compact topological space and (X,ϕN) be an FC-space. Let T : Y → 2X

be a Φ-mapping. Then there exists a continuous function f : Y → X such that for each y ∈ Y ,f(y) ∈ T(y); that is, T has a continuous selection.

Proof. Since T is a Φ-mapping, there exists a companion mapping F : Y → 2X such that

(i) for each y ∈ Y , for each N = {x0, . . . , xn} ∈ 〈X〉, and for any {xi0 , . . . , xik} ∈N

⋂F(y), ϕN(Δk) ⊂ T(y);

(ii) Y =⋃x∈X cint F−1(x).

Since Y be a compact, there exists M = {x0, . . . , xm} ∈ 〈X〉 such that Y =⋃mi=0 cint F

−1(xi). Let {ψi}mi=0 be the continuous partition of unity subordinated to the opencovering {cint F−1(xi)}mi=0 of Y , then for each i ∈ {0, 1, . . . , m} and y ∈ Y , we have

ψi(y)/= 0 ⇐⇒ y ∈ cint F−1(xi) ⊂ F−1(xi) =⇒ xi ∈ F

(y). (3.1)

Define a mapping ψ : Y → Δn by ψ(y) = Σmi=0ψi(y)ei, then ψ is continuous and for each

y ∈ Y , ψ(y) = Σj∈J(y)ψj(y)ej ∈ ΔJ(y), where J(y) = {j ∈ {0, 1, . . . , m} : ψj(y)/= 0}. By (3.1), wehave {xj : j ∈ J(y)} ∈ 〈F(y)〉

⋂{x0, . . . , xm}. From the definition ofΦ-mapping, we obtain for

each y ∈ Y , ϕN(ΔJ(y)) ⊂ T(y). It follows that

f(y)= ϕN ◦ ψ

(y)∈ ϕN

(ΔJ(y)

)⊂ T

(y). (3.2)


This shows that for each y ∈ Y , f(y) ∈ T(y), that is, T has a continuous selection. Thiscompletes the proof.

Lemma 3.4. Let (X,ϕN) be an FC-space and let Y be a topological space. If T : X → 2Y is aΦ-mapping, then T ∈ KKM(X,Y ).

Proof. Since T is a Φ-mapping, we have that for each N = {x0, . . . , xn} ∈ 〈X〉 and ϕN(Δn)in X, T |ϕN(Δn) : ϕN(Δn) → Y is also a Φ-mapping. By Lemma 3.3, T |ϕN(Δn) has a continuousselection function. Then T |ϕN(Δn) ∈ KKM(ϕN(Δn), Y ). It follows from Lemma 3.1 that T ∈KKM(X,Y ). This completes the proof.

Lemma 3.5. Let Y andZ be two topological spaces and (X,ϕN) be an FC-space. If T ∈ KKM(X,Y )and f ∈ C(Y,Z), then fT ∈ KKM(X,Z).

Proof. Let F : X → 2Z be a generalized F-KKM mapping with respect to fT such thatFx is closed in Z for each x ∈ X. Take N = {x0, . . . , xn} ∈ 〈X〉 and for any {ei0 , . . . , eik} ⊂{e0, . . . , en}, we have fT(ϕN(Δk)) ⊂

⋃kj=0 F(xij ), and hence T(ϕN(Δk)) ⊂

⋃kj=0 f

−1F(xij ). Sof−1F is a generalized F-KKM mapping with respect to T . Since T ∈ KKM(X,Y ), then{f−1Fx : x ∈ X} has the finite intersection property, and hence {Fx : x ∈ X} has also thefinite intersection property and fT ∈ KKM(X,Z). This completes the proof.

Remark 3.6. Lemmas 3.3, 3.4, and 3.5 generalize Lemmas 2, 3, and 5 in [5], respectively.

Corollary 3.7. Let (X,U, ϕN) be a locally FC-space and T ∈ KKM(X,Y ) is compact and closed.Then T has a fixed point in X.

