The Study of KKM Theorems With Applications to
Vector Equilibrium Problems and Implict Vector
Variational Inequalities Problems
LAI-JIU LIN and HSIU-LI CHEN
Department of Mathematics, National Changhua University of Education, Changhua 50058, Taiwan E-mail: [email protected]
(Received 17 July 2003; accepted in revised form 8 August 2004)
Abstract. In this paper, we establish some equivalence relations between coincidence theorems and KKM-type theorems. We also obtain some new coincidence theorems and fixed point theorems. Applying the KKM-type theorems we obtain the existence theorems of generalized vector equilibrium problems. From these results, some existence theorems of generalized vector implicit variational inequality problems are established in this paper.
Key words: CðxÞ-quasiconvex and CðxÞ-quasiconvexlike, Equilibrium problem, Implict vector variational inequality, KKM property (mapping), Properly quasimonotone, Transfer open (closed), upper (lower) semicontinuous
1. Introduction
In 1929, Knaster et al. [21] established the well-known KKM theorem. Since then, there were many generalizations and applications of KKM the-orem; see for example [9, 11, 15–,17, 23, 24, 28–30, 32–39, 42, 44]. Border [8] showed the equivalence relations between Brouwer fixed point theorem, KKM theory and geometric form of minimax theorem, Tarafdar [42] established the equivalence relation between fixed point theorem and KKM theorem, Park [34] studied the equivalence theorems between some KKM theorem, matching theorem, coincidence theorem, and minimax inequality. Recently Lin et al. [28] Lin and Wan [29], Chang and Yen [15] established some generalized KKM theorems and coincidence theorems. In the first part of this paper, we want to establish the equivalence relations between the generalized KKM theorems and coincidence theorems. We establish some new coincidence theorem and fixed point theorem. Our result on fixed point theorem include the results of Tarafdar [17, 41, 42] as a special cases. The coincidence theorem we establish also include recent result of Djafari-Rouhani et al. [17] as a special case.
In the second part of this paper, we use the generalized KKM theorem in this paper to establish the existence theorems of equilibrium problems.
Let X be a nonempty subset of a topological vector space (in short t.v.s.) E, and F : X X ! R be a real valued bifunction such that Fðx; xÞP0 for all x 2 X. Then the scalar equilibrium Problem (in short, EP) is to find y2 X such that
Fðx; yÞP0 for all x2 X:
The EP has many applications in mathematical physics, economics, game theory, and operation research, etc. It contains several problems like opti-mization, variational inequality, complementarity, Nash equilibrium, and fixed point problems; for detail, see for example [7].
(A) If Z is a t.v.s. with order cone C; that is a closed convex pointed cone, and F : X X ! Z, then the equilibrium problem (EP) can be gener-alized in the following ways:
find y2 X such that
(1) Fðx; yÞ 2 C for all x 2 X; or (2) Fðx; yÞ 62 ()Int C) for all x 2 X.
In these cases, (EP) are called vector equilibrium problem (in short, VEP). These problems contains vector optimization, vector variational inequality problem and vector Nash problem as special cases.
(B) Let F : X X Z and C : X Z be multivalued maps such that CðxÞ is a closed convex pointed cone for each x 2 X, (VEP) can be general-ized in the following forms.
VEP (1): find y2 X such that Fðx; yÞ CðyÞ for all x 2 X. VEP (2): find y2 X such that Fðx; yÞ \ CðyÞ 6¼ ; for all x 2 X. VEP (3): find y2 X such that Fðx; yÞ 6 ()IntCðyÞÞ for all x 2 X. VEP (4): find y2 X such that Fðx; yÞ \ ð)IntC ðyÞÞ ¼ ; for all x 2 X. Recently, the equilibrium problems in both scalar and vector cases have been extensively studied in many literatures, see [2–6, 12–14, 18–20, 22, 25, 27–29, 31, 40].
In this paper, we consider the above four types of equilibrium problems when F is defined on the product of two different spaces and CðyÞ is not necessary a closed convex cone for each y2 Y. Bianchi and Pini [6] first considered these types of equilibrium problem when CðyÞ is a constant set for all y2 Y. Recently Lin and Wan [29] studied the above four types of equilibrium problems when CðyÞ is not necessary a cone and is not a con-stant set. In this paper , we continue the study of Lin and Wan [29]. We study the above four types of equilibrium problems by applying the KKM theorem in Section 3.
As applications of our results, we study the following generalized vector equilibrium problems:
GVEP (1): Find y2 X such that
gðx; y; uÞ Cð yÞ for all u2 /ð yÞ and x 2 X; GVEP (2): Find y2 X such that for each x 2 X
there exists u2 /ðyÞ with gðx; y; uÞ \ CðyÞ 6¼ ;; GVEP (3): Find y2 X such that for each x 2 X
there exists u2 /ðyÞ with gðx; y; uÞ 6 ðIntCðyÞÞ; GVEP (4): Find y2 X such that
gðx; y; uÞ \ ðIntCðyÞÞ ¼ ; for all u2 /ðyÞ and all x 2 X;
where g : X X D Z; / : X D and D is a nonempty subset of topo-logical space Y . Recently Fu and Wan [19] studied the existence theorems of (GVEP (3)).
If g is a single valued function, the above four equilibrium problems are reduced to the following four types of implicit vector variational inequali-ties problems.
GVEPð1Þ0: Find y2 X such that
gðx; y; uÞ 2 CðyÞ for all u 2 /ðyÞ and all x 2 X; GVEP ð2Þ0: Find y2 X such that for each x 2 X
there exists u2 /ðyÞ with gðx; y; uÞ 2 CðyÞ; GVEP ð3Þ0: Find y2 X such that for each x 2 X
there exists u2 /ðyÞ with gðx; y; uÞ 62 IntCðyÞ; GVEP ð4Þ0: Find y2 X such that
gðx; y; uÞ 62 IntCðyÞ for all u 2 /ðyÞ and all x 2 X:
Let E be a t.v.s., LðE; ZÞ ¼ fTjT : E ! Zg is a continuous linear operatorg, let u 2 LðE; ZÞ; y 2 E, we denote hu; yi the evaluation of u at y. If gðx; y; uÞ ¼ hu; yi þ hðx; yÞ, then the above equilibrium problem contains the mixed variational inequality problem recently studied by Khanh and Luu [22]. In this paper, we apply the existence theorems of VEP to study the existence theorems of GVEP for both the cases that g is a multivalued map and g is a single valued function.
