R E S E A R C H
Open Access
Existence of solutions for generalized vector
quasi-equilibrium problems in abstract
convex spaces with applications
Wei-bing Zhang
1, Shu-qiang Shan
1and Nan-jing Huang
1,2**Correspondence:
1Department of Mathematics,
Sichuan University, Chengdu, Sichuan 610064, P.R. China
2Department of Mathematics,
Sichuan University of Science & Engineering, Zigong, Sichuan 643000, P.R. China
Abstract
In this paper, several kinds of generalized vector quasi-equilibrium problems are introduced and studied in abstract convex spaces. Using the properties of
-convex andKC-maps, some sufficient conditions are given to guarantee the existence of solutions in connection with these generalized vector quasi-equilibrium problems. As applications, some existence theorems of solutions for the generalized semi-infinite programs with vector quasi-equilibrium constraints are also given.
Keywords: abstract convex space; generalized vector quasi-equilibrium problem; generalized semi-infinite program; set-valued mapping; KKM mapping
1 Introduction
It is well known that the vector quasi-equilibrium problem is an important general-ization of the vector equilibrium problem which provides a unified model for vector quasi-variational inequalities, vector quasi-complementarity problems, vector optimiza-tion problems and vector saddle point problems. In , Fu [] established the existence theorems for the generalized vector quasi-equilibrium problems and the set-valued vec-tor equilibrium problems. In , Ansari and Fabián [] considered a generalized vecvec-tor quasi-equilibrium problem with or without involving-condensing mappings and proved the existence of its solution in real topological vector spaces. In , Liet al.[] studied the existence of solutions for two classes of generalized vector quasi-equilibrium prob-lems. Recently, Linet al.[] introduced and studied a class of generalized vector quasi-equilibrium problems involving pseudomonotonicity hemicontinuity mappings under dif-ferent conditions in topological vector spaces. Linet al.[] proved the existence of equilib-ria for generalized abstract economy with a lower semicontinuous constraint correspon-dence and a fuzzy constraint corresponcorrespon-dence defined on a noncompact/nonparacompact strategy set. They also considered a systems of generalized vector quasi-equilibrium prob-lems in topological vector spaces. Very recently, Yang and Pu [] studied the existence and essential components in connection with the set of solutions for the system of strong vec-tor quasi-equilibrium problems. Fu and Wang [] considered the generalized strong vecvec-tor quasi-equilibrium problems with domination structure. On the other hand, Ding [] stud-ied the existence of solutions for generalized vector quasi-equilibrium problems in locally
G-convex spaces. Balaj and Lin [] investigated existence of solutions for the generalized equilibrium problems inG-convex spaces.
The abstract convex space, introduced by Park [] in , includes the convex subset of a topological vector space, the convex space, theH-space, and theG-convex space as special cases. Moreover, Park [] investigated the property of the abstract convex spaces and showed some applications. Recently, several authors have focused on the studies con-cerned with the set-valued maps and optimization problems in abstract convex spaces with applications. For instance, Choet al.[] studied some coincidence theorems and minimax inequalities in abstract convex spaces. Yanget al.[] proved some maximal ele-ment theorems for set-valued maps in abstract convex spaces with applications. Yang and Huang [] gave some coincidence theorems for compact and noncompactKC-maps in abstract convex spaces with applications. Lu and Hu [] established a new collectively fixed point theorem in noncompact abstract convex spaces with applications to equilibria for generalized abstract economies. Park [] gave some comments on fixed points, max-imal elements, and equilibria of economies in abstract convex spaces. Yang and Huang [] studied the existence of solutions for the generalized vector equilibrium problems in abstract convex spaces. At the end of the paper [], Yang and Huang pointed out that it is an interesting and important work to study some types of generalized vector quasi-equilibrium problems with moving cones in topological spaces. To the best of our knowl-edge, it seems that there is no work concerned with the study of the generalized vector quasi-equilibrium problems in abstract convex spaces. Therefore, it is natural and inter-esting to study some generalized vector quasi-equilibrium problems in abstract convex spaces under some suitable conditions.
On the other hand, we know that semi-infinite programs are constrained optimization problems in which the number of decision variables is finite, but the number of constraints is infinite. Since John [] initiated semi-infinite programming precisely to deduce impor-tant results about two such geometric problems: the problems of covering a compact body in finite dimensional spaces by the minimum-volume disk and the minimum-volume ellip-soid, many researchers have been investigated the theory, applications and methods for the semi-infinite programming (see, for example, [–]). As a generalization of semi-infinite programming, the generalized semi-infinite programming has been become a vivid field of active research in mathematical programming in recent years due to its important ap-plications to numerous real-life problems such as Chebyshev approximation, design cen-tering, robust optimization, optimal layout of an assembly line, time minimal control, and disjunctive optimization (see [] and the references therein). Therefore, it is important and interesting to study the existence of solutions concerned with some generalized semi-infinite programs with vector quasi-equilibrium constraints in abstract convex spaces.