Proof. Since T is compact, we have K = T(X) is compact in X. Without loss of generality,assume that {Vi}i∈I be a basis of symmetric open entourages for the uniformity U. It followsfrom the Lemma 3.2 that for each V ∈ {Vi}i∈I there exists xi ∈ X such that T(xi)

⋂V [xi]/= ∅.

Hence, for each V ∈ {Vi}i∈I there exists yi ∈ T(xi) and yi ∈ V [xi]. Since {yi} ∈ K and K iscompact, we may assume that {yi} converges to some y ∈ K, and then xi also converges toy. Since T is closed, we have y ∈ T(y). This completes the proof.

Theorem 3.8. Let (X,ϕN) be a bounded FC-space. Assume that T : X → 2X be a k-set contractionmap, 0 < k < 1. Then X contains a precompact FC-subspace.

Proof. Since T is a k-set contraction map, 0 < k < 1, there exists P ∈ P such that for each p ∈ P ,we have αp(T(A)) ≤ kαp(A) for each subset A of X. Take y ∈ X. Let

X0 = X, X1 = FC

(

T(X0)⋃{

y})

,

Xn+1 = FC

(

T(Xn)⋃{

y})

for each n ∈N.

(3.3)

Then

(1) Xn is FC-subspace of X for each n ∈N,

(2) Xn+1 ⊂ Xn for each n ∈N,


(3) T(Xn) ⊂ Xn+1 for each n ∈N,

(4) αp(Xn+1) ≤ αp(T(Xn)) ≤ kαp(Xn) ≤ · · · ≤ kn+1αp(X0) for each n ∈N.

Thus αp(Xn) → 0, as n → ∞, and hence X∞ =⋂n≥1Xn is a nonempty precompact

FC-subspace of X. This completes the proof.

Remark 3.9. Theorem 3.8 generalized [5, Theorem 1] from a nonempty bounded convexsubset of a Hausdorff topological vector space to a bounded FC-space. In the process of theproof Theorem 3.8, we call the set X∞ a precompact-inducing FC-subspace of X.

The following result is a fixed point theorem for a k-set contraction map on the familyKKM(X,Y ), which does not need to be a compact map.

Theorem 3.10. Let X be a bounded FC-subspace of a locally FC-space (E,U, ϕN), and let T ∈KKM(X,X) be a k-set contraction map, 0 < k < 1 and closed with T(X) ⊂ X. Then T has a fixedpoint in X.

Proof. By the same process of the proof Theorem 3.8, we get a precompact-inducing FC-subspace X∞ of X. Since T(X) ⊂ X and T(Xn+1) ⊂ T(Xn) ⊂ T(X) for each n ∈ N, wehave T(Xn+1) ⊂ T(Xn) ⊂ X for each n ∈ N. Since αp(T(Xn)) → 0 as n → ∞, we haveT(X∞) =

⋂n≥1 T(Xn) is a nonempty compact subset of X.

Since T ∈ KKM(X,X) and X∞ is a nonempty FC-subspace of X, by Lemma 3.1, wehave T |X∞

∈ KKM(X∞, X).Let {Vi}i∈I be a basis of symmetric open entourages for the uniformity U, then there

exists V ∈ {Vi}i∈I such that V ∈ U. We now claim that for Ui ∈ U there exists xi ∈ X∞ suchthat T(xi)

⋂(xi +Ui)/= ∅. If it is false, then there exists U ∈ U such that T(x)

⋂(x +U) = ∅ for

all x ∈ X∞. Let K = T(X∞). Define F : X∞ → 2K by

F(x) = K \(

x +12U

)

for each x ∈ X∞, (3.4)

then F(x) is a compact for each x ∈ X∞. Next, let V [x] = x + (1/2)U, then F(x) = K \ V [x],we prove that F is a generalized F-KKM mapping with respect to T |X∞