2. Preliminaries
Let X and Y be nonempty sets. A multivalued map T : XY is a function from X into power set of Y. Let x2 X; B Y and y 2 Y, we define x2 TðyÞ if and only if y 2 TðxÞ; TðBÞ ¼ fx 2 X : TðxÞ \ B 6¼ ;g; TþðBÞ ¼ fx 2 X : TðxÞ Bg and T1ðBÞ will denote either TðBÞ or TþðBÞ. For topological space E; A E; A is said to be compactly open (compactly closed) if for every nonempty compact subset K of E; A\ K is open (closed) in K.
Let X and Y be two topological spaces, T : X Y; T is said to be trans-fer open [43], if for every x2 X; y 2 TðxÞ, there exists an x02 X such that y2 int Tðx0Þ; transfer closed [43], if for every x 2 X; y 62 TðxÞ, there exists an x0 2 X such that y 62 clTðx0ÞÞ; compact if TðXÞ is compact; upper semi-continuous (in short u.s.c.) (resp. lower semisemi-continuous (in short l.s.c.)) at x2 X, if for every open set U in Y with TðxÞ U (resp. TðxÞ \ U 6¼ ;), there exists an open neighborhood VðxÞ of x such that Tðx0Þ U (resp. Tðx0Þ \ U 6¼ ;Þ for all x02 VðxÞ; T is said to be u.s.c. (resp. l.s.c.) on X if T is u.s.c. at every point of X.
LEMMA 2.1 [30]. Let X and Y be topological spaces and G : X Y be mul-tivalued map. Then
(1) G is transfer closed if and only if Tx2XGðxÞ ¼Tx2XclGðxÞ;
(2) G is transfer open if and only if F : X Y defined by FðxÞ ¼ YnGðxÞ for all x2 X is transfer closed;
(3) G is transfer open if and only if Sx2XGðxÞ ¼Sx2XintGðxÞ.
LEMMA 2.2 [26]. Let X and Y be topological spaces, T : X Y be a multi-valed map. Then the following statements are equivalent:
(1) T: Y X is transfer open and TðxÞ 6¼ ; for all x 2 X; (2) X¼ [y2YintTðyÞ.
LEMMA 2.3 [40]. Let X and Y be topological spaces, T : X Y be a multi-valued map. Then T is l.s.c. at x2 X if and only if for any y 2 TðxÞ and any net fxag in X converges to x, there exists a net fyag such that ya2 TðxaÞ with ya! y.
A convex space [23] X is a nonempty convex set in a vector space with any topology that induces the Euclidean topology on the convex hull of its finite subsets.
Let X be a convex space and Y be a topological space. If S; T : X Y are multivalued maps such that for each N2 hXi; T(coN) SðNÞ, then S is said to be a generalized KKM mapping w.r.t. T; the multivalued map T : X Y is said to have the KKM property [15] if S : X Y is a
generalized KKM mapping w.r.t. T such that the family fSðxÞ : x 2 Xg has the finite intersection property. We denote by KKMðX; YÞ [15] the family of all multivalued maps from X into Y having the KKM property. We denote by KðX; YÞ the family of all u.s.c. multivalued maps with compact convex values and VðX; YÞ the family of all u.s.c. multivalued maps with compact acyclic values. Any convex set in a Hausdorff t.v.s. is acyclic. Then KðX; YÞ VðX; YÞ. In [15], Chang and Yen showed that VðX; YÞ KKM ðX; YÞ.
LEMMA 2.4 [26]. Let X be a convex space and Y be a topological space, T; S : XY be multivalued maps and F; H : Y X be defined by FðyÞ ¼ XnTðyÞ and HðyÞ ¼ XnSðyÞ. Then the following two statements are equivalent:
(1) for each y2 Y, A 2 hFðyÞi implies coA HðyÞ; (2) for each A2 hXi; SðcoAÞ TðAÞ.
DEFINITION 2.1 [20]. Let X be a convex subset of a t.v.s and Z be a Hausdorff t.v.s.. Let C : X Z and F : X X Z be multivalued maps. Given any finite subset N¼ fx1; x2; . . . ; xng in X and any x 2 coN,
(1) F is said to be strong type I C-diagonally quasiconvex in the first argument if for some xi in N,
Fðxi; xÞ CðxÞ;
(2) F is said to be strong type II C-diagonally quasiconvex in the first argument if for some xi in N,
Fðxi; xÞ \ CðxÞ 6¼ ;;
(3) F is said to be weak type I C-diagonally quasiconvex in the first argu-ment if for some xi in N,
Fðxi; xÞ \ ðIntCðxÞÞ ¼ ;;
(4) F is said to be weak type II C-diagonally quasiconvex in the first argument if for some xi in N,
Fðxi; xÞ 6 ðIntCðxÞÞ:
THEOREM 2.1 [1]. Let X and Y be Hausdorff topological spaces, T : X Y be a multivalued map.
(2) If X is compact and T is an u.s.c. multivalued map with compact values, then TðXÞ is compact.
THEOREM 2.2 [25]. Let E1; E2 and Z be Hausdorff t.v.s., X and Y be non-empty subsets of E1 and E2, respectively, F : X Y X Z and S : X X.
(a) If both S and F are l.s.c., then T : X Y Z which is defined by Tðx; yÞ ¼ [u2SðxÞFðx; y; uÞ ¼ Fðx; y; SðxÞÞ
is l.s.c on X Y; and
(b) If both S and F are u.s.c. multivalued maps with compact values, then T is an u.s.c. multivalued map with compact values.
3. Some Equivalent KKM-type Theorems and Coincidence Theorems
In this section, we assume that X is a convex space, Y is a Hausdorff topo-logically space, T : X Y: S; G : X Y are multivalued maps. We start from the following two new results of [28, 29].
THEOREM 3.1 (Theorem 2.6 [28]). Suppose that T2 KKMðX; YÞ and that
(1) for each y2 Y; A 2 hPðyÞi implies coA QðyÞ;
(2) P: X Y is transfer open and for all y 2 Y; PðyÞ is nonempty; (3) for each compact subset A of X; TðAÞ is compact; and
(4) there exists a nonempty compact subset K of Y such that for each N2 hXi, there exists a compact convex subset LN of X containing N such that
TðLNÞnK [fintPðxÞ : x 2 LNg:
Then there exists ðx; yÞ 2 X Y such that y2 TðxÞ and x2 QðyÞ.