The main purpose of this paper is to study several classes of generalized vector quasi-equilibrium problems in abstract convex spaces with applications to generalized semi-infinite programs. We give some sufficient conditions to guarantee the existence of solu-tions for these generalized vector quasi-equilibrium problems in abstract convex spaces. As applications, we give some existence theorems of solutions for the generalized semi-infinite programs under suitable conditions.
2 Preliminaries
LetX,Y be two nonempty sets. A set-valued mappingT:X⇒Y is a mapping fromX
by
T–(y) =x∈X:y∈T(x).
An abstract convex space (X,D,) consists of a nonempty setX, a nonempty setD, and a set-valued mapping:D⇒Xwith nonempty values, whereDdenotes the set of all nonempty finite subset of a setD. If for eachA∈ Dwith the cardinality|A|=n+ , there exists a continuous function φA:n→(A) such thatJ∈ AimpliesφA(J)⊆
(J), wherenis the standardn-simplex andJthe face ofncorresponding toJ∈ A,
then the abstract convex space reduces to theG-convex space. LetA:=(A) forA∈ D.
WhenD⊂X, the space is defined by (X⊇D,). In this case, a subsetMofXis said to be
-convex if, for anyA∈ M∩D, we haveA⊆M. In the caseX=D, let (X,) := (X,X,).
It is easy to see that any vector spaceYis an abstract convex space with:=co, where co denotes the convex hull in the vector spaceY. Next we give more examples as follows.
Example .([]) LetEbe a topological vector space with a neighborhood systemV of its origin. A subsetX ofE is said to be almost convex (see [] for more details) if for anyV∈V and for any finite subsetA={x,x, . . . ,xn}ofX, there exists a subsetB= {y,y, . . . ,yn}ofXsuch thatyi–xi∈Vfor alli= , , . . . ,nandcoB⊂X. LetA=coBfor
anyA∈ X. Then (X,) is aG-convex space and hence an abstract convex space.
Example . ([]) Usually, a convex space (E,C) in the classical sense consists of a nonempty setEand a familyCof subsets ofEsuch thatE itself is an element ofCand
Cis closed under arbitrary intersection. For any given subsetX⊂E, theC-convex hull of
Xis defined as by
CoCX={Y∈C:X⊂Y}.
We say thatX isC-convex ifX=CoCX. Consider the mapping:E⇒Edefined by
A=CoCA. Then (E,) is an abstract convex space.
Example . Let (M,d) be a pseudo-metric space, that is,d:M×M→[, +∞) such that, for everyx,y,z∈M,
(i) d(x,x) = ; (ii) d(x,y) =d(y,x);
(iii) d(x,z)≤d(x,y) +d(y,z).
For anyA∈ M, define a set-valued mapping:M⇒Mby
A=(A) =
{B:Bis a closed ball containingA}.
Then it is easy to see that (M,) is an abstract convex space.
Let (X,) be an abstract convex space andVbe a real topological vector space. LetEbe a nonempty subset ofX. Assume thatS:E⇒EandB:E⇒Eare two set-valued mappings. Suppose thatF:X×X×X⇒VandC:X⇒Vare two set-valued mappings such that for eachx∈X,C(x) is a closed convex cone withintC(x)=∅, hereintC(x) denotes the interior ofC(x). In this paper, we will consider the following generalized vector quasi-equilibrium problems in abstract convex spaces.
•(GVQEP) Findx˜∈Esuch that
˜
x∈S(x˜) and F(x˜,y,z)⊆C(x˜), ∀y∈S(x˜),∀z∈B(x˜).
We would like to mention that (GVQEP) was considered by Linet al.[] in topologi-cal vector spaces. WhenS(x) =B(x) =Efor allx∈E, (GVQEP) was considered by Yang and Huang [] in abstract convex spaces and by Balaj and Lin [] inG-convex spaces, respectively.
•(GVQEP) Findx˜∈Eandz˜∈B(x˜) such that
˜
x∈S(x˜) and F(x˜,y,z˜)⊆C(x˜), ∀y∈S(x˜).
WhenC(x) was replaced by –C(x), (GVQEP) was considered by Li and Li [] in topo-logical vector spaces. IfS(x) =Efor allx∈E, then (GVQEP) was investigated by Fu and Wang [] in topological vector spaces.
•(GVQEP) Findx˜∈Esuch that
˜
x∈S(x˜) and F(x˜,y,z)∩–intC(x˜) =∅, ∀y∈S(x˜),∀z∈B(x˜).
We note that (GVQEP) was considered by Linet al.[] in topological vector spaces. WhenS(x) =B(x) =E for allx∈E, (GVQEP) was studied by Yang and Huang [] in abstract convex spaces and by Balaj and Lin [] inG-convex spaces, respectively.