.Suppose, on contrary, F is not a generalized F-KKM mapping with respect to T |X∞

.Then there existsN = {x0, . . . , xn} ∈ 〈X∞〉 and {ei0 , . . . , eik} ⊂ {e0, . . . , en} such that

T |X∞

(ϕN(Δk)

)/⊂

k⋃

j=0

F(xij

), (3.5)

where Δk = co({eij : j = 0, . . . , k}). Hence there exist u ∈ ϕN(Δk) and v ∈ T |X∞(u) ⊂ T(X∞) =

K such that v /∈⋃kj=0 F(xij ). From the definition of F it follows that v ∈ V [xij ] for all j ∈

{0, . . . , k}. Noting that V ∈ U is symmetric, we have {xij : j = 0, . . . , k} ⊂ V [v]. Since V [v] isFC-subspace of X, we have u ∈ ϕN(Δk) ⊂ V [v]. By the symmetry of V , we obtain v ∈ V [u]and v ∈ T |X∞(u)

⋂V [u] ⊂ T |X∞

(u)⋂U[u] ⊂ T |X∞

(u)⋂(u + U) which contradicts the fact

T(x)⋂(x + U) = ∅ for all xi ∈ X∞. Therefore F is a generalized F-KKM mapping with

respect to T |X∞.


Since T |X∞∈ KKM(X∞, X) and F is a generalized F-KKM mapping with respect to

T |X∞, the family {F(x) : x ∈ X∞} has the finite intersection property, and so we conclude

that⋂x∈X∞

F(x)/= ∅. Choose η ∈⋂x∈X∞

F(x), then η ∈ K \ V [x] for all x ∈ X∞. Since η ∈⋂x∈X∞

F(x) ⊂ K = T(X∞) ⊂ X∞ + (1/4)U, hence there exists x0 ∈ X∞ such that η ∈ x0 +(1/2)U = V [x0]. But η ∈ K \ V [x0], a contradiction. Therefore, we have prove that for eachUi ∈ {Vi}i∈I , there exists xi ∈ X∞ such that T(xi)

⋂(xi+Ui)/= ∅. Let yi ∈ T(xi)

⋂(xi+Ui). Since

{yi} ⊂ K and K is compact, we may assume that {yi} converges to some y ∈ K, and then xialso converges to y. Since T is closed, we have y ∈ T(y). This completes the proof.

By Lemma 3.4 and Theorem 3.10, we can get the following result immediately.

Corollary 3.11. Let X be a bounded FC-subspace of a locally FC-subspace (E,U, ϕN), and let T :X → 2X be aΦ-mapping, k-set contraction, 0 < k < 1 and closed with T(X) ⊂ X. Then T has a fixedpoint in X.

Remark 3.12. Theorem 3.10 generalizes [5, Theorem 2] from a nonempty bounded convexsubset of a locally convex space to a bounded FC-subspace of a locally FC-space and [8,Theorem 3.10]. Corollary 3.11 generalizes [5, Corollary 1] in several aspects.

4. Applications

By Definition 4.3 of Ding [13], we have the following definition.

Definition 4.1. Let (X,ϕN) be an FC-space. Y be topological space. f : X × Y → R is said tobe FC-quasiconvex (resp., FC-quasiconcave) if for each y ∈ Y and for each λ ∈ R, the set{x ∈ X : f(x, y) < λ} (resp., {x ∈ X : f(x, y) > λ}) is an FC-subspace of X.

Definition 4.2 (see [5]). Let X and Y be two topological spaces, and let f : X × Y → R ∪{−∞,+∞} be a function. Then f is said to be transfer compactly lower semicontinuous (inshort, transfer compactly l.s.c) in y if for each y ∈ Y and γ ∈ Rwith y ∈ {u ∈ Y : f(x, u) > γ},there exists x ∈ X such that y ∈ cint{u ∈ Y : f(x, u) > γ}. f is said to be transfer compactlyu.s.c in y if −f is transfer compactly l.s.c in y.