THEOREM 3.2 (Theorem 3.6 [29]). Suppose that T2 KKM(X; YÞ and that (1) for each x2 X; TðxÞ SðxÞ;
(2) for each A2 hXi; SðcoAÞ ðAÞ; (3) G : X Y is transfer closed;
(4) for each compact subset A of X; TðAÞ is compact; and
(5) there exists a nonempty compact subset K of Y such that for each N2< X >, there exists a compact convex subset LN of X containing N such that
TðLNÞ \ \
Then Tx2XGðxÞ 6¼ ;:
Proof. Suppose that \x2XGðxÞ ¼ ;: Then for any y 2 Y , there exists x 2 X such that y62 GðxÞ. Now we define P0; Q0 : Y X by P0ðyÞ ¼ XnGðyÞ and Q0ðyÞ ¼ XnSðyÞ for all y 2 Y. Then for y 2 Y, there exists x 2 X such that x2 P0ðyÞ. By (3), ðP0Þ: X Y is transfer open. By (2) and Lemma 2.4, for each y2 Y; A 2 hP0ðyÞi implies coA Q0ðyÞ. By (5), TðLNÞnK SfintðP0ÞðxÞ : x 2 LNg. Then by Theorem 3.1 that there exists ðx; yÞ 2 X Y such that y2 TðxÞ and x2 Q0ðyÞ. Then y62 SðxÞ. This
contradicts (1). Therefore\x2XGðxÞ 6¼ ;. (
REMARK 3.1. Theorems 3.1 and 3.2 are equivalent.
Proof. We want to show that Theorem 3.2 implies Theorem 3.1. Under the assumptions of Theorem 3.1. Suppose that for all x2 X; TðxÞ \ QðxÞ ¼ ;. This implies TðxÞ YnQðxÞ. Let H; S : X Y be defined by HðxÞ ¼ YnPðxÞ, and SðxÞ ¼ YnQðxÞ for x 2 X: Then
(1) for each x2 X; TðxÞ SðxÞ. (2) H : X Y is transfer closed.
Since for each y2 Y; A 2 hPðyÞi implies coA QðyÞ, it follows from Lemma 2.4 that for each A2 hXi implies SðcoAÞ HðAÞ. By (4), TðLNÞ \TTfclHðxÞ : x 2 LNg K: Hence by Theorem 3.2, we get T
x2XHðxÞ 6¼ ;. That is for all x 2 X, there exists y 2 HðxÞ ¼ YnPðxÞ. Then for all x2 X; y 62 PðxÞ i.e. x 62 PðyÞ. Thus PðyÞ ¼ ;. This leads to a con-tradiction. Therefore, there exists x2 X; TðxÞTQðxÞ 6¼ ;: This shows that ðx; yÞ 2 X Y such that y2 TðxÞ and x2 QðyÞ. In Theorem 3.2, we see
that Theorem 3.1 implies Theorem 3.2. (
The following Theorem is a special case of Theorem 3.2, but it is equiva-lent to Theorem 3.2.
THEOREM 3.3. Suppose that T2 KKMðX; YÞ and that (1) for each x2 X; TðxÞ SðxÞ;
(2) for each x2 X; Gx is closed in Y; (3) for any N2< X >; SðcoNÞ GðNÞ;
(4) for each compact subset A of X; TðAÞ is compact; and
(5) there exists a nonempty compact subset K of Y such that for each N2 hXi, there exists a compact convex subset LN of X containing N such that
TðLNÞ \ \
ThenTfGx : x 2 Xg 6¼ ;:
Proof. Since G has closed values, then G : X Y is transfer closed. Hence
by Theorem 3.2, we get TfGx : x 2 Xg 6¼ ;: (
THEOREM 3.4. Theorems 3.2 and 3.3 are equivalent.
Proof. Under the assumptions of Theorem 3.2. Let M : X Y be defined by
MðxÞ ¼ fy 2 Y : y 2 clGðxÞg for x 2 X:
Then for each x2 X; MðxÞ ¼ clGðxÞ is closed. By (4) and Lemma 2.1, TðLNÞ \
T
TfclGðxÞ : xLNg ¼ TðLNÞ \ T
fMðxÞ : x 2 LNg K: By Theo-rem 3.3, Tx2XMðxÞ 6¼ ;: That is Tx2XclGðxÞ 6¼ ;: Since G : X Y is transfer closed, by Lemma 2.1, Tx2XGðxÞ ¼Tx2XclGðxÞ 6¼ ;: ( The following theorem is a special case of Theorem 3.1, but it is equiva-lent to Theorem 3.1.
THEOREM 3.5. Suppose that T2 KKMðX; YÞ and that (1) for each y2 Y; A 2 hPðyÞi implies coA QðyÞ;
(2) P: X Y; PðxÞ is open for all x 2 X and for all y 2 Y; PðyÞ is nonempty;
(3) for each compact subset A of X; TðAÞ is compact ; and
(4) there exists a nonempty compact subset K of Y such that for each N2< X >, there exists a compact convex subset LN of X containing N such that
TðLNÞnK [fPðxÞ : x 2 LNg:
Then there exists ðx; yÞ 2 X Y such that y2 TðxÞ and x2 QðyÞ.
THEOREM 3.6. Theorems 3.1 and 3.5 are equivalent.
Proof. It is clear that Theorem 3.1 implies Theorem 3.5. Under the assump-tions of Theorem 3.1. By (2), Y¼ [x2XintPðxÞ, therefore, for each y 2 Y , there exists x2 X such that y 2 intPðxÞ. Let H : Y X be defined by
HðyÞ ¼ fx 2 X : y 2 intPðxÞg for y2 Y: :
Then HðxÞ ¼ intPðxÞ is open and for each y 2 Y; HðyÞ 6¼ ;: It is easy to see that HðyÞ PðyÞ for all y 2 Y . By (1), for each y 2 Y; A 2 hHðyÞi implies coA QðyÞ. By (4), for each y 2 TðLNÞnK, there exists x 2 LN
such that y2 intPðxÞ. Therefore x 2 HðyÞ and y 2 HðxÞ. Hence TðLNÞnK SfHðxÞ: x 2 LNg. By Theorem 3.5, there exists ðx; yÞ 2 X Y such that y2 TðxÞ and x2 QðyÞ. ( By Theorem 3.1, we obtain the following coincidence theorem which contains many fixed point theorems and coincidence theorems as special cases.
THEOREM 3.7. Suppose the conditions (2) and (4) in Theorem 3.2 are replaced by (2)0 and(4)0 respectively, where
(2)0 for each x2 X; PðxÞ contains an open set O
x Y and [x2XOx ¼ Y; (4)0there exists a nonempty compact subset K of Y such that for each
N2 hXi, there exists a compact convex subset LN of X containing N such that
TðLNÞnK [fOx : x2 LNg:
Then there exists ðx; yÞ 2 X Y such that y2 TðxÞ and x2 QðyÞ.