•(GVQEP) Findx˜∈Eandz˜∈B(x˜) such that
˜
x∈S(x˜) and F(x˜,y,z˜)∩–intC(x˜) =∅, ∀y∈S(x˜).
We note that (GVQEP) was considered by Li and Li [] in topological vector spaces.
•(GVQEP) Findx˜∈Esuch that
˜
x∈S(x˜) and F(x˜,y,z)–intC(x˜), ∀y∈S(x˜),∀z∈B(x˜).
WhenS(x) =B(x) =Efor allx∈E, (GVQEP) was investigated by Yang and Huang [] in abstract convex spaces and by Balaj and Lin [] inG-convex spaces, respectively.
•(GVQEP) Findx˜∈Eandz˜∈B(x˜) such that
˜
x∈S(x˜) and F(x˜,y,z˜)–intC(x˜), ∀y∈S(x˜).
•(GVQEP) Findx˜∈Esuch that
˜
x∈S(x˜) and F(x˜,y,z)∩C(x˜)=∅, ∀y∈S(x˜),∀z∈B(x˜).
WhenS(x) =B(x) =E for allx∈E, (GVQEP) was studied by Yang and Huang [] in abstract convex spaces and by Balaj and Lin [] inG-convex spaces, respectively.
•(GVQEP) Findx˜∈Eandz˜∈B(x˜) such that
˜
x∈S(x˜) and F(x˜,y,z˜)∩C(x˜)=∅, ∀y∈S(x˜).
WhenS(x) =B(x) =Efor allx∈E, (GVQEP) was considered by Lin [] in topological vector spaces.
We would like to point out that, for a suitable choice of the spaces E,X, V and the mappingsS,B,F,C, one can obtain a number of well-known insights into the generalized vector quasi-equilibrium problem [, , , , , ], the generalized vector equilibrium problem [, , ], the vector equilibrium problem, and the vector variational inequality problem [, ] as special cases of the problems (GVQEP)-(GVQEP).
Furthermore, assume thath:X⇒Lis a set-valued mapping, whereLis a real topological vector space ordered by a closed convex pointed coneH⊆LwithintH=∅. It is clear that the existence of solutions for problems (GVQEP)-(GVQEP) is closely analogous to the existence of solutions in connection with the following generalized semi-infinite programs with generalized vector quasi-equilibrium constraints:
•(GSIP) Generalized semi-infinite program with constraint (GVQEP):
wMinHh(K),
where
K=x∈E:x∈S(x),F(x,y,z)∩–intC(x) =∅,∀y∈S(x),∀z∈B(x).
WhenS(x) =B(x) =Efor allx∈E, (GSIP) was considered by Yang and Huang [] in abstract convex spaces.
•(GSIP) Generalized semi-infinite program with constraint (GVQEP):
wMinHh(K),
where
K=x∈E:x∈S(x),∃z∈B(x),F(x,y,z)⊆C(x),∀y∈S(x).
Some special cases of (GSIP) were considered by Lin [] in topological vector spaces.
•(GSIP) Generalized semi-infinite program with constraint (GVQEP):
wMinHh(K),
where
WhenS(x) =B(x) =Efor allx∈E, (GSIP) was studied by Yang and Huang [] in abstract convex spaces.
•(GSIP) Generalized semi-infinite program with constraint (GVQEP):
wMinHh(K),
where
K=x∈E:x∈S(x),∃z∈B(x),F(x,y,z)∩–intC(x) =∅,∀y∈S(x).
We would like to mention that some special cases of (GSIP) were studied by Lin [] in topological vector spaces.
•(GSIP) Generalized semi-infinite program with constraint (GVQEP):
wMinHh(K),
where
K=x∈E:x∈S(x),F(x,y,z)–intC(x),∀y∈S(x),∀z∈B(x).
WhenS(x) =B(x) =Efor allx∈E, (GSIP) was investigated by Yang and Huang [] in abstract convex spaces.
•(GSIP) Generalized semi-infinite program with constraint (GVQEP):
wMinHh(K),
where
K=x∈E:x∈S(x),∃z∈B(x),F(x,y,z)–intC(x),∀y∈S(x).
We note that some special cases of (GSIP) were considered by Lin [] in topological vector spaces.
•(GSIP) Generalized semi-infinite program with constraint (GVQEP):
wMinHh(K),
where
K=x∈E:x∈S(x),F(x,y,z)∩C(x)=∅,∀y∈S(x),∀z∈B(x).
WhenS(x) =B(x) =Efor allx∈E, (GSIP) was studied by Yang and Huang [] in abstract convex spaces.
•(GSIP) Generalized semi-infinite program with constraint (GVQEP):
where
K=x∈E:x∈S(x),∃z∈B(x),F(x,y,z)∩C(x)=∅,∀y∈S(x).
It is worth mentioning that (GSIP) can be considered as a generalization of the general-ized vector semi-infinite programming introduced and studied by Lin [] in topological vector spaces.