Now, we establish the following KKM-type theorem for a k-set contraction map.

Theorem 4.3. Let (Y, ϕN) be a bounded FC-space and let X be a topological space. If T, F : Y → 2X

are two set-valued mappings satisfying the following:

(i) T ∈ KKM(Y,X) is a k-set contraction map, 0 < k < 1, with T(Y ) ⊂ X,

(ii) for any y ∈ Y , F(y) is compactly closed in X,

(iii) F is a generalized F-KKM mapping with respect to T .

Then

T(Y∞)⋂(⋂{

F(y): y ∈ Y∞

})/= ∅, (4.1)

where Y∞ is the precompact-inducing FC-subspace of Y .


Proof. Let {Vi}i∈I be a basis of symmetric open entourages for the uniformity U. By thesame process of the proof Theorem 3.10, we get a compact subset T(Y∞) of Y , and T |Y∞ ∈KKM(Y∞, X), since T ∈ KKM(Y,X).

DefineH : Y∞ → 2X by

H(y)= T(Y∞)

⋂F(y)

for each y ∈ Y∞. (4.2)

By condition (ii),H(y) is compact inX, for each y ∈ Y∞. We now claim thatH is a generalizedF-KKMmapping with respect to T |Y∞

. LetN ∈ 〈Y∞〉 and for each {ei0 , . . . , eik} ⊂ {e0, . . . , en}.By condition (iii), T |Y∞

(ϕN(Δk)) ⊂ T(Y∞)⋂(⋃kj=0 F(yij )) =

⋃kj=0H(yij ). Thus, we have shown

that H is a generalized F-KKM mapping with respect to T |Y∞. Since T |Y∞

∈ KKM(Y∞, X),the family {H(y) : y ∈ Y∞} has the finite intersection property. And, since H(y) is compact,⋂y∈Y∞

H(y)/= ∅, that is, T(Y∞)⋂(⋂{F(y) : y ∈ Y∞})/= ∅. This completes the proof.

Theorem 4.4. Let (Y, ϕN) be a bounded FC-space and let X be a topological space. If T, F : Y → 2X

are two set-valued mappings satisfying the following:

(i) T ∈ KKM(Y,X) is a k-set contraction map, 0 < k < 1, with T(Y ) ⊂ X,

(ii) for any y ∈ Y , F(y) is transfer compactly closed in X,

(iii) F is a generalized F-KKM mapping with respect to T .

Then

T(Y∞)⋂(⋂{

F(y): y ∈ Y∞

})/= ∅, (4.3)

where Y∞ is the precompact-inducing FC-subspace of Y .

Proof. Define a mapping ccl F : Y∞ → 2X by (ccl F)(y) = ccl F(y) for each y ∈ Y∞, it is easyto see that ccl F is also a generalized F-KKM mapping with respect to T |Y∞

with compactlyclosed values. By the same process of the proof Theorem 3.10, we get a compact subset T(Y∞)of Y . By Theorem 4.3, T(Y∞)

⋂(⋂{ccl F(y) : y ∈ Y∞})/= ∅. And since for any y ∈ Y , F(y) is

transfer compactly closed in X, by Lemma 2.2 [9], we have T(Y∞)⋂(⋂{F(y) : y ∈ Y∞}) =

T(Y∞)⋂(⋂{ccl F(y) : y ∈ Y∞})/= ∅. This completes the proof.

Remark 4.5. Theorem 4.3 generalizes [5, Theorem 3] from a nonempty bounded convexsubset of a Hausdorff topological vector space to a bounded FC-space and [9, Theorem 3.1].Theorem 4.4 generalizes [5, Theorem 3] in several aspects and [9, Theorem 3.2].

The following results are the generalization of the Ky Fan matching theorem andcoincidence theorems.