Proof. Since Ox PðxÞ; Ox is open, and [x2XOx¼ Y. Then [x2XintPðxÞ ¼Sx2XOx¼ Y . By Lemma 2.2, P: X Y is transfer open and for all y2 Y; PðyÞ is nonempty. By (4)0, TðLNÞnK [fOx : x2 LNg [fintPðxÞ : x 2 L
Ng. By Theorem 3.1, there exists ðx; yÞ 2 X Y such
that y2 TðxÞ and x2 QðyÞ. (
If TðxÞ ¼ fxg and PðxÞ ¼ QðxÞ for all x 2 X, then Theorem 3.7 is reduced to the following fixed point theorem.
COROLLARY 3.1. Suppose that
(1) for each x2 X; A 2< PðxÞ > implies coA PðxÞ; (2) for each x2 X; PðxÞ contains an open set O
x X and [2XOx ¼ X; and
(3) there exists a nonempty compact subset K of Y such that for each N2 hXi, there exists a compact convex subset LN of X containing N such that
LNnK [fOx : x2 LNg:
Then there exists x2 X such that x2 PðxÞ:
THEOREM 3.8. Theorems 3.1 and 3.7 are equivalent.
Proof. In Theorem 3.7, we see that Theorem 3.1 implies Theorem 3.7. Under the assumptions of Theorem 3.1. By (2), X¼ [x2XintPðxÞ. Let
Ox ¼ intPðxÞ. Then Ox is open , Ox PðxÞ, and X ¼ [x2XOx. By (4), TðLNÞnK [fintPðxÞ : x 2 LNg ¼ [fOx : x2 LNg. Then Theorem 3.1
follows from Theorem 3.7. (
REMARK 3.2. Corollary 3.2 contains Theorem 1 [41], Theorem 1 [42] and Theorem 2.1 [21] as special cases.
REMARK 3.3. Theorems 3.1–3.3, 3.5 and 3.7 are equivalent.
4. Existence Results of General Vector Equilibrium Problems
When dealing with equilibrium problems, the definition of properly quasi-monotone bimap is frequently used (see [5, 6]).
DEFINITION 4.1 [6] Let X; Z be t.v.s., Y a topological space. Let T : X Y; F : X Y Z and C : Y Z be multivalued maps. F is said to be properly quasimonotone relatively to T on X Y if the map G : X Y ,
GðxÞ ¼ fy 2 Y : ðx; yÞ 2 F1ðCðyÞÞg: is a KKM mapping w.r.t. T.
The following Proposition gives a sufficient condition for the properly quasi-monotonicity.
PROPOSITION 4.1. Let X; Z be t.v.s., Y be a topological space. Let T : X Y; F : X Y Z and C : Y Z: Assume that
(1) for all x2 X and y 2 TðxÞ; ðx; yÞ 2 F1ðCðyÞÞ; and
(2) for any y2 Y; BðyÞ ¼ fx 2 K : ðx; yÞ 62 F1ðCðyÞÞgis convex. Then F is properly quasimonotone relative to T.
Proof. Let G : X Y be defined by
GðxÞ ¼ fy 2 Y : ðx; yÞ 2 F1ðCðyÞÞg for x 2 X:
Suppose F is not properly quasimonotone relative to T. Then there exists a finite subset N¼ fx1; x2; . . . ; xng in X such that TðcoNÞ 6 GðNÞ. There exist
x2 coN and y2 TðxÞ such that y62 GðxiÞ for all i ¼ 1; 2; . . . ; n: Then ðxi; yÞ 62 F1ðCðyÞÞ for all i ¼ 1; 2; . . . ; n: Hence xi2 BðyÞ for all i ¼ 1; 2; . . . ; n: By (2), x2 BðyÞ. Thus ðx; yÞ 62 F1ðCðyÞÞ. This contradicts to condition (1).
Therefore, F is properly quasimonotone relative to T. (
REMARK 4.1. There are similar results in [5, 6]. In Proposition 1.1 [5], CðyÞ ¼ C for all y 2 Y .
In this section, unless otherwise specify, we assume that X is a convex space, Y a Hausdorff topological space and Z is a Hausdorff t.v.s. We assume T : X Y; F : X Y Z and C : Y Z are multivalued maps. Applying Proposition 4.1 and the KKM-type theorems in Section 3, we establish the existence theorems of the four types of (VEP).
THEOREM 4.1. Let T2 KKMðX; YÞ and suppose that (1) for any x2 X; Fðx; Þ is l.s.c.and C : Y Z is closed; (2) for all x2 X and y 2 TðxÞ; Fðx; yÞ CðyÞ;
(3) for any y2 Y; BðyÞ ¼ fx 2 X : Fðx; yÞ 6 CðyÞgÞ is convex; (4) for each compact subset A of X; TðAÞ is compact; and
(5) there exists a nonempty compact subset K of Y such that for each N2 hXi, there exists a compact convex subset LN of X containing N such that for each y2 TðLNÞnK, there exists x 2 LN with Fðx; yÞ 6 CðyÞ. Then there exists y2 Y such that Fðx; yÞ CðyÞÞ for all x 2 X.
Proof. Let G : X Y be defined by
GðxÞ ¼ fy 2 Y : Fðx; yÞ CðyÞg for x 2 X:
Take F1:¼ Fþ in Proposition 4.1. By assumption (2) and (3), for any N2 hXi; TðcoNÞ GðNÞ. For any x 2 X; GðxÞ is closed; Indeed let
y2 GðxÞ, then there exists a net fyag in GðxÞ such that ya converges to y. Since ya2 GðxÞ; Fðx; yaÞ CðyaÞ. Let z2 Fðx; yÞ. Since for any x2 X; Fðx; Þ is l.s.c., by Lemma 2.3, there exists a net fzag such that za2 Fðx; yaÞ with fzag converges to z. Thus za2 CðyaÞ. Since C is closed, z2 CðyÞ. Hence Fðx; yÞ CðyÞ. That is y2 GðxÞ. Therefore GðxÞ is closed. By Theorem 3.3, Tx2XGðxÞ 6¼ ;. Therefore, there exists y2 Y such that
Fðx; yÞ CðyÞÞ for all x 2 X. (
COROLLARY 4.1. In Theorem 4.1, if T2 KKMðX; YÞ is replaced by T is an u.s.c. multivalued map with nonempty compact convex values and suppose that conditions, (1–3) and (5) of Theorem 4.1 hold.
