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NONEMPTY INTERSECTION THEOREMS MINIMAX THEOREMS
WITH APPLICATIONS IN GENERALIZED
INTERVAL SPACES
DA-CHENG WANG
(Received 7 August 2000)
Abstract.We establish some new topological types of nonempty intersection theorems in more general topological spaces without linear structure. As applications, we utilize results to study the minimax problems, coincidence problems, and economy equilibrium problems in generalized interval spaces and some new results are obtained.
2000 Mathematics Subject Classification. 49K35, 54D20, 54D05, 54H25.
1. Introduction and preliminaries. Recently, some versions of Fan’s KKM theorem and Fan’s minimax theorem in topological spaces without linear structure have been considered by Cheng and Lin [2], Lin et al. [10,11], and Tian [14]. The purpose of this paper is first to introduce and establish a class of more general topological space-generalized interval space, which has no linear structure and containsH-space, inter-val space as its special cases, and so as it contains strong interinter-val space [2], convex space, contractible space, and topological vector space as its special cases. With the aid of this kind of topological frameworks some new topological types of nonempty intersection theorems are established inSection 2. As applications, we utilize these results to obtain some minimax theorems and coincidence theorems in generalized interval space. The results presented in this paper not only contain many recent results of Stacho [13], Kindler and Trost [8], Cheng and Lin [2], Brezis et al. [1], Tuy [7], Komornik [9], Geraghty and Lin [4], Sion [12], Wu [15], Hartung [6], and the fa-mous Neumann’s saddle theorem as its special cases, but also improve and extend the corresponding results of [1,2,4,5,6,7,8,9,10,11,12,14].
For the sake of convenience, we first give some definitions and notations.
LetX and Y be two Hausdorff topological spaces. We denote byC(X, Y ) the set of all continuous mappings fromXtoY. LetLBbe a subset ofX, we denote by clB
the closure ofB. IfMis a nonempty subset ofX, we denote byD(M),(M), and 2X the families of all nonempty finite subsets, all nonempty connected subsets, and all subsets ofM, respectively. We denoteR=(−∞,+∞).
Definition1.1(see [2]). LetEbe a linear ordered space. We callE a completely
dense linear space, if each subset ofEhas a least upper bound and for anyZ1, Z2∈E
withZ1< Z2, there existsz∈Esuch thatZ1< z < Z2.
Definition1.2. LetX be a topological space,X is called a generalized interval
Γ(ᐄ1,ᐄ2)=Γ(ᐄ2,ᐄ1), andΓ(ᐄ1,ᐄ2)is called a generalized interval associated with ᐄ1,ᐄ2.
Definition1.3. LetXbe a generalized interval space,Y a topological space, and
Ea completely dense linear space. A subsetBofXis calledT-convex set, if for any ᐄ1,ᐄ2∈B, we haveΓ(ᐄ1,ᐄ2)⊂B. A mappingf:X→E is calledT-quasi-concave, if
for anyᐄ1,ᐄ2∈X, there existᐅ1,ᐅ2∈Γ(ᐄ1,ᐄ2)such thatf (ᐅi)≥f (ᐄi),i=1,2, and for anyz∈E, the set{x∈X:f (x)≥z}isT-convex. We callF :X→2Y aT-KKM mapping, if for anyᐄ1,ᐄ2∈Xwe haveF (x)⊂F (ᐄ1)∪F (ᐄ2)for allx∈Γ(ᐄ1,ᐄ2), and
there existᐅ1,ᐅ2∈Γ(ᐄ1,ᐄ2)such thatF (ᐅi)⊂F (ᐄi),i=1,2.
Remark1.4. For Definitions1.2and1.3it is easy to see that both interval space and
H-space are the special cases of generalized interval space, and so as all the strong interval space [2], convex space, contractible space, and topological vector space are the special cases of generalized interval space. ForH-space, theH-convex set isT -convex. For topological linear space, a convex subset isT-convex, a quasi-concave function isT-quasi-concave.
Remark1.5. It is obvious that a nonemptyT-convex subset in a generalized interval
space is also a generalized interval space.
Definition 1.6 (see [14]). Let X and Y be two topological spaces. A mapping
F:X→2Y is called transfer closed valued, if for eachx∈X,y∈F (x)implies that there exists a pointᐄ•∈Xsuch that y∈clF (ᐄ•).
Lemma 1.7. Let X be a generalized interval space, E a completely dense linear
ordered space, and f :X→E a mapping. Then f is T-quasi-concave if and only if for anyᐄ1,ᐄ2∈X, there existᐅ1,ᐅ2∈Γ(ᐄ1,ᐄ2)such thatf (ᐅi)≥f (ᐄi),i=1,2, and f (x)≥min{f (ᐄ1), f (ᐄ2)}for allx∈Γ(ᐄ1,ᐄ2).
Proof. The proof follows fromDefinition 1.3immediately.
Lemma1.8. LetXbe a generalized interval space,Y a topological spaces, andE a
completely dense linear ordered space. Letα∈Eandϕ:X×X→EbeT-quasi-concave inX. If we define a mappingF:X→2Y byF (x)= {y∈Y:ϕ(x, y)≤α}for allx∈X, thenF is aT-KKM mapping.
Proof. Sinceϕ is T-quasi-concave in X, for any given ᐄ1,ᐄ2 ∈X and for any
x∈Γ(ᐄ1,ᐄ2)we haveϕ(x, y)≥min{ϕ(ᐄ1, y), ϕ(ᐄ2, y)}for anyy∈Y, and there existᐄ1•,ᐄ2•∈Γ(ᐄ1,ᐄ2)such thatϕ(ᐄi•, y)≥ϕ(ᐄi, y),i=1,2 for ally∈Y. Hence F (ᐄi•)⊂F (ᐄi),i=1,2, for allx∈Γ(ᐄ1,ᐄ2), we haveF (x)⊂2i=1F (ᐄi). This implies thatF is aT-KKM mapping. LetX, Y be two topological spaces, F :X→2Y. In the sequel we denote cl{F (x)} =clF (x).