Theorem 4.6. Let (X,ϕN) be a bounded FC-space. If T,H : X → 2X are two set-valued mappingssatisfying the following:

(i) T ∈ KKM(X,X) is a k-set contraction map, 0 < k < 1, with T(X) ⊂ X,

(ii) for any x ∈ X,H(x) is compactly open in X,

(iii) for the precompact-inducing FC-subspace X∞ of X, T(X∞) ⊂ H(X∞).


Then for the precompact-inducing FC-subspace X∞ of X satisfying the following condition:

T(X∞)⋂(⋂

{H(x) : x ∈M})/= ∅ for some M ∈ 〈X∞〉. (4.4)

Proof. Let {Vi}i∈I be a basis of symmetric open entourages for the uniformity U. By thesame process of the proof Theorem 3.10, we get a compact subset T(X∞) of X, and T |X∞ ∈KKM(X∞, X), since T ∈ KKM(X,X).

We claim that there exists M ∈ 〈X∞〉 such that T(X∞)⋂(⋂{H(x) : x ∈ M})/= ∅.

On the contrary, assume that T(X∞)⋂(⋂{H(x) : x ∈ M}) = ∅ for any M ∈ 〈X∞〉, then

T(X∞) ⊂⋂x∈MHc(x). Since X∞ is FC-subspace of X,M ∈ 〈X∞〉 and for any {xi0 , . . . , xik} ⊂

M, we have T(ϕN(Δk)) ⊂ T(X∞) ⊂⋂x∈MHc(x), where Δk = co({eij : j = 0, . . . , k}). This

implies thatHc is a generalized F-KKMmapping with respect to T . By condition (ii), for anyx ∈ X, Hc(x) is compactly closed in X. Follows Theorem 4.3, we have T(X∞)

⋂(⋂{Hc(x) :

x ∈ X∞})/= ∅, which implies T(X∞)/⊆⋃x∈X∞

H(x), a contradiction to condition (iii). Thiscompletes the proof.

Theorem 4.7. Let (X,U, ϕN) be a locally FC-space. Assume that

(i) T ∈ KKM(X,X) is a k-set contraction map, 0 < k < 1, with T(X) ⊂ X,

(ii) F : X → 2X is a Φ-mapping.

Then there exists (x, y) ∈ X ×X such that y ∈ T(x) and x ∈ F(y).

Proof. By the same process of the proof Theorem 3.10, we get a compact subset T(X∞) of X,and T |X∞ ∈ KKM(X∞, X), since T ∈ KKM(X,X).

Let K = T(X∞) is compact. Then F|K is a Φ-mapping, and by Lemma 3.3, F|K has acontinuous selection f : K → X. So, by Lemma 3.5, we have fT |X∞

∈ KKM(X∞, X) and so,it follows from Theorem 3.10 that there exists x ∈ T(X∞) such that x ∈ fT(x) ⊂ FT(x); thatis, there exists y ∈ T(x) such that x ∈ F(y). This completes the proof

Theorem 4.8. Let (X,U, ϕN) be a locally FC-space, and let Y be a topological space. Assume that

(i) T ∈ KKM(X,Y ) is compact and closed,

(ii) F : Y → 2X is a Φ-mapping.

Then there exists (x, y) ∈ X × Y such that y ∈ T(x) and x ∈ F(y).

Proof. Since T is compact, we haveK = T(X) is compact in Y . By condition (ii), we have F|K isalso a Φ-mapping. By Lemma 3.3, F|K has a continuous selection f : K → X. So, by Lemma3.5, we have fT ∈ KKM(X,X), and so, it follows from Corollary 3.7 that there exists x ∈ Xsuch that x ∈ fT(x) ⊂ FT(x); that is, there exists y ∈ T(x) such that x ∈ F(y). This completesthe proof.

As a consequence of the above Theorem 4.6, we have the following generalizedvariational inequality.