Then there exists y2 Y such that Fðx; yÞ CðyÞÞ for all x 2 X.
Proof. Since T is u.s.c. with nonempty compact convex values, T2 KðX; YÞ KKMðX; YÞ [15]. If A is a compact subset of X, then by Theorem 2.6, TðAÞ and TðAÞ are compact and Corollary 4.1 follows from
Theorem 4.1. (
COROLLARY 4.2. Suppose that X is a Hausdorff convex space,
F : X X Z and C : X Z is multivalued maps satisfying the following conditions:
(1) for any x2 X; Fðx; Þ is l.s.c. and C is closed; (2) for all x2 X; Fðx; xÞ CðxÞ;
(3) for any y2 X; BðyÞ ¼ fx 2 X : Fðx; yÞ 6 CðyÞg is convex; and
(4) there exists a nonempty compact subset K of Y such that for each N2 hXi, there exists a compact convex subset LN of X containing N such that for each y2 LNnK; there exists x 2 LN with Fðx; yÞ 6 CðyÞ. Then there exists y2 X such that Fðx; yÞ CðyÞ for all x 2 X.
Proof. Take TðxÞ ¼ fxg in Corollary 4.1. (
THEOREM 4.2. Let X be a Hausdorff convex space.
Suppose that conditions(1) and (4) of Corollary 4.2 and (20), where
(20) F : X X Z is strong type I C-diagonally quasiconvex in the first argument.
Then there exists y2 X such that Fðx; yÞ CðyÞÞ for all x 2 X.
Proof. Let TðxÞ ¼ SðxÞ ¼ fxg for all x 2 X and G : X X be defined by GðxÞ ¼ fy 2 X : Fðx; yÞ CðyÞg for x 2 X:
Suppose that there exists N¼ fx1; x2; . . . ; xng 2 hXi such that coN¼ TðcoNÞ 6 GðNÞ. Then there exists x2 coN such that x62 GðxiÞ for all i¼ 1; 2; . . . ; n. That is Fðxi; xÞ 6 CðxÞ for all i ¼ 1; 2; . . . ; n. But F is strong type I P-diagonally quasiconvex in the first argument (20), Fðx
i; xÞ CðxÞ for some xi 2 N. This leads to a contradiction. Hence for each N 2 hXi; coN GðNÞ. By (1), GðxÞ is closed for all x 2 X. Therefore the conclusion of Theorem 4.2 follows
from Theorem 3.3. (
DEFINITION 4.2. Let C : X Z be a multivalued map such that CðyÞ is convex cone for all y2 Y . We say that
(a) For each y2 Y; x Fðx; yÞ is CðyÞ-quasiconvex [20] if x1; x22 X, and k2 ½0; 1, then either
Fðx1; yÞ Fðkx1þ ð1 kÞx2; yÞ þ CðyÞ: or Fðx2; yÞ Fðkx1þ ð1 kÞx2; yÞ þ CðyÞ:
(b) For each y2 Y; x Fðx; yÞ is CðyÞ-quasiconvex-like [3] if for x1; x22 X, and k 2 ½0; 1, then either
Fðkx1þ ð1 kÞx2; yÞ Fðx1Þ CðyÞ: or Fðkx1þ ð1 kÞx2; yÞ Fðx2Þ CðyÞ:
REMARK 4.2. If CðyÞ is a convex cone and Fð; yÞ is CðyÞ-quasiconvex, then the set BðyÞ ¼ fx 2 X : Fðx; yÞ 6 CðyÞg is convex.
Proof. (a) To prove BðyÞ ¼ fx 2 X : Fðx; yÞ 6 CðyÞg is convex.
Let x1; x22 BðyÞ, then Fðx1; yÞ 6 CðyÞ and Fðx1; yÞ 6 CðyÞ. We want to show that kx1þ ð1 kÞx22 BðyÞ for all k 2 ½0; 1. Suppose there exists k02 ½0; 1 such that Fðk0x1þ ð1 k0Þx2; yÞ CðyÞ. Since Fð; yÞ is CðyÞ-quasiconvex, either Fðx1; yÞ CðyÞ or Fðx2; yÞ CðyÞ. This leads to a con-tradiction. Hence for all k2 ½0; 1; Fðkx1þ ð1 kÞx2; yÞ 6 CðyÞ and BðyÞ is convex.
By using Theorem 3.3 and Theorem 4.1, we establish the following theo-rem.
THEOREM 4.3. Suppose that T2 KKMðX; YÞ, (1), (4), (5) of Theorem 4.1 and
(a) for all x2 X and y 2 TðxÞ; Aðx; yÞ CðyÞ;
(b) for all ðx; yÞ 2 X Y; Aðx; yÞ CðyÞ implies Fðx; yÞ CðyÞ; (c) for any y2 Y; BðyÞ ¼ fx 2 X : Aðx; yÞ 6 CðyÞg is convex;
Then there exists y2 Y such that Fðx; yÞ CðyÞ; for all x 2 X .
Proof. Let G; H : X Y be defined by GðxÞ ¼ fy 2 Y : Aðx; yÞ CðyÞg and HðxÞ ¼ fy 2 Y : Fðx; yÞ CðyÞg for x 2 X: By (1), HðxÞ is closed for all x2 X. By (b), for any x 2 X; GðxÞ HðxÞ. By (a) ,(c) and Proposition 4.1 that for all N2 hXi; TðcoNÞ GðNÞ. By (c), TðLNÞ\
T
fHðxÞ : x 2 LNg K. Then by Theorem 3.3, Tx2XHðxÞ 6¼ ;, i.e. there exists y2 Y such that
Fðx; yÞ CðyÞ, for all x 2 X. (
DEFINITION 4.3 [18]. Let H : X Z be a multivalued map. H is said to be properly quasiconvex if for every x; y2 K; t 2 ½0; 1, and u 2 HðxÞ; v 2 HðyÞ, there exists z2 Hðtx1þ ð1 tÞx2Þ such that either zOu or zOv.
LEMMA 4.1 [18]. Let H : X Z be a multivalued mapping. Then H is prop-erly quasiconvex if and only if for any xi2 K; zi2 HðxiÞ; ti >0, for i¼ 1; 2; . . . ; n;Pn
i¼1ti¼ 1, there exist z 2 Fðt1x1þ þ tnxnÞ and some i such that zOzi.