In brief, for suitable choice of the spacesL,V,X,Eand the mappingsS,B,F,C,h, one can obtain a number of known the generalized semi-infinite program [], the mathematical program with equilibrium constraint [], the generalized semi-infinite program [], the generalized vector semi-infinite programming [], and the vector optimization problem [–] as special cases from the problems (GSIP)-(GSIP).
Now, we recall some useful definitions and lemmas as follows.
Definition . LetK⊆V be a nonempty set andC⊆V be the closed convex pointed cone withintC=∅. The set of all weak minimal points ofKwith respect to the ordering coneCis defined as
wMinC(K) =
x∈K: (x–K)∩intC=∅.
Definition . Let (X,D,) be an abstract convex space andZbe a set. For a set-valued mappingT:X⇒Zwith nonempty values, if a set-valued mappingG:D⇒Zsatisfies
T(N)⊆G(N) :=
y∈N
G(y) for allN∈ D,
thenGis called a KKM mapping with respect toT. A KKM mappingG:D⇒Xis a KKM mapping with respect to the identity mappingIX.
A set-valued mappingF:X⇒Zis called to be aKC-map if, for any closed-valued KKM mappingG:D⇒Zwith respect toF, the family{G(y)}y∈Dhas the finite intersection
prop-erty. We denote
KC(X,Z) :={F:FisKC-map}.
Definition .([]) LetXandY be two topological spaces. A set-valued mappingF:
X⇒Yis said to be
(i) upper semicontinuous (u.s.c.) atxif for any open setV⊇F(x), there is an open
neighborhoodOxofxsuch thatF(x)⊆V for eachx∈Ox,
(ii) lower semicontinuous (l.s.c.) atxif for any open setV∩F(x)=∅, there is an open
neighborhoodOxofxsuch thatF(x)∩V=∅for eachx∈Ox, (iii) continuous atxif it is both upper and lower semicontinuous atx,
(iv) upper semicontinuous (lower semicontinuous or continuous) onXif it is upper semicontinuous (lower semicontinuous or continuous) at everyx∈X,
(v) closed if and only if its graphGraph(F) :={(x,y)∈X×Y:y∈F(x)}is closed.
(i) IfYis compact,thenFis closed if and only if it is upper semicontinuous,
(ii) ifXis a compact space andFis a u.s.c.mapping with compact values,thenF(X)is a compact subset ofY.
Lemma .([]) Let X and Y be two topological spaces and F:X⇒Y be upper semi-continuous and F(x)is compact.Then for any net{xα} ⊂X with xα→x and yα∈F(xα),
there exists a subnet{yβ} ⊂yαsuch that yβ→y∈F(x).
Lemma .([]) Let X and Y be two topological spaces and F:X⇒Y be lower semi-continuous at x∈X if and only if for any y∈F(x)and any net{xα}with xα→x,there is a
net{yα}such that yα∈F(xα)and yα→y.
Lemma . ([]) Let(X,D,)be an abstract convex space, Z a set,and T:X⇒Z a set-valued mapping.Then F∈KC(X,Z)if and only for any G:D⇒Z satisfying
(i) Gis closed-values;
(ii) F(N)⊆G(N)for anyN∈ D, we have
F(E)G(y) :y∈N=∅
for each N∈ D.
Lemma .([]) Assume that A is a nonempty compact subset of a real topological vector space V and D is a closed convex cone in V with D=V.Then one haswMinDA=∅.
An abstract convex space with any topology is called an abstract convex topological space. In the rest of this paper, let (X,) be an abstract convex Hausdorff topological space andEbe a nonempty compact subset ofX. LetVbe a topological vector spaces. Assume thatT:X⇒X,B:E⇒E,S:E⇒E,F:E×E×E⇒VandQ:E⇒Vare five set-valued mappings. Letρbe a binary relation on V andρc be the complementary relation ofρ.
Letαbe any of the quantifiers∀,∃, andα¯be the other of the quantifiers∀,∃.
3 Main results
In order to show the existence of solutions for the vector quasi-equilibrium problems (GVQEP)-(GVQEP), we first give the following general result.
Theorem . Suppose that the following conditions are satisfied: (i) T∈KC(X,X);
(ii) for eachy∈E,the set{x∈E: (α¯)z∈B(x),ρc(F(x,y,z),Q(x))}is open inE;
(iii) G={x∈E:x∈/S(x)}is open inE;
(iv) for eachx∈E,S(x)is nonempty-convex,S–(y)is open for ally∈E;
(v) for each(x,y)∈E×Ewithx∈T(y)such thaty∈/S(x).
Then there existsx˜∈S(x˜)such that(α)z∈B(x˜),ρ(F(x˜,y,z),Q(x˜))for any y∈S(x˜).
Proof For anyx∈E, defineA:E⇒Eby
From the definition ofA(x), one has
A–(y) =x∈E: (α¯)z∈B(x),ρcF(x,y,z),Q(x). DefineP:E⇒Eby
P(x) =
S(x)∩A(x), x∈E\G; S(x), x∈G.