2. Some nonempty intersection theorems
Theorem 2.1. Let X be a generalized interval space,Y a topological space, and
F :X →2Y an upper semi-continuous T-KKM mapping. Suppose that the following conditions are satisfied:
(2) for anyA∈D(X),x∈AF (x)∈(Y ),
then
(1) {F (x):x∈X}has the finite intersection property;
(2) if there exists anA0∈D(X)such thatx∈AclF (x)is compact, then
x∈A
clF (x)≠∅. (2.1)
Proof. LettingΩ= {F (x):x∈X}, and by using induction we prove thatΩhas
the finite intersection property.
By condition (1), for anyF (x) inΩ, x∈X is nonempty. Suppose that for anyn
elements inΩ,n≥2, their intersection is nonempty, next we prove that for anyn+1 elements inΩ, their intersection is also nonempty. Suppose the contrary, then there existᐄ1,ᐄ2, . . . ,ᐄn+1∈Xsuch thatni=1+1F (ᐄi)= ∅. LettingH=
n+1
i=1F (ᐄi), then
2
i=1
H∩Fᐄi
= ∅. (2.2)
By condition (2) and the assumption of induction, for anyx∈X
H∩F (x)is nonempty and connected. (2.3)
SinceFis aT-KKM mapping,F (x)⊂2i=1F (ᐄi)for allx∈Γ(ᐄ1,ᐄ2)and there exist ᐄ1•,ᐄ2•∈Γ(ᐄ1,ᐄ2)such thatF (ᐄi•)⊂F (ᐄi),i=1,2. Hence,
H∩F (x)⊂ 2
i=1
H∩F (ᐄi), ∀x∈Γ
ᐄ1,ᐄ2
, (2.4)
lettingDj= {x∈Γ(ᐄ1,ᐄ2):H∩F (x)⊂H∩F (ᐄi)}, j=1,2, thenᐄi•∈Di, i=1,2. It follows from (2.2), (2.3), (2.4), and condition (1) that D1∪D2=Γ(ᐄ1,ᐄ2). Since Γ(ᐄ1,ᐄ2) is connected, we know that either clD1∩D2 or clD2∩D1 is nonempty. Without loss of generality, we can assume clD2∩D1≠ ∅. Hence there exists an
x0∈clD2∩D1. By (2.2), we have(H∩F (ᐄ0))∩(H∩F (ᐄ2))= ∅, and it is easy to prove that there exists an open setUcontainingF (ᐄ0)such that(H∩U )∩(H∩F (ᐄ2))= ∅. In fact, ifF is open valued, it is sufficient to take U=F (ᐄ0); ifF is closed valued, takingU=Y\(H∩F (ᐄ2)), thenUis an open set inY and
(H∩U )∩H∩Fᐄ2= ∅, U⊃H∩Fᐄ0≠∅. (2.5)
Moreover, for anyy∈U, theny∈H∩F (ᐄ2)andy∈H∩F (ᐄ0), and soy∈F (ᐄ0). HenceF (ᐄ0)⊂U. This means thatUsatisfies the desired condition.
By the upper semi-continuity ofF, there exists an open neighborhoodVofᐄ0such
that for anyx∈V,F (x)⊂U. Sinceᐄ0∈clD2,V∩D2≠∅. Takingᐄ•∈V∩D2, we
haveH∩F (ᐄ•)⊂H∩F (ᐄ2),H∩F (ᐄ•)⊂H∩U. Hence we have(H∩F (ᐄ2))(H∩U )⊃ H∩F (ᐄ•)≠∅. This contradicts (2.5), thereforeΩhas the finite intersection property.
In addition, if there exists anA0∈D(M)such thatx∈A
0clF (x)is compact. Letting K=x∈A0clF (x), it follows from the finite intersection property ofΩandA0being
finite that{clF (x)∩K:x∈X}is a family of closed set in the compact setKand has the finite intersection property, thereforex∈X(clF (x)∩K)=
Theorem 2.2. Let X be a generalized interval space, Y a topological space,
F:X→2Y a mapping with nonempty values and satisfy the following conditions: (1) F is transfer closed valued;
(2) clF:X→2Y is an upper semi-continuousT-KKM mapping; (3) for eachA∈D(X),x∈XclF (x)∈(Y );
(4) there exists anA0∈D(X)such thatx∈A0clF (x)is compact, then
x∈XF (x)≠∅.
Proof. First we prove x∈XF (x) = x∈XclF (x). In fact, since x∈XF (x) ⊂
x∈XclF (x), it is sufficient to prove that
x∈XF (x)⊃
x∈XclF (x). Suppose the con-trary, then there existsY0∈x∈XclF (x)such thaty0∈
x∈XF (x). Hence there ex-ists anx0∈X such thaty0∈F (x). By condition (1), there existsx∈X such that
y0∈clF (x). This contradicts the choicey0. Hence the desired conclusion is proved.
It is obvious that clF : X → 2Y satisfies all the conditions in Theorem 2.1. By Theorem 2.1, we know thatx∈XF (x)=
x∈XclF (x)≠∅.