THEOREM 4.4. In Theorem 4.1, if condition (3) is replaced by (3)0, then we have the same conclusion, where
Proof. Let G : X Y be defined by
GðxÞ ¼ fy 2 Y : Fðx; yÞ CðyÞg for x 2 X:
By (2), (3)0 and following the same arguments as in Theorem 2 of [18], we can show that for all N2 hXi; TðcoNÞ GðNÞ. Then the conclusion of
Theorem 4.4 follows from Theorem 3.3. (
REMARK 4.3. Theorem 4.4 is different from Theorem 2 [18]. In Theorem 2 [18], the multivalued map T : X D is u.s.c. with nonempty compact con-vex values, X is a nonempty compact concon-vex set, and GðxÞ is closed for all x2 X.
THEOREM 4.5. Suppose T2 KKMðX; YÞ and
(1) for any x2 X; y Fðx; yÞ is u.s.c. with compact values and C : Y Z is closed;
(2) for all x2 X and y 2 TðxÞ; Fðx; yÞ \ CðyÞ 6¼ ;;
(3) for any y2 Y; BðyÞ ¼ fx 2 X : Fðx; yÞ \ CðyÞ ¼ ;g is convex; (4) for each compact subset A of X; TðAÞ is compact; and
(5) there exists a nonempty compact subset K of Y such that for each N2 hXi, there exists a compact convex subset LN of X containing N such that for each y2 TðLNÞnK, there exists x 2 LN with Fðx; yÞ \ CðyÞ ¼ ;. Then there exists y2 Y such that Fðx; yÞ \ CðyÞ 6¼ ;; for all x 2 X.
Proof. Let G : X Y be defined by
GðxÞ ¼ fy 2 Y : Fðx; yÞ \ CðyÞ 6¼ ;g for x 2 X:
Take F1:¼ F in Proposition 4.1. By (2), (3) and Proposition 4.1, for any N2 hXi; TðcoNÞ GðNÞ. By (1) and following the same argument as in [3], GðxÞ is closed for each x 2 X. Then by Theorem 3.3, \x2XGðxÞ 6¼ ;. Therefore, there exists y2 Y such that Fðx; yÞ \ CðyÞ 6¼ ;
for all x2 X. (
REMARK 4.4. Following the method of Ansari and Yao [3], we can show that if for each y2 Y; Fð; yÞ is CðyÞ-quasiconvex-like and CðyÞ is a convex cone for each y2 Y . Then the set BðyÞ ¼ fx 2 X : Fðx; yÞ \ CðyÞ ¼ ;g is convex.
THEOREM 4.6. Let X be a Hausdorff convex space, let F : X X Z and C : X Z be multivalued maps. Suppose that
(1) of Theorem 4.5 and
(2)0F is strong type II C-diagonally quasiconvex in the first argument; and
(3)0there exists a nonempty compact subset K of X such that for each N2 hXi, there exists a compact convex subset LN of X containing N such that for each y2 LNnK, there exists x2 LN with Fðx; yÞ \ CðyÞ ¼ ;.
Then there exists y2 X such that Fðx; yÞ \ CðyÞ 6¼ ;, for all x 2 X.
Proof. Let TðxÞ ¼ fxg for x 2 X and G : X X be defined by GðxÞ ¼ fy 2 X : Fðx; yÞ \ CðyÞ 6¼ ;g for x 2 X:
Suppose that there exists N¼ fx1; x2; . . . ; xng 2 hXi such that coN¼ TðcoNÞ 6 GðNÞ. Then there exists x2 coN such that x62 GðxiÞ for all i¼ 1; 2; . . . ; n. That is Fðxi; xÞ \ CðxÞ ¼ ; for all i ¼ 1; 2; . . . ; n. But by (2)0, Fðxi; xÞ \ CðxÞ 6¼ ; for some xi2 N. This leads to a contradiction. Hence for each N2 hXi; coN GðNÞ. Therefore the conclusion of Theorem 4.6 follows from Theorem 3.3.
Applying Remark 3.3 of Lin [28], we establish the following theorem
THEOREM 4.7. Suppose that T2 KKMðX; YÞ and
(1) for any x2 X; y Fðx; yÞ is u.s.c. with compact values and W : Y Z is u.s.c., where WðyÞ ¼ Zn ()IntCðyÞÞfor all y 2 Y ;
(2) for all x2 X and y 2 TðxÞ; Fðx; yÞ 6 ðIntCðyÞÞ;
(3) for any y2 Y; BðyÞ ¼ fx 2 X : Fðx; yÞ ()IntCðyÞÞg is convex; (4) for each compact subset A of X, TðAÞ is compact; and
(5) there exists a nonempty compact subset K of Y such that for each N2 hXi, there exists a compact convex subset LN of X containing N such that for each y2 TðLNÞnK, there exists x2 LN with Fðx; yÞ ðIntCðyÞÞ.
Then there exists y2 Y such that Fðx; yÞ 6 ()IntCðyÞÞ for all x 2 X. Proof. Let G : X Y be defined by
GðxÞ ¼ fy 2 Y : Fðx; yÞ 6 ðIntCðyÞg for x 2 X: By (2), (3) and Proposition 4.1, for all N2 hXi; TðcoNÞ GðNÞ.
By (1) and following the same argument as in Theorem 2.1 of Ansari and Yao [3], for each x2 X; GðxÞ is closed. By Theorem 3.3 , \x2XGðxÞ 6¼ ;. Therefore, there exists y2 Y such that Fðx; yÞ 6 ðIntCðyÞÞ, for all x 2 X. (
REMARK 4.5. Following the method of Ansari and Yao [3], we can show that if F is CðyÞ-quasiconvexlike and CðyÞ is a convex cone for each y 2 Y, then the set BðyÞ ¼ fx 2 X : Fðx; yÞ ðIntCðyÞÞg is convex.
THEOREM 4.8. Let X be a Hausdorff convex space, let F : X X Z and C : X Z be multivalued maps. Suppose that
(1) of Theorem 4.7 and
(2)0F is weak type II C-diagonally quasiconvex in the first argument for each fixed y; and
(3)0there exists a nonempty compact subset K of X such that for each N2 hXi, there exists a compact convex subset LN of X containing N such that for each y2 LNnK, there exists x 2 LN with Fðx; yÞ ðIntCðyÞÞ. Then exists y2 X such that Fðx; yÞ 6 ðIntCðyÞÞ for all x 2 X.
Proof. Let TðxÞ ¼ fxg for x 2 X and G : X X be defined by GðxÞ ¼ fy 2 X : Fðx; yÞ 6 ðIntCðyÞÞg for x2 X:
By (1) and following the same argument as in Theorem 4.1, GðxÞ is closed. By (2)0 and follows the same argument as in Theorem 4.2, we show that for each N2 hXi; coN ¼ TðcoNÞ GðNÞ. Therefore the conclusion of The-orem 4.8 follows from TheThe-orem 3.3.