()
LetM(y) =E\P–(y). We show thatM(y) is closed for ally∈E. In fact, it follows from ()
that
P–(y) =x∈E\G:y∈S(x)∩A(x)
∪x∈G:y∈S(x)
=x∈E\G:x∈S–(y)∩A–(y)
∪x∈G:x∈S–(y)
=(E\G)∩S–(y)∩A–(y)
∪G∩S–(y)
=S–(y)∩G∪A–(y)
.
SinceS–(y),A–(y), andG
are open, we know thatP–(y) is open and soM(y) is closed.
We show thatMis a KKM mapping with respect toT. Suppose thatMis not a KKM mapping with respect toT. Then there exist a finite subsetNand a pointx∈Esuch that x∈T(N)\M(N). This shows that there exists a pointy∈N such thatx∈T(y),x∈ P–(y) for anyy∈N, and soN⊂P(x)⊂S(x). SinceS(x) is-convex andN∈ S(x), we
know thaty∈N⊂S(x), which is a contradiction. It follows thatMis a KKM mapping
with respect toT.
It follows from Lemma . thatMhas finite intersection property. From the facts that
M(y)⊂Eis closed andEis compact, we know thatM(y) is compact for anyy∈Eand so
y∈E
M(y)=∅.
Thus, there exists a pointx˜∈Esuch that
˜ x∈
y∈E
M(y) =E\ y∈E
P–(y).
This implies thatx˜∈/P–(y) for ally∈Eand soP(x˜) =∅.
Ifx˜∈G, then it is easy to see thatS(x˜) =P(x˜) =∅, which is a contradiction. Therefore,
we have
˜
x∈E\G withS(x˜)∩A(x˜) =P(x˜) =∅
and so
˜
x∈S(x˜) and y∈/A(x˜),∀y∈S(x˜),
Remark . By Lemma ., it is easy to see that the condition (iii) can be replaced by the following condition:
(iii) S:E⇒Eis a u.s.c. set-valued mapping.
Next we give some existence theorems in connection with the solution of the vector quasi-equilibrium problems (GVQEP)-(GVQEP).
Theorem . Assume that the conditions(i), (iii), (iv),and(v)in Theorem.are satisfied.
Moreover,suppose that
(a) for eachy∈E,F(·,y,·)is l.s.c.andCis closed; (b) Bis l.s.c.
Then there existsx˜∈E such thatx˜∈S(x˜)and F(x˜,y,z)⊂C(x˜)for all y∈S(x˜)and z∈B(x˜).
Proof Let
A(x) =y∈E:∃z∈B(x),F(x,y,z)C(x).
We show that
A–(y) =x∈E:∃z∈B(x),F(x,y,z)C(x)
is open. Let{xα} ⊆E\A–(y) be a net withxα→x. Then
Fxα,y,z⊆C(xα), ∀z∈B(xα).
SinceBandF(·,y,·) are l.s.c., by Lemma ., for anyz∈B(x) andv∈F(x,y,z), there exist zα∈B(xα) andvα∈F(xα,y,zα) such thatzα→zandvα→v. Now the closedness ofCwith
vα∈C(xα) shows thatv∈C(x) and soF(x,y,z)⊆C(x) for anyz∈B(x). This shows that x∈E\A–(y) and soE\A–(y) is closed. Thus,A–(y) is open. It follows from Theorem .
that there existsx˜∈Esuch thatx˜∈S(x˜) andF(x˜,y,z)⊆C(x˜) for ally∈S(x˜) andz∈B(x˜).
This completes the proof.
Remark . Theorem . can be considered as a generalization of Theorem . in [] under different conditions from the topological vector space to the abstract convex space.
Corollary . Assume that the conditions(i), (iii), (iv),and(v)in Theorem.are satisfied with B=S.Suppose that,for each y∈E,F(·,y)is l.s.c.and C is closed.Then there exists ˜
x∈E such thatx˜∈S(x˜)and F(x˜,y)⊆C(x˜)for all y∈S(x˜).
Proof The proof is similar to that of Theorem . and so we omit it here.
Remark . WhenS(x) =Efor allx∈E, Corollary . was given by Theorem of Yang and Huang [] under quite different conditions.
Theorem . Assume that the conditions(i), (iii), (iv),and(v)in Theorem.are satisfied.
Moreover,suppose that
(b) Bis u.s.c.andB(x)is compact for eachx∈E.
Then there existx˜∈E and˜z∈B(x˜)such thatx˜∈S(x˜)and F(x˜,y,z˜)⊆C(x˜)for all y∈S(x˜).
Proof Let
A(x) =y∈E:∀z∈B(x),F(x,y,z)C(x).