Theorem 2.3. Let X be a generalized interval space, Y a topological space,S ∈
C(X, Y )a given mapping, andF:X→2Ya mapping satisfying the following conditions: (1) F is transfer closed valued;
(2) clF:X→2Y is an upper semi-continuousT-KKM mapping;
(3) there exists a compact setLinY and a subsetKinXsuch that for anyA∈D(X) there exists a compactT-convex subsetD⊂XwithKA⊂Dsuch that
(a) for anyx∈D,clF (x)∩s(D)≠∅;
(b) for any finite set{x1, x2, . . . , xn} ⊂D,ni=1(clF (xi)∩s(D))∈(Y ); (c) x∈D(clF (x)∩s(D))⊂L,
thenx∈XF (x)≠∅.
Proof. By condition (1),x∈XF (x)=x∈XclF (x). Hence in order to prove the
conclusion of the theorem, it suffices to prove thatx∈XclF (x)≠∅. SinceLis com-pact, for the purpose, it suffices to prove that{clF (x)∩L: x∈X} has the finite intersection property. For anyA∈ D(X), let D be the set satisfying condition (3). Hence we have
x∈A
clF (x)∩L⊃ x∈D
clF (x)∩s(D). (2.6)
LettingY0=s(D), by the continuity ofsand compactness ofD, we know thatY0is a compact subset ofY.
Next we prove that x∈D(clF (x)∩Y0)≠∅. In fact, letting G(x)=clF (x)∩Y0, x∈D, by condition (3)(a), for anyx∈X,G(x)is nonempty compact. SinceDisT -convex,Dis a generalized interval space. It is easy to prove thatG:D→2Y satisfies all conditions inTheorem 2.1. ByTheorem 2.1,{G(x):x∈X}has the finite intersection property. Therefore, we havex∈DG(x)≠∅, that is,
x∈D(clF (x)∩Y0)≠∅. By (2.6),
x∈A(clF (x)∩L)≠∅. This implies that{clF (x)∩L:x∈X}has the finite intersection property.
Theorem2.4. LetXbe a generalized interval space,Ya compact topological space,
(1) Gis transfer closed valued;
(2) clG:X→2Y is aT-KKM mapping; (3) for anyA∈D(X),x∈AclG(x)∈(Y ); (4) clGis closed graph,
thenx∈XG(x)≠∅.
Proof. Since clGis a closed graph andYis compact, by using Fan [3, Lemma 2] we
know that clGis upper semi-continuous. The conclusion follows fromTheorem 2.2 immediately.
3. Applications to minimax problems. As applications, we use the results obtained inSection 2to give some minimax theorems in generalized interval spaces.
Theorem3.1. LetXbe a generalized interval space,Ya compact topological space,
Ea completely dense linear ordered space, andϕ:X×X→Ea lower semi-continuous function satisfying the following conditions:
(1) ϕ(x, y)isT-quasi-concave inx;
(2) for anyA∈D(X)and for anyα∈E,x∈A{y∈Y :ϕ(x, y)≤α} ∈(Y ),
thenα•=supx∈Xinfy∈Yϕ(x, y)=infy∈Ysupx∈Xϕ(x, y)=α•.
Proof. It is obvious thatα•≤α•.
Now we prove thatα•≥α•. Without loss of generality, we can assume thatα• is
not the greatest element inE. For anyx∈Xand for any givenα > α•, let
G(x, α)=y∈Y:ϕ(x, y)≤α. (3.1)
Next we prove that{G(x, α):x∈X, α > α•}has the finite intersection property. In
fact, for anynpointsx1, . . . , xn∈Xand anynelementsα1, . . . , αn∈Ewithαi> α•, i=1, . . . , n, sinceEis a linear ordered space, we can assume thatαn≥ ··· ≥α1> α•.
By the density ofE, there exists an ¯α∈Esuch thatα1>α > α¯ •. Letting
F (x)=G(x,α),¯ x∈X, (3.2)
by the lower semi-continuity ofϕand the choice of ¯α∈E, we know thatF:X→2Y is nonempty closed valued. This implies thatFsatisfies condition (1) inTheorem 2.1. From condition (2),F satisfies condition (2) inTheorem 2.1. In view of condition (1) andLemma 1.7, we know thatFis aT-KKM mapping. In addition, sinceϕ:X×Y→E
is lower semi-continuous,F is closed graph. By Fan [3, Lemma 2],F is upper semi-continuous. ThereforeF satisfies all the conditions inTheorem 2.1. ByTheorem 2.1,
{F (x): x ∈X} has the finite intersection property, and so ni=1F (xi)≠ ∅. Since n
i=1F (xi) ⊂ n
i=1G(xi, αi), {G(x, α) : x ∈ X, α > α•} has the finite intersection
property.
On the other hand, for anyx∈X and for anyα > α•, G(x, α)is closed. SinceY
is compact,x∈X,α>α•G(x, α)≠∅. Hence, there exists ¯y G(x, α)for allx∈X and for allα > α•, that is,ϕ(x,y)¯ ≤αfor all x∈Xand for allα > α•. Hence, we have infy∈Ysupx∈Xϕ(x, y)≤α, for allα > α•.
Remark3.2. Theorem 3.1not only contains Theorem 1 in Shacho [13] as its special
case but also relaxes its continuous condition.
Theorem 3.3. LetX be a generalized interval space, Y a topological space, E a
completely dense linear ordered space, andϕ,φ:X×X→Etwo functions satisfying the following conditions:
(1) ϕ(x, y)≤φ(x, y)for all(x, y)∈X×Y;
(2) ϕ(x, y)isT-quasi-concave inx, and it is upper semi-continuous on any gener-alized interval ofX;
(3) ϕ(x, y)is lower semi-continuous inyand for anyA∈D(X), the setx∈A{y∈
Y:ϕ(x, y) < α}is connected for allα∈E;
(4) there exists anA0∈D(X)and az0>supx∈Xinfy∈Yφ(x, y)such thatx∈A
0{y∈ Y:ϕ(x, y)≤z0}is compact,
theninfy∈Ysupx∈Xϕ(x, y)≤supx∈Xinfy∈Yφ(x, y).