THEOREM 4.9. Suppose that T2 KKMðX; YÞ, (4) of Theorem 4.7 and (1)0 for any x2 X; y Fðx; yÞ is l.s.c. and W : Y Z is u.s.c, where
WðyÞ ¼ ZnðIntCðyÞÞ for all y 2 Y;
(2)0 for all x2 X and y 2 TðxÞ; Fðx; yÞ \ ðIntCðyÞÞ ¼ ;;
(3)0 for any y2 Y; BðyÞ ¼ fx 2 X : Fðx; yÞ \ ð)CðyÞÞ 6¼ ;gis convex; and (5)0 there exists a nonempty compact subset K of Y such that for each
N2 hXi,there exists a compact convex subset LN of X containing N such that for each y2 TðLNÞnK, there exists x2 LN with Fðx; yÞ \ ðIntCðyÞÞ 6¼ ;:
Then there exists y2 Y such that Fðx; yÞ \ ðIntCðyÞÞ ¼ ; for all x 2 X. Proof. Let G : X Y be defined by
GðxÞ ¼ fy 2 X : Fðx; yÞ \ ðIntCðyÞÞ ¼ ;g for x 2 X: By (2)0 (3)0 and Proposition 4.1, for all N2 hXi; TðcoNÞ GðNÞ.
By (1)0 GðxÞ is closed for all x 2 X. By Theorem 3.3, there exists y2 Y such that Fðx; yÞ \ ðIntCðyÞÞ ¼ ; for all x 2 X .
THEOREM 4.10. Let X be a Hausdorff convex space, let F : X X Z and C : X Z be multivalued maps. Suppose that
(1) for any x2 X; y Fðx; yÞ is l.s.c. and W : Y Z is u.s.c., where WðyÞ ¼ ZnðIntCðyÞÞ for all y 2 Y;
(3) there exists a nonempty compact subset K of X such that for each N2 hXi, there exists a compact convex subset LNof X containing N such that for each y2 LNnK, there exists x2 LN with Fðx; yÞ\ ðIntCðyÞÞ 6¼ ;:
Then there exists y2 X such that Fðx; yÞ \ ðIntCðyÞÞ ¼ ; for all x 2 X.
Proof. Let TðxÞ ¼ fxg for x 2 X and G : X X be defined by GðxÞ ¼ fy 2 X : Fðx; yÞ \ ðIntCðyÞÞ ¼ ;g for x 2 X:
By (1), GðxÞ is closed for all x 2 X. With the similar argument as Theorem 4.2, we show that for each N2 hXi; coN ¼ TðcoNÞ GðNÞ. Therefore
The-orem 4.10 follows from TheThe-orem 3.3. (
The rest of this section, let X be a Hausdorff convex space, Y a Hausdorff topological space, D be a nonempty subset of Y; Z be a Haus-dorff t.v.s. As simple consequences of (VEP), we establish the existence the-orems of (GVEP).
THEOREM 4.11. Suppose that g : X X D Z; / : X D and
(1) for each fixed x2 X; ðy; uÞ gðx; y; uÞ and / are l.s.c. and C is closed;
(2) for each y2 X and u 2 /ðyÞ; gðy; y; uÞ CðyÞ;
(3) for each fixed y2 X; x gðx; y; /ðyÞÞ is CðyÞ-quasiconvex; and (4) there exists a nonempty compact subset K of X such that for each
N2 hXi, there exists a compact convex subset LN of X containing N such that for each y2 LNnK, there exists x 2 LN and u2 /ðyÞ such that gðx; y; uÞ 6 CðyÞ.
Then there exists y2 X such that gðx; y; uÞ CðyÞÞ for all u 2 /ðyÞ and all x2 X.
Proof. Let Fðx; yÞ ¼ gðx; y; /ðyÞÞ ¼ [u2/ðyÞgðx; y; uÞ. By (1) and Theorem 2.2, Fðx; yÞ is l.s.c. By (3), the multivalued map x Fðx; yÞ is CðyÞ-quasiconvex. Then by Remark 4.2, the set fx 2 X : Fðx; yÞ 6 CðyÞg is convex. By Corollary 4.2, then there exists y2 X such that
gðx; y; uÞ CðyÞ for all x 2 X and u 2 /ðyÞ. (
In Theorem 4.11, if g : X X D Z is a single value function, we have the following existence theorem of generalized vector implicit vector variational inequality.
COROLLARY 4.3. Suppose that D; g and / be the same as Theorem 4.11 and
(1) / is l.s.c. and for each fixed x2 X; ðy; uÞ ! gðx; y; uÞ is a continuous function and C is closed and CðyÞ is a convex cone for each y 2 Y; (2) for each y2 X and u 2 /ðyÞ; gðy; y; uÞ 2 CðyÞ;
(3) for each fixed y2 X; x gðx; y; /ðyÞÞis CðyÞ-quasiconvex; and
(4) there exists a nonempty compact subset K of X such that for each N2< X >, there exists a compact convex subset LN of X containing N such that for each y2 LNnK, there exists x 2 LN; u2 /ðyÞ such that gðx; y; uÞ 2 CðyÞ.
Then there exists y2 X such that gðx; y; uÞ 2 CðyÞÞ for all u 2 /ðyÞ and all x2 X.
THEOREM 4.12. Suppose that g : X X D Z; / : X D and
(1) for each fixed x2 X; ðy; uÞ gðx; y; uÞand / are u.s.c. with compact values and C : Y Z is closed;
(2) for all y2 X and u 2 /ðyÞ; gðy; y; uÞ \ CðyÞ 6¼ ;;
(3) for each fixed y2 X; x gðx; y; /ðyÞÞis CðyÞ-quasiconvex-like and CðyÞ is a closed convex cone for each y 2 Y; and
(4) there exists a nonempty compact subset K of X such that for each N2 hXi, there exists a compact convex subset LNof X containing N such that for each y2 LNnK, there exists x 2 LN with gðx; y; /ðyÞÞ\ CðyÞ ¼ ;.