We first show that
A–(y) =x∈E:∀z∈B(x),F(x,y,z)C(x)
is open. Let{xα} ⊆E\A–(y) be a net withxα→x. Then there existszα∈B(xα) such thatF(xα,y,zα)⊆C(xα). SinceBis u.s.c. with compact values, by Lemma ., there ex-ists a subset net of{zα}, denoted again by{zα}, such thatzα→z∈B(x). The fact that F(·,y,·) is l.s.c. together with Lemma . shows that, for anyv∈F(x,y,z), there exists vα∈F(xα,y,zα) such thatvα→v. Sincevα∈C(xα) andCis closed, we know thatv∈C(x)
and soF(x,y,z)⊆C(x) for somez∈B(x). This implies thatx∈E\A–(y) and so E\A–(y) is closed. Thus,A–(y) is open. It follows from Theorem . that there exist ˜
x∈Eandz˜∈B(x˜) such thatx˜∈S(x˜) andF(x˜,y,˜z)⊆C(x˜) for anyy∈S(x˜). This completes
the proof.
Remark . WhenS(x) =E for allx∈E, the existence of the solutions for generalized vector quasi-equilibrium was studied in Theorem . of [] in real Hausdorff topological vector spaces.
Theorem . Assume that the conditions(i), (iii), (iv),and(v)in Theorem.are satisfied.
Moreover,suppose that
(a) for eachy∈E,F(·,y,·)is l.s.c.,C(x)is a set-valued mapping with a nonempty interior for eachx∈E,the mappingW:E⇒V,defined byW(x) =V\(–intC(x)),is closed; (b) Bis l.s.c.
Then there existsx˜∈E such thatx˜∈S(x˜)and F(x˜,y,z)∩(–intC(x˜)) =∅for all y∈S(x˜)and z∈B(x˜).
Proof Let
A(x) =y∈E:∃z∈B(x),F(x,y,z)∩–intC(x)=∅.
We prove that
A–(y) =x∈E:∃z∈B(x),F(x,y,z)∩–intC(x)=∅
is open. Let{xα} ⊆E\A–(y) be a net withxα→x. Then
Fxα,y,z∩–intC(xα)=∅, ∀z∈B(xα)
and so
Similar to the proof of Theorem ., we get
F(x,y,z)⊆W(x) =V\
–intC(x)
.
This shows thatx∈E\A–(y) and soE\A–(y) is closed. Thus,A–(y) is open. It follows
from Theorem . that there existsx˜∈Esuch thatx˜∈S(x˜) and
F(x˜,y,z)∩–intC(x˜)=∅, ∀y∈S(x˜),∀z∈B(x˜).
This completes the proof.
Remark . Theorem . can be considered as a generalization of Theorem . in [] under different conditions from the topological vector space to the abstract convex space.
Corollary . Assume that the conditions(i), (iii), (iv),and(v)in Theorem.are satisfied with S=B.Moreover,suppose that
(a) for eachy∈E,F(·,y)is l.s.c.,C(x)has a nonempty interior for eachx∈E,the map
W:E⇒V,defined byW(x) =V\(–intC(x)),is closed.
Then there existsx˜∈E such thatx˜∈S(x˜)and F(x˜,y)∩(–intC(x˜)) =∅for all y∈S(x˜).
Proof The proof is similar to that of Theorem . and so we omit it here.
Remark . WhenS(x) =Efor allx∈E, Corollary . was given by Theorem of Yang and Huang [] under quite different conditions.
Theorem . Suppose the conditions(i), (iii), (iv),and(v)in Theorem.are satisfied.
Moreover,suppose that
(a) for eachy∈E,F(·,y,·)is l.s.c.,C(x)has a nonempty interior for eachx∈E,and the mappingW:E⇒V,defined byW(x) =V\(–intC(x)),is closed;
(b) Bis u.s.c.andB(x)is compact for eachx∈E.
Then there existx˜∈E andz˜∈B(x˜)such thatx˜∈S(x˜)and F(x˜,y,˜z)∩(–intC(x˜)) =∅for all y∈S(x˜).
Proof Let
A(x) =y∈E:∀z∈B(x),F(x,y,z)∩–intC(x)=∅.
We show that
A–(y) =x∈E:∀z∈B(x),F(x,y,z)∩–intC(x)=∅
is open. Let{xα} ⊆E\A–(y) be a net withxα→x. Then
F(xα,y,zα)∩–intC(xα)=∅
for somezα∈B(xα), that is,
Using similar arguments to the proof of Theorem ., we have
F(x,y,z)⊆W(x) =V\
–intC(x)
for somez∈B(x). This shows thatx∈E\A–(y) and soE\A–(y) is closed. Thus,A–(y)
is open. It follows from Theorem . that there existx˜∈Eandz˜∈B(x˜) such thatx˜∈S(x˜) and
F(x˜,y,z˜)∩–intC(x˜)=∅, ∀y∈S(x˜).
This completes the proof.
Remark . WhenEis a nonempty convex compact of a topological vector space, Li and Li [] studied the existence of solutions for (GVQEP).