Proof. Lettingz•=supx∈Xinfy∈Yφ(x, y), by the completeness ofE, we know
thatz•exists. From condition (4),z•is not a largest element inE.
For anyx∈X and for anyz > z•, letF (x, z)= {y∈Y :ϕ(x, y)≤z}. It follows from conditions (1) and (3) thatF (x, z)is a nonempty closed set. By condition (4),
x∈AF (x, z)is compact. DenoteΩ= {F (x, z):x∈X, z∈Z}. Next we prove thatΩ has the finite intersection property. In fact, we have proved that each element inΩ is nonempty. Suppose for anynelements inΩtheir intersection is nonempty. Now we prove that anyn+1 elements inΩtheir intersection is nonempty. Suppose the contrary, then there existᐄ1, . . . ,ᐄn,ᐄn+1inX andz1, . . . , zn, zn+1inE withzi> z•, i= 1, 2, . . . , n+1 such that ni=+11F (xi, zi)= ∅. Without loss of generality, we can assume thatz•< z1≤ ··· ≤zn≤zn+1. By the density ofE, there exists a ¯z∈Esuch
thatz1>z > z¯ •. Letting
T (x, z)=y∈Y:ϕ(x, y) < z, ∀x∈X,∀z > z•;
T (x)=Tx, z1, ∀x∈X;H= n+1
i=3 Txi
. (3.3)
By the assumption of the induction, for anyx∈X,
H∩T (x)⊃ n+1
i=3 Fxi; ¯z
∩Fx; ¯z
≠∅. (3.4)
It follows from condition (2) and (3), forx1, x2∈X there exist x1, x2 ∈Γ(x1, x2)
such thatT (xi)⊂T (xi), i=1,2 andT (x)⊂
2
i=1T (xi)for allX∈Γ(x1, x2). Hence we have
H∩T (x)⊂ 2
i=1
H∩Txi, ∀x∈Γx1, x2. (3.5)
By condition (3),
2
i=1
clH∩Txi
⊂ n+1
i=1 Fxi, zi
≠∅. (3.6)
H∩T (x)⊂H∩T (xi)},i=1,2, from (3.5) andT (xi)⊂T (xi),i=1,2, we know that xi∈Ii,i=1,2 andI1I2=Γ(x1, x2). SinceΓ(x1, x2)is connected, eitherI1∩clI2or clI1∩I2 is nonempty. Without loss of generality, we can assume thatI1∪clI2≠∅. Takingx0∈I1∩clI2, then there exists a net{xγ}γ∈θ ⊂I2 such that xγ →x0 and H∩T (x1)⊃H∩T (x0)≠∅. Takingy0∈H∩T (x0), we have
ϕ(x0, y0) < z1. (3.7)
It follows from (3.6) thaty0∈H∩T (x2). Hence,y0∈H∩T (xγ)for allγ∈θ, that is, ϕ(xγ, y0)≥z1for allγ∈θ.
By condition (2) andxγ→x0, we haveϕ(x0, y0)≥z1. This contradicts (3.7). There-foreΩhas finite intersection property.
By condition (4), K = x∈A0F (x, z0) is compact and {F (x, z)∩K : x ∈ X, z∈z•}has the finite intersection property, therefore we have x∈X,z∈z•F (x, z)=
x∈X, z∈z•(F (x, z)∩K)≠∅, and so infy∈Ysupx∈Xϕ(x, y)≤z•. This completes the proof.
Remark3.4. Theorem 3.3improves and extends Theorem 1 in Lin and Quan [10].
Corollary3.5. LetX be a generalized interval space,Y a topological space,E a
completely dense linear ordered space, and f :X×X →E a function satisfying the following conditions:
(1) f (x, y)isT-quasi-concave inx and it is upper semi-continuous on any gener-alized interval ofX;
(2) f (x, y)is lower semi-continuous inyand for anyA∈D(X), the set
x∈A
y∈Y:f (x, y) < α∈ (Y ), ∀α∈E; (3.8)
(3) there exist anA0∈D(X)and az0>supx∈Xinfy∈Yf (x, y)such that
x∈A0
y∈Y:f (x, y)≤z0∈ (Y ), (3.9)
theninfy∈Ysupx∈Xf (x, y)=supx∈Xinfy∈Yf (x, y).
Proof. It is obvious that infy∈Ysupx∈Xf (x, y)≥supx∈Xinfy∈Yf (x, y). On the other
hand, byTheorem 3.3 we know that infy∈Ysupx∈Xf (x, y)≤supx∈Xinfy∈Yf (x, y). This completes the proof.
Remark3.6. The corresponding results in Kindler and Trost [8], Tuy [7], Cheng
and Lin [2], Brezis et al. [1], Komornik [9], Geraghty and Lin [4], Stacho [13], Sion [12], Wu [15], and Hartung [6] all are the special cases ofCorollary 3.5.
Corollary3.7. LetXbe a compact generalized interval space,Ea completely dense
linear ordered space, andϕ,φ:X×X→E,θ∈E. If the following conditions are satisfied:
(1) for any(x, y)∈X×X,ϕ(x, y)≤φ(x, y);
(3) ϕ(x, y)is lower semi-continuous iny, and for anyA∈D(X)and anyα∈E,
the set
x∈A
y∈X:f (x, y) < α∈ (X), (3.10)
then
(1) there existsx¯∈Xsuch thatφ(x,¯x) > θ, or¯
(2) there existsy¯∈Xsuch thatϕ(x,y)¯ ≤θfor allx∈X.