Then there exists y2 X such that for each x 2 X, there exists u 2 /ðyÞwith gðx; y; uÞ \ CðyÞ 6¼ ;:
Proof. Let Fðx; yÞ ¼ gðx; y; /ðyÞÞ ¼ [u2/ðyÞgðx; y; uÞ. By (1) and Theorem 2.7, for each x2 X; Fðx; yÞ is u.s.c. with compact values. Since for each y2 X; x Fðx; yÞ is CðyÞ-quasiconvex-like and CðyÞ is a closed convex cone, by (3) and Remark 4.4, the set fx 2 X : Fðx; yÞ \ CðyÞ 6¼ /g is con-vex. By Theorem 4.5 with TðxÞ ¼ fxg for all x 2 X, there exists y2 X such that Fðx; yÞ ¼ gðx; y;/ðyÞÞ \ CðyÞ 6¼ ; for all x 2 X. Therefore for each x2 X, there exists u 2 /ðyÞ such that gðx; y; uÞ \ CðyÞ 6¼ ;: (
For the special cases of Theorem 4.12, we have the following existence theorem of generalized implicit vector variational inequality.
COROLLARY 4.4. Let g : X X D ! Z be a function satisfying the following conditions:
(1) for each fixed x2 X; ðy; uÞ ! gðx; y; uÞ is continuous, / is u.s.c. with compact values and C is closed;
(3) for each fixed y2 X; x gðx; y; /ðyÞÞ is CðyÞ-quasiconvex-like and CðyÞ is a closed convex cone for each y 2 Y; and
(4) there exists a nonempty compact subset K of X such that for each N2 < X > there exists a compact convex subset LN of X containing N such that for each y2 LNnK, there exists x 2 LN such that gðx; y; uÞ 62 CðyÞ for all u 2 /ðyÞ.
Then there exists y2 X such that for each x 2 X, there exists u 2 /ðyÞsuch that gðx; y; uÞ 2 CðyÞ.
THEOREM 4.13. Suppose that g; h : X X D Z; / : X D and
(1) for each fixed x2 X; ðy; uÞ gðx; y; uÞ and / are u.s.c. with compact values and W : X Z is u.s.c., where WðyÞ ¼ ZnðIntCðyÞÞ for all y2 Y;
(2) for all y2 X, there exists u 2 /ðyÞ such that gðy; y; uÞ 6 IntCðyÞ; (3) for each fixed y2 X; x gðx; y; /ðyÞÞ is CðyÞ-quasiconvex-like and
CðyÞ is a closed convex cone for each y 2 Y; and
(4) there exists a nonempty compact subset K of X such that for each N2 hXi, there exists a compact convex subset LN of X containing N such that for each y2 LNnK, there exists x 2 LN with gðx; y; /ðyÞÞ ()IntCðyÞÞ.
Then there exists y2 X such that for each x 2 X, there exists u 2 /ðyÞ with gðx; y; uÞ 6 ()IntCðyÞÞ.
Proof. Let Fðx; yÞ ¼ gðx; y; /ðyÞÞ ¼ [u2/ðyÞgðx; y; uÞ. Let G : X X be defined by
GðxÞ ¼ fy 2 X : Fðx; yÞ ¼ gðx; y; /ðyÞÞ 6 ðIntCðyÞÞg for x2 X: By Theorem 2.2 and Theorem 4.7, we can prove that Theorem 4.13. ( For the special case of Theorem 4.13, we have the following existence theorem of generalized vector implicit variational inequality.
COROLLARY 4.5. Suppose that g : X X D ! Z and / : X Z and
(1) for each fixed x2 X; ðy; uÞ ! gðx; y; uÞ is continuous, / is u.s.c. with compact values and W : X Z is u.s.c., where WðyÞ ¼ ZnðIntCðyÞÞ for all y2 Y;
(2) for all y2 X there exist u 2 /ðyÞsuch that gðy; y; uÞ 62 ðIntCðyÞÞ; (3) for each fixed y2 X; x gðx; y; /ðyÞÞis CðyÞ-quasiconvex-like and
(4) there exists a nonempty compact subset K of X such that for each N2 hXi there exists a compact convex subset LN of X containing N such that for each y2 LNnK, there exists x 2 LN with gðx; y; uÞ 2 ðIntCðyÞÞ for all u 2 CðyÞ.
Then there exists y2 X such that for each x 2 X, there exists u 2 /ðyÞ with gðx; y; uÞ 62 ()Int CðyÞÞ.
THEOREM 4.14. Suppose that g : X X D Z; / : X D and
(1) for each fixed x2 X, the multivalued maps ðy; uÞ gðx; y; uÞ and / are l.s.c. and W : X Z is u.s.c., where WðyÞ ¼ Zn()IntCðyÞÞfor all y 2 Y; (2) gðx; y; /ðyÞÞ is weak type I C-diagonally quasiconvex in the first
argu-ment; and
(3) there exists a nonempty compact subset K of X such that for each N2< X >, there exists a compact convex subset LN of X containing N such that for each y2 LNnK, there exists x2 LN with gðx; y; /ðyÞÞ \ ðIntCðyÞÞ 6¼ ;:
Then there exists y2 X such that gðx; y; uÞ\()IntCðyÞÞ ¼ ; for all u 2 /ðyÞ and all x2 X.
Proof. Let Fðx; yÞ ¼ gðx; y; /ðyÞÞ ¼ [u2/ðyÞgðx; y; uÞ. Let G : X X be defined by
GðxÞ ¼ fy 2 X : Fðx; yÞ ¼ gðx; y; /ðyÞÞ \ ðIntCðyÞÞ ¼ ;g for x2 X:
Theorem 4.14 follows from Theorems 2.7 and 4.10.
COROLLARY 4.6. Suppose that g : X X D ! Z; / : X Z and
(1) for each fixed x2 X; ðy; uÞ ! gðx; y; uÞ is continuous, / is l.s.c. and W : X Z is u.s.c., where WðyÞ ¼ Zn (IntCðyÞÞ for all y 2 Y; (2) gðx; y; /ðyÞÞ is weak type I C-diagonally quasiconvex in the first
argu-ment; and
(3) there exists a nonempty compact subset K of X such that for each N2< X >, there exists a compact convex subset LN of X containing N such that for each y2 LNnK, there exists x 2 LN; u2 /ðy) with gðx; y; uÞ 2 ()IntCðyÞÞ.
Then there exists y2 X such that gðx; y; uÞ 62 ()IntCðyÞÞ for all u 2 /ðyÞ and all x2 X.
REMARK 4.6.
(1) Corollaries 4.4 and 4.5 are different from Theorems 3 and 4 [19]. (2) If gðx; y; uÞ ¼< u; gðy; xÞ > þhðx; yÞ
where h : X X ! Z and g : X X ! X; u 2 LðX; ZÞ ¼ fTjT : X ! Z is a continuous linear operatorg. Then Corollaries 4.5 and 4.6 are the exis-tence theorems of mixed generalized vector variational inequality problems recently studied by Khanh and Luu. [22].
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