Theorem . Assume that the conditions(i), (iii), (iv),and(v)are satisfied in Theorem..
Moreover,suppose that
(a) for eachy∈E,F(·,y,·)is u.s.c.with compact valued onE×E×EandC(x)has a nonempty interior for eachx∈E,the mappingW:E⇒V,defined by
W(x) =V\(–intC(x)),is closed; (b) Bis l.s.c.
Then there existsx˜∈E such thatx˜∈S(x˜)and F(x˜,y,z)–intC(x˜)for all y∈S(x˜)and z∈B(x˜).
Proof Let
A(x) =y∈E:∃z∈B(x),F(x,y,z)⊆–intC(x).
We prove that
A–(y) =x∈E:∃z∈B(x),F(x,y,z)⊆–intC(x)
is open. Letxα∈E\A–(y) be a net withxα→x. Then
Fxα,y,z–intC(xα)
for anyz∈B(xα) and so there existsvα∈V such that
vα∈F
xα,y,z\–intC(xα).
SinceBis l.s.c., by Lemma ., for anyz∈B(x), there existszα∈B(xα) such thatzα→z. SinceF(·,y,·) is u.s.c. with compact valued, by Lemma ., there exists a subset net of{vα}, denoted again by {vα}, such that vα→v∈F(x,y,z). On the other hand, the fact that vα∈/–intC(xα) shows thatvα∈W(xα). Now the closedness ofW shows thatv∈W(x)
and sov∈/ –intC(x). ThusF(x,y,z)–intC(x) for anyz∈B(x). This implies that x∈E\A–(y) and soE\A–(y) is closed. Thus,A–(y) is open. It follows from Theorem .
that there existsx˜∈Esuch thatx˜∈S(x˜) andF(x˜,y,z)–intC(x˜) for ally∈S(x˜) andz∈
Corollary . Assume that the conditions(i), (iii), (iv),and(v)in Theorem.are satisfied with S=B.Moreover,suppose that
(a) for eachy∈E,F(·,y)is u.s.c.with compact valued onE×EandC(x)has a nonempty interior for eachx∈E,the mappingW:E⇒V,defined by
W(x) =V\(–intC(x)),is closed.
Then there existsx˜∈E such thatx˜∈S(x˜)and F(x˜,y)–intC(x˜)for all y∈S(x˜).
Proof The proof is similar to that of Theorem . and so we omit it here.
Remark . WhenS(x) =Efor allx∈E, Corollary . was given by Theorem of Yang and Huang [] under quite different conditions.
Theorem . Assume that the conditions(i), (iii), (iv),and(v)are satisfied in Theorem..
Moreover,suppose that
(a) for eachy∈E,F(·,y,·)is u.s.c.with compact valued onE×E×EandC(x)has a nonempty interior for eachx∈E,the mappingW:E⇒V,defined by
W(x) =V\(–intC(x)),is closed.
(b) Bis u.s.c.andB(x)is compact for eachx∈E.
Then there existx˜∈E andz˜∈B(x˜)such thatx˜∈S(x˜)and F(x˜,y,z˜)–intC(x˜)for all y∈S(x˜).
Proof Let
A(x) =y∈E:∀z∈B(x),F(x,y,z)⊆–intC(x).
We prove that
A–(y) =x∈E:∀z∈B(x),F(x,y,z)⊆–intC(x)
is open. Letxα∈E\A–(y) be a net withxα→x. ThenF(xα,y,zα)–intC(xα) for some zα∈B(xα) and so there existsvα∈Vsuch that
vα∈F(xα,y,zα)\
–intC(xα).
SinceBis u.s.c. with compact valued, by Lemma ., there exists a subnet of{zα}, denoted again by{zα}, such thatzα→z∈B(x). The fact thatF(·,y,·) is u.s.c. with compact valued
together with Lemma . shows that there exists a subset net of{vα}, denoted again by{vα}, such thatvα→v∈F(x,y,z). On the other hand, it is easy to see thatvα∈W(xα). Since
Wis closed, we know thatv∈W(x) and sov∈/–intC(x). ThusF(x,y,z)–intC(x)
for somez∈B(x) and sox∈E\A–(y). This implies thatE\A–(y) is closed and so A–(y) is open. It follows from Theorem . that there existx˜∈Eandz˜∈B(x˜) such that
˜
x∈S(x˜) andF(x˜,y,z˜)–intC(x˜) for ally∈S(x˜). This completes the proof.
Remark . Theorem . can be considered as a generalization of Theorem . in [, ] under different conditions from the topological vector space to the abstract convex space.
Theorem . Suppose the conditions(i), (iii), (iv),and(v)in Theorem.are satisfied.
Moreover,assume that
(a) for eachy∈E,F(·,y,·)is u.s.c.with compact valued onE×E×EandCis closed; (b) Bis l.s.c.