Proof. Suppose that (1) is not true, then for anyx∈X we haveφ(x, x)≤θ. By
condition (2), infy∈Yφ(x, y)is upper semi-continuous inx. Moreover, byTheorem 3.3 and the compactness ofXthere exists an ¯x∈Xsuch that
inf y∈X
sup x∈X
ϕ(x, y)≤sup x∈X
inf y∈X
φ(x, y)=inf y∈Xφ
¯
x, y≤φx,¯ x¯≤θ. (3.11)
SinceXis compact, by condition (3) there exists a ¯y∈Xsuch that
ϕ(x,y)¯ ≤sup x∈X
ϕ(x,y)¯ =inf y∈X
sup x∈X
ϕ(x, y), ∀x∈X. (3.12)
Corollary3.8. LetXbe a compact generalized interval space,Y a compact
topo-logical space,Ea completely dense linear ordered space, andf:X×Y→Ea mapping satisfying the following conditions:
(1) xf (x, y)isT-quasi-concave and upper semi-continuous;
(2) yf (x, y)is lower semi-continuous;
(3) for anyα∈E, anyA∈D(X),x∈A{y∈Y:f (x, y) < α} ∈(Y ),
then there exists a saddle point(x,¯y)¯ ∈X×Y such thatf (x,y)¯ ≤f (x,¯y)¯ ≤f (x, y)¯
for allx∈Xand for ally∈Y.
Proof. If supx∈Xinfy∈Yf (x, y)is a greatest element inE, then it is obvious that
inf y∈Y
sup x∈X
f (x, y)=sup x∈X
inf y∈Y
f (x, y). (3.13)
If supx∈Xinfy∈Yf (x, y)is not a greatest element inE, thenf satisfies all the condi-tions inCorollary 3.5obviously. ByCorollary 3.5, we also have
inf y∈Y
sup x∈X
f (x, y)=sup x∈X
inf y∈Y
f (x, y). (3.14)
In view of the compactness ofXandY, there exist ¯x∈Xand ¯y∈Y such that
sup x∈X
inf y∈Y
f (x, y)=inf
y∈Yf (x, y)¯ ≤f (x, y),¯ ∀y∈Y ,
inf y∈Y
sup x∈X
f (x, y)=sup x∈X
f (x,y)¯ ≥f (x,y),¯ ∀x∈X. (3.15)
Therefore, we havef (x, y)¯ ≥f (x,y)¯ for ally∈Y and for allx∈X.
Takingx=x¯in the preceding inequality, we havef (x, y)¯ ≥f (x,¯y)¯ for ally∈Y. Takingy=y¯ in the preceding inequality, we havef (x,¯ y)¯ ≥f (x,y)¯ for allx∈X. Therefore, we have
Corollary3.9(von Neumann). LetC,Fbe two Hausdorff topological vector spaces,
X⊂C,Y⊂F two compact convex subsets, andφ:X×Y →Ra function satisfying the following conditions:
(1) for anyx∈X,φ(x, y)is an upper semi-continuous concave function iny;
(2) for anyy∈Y,φ(x, y)is a lower semi-continuous convex function inx, then there exists a saddle point(x,¯y)¯ ∈X×Y ofϕ.
Proof. For anyx1, x2∈Xand for ally1, y2∈Y, letting
Γx1, x2=cox1, x2, Γy1, y2=coy1, y2, (3.17)
then ϕ satisfies all the conditions in Corollary 3.8. Therefore, the conclusion of Corollary 3.9follows fromCorollary 3.8immediately.
Corollary3.10. LetXbe a compact generalized interval space,f:X×X→R, and
h:X→Rtwo functions satisfying the following conditions:
(1) for anyx∈X,f (x, x)≤0;
(2) xf (x, y)is upper semi-continuous andyf (x, y)is lower semi-continuous; his upper semi-continuous;
(3) f (x, y)+h(x)isT-quasi-concave inx;
(4) for anyA∈D(X)and for anyα∈Rthe setx∈A{y∈Y:f (x, y)−h(x) < α} ∈ (X),
then there exists ay¯∈Xsuch thatf (x,y)¯ ≤h(y)¯ −h(x)for allx∈X.
Proof. Lettingg(x, y)=f (x, y)−h(y)+h(y), it is easy to prove thatgsatisfies
all the conditions inCorollary 3.8. Hence, there existx•, y•∈Xsuch thatg(x, y•)≤ g(x•, y) for all x, y ∈ X. Hence g(x, y•) ≤ g(x•, x•) ≤ 0 for all x ∈ X, that is,
f (x, y•)≤h(y•)−h(x)for allx∈X.
Theorem3.11. LetX be a generalized interval space,Y a topological space,E a
completely dense linear ordered space, andφ, ϕ:X×X→Etwo mappings satisfying the following conditions:
(1) for any(x, y)∈X×Y,ϕ(x, y)≤φ(x, y);
(2) x ϕ(x, y)is T-quasi-concave and upper semi-continuous,y ϕ(x, y) is lower semi-continuous;
(3) φ(x, y)is upper semi-continuous inx;
(4) there exist a compact K of X and a nonempty subsetY0 of Y satisfying the following condition: for anyB∈D(Y )there exists a compact setM containing
Y0B such that for anyA∈ D(X)and for anyα∈E the setx∈A{y∈M: ϕ(x, y) < α} ∈(Y ), and
sup x∈K
inf y∈Y0
φ(x, y)≤sup x∈X
inf y∈Y
φ(x, y), (3.18)
theninfy∈Ysupx∈Xϕ(x, y)≤supx∈Xinfy∈Yφ(x, y).