Corollary 4.9. Let (X,ϕN) be a bounded FC-space, and let T ∈ KKM(X,X) be a k-set contractionmap, 0 < k < 1, with T(X) ⊂ X. If ϕ, ψ : X × X → R ∪ {−∞,+∞} are two real-valued mappingssatisfying the following:


(i) ψ(x, y) ≤ 0 for each (x, y) ∈ GT ,

(ii) for fixed x ∈ X, the mapping y �→ ϕ(x, y) is lower semicontinuous on K for each compactsubset K of X,

(iii) for fixed y ∈ X, for each N = {x0, . . . , xn} ∈ 〈X〉 and N1 = {xi0 , . . . , xik} ⊂ N, x ∈ϕN(Δk) such that ψ(x, y) ≤ 0 implies that there exists x ∈N1 such that ϕ(x, y) ≤ 0.

Then for the precompact-inducing FC-subspace X∞ of X, there exists y ∈ X∞ such thatϕ(x, y) ≤ 0 for each x ∈ X∞.

Proof. Define F, S : X → 2X by

S(x) ={y ∈ X : ψ

(x, y

)≤ 0

}for each x ∈ X,

F(x) ={y ∈ X : ϕ

(x, y

)≤ 0

}for each x ∈ X.

(4.5)

By condition (i), we haveGT ⊂ GS, and by condition (ii), F(x) is compactly closed for each x ∈X. The condition (iii) implies that for each N = {x0, . . . , xn} ∈ 〈X〉 and N1 = {xi0 , . . . , xik} ⊂N, S(ϕN(Δk)) ⊂

⋃kj=0 F(xij ), and then T(ϕN(Δk)) ⊂

⋃kj=0 F(xij ); that is, F is a generalized

F-KKM mapping with respect to T .Thus, all conditions of Theorem 4.3 are satisfied. By Theorem 4.3, and for the

precompact-inducing FC-subspace X∞ of X, we have that T(X∞)⋂(⋂{F(x) : x ∈ X∞})/= ∅.

Let y ∈ T(X∞)⋂(⋂{F(x) : x ∈ X∞}), and hence we have ϕ(x, y) ≤ 0 for each x ∈ X∞. This

completes the proof.

Applying Theorem 4.8 and Lemma 3.4, we have the following result.

Corollary 4.10. Let (X,U, ϕN) be a locally FC-space, and let Y be a topological space. If T : X → 2Y ,F : Y → 2X are two Φ-mappings, and T is compact and closed, then there exists (x, y) ∈ X × Y suchthat y ∈ T(x) and x ∈ F(y).

From Corollary 4.10, we have the following result.

Corollary 4.11. Let (X,U, ϕN) be a locally FC-space, and let (Y, ϕM) be a compact FC-space. Iff, g, p, q : X × Y → R are four real-valued functions, and a, b be two real numbers. Suppose thefollowing conditions hold:

(i) g(x, y) ≤ f(x, y) and p(x, y) ≤ q(x, y) for all x ∈ X, y ∈ Y ,(ii) for each x ∈ X, f be FC-quasiconcave on Y and for each y ∈ Y, p be FC-quasiconvex onX,

(iii) for each x ∈ X, g be transfer compactly lower semicontinuous in x and for each y ∈ Y , qbe transfer compactly upper semicontinuous in y,

(iv) f is upper semicontinuous on X × Y .

Then one of the following statements holds:

(1) there exists μ ∈ X such that g(μ, y) ≤ a for each y ∈ Y ,(2) there exists ν ∈ Y such that q(x, ν) ≥ b for each x ∈ X,

(3) there exists (μ, ν) ∈ X × Y such that f(μ, ν) ≥ a and p(μ, ν) ≤ b.


Proof. Let S, T : X → 2Y andH,F : Y → 2X be defined by

S(x) ={y ∈ Y : g

(x, y

)− a > 0

}for each x ∈ X,

T(x) ={y ∈ Y : f

(x, y

)− a ≥ 0

}for each x ∈ X,

H(y)={x ∈ X : q

(x, y

)− b < 0

}for each y ∈ Y,

F(y)={x ∈ X : p

(x, y

)− b ≤ 0

}for each y ∈ Y.