Proof. Lettingz•=infy∈Ysupx∈Xϕ(x, y),z•=supx∈Xinfy∈Y∅(x, y). From the
completeness ofE, we know thatz•andz•both exist. Ifz•> z•, then it follows from
the density ofEthat there exists a ¯z∈Esuch thatz•>z > z¯ •. Let
By condition (3),F (y)is closed for ally∈Y. From condition (4), for anyy1, . . . , yn∈Y there exists a compact setMcontainingY0{y1, . . . , yn}. It is easy to see thatφ, ϕ: X×M→Esatisfy all the conditions inTheorem 3.3. ByTheorem 3.3, we know
inf y∈M
sup x∈X
ϕ(x, y)≤sup x∈X
inf y∈M
φ(x, y). (3.20)
On the other hand, since infy∈Msupx∈Xϕ(x, y)≥infy∈Ysupx∈Xϕ(x, y)=z•>z¯, supx∈Xinfy∈Mφ(x, y) >z¯, and so
y∈MF (y)≠∅. Hence we have
y∈Y0F (y)⊃
y∈MF (y)≠∅. By condition (4), we have supx∈Kinfy∈Y0φ(x, y)≤z• <z¯. Hence,
for allx∈X\Kwe have infy∈Y0φ(x, y)≤z•<z¯, and so there exists ay0∈Y0such
thatφ(x, y0) <z¯, that is,x∈F (y0). Hencex∈y∈y0F (y)=D. This implies that D⊂K. SinceKis compact andDis closed, thenDis a compact set.
Summing up the above arguments, we have proved that{F (y)∩D:y∈Y}is a family of compact sets inDhaving the finite intersection property. Hence, we have
y∈Y
F (y)= y∈Y
F (y)D≠∅. (3.21)
Thereforez•=supx∈Xinfy∈Yφ(x, y)≥z¯. This contradicts the choice of ¯z. Therefore z•≤z•. This completes the proof.
Remark3.12. Theorem 3.11improves the corresponding results in Ha [5] and Lin
and Quan [10]. FromTheorem 3.11we can obtain the following results.
Corollary3.13. LetX be a generalized interval space,Y a topological space,E
a completely dense linear ordered space, andϕ:X×Y→Ea mapping satisfying the following conditions:
(1) ϕ(x, y)isT-quasi-concave and upper semi-continuous inxand it is lower semi-continuous iny;
(2) there exist a compact subsetKinXand a nonempty subsetY0inY satisfying the following condition: for anyB∈D(Y )there exists a compact setM containing
Y0Bsuch that for anyA∈D(X)and for anyα∈E,x∈A{y∈M:ϕ(x, y) < α} ∈(Y )and
sup x∈K
inf y∈Y0
ϕ(x, y)≤sup x∈X
inf y∈Y
ϕ(x, y), (3.22)
theninfy∈Ysupx∈Xϕ(x, y)=supx∈Xinfy∈Yϕ(x, y).
Theorem3.14. LetX be a generalized interval space,Y a topological space,E a
completely dense linear ordered space,ϕ:X×Y→Elower semi-continuous, andφ:X× Y→Eupper semi-continuous inx. Suppose that the following conditions are satisfied:
(1) ϕ(x, y)isT-quasi-concave inxandϕ(x, y)≤φ(x, y)for all(x, y)∈X×Y;
(2) there exist a compact subsetKinXand a nonempty subsetY0inY satisfying the following condition: for anyB∈D(Y )there exists a compact setM containing
Y0Bsuch that for anyA∈D(X)and for allα∈E,x∈A{y∈M:ϕ(x, y)≤ α} ∈(Y )and
sup x∈K
inf y∈Y0
φ(x, y)≤sup x∈X
inf y∈Y
theninfy∈Ysupx∈Xϕ(x, y)≤supx∈Xinfy∈Yφ(x, y).
Proof. Lettingz•=infy∈Ysupx∈X, ϕ(x, y),z•=supx∈Xinfy∈Yφ(x, y), ifz•> z•,
then by the density ofE, there exists a ¯z∈Esuch thatz•>z > z¯ •. DefiningF (y)= {x∈X:φ(x, y)≥z¯},y∈Y, sincexφ(x, y)is upper semi-continuous, we know thatF (y)is closed for ally∈Y. DefiningD=y∈Y0F (y), thenDis closed.
On the other hand, for any finite set{y1, . . . , yn} ⊂Y, it follows from condition (2) that there exists a compact subsetM which containsY0{y1, . . . , yn}. Obviously, we can prove thatϕ:X×M→Esatisfies all the conditions inTheorem 3.1, we have
inf y∈M
sup x∈X
ϕ(x, y)=sup x∈X
inf y∈M
ϕ(x, y), (3.24)
and so we have
sup x∈X
inf y∈M
φ(x, y)≥sup x∈X
inf y∈M
ϕ(x, y)= inf y∈M
sup x∈X
ϕ(x, y)
≥inf y∈Y
sup x∈X
ϕ(x, y)=z•>¯z. (3.25)
By the same method as inTheorem 3.11, we can provez•≥z¯. This contradicts the
choice of ¯z. Therefore we havez•≤z•. This completes the proof.
FromTheorem 3.1we can obtain the following theorem.
Theorem3.15. LetXbe a compact generalized interval space,Ea completely dense
linear ordered space, andϕ:X×X→Ea lower semi-continuous function satisfying the following conditions:
(1) xϕ(x, y)isT-quasi-concave;
(2) for anyA∈D(X)and for anyα∈E,x∈A{y∈X:ϕ(x, y)≤α} ∈(X),
then
(1) there exists anx¯∈Xsuch thatϕ(x,¯ x) > θ; or¯
(2) there exists any¯∈Xsuch thatϕ(x,y)¯ ≤θfor allx∈X.
Proof. Suppose the conclusion (1) is not true, then for anyx∈Xwe haveϕ(x, x) ≤θ. ByTheorem 3.1the following equation holds:
sup x∈X
inf y∈X
ϕ(x, y)=inf y∈X
sup x∈X
ϕ(x, y). (3.26)
Since for anyx∈X, infy∈Xϕ(x, y)≤ϕ(x, x)≤θ, we have supx∈Xinfy∈Xϕ(x, y)≤ θ, and so we have infy∈Xsupx∈Xϕ(x, y)≤θ. By the compactness ofXand the lower semi-continuity of supx∈Xϕ(x, y)iny, there exists a ¯y∈Xsuch that
sup x∈X
ϕ(x,y)¯ = inf y∈X
sup x∈X
ϕ(x, y)≤θ, (3.27)
that is,ϕ(x,y)¯ ≤θfor allx∈X. Therefore the conclusion (2) is true.
4. Application to coincidence problem and fixed point problem
Theorem4.1. Let X be a compact generalized interval space,Y a topological space,
following conditions are satisfied:
(1) for anyx∈X,s−1F (x)≠∅andF−1(s(x))is open;
(2) for anyA∈D(X),X\(x∈AF−1(s(x)))∈(X); (3) for anyx1, x2∈Xand for anyy∈2
i=1F−1(s(xi)),s(Γ(x1, x2))⊂F (y)there existx1, x2∈Γ(x1, x2)such thatF−1(s(xi))⊂F−1(s(xi)),i=1,2,
then there exists anx¯∈Xsuch thats(x)¯ ∈F (x).¯
Proof. For any(x, z)∈X×X, let
f (x, z)=
0, s(x)∈F (x),
1, s(x)∈F (x). (4.1)
If the conclusion is not true, then for anyx∈Xwe havef (x, x)=0. Since for anyx∈Xand for anyr∈R,
z∈X:f (x, z)≤r=
∅, r <0, F•s(x), 0≤r <1,
X, r≥1,
(4.2)
whereF•(x)=X|F−1(x), it follows from conditions (1) and (2) thatf (x, z)is lower semi-continuous inzand for anyA∈D(X)and for anyr∈R,x∈A{z∈X:f (x, z) < r} ∈(X). On the other hand, since for anyz∈Xand for anyr∈R,
x∈X:f (x, z)≥r=
X, r≤0,
s−1F (z), 0< r≤1, ∅, r >1,
(4.3)
andF is a mapping with closed values,f (x, z)is upper semi-continuous inx. LettingG(x)=F•(s(x)), thenGis a mapping fromXinto 2Xand for any(x, z)∈ X×X,
f (x, z)=
0, z∈G(x),
1, z∈G(x). (4.4)
Next we prove thatGis aT-KKM mapping. In fact, from condition (3) for anyx1, x2∈ X, we haves(Γ(x1, x2))⊂F (z)for allz∈2
i=1F−1(s(xi)). Hence for anyx∈Γ(x1, x2), and for anyz∈2i=1F−1(s(xi)),s(x)∈F (z).
Therefore, for all x ∈ Γ(x1, x2) and for any z ∈ 2
i=1F−1(s(xi)), we have z ∈ F−1(s(x)). This implies that2
i=1F−1(s(xi))⊂F−1(s(x))for allx∈Γ(x1, x2). There-fore,G(x)⊂2i=1G(xi)for allx∈Γ(x1, x2). Again by condition (3) there existx1, x2∈ Γ(x1, x2)such thatF−1(s(xi))⊂F−1(s(xi)),i=1,2. ThusG(xi)⊂G(xi),i=1,2. This shows thatGis aT-KKM mapping. ByLemma 1.8it is easy to prove thatf (x, y)is aT-quasi-concave mapping inx. SinceXis compact,f satisfies all the conditions in Corollary 3.7. ByCorollary 3.7there exists a ¯z∈Xsuch thatf (x,z)¯ ≤0 for allx∈X, and sof (x,z)¯ =0 for allx∈X, that is,s(x)∈F (z)¯ for allx∈X. Hencex∈s−1F (z)¯
for allx∈X, and sos−1F (z)¯ = ∅.
FromTheorem 4.1we obtain the following fixed point theorem.
Corollary4.2. LetX be a generalized interval space,C be a compactT-convex
subset ofX, andF :C→2C be a set-valued mapping with nonempty closed values. If the following conditions are satisfied:
(1) for anyx∈C,F−1(x)is open inC;
(2) for anyA∈D(C),C\(x∈AF−1(x))∈(C);
(3) for anyx1, x2∈X,Γ(x1, x2)⊂F (y)for ally∈2i=1F−1(xi), and there exist x1, x2∈Γ(x1, x2)such thatF−1(xi)⊂F−1(xi),i=1,2,
then there exists anx¯∈Csuch thatx¯∈F (x).¯
Theorem4.3. LetXbe a generalized interval space,Ya topological space,F:X→2Y
ands∈C(X, Y )two mappings satisfying the following conditions:
(1) for anyx∈X,F−1(s(x))is nonempty closed andX\s−1(F (x))isT-convex;
(2) for anyA∈D(X),x∈As−1(F (x))∈(X);
(3) F−1:Y→2Xis upper semi-continuous and for anyx1, x2∈X, there existx 1, x2∈ Γ(x1, x2)such thatF−1(s(xi))⊂F−1(s(xi)),i=1,2;
(4) there exists anA0∈D(X)such that x∈A
0F−1(s(x))is compact, then there exists anx¯∈Xsuch thats(x)¯ ∈F (x).¯
Proof. LettingG(x)=F−1(s(x)), thenG:X→2X is a nonempty mapping and
closed valued andGsatisfies condition (2) inTheorem 2.1.