(4.6)

By the condition (iii), S−1 is transfer compactly open valued on Y and H−1 is transfercompactly open valued on X. By the condition (i), we have S(x) ⊂ T(x) for each x ∈ Xand H(y) ⊂ F(y) for each y ∈ Y . By the condition (ii), T(x) is FC-subspace of Y for eachx ∈ X, and so F(x) is FC-subspace of X for each y ∈ Y .

Suppose that the Statements (1) and (2) are false. Then S(x)/= ∅ for each x ∈ X andH(y)/= ∅ for each y ∈ Y . So, we conclude that T is a Φ-mapping with a companion mappingS and F is a Φ-mapping with a companion mapping H. By the Condition (iv), T is closed.Thus, all conditions of Corollary 4.9 are satisfied. By Corollary 4.9, there exists (μ, ν) ∈ X × Ysuch that ν ∈ T(μ) and μ ∈ F(ν); that is, f(μ, ν) ≥ a and p(μ, ν) ≤ b. This completes theproof.

From Corollary 4.11, we have the following minimax theorem.

Corollary 4.12. Let (X,U, ϕN) be a locally FC-space, and let (Y, ϕM) be a compact FC-space. Iff, g, p, q : X × Y → R are four real-valued functions, and let a, b be two real numbers. Suppose thefollowing conditions hold

(i) g(x, y) ≤ f(x, y) ≤ p(x, y) ≤ q(x, y) for all x ∈ X, y ∈ Y ,

(ii) for each x ∈ X, let f be FC-quasiconcave on Y and for each y ∈ Y , let p be FC-quasiconvexon X,

(iii) for each x ∈ X, let g be transfer compactly lower semicontinuous in x and for each y ∈ Y ,let q be transfer compactly upper semicontinuous in y,

(iv) f is upper semicontinuous on X × Y .

Then

infx∈X

supy∈Y

g(x, y

)≤ sup

y∈Yinfx∈X

q(x, y

). (4.7)

Proof. Let ε > 0 and let

a = infx∈X

supy∈Y

g(x, y

)− ε, b = sup

y∈Yinfx∈X

q(x, y

)+ ε. (4.8)


Then for each x ∈ X, there exists y ∈ Y such that g(x, y) > a, and for each y ∈ Y , there existsx ∈ X such that q(x, y) < b. Therefore, Conclusions (1) and (2) of Corollary 4.11 are false. Sothere exist (μ, ν) ∈ X × Y such that f(μ, ν) ≥ a and p(μ, ν) ≤ b; that is

f(μ, ν

)≥ inf

x∈Xsupy∈Y

g(x, y

)− ε, p

(μ, ν

)≤ sup

y∈Yinfx∈X

q(x, y

)+ ε. (4.9)

So, by Condition (i), we have

infx∈X

supy∈Y

g(x, y

)− ε < sup

y∈Yinfx∈X

q(x, y

)+ ε. (4.10)

Since ε is arbitrary positive number, by letting ε ↓ 0, we get

infx∈X

supy∈Y

g(x, y

)≤ sup

y∈Yinfx∈X

q(x, y

). (4.11)

This completes the proof.

Remark 4.13. Theorem 4.6 (resp., Theorems 4.7 and 4.8, Corollaries 4.9, 4.10, 4.11, and 4.12)generalize Theorem 4 (resp., Theorems 7, 6, 5, 8, 9, and 10) in [5] in several aspects.

Acknowledgment

The author’s research was supported by the Scientific Research Foundation of CUIT underGrant KYTZ201114.

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[11] X. P. Ding, “System of generalized vector quasi-equilibrium problems in locally FC-space,” ActaMathematica Sinica. In press.

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