On the other hand, for any x∈X, letW be an open set containingG(x). Since
F−1 is upper semi-continuous, there exists an open neighborhood V of s(x) such that for anyy ∈V, we have F−1(y)⊂W. Sinces ∈C(x, y), there exists an open
neighborhoodUofxsuch that for anyz∈Uwe haves(z)∈V, and so forz∈U, we haveG(z)=F−1(s(z))⊂W. This showsGis an upper semi-continuous mapping.
Next we prove that G:X→2X is aT-KKM mapping. In fact, for anyx1, x2∈X and for any x∈Γ(x1, x2), if z∈2i=1G(xi), then z∈F−1(s(xi)), i=1,2, that is, s(xi)∈F (z),i=1,2, and so{x1, x2} ⊂X\s−1F (z). By the assumption thatX\s−1F (z) isT-convex, henceΓ(x1, x2)⊂X\s−1F (z), and soz∈F−1(s(x))for allx∈Γ(x1, x2).
This implies that
G(x)=F−1s(x)⊂ 2
i=1 Gxi
. (4.5)
Again by condition (3), there existx1, x2∈Γ(x1, x2)such that G(xi)⊂G(xi),i= 1,2. This means thatGis aT-KKM mapping.
In view of condition (4), we know thatGsatisfies all the conditions inTheorem 2.1. ByTheorem 2.1x∈XG(x)≠∅, and so there exists an ¯x∈Xsuch that ¯x∈G(x)for allx∈X. Hence ¯x∈G(x)¯ , that is, ¯x∈F−1(s(x))¯ , and sos(x)¯ ∈F (x)¯ .
Theorem4.4. LetK be a generalized interval space,F :X→2X a mapping with
nonempty values. IfXis normal and the following conditions are satisfied:
(1) F is transfer closed valued;
(2) Fis almost upper semi-continuous, that is, for anyx∈Xand for any open setV containingF (x), there exists an open neighborhoodUofxsuch thatF (y)⊂clV for ally∈U;
(4) there exists anA0∈D(X)such thatx∈A0clF (x)is compact;
(5) for anyx∈X,X\F−1(x)=F•(x)isT-convex, thenF has a fixed point inX.
Proof. For anyx∈X, ifWis any open set containing clF (x), by the normality of
X, there exists an open setV containing clF (x)such that clV ⊂W. It follows form the almost upper semi-continuity ofF that there exists an open neighborhoodU of
xsuch thatF (y)⊂clV for allY∈U. Hence clF (y)⊂clV⊂W. This means thatF is upper semi-continuous.
Next we prove that F is a T-KKM mapping. In fact, for any x1, x2∈ X if X ∈
2
i=1F (xi), thenx1, x2 ∈F−1(y). Since F•(y) is T-convex, Γ(x1, x2)⊂F•(y), and so for anyx∈Γ(x1, x2) we havex∈F−1(y), that is,y∈F (x). Therefore,F (x)⊂
2
i=1F (xi)for allx∈Γ(x1, x2), and so clF (x)⊂2i=1clF (xi)for allx∈Γ(x1, x2). In view of condition (5) we know thatFis aT-KKM mapping.
Moreover, it is easy to prove that all conditions inTheorem 2.2 are satisfied. By Theorem 2.2,x∈XF (x)≠∅. Hence there exists an ¯x∈Xsuch that ¯x∈F (x)for all x∈X, and so we have ¯x∈F (x)¯ . This completes the proof.
5. Application to economy equilibrium problem
Theorem5.1. LetXbe a generalized interval space,A,B,P:X→2X. If the following
conditions are satisfied:
(1) for anyx∈X,A(x)⊂B(x),x∈P (x),A(x), andP (x)are nonempty closed sets;
(2) AandPare upper semi-continuous;
(3) for anyy∈X,A•(y)P•(y)andA•(y){x∈X:x∈clB(x)}areT-convex sets;
(4) for anyM,N∈D(X),(x∈MA(x))∩(x∈NP (x))∈(X); (5) there existsM0∈D(X)such thatx∈M
0A(x)is compact,
then (X, A, B, P) has a Shafer-Sonneinschein equilibrium, that is, there exists anx¯∈X such thatx¯∈clB(x)¯ andA(x)¯ ∩P (x)¯ ≠∅.
Proof. Suppose the contrary, for anyx∈Xwe havex∈clB(x)orA(x)∩P (x)=∅.
Letting
D=x∈X:x∈clB(x),
F (x)=
A(x)∩P (x), x∈D,
A(x), x∈D.
(5.1)
By condition (1) and (2),F is an upper semi-continuous mapping with nonempty closed values. For anyy∈X,
F−1(y)=x∈X:y∈F (x)
=x∈X:x∈A−1(y)∩P−1(y)∪x∈D:x∈A−1(y), F•(y)=x∈X:x∈A•(y)∪P•(y)∩x∈X:x∈A•(y)∪D
=A•(y)∪P•(y)∩A•(y)∪D.
By condition (3),F•(y)is aT-convex set; and by condition (4),x∈MF (x)∈(X). It is obvious thatF satisfies all conditions inTheorem 2.1. ByTheorem 2.1, there exists a ¯y∈X such that ¯y∈F (y)¯ . If ¯y∈D, then ¯y∈A(y)¯ ⊂B(y)¯ which is impossible; if ¯y∈D, then ¯y∈P (y)¯ which contradicts condition (1). Therefore, there exists an ¯
x∈Xsuch that ¯x∈clB(x)¯ andA(x)¯ ∩P (x)¯ ≠∅.
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