There existsx˜∈E such thatx˜∈S(x˜)and F(x˜,y,z)∩C(x˜)=∅for all y∈S(x˜)and z∈B(x).
Proof Let
A(x) =y∈E:∃z∈B(x),F(x,y,z)∩C(x) =∅.
We show that
A–(y) =x∈E:∃z∈B(x),F(x,y,z)∩C(x) =∅
is open. Let{xα} ⊆E\A–(y) be a net withxα→x. Then
Fxα,y,z∩C(xα)=∅, ∀z∈B(xα).
It follows that there existsvα∈F(xα,y,z)∩C(xα). SinceBis l.s.c., by Lemma ., there existszα∈B(xα) such thatzα→zfor anyz∈B(x). By the fact thatF(·,y,·) is u.s.c. with
compact valued, there exists a subset of{vα}, denoted again by{vα}, such thatvα→v∈ F(x,y,z). Sincevα∈C(xα) andCis closed, we know thatv∈C(x) and sov∈F(x,y,z)∩ C(x). Thus,
F(x,y,z)∩C(x)=∅, ∀z∈B(x).
This shows thatx∈E\A–(y) and soE\A–(y) is closed. Thus,A–(y) is open. By
Theo-rem ., there existsx˜∈Esuch thatx˜∈S(x˜) and
F(x˜,y,z)∩C(x)=∅, ∀y∈S(x˜),∀z∈B(x˜).
This completes the proof.
Corollary . Assume that the conditions(i), (iii), (iv),and(v)in Theorem.are satisfied with S=B.Moreover,suppose that,for each y∈E,F(·,y)is u.s.c.with compact valued on E×E and C is closed.Then there existsx˜∈E such thatx˜∈S(x˜)and F(x˜,y)∩C(x˜)=∅for all y∈S(x˜).
Proof The proof is similar to that of Theorem . and so we omit it here.
Remark . WhenS(x) =Efor allx∈E, Corollary . was given by Theorem of Yang and Huang [] under some different conditions.
Theorem . Suppose the conditions(i), (iii), (iv),and(v)are satisfied in Theorem..
Moreover,assume that
(a) for eachy∈E,F(·,y,·)is u.s.c.with compact valued onE×E×EandCis closed; (b) Bis u.s.c.andB(x)is compact for eachx∈E.
Proof Let
A(x) =y∈E:∀z∈B(x),F(x,y,z)∩C(x) =∅.
We prove that
A–(y) =x∈E:∀z∈B(x),F(x,y,z)∩C(x) =∅
is open. Let{xα} ⊆E\A–(y) be a net withxα→x. Then
F(xα,y,zα)∩C(xα)=∅
for somezα∈B(xα), that is, there existsvα∈F(xα,y,zα)∩C(xα). SinceBis u.s.c. andB(x) is compact, it follows from Lemma . that there exists a subset of{zα}, denoted again by {zα}, such thatzα→z∈B(x). Similar to the proof of Theorem ., we can prove
thatF(x,y,z)∩C(x)=∅for somez∈B(x). This shows thatx∈E\A–(y) and so E\A–(y) is closed. Thus,A–(y) is open. It follows from Theorem . that there exist
˜
x∈Eandz˜∈B(x˜) such thatx˜∈S(x˜) and
F(x˜,y,z˜)∩C(x˜)=∅, ∀y∈S(x˜).
This completes the proof.
4 Applications to the generalized semi-infinite programs
In this section, by the results presented in Section , we give some existence theorems of solutions to the generalized semi-infinite programs. LetLbe a real topological vector space ordered by a closed convex pointed coneH⊆LwithintH= ∅andh:X⇒Lbe a u.s.c. mapping with compact values.
Theorem . Suppose that all the conditions of Theorem.are satisfied.Moreover, as-sume that F(·,·,·)and S are l.s.c.Then there is a solution to the problem
wMinHh(K),
where
K=x∈E:x∈S(x),F(x,y,z)⊆C(x),∀y∈S(x),∀z∈B(x).
Proof Theorem . shows thatK=∅. From Lemma ., it is sufficient to show thath(K) is compact. Sincehis u.s.c. andK⊆E, by Lemma ., we only need to prove thatK is closed. Let{xα} ⊆Kbe a net withxα→x. Thenxα∈S(xα) and
Fxα,y,z⊆C(xα), ∀y∈S(xα),∀z∈B(xα).
SinceSandBare l.s.c., for anyy∈S(x) andz∈B(x), it follows from Lemma . that there
ofFand Lemma ., for anyv∈F(x,y,z), there existsvα∈F(xα,yα,zα) such thatvα→v. Now the closedness ofCwithvα∈C(xα) shows thatv∈C(x) and soF(x,y,z)⊆C(x)
for ally∈S(x) andz∈B(x). Moreover, the closedness ofE\Gshows thatx∈S(x).
Thus,Kis closed. This completes the proof.
Corollary . Suppose that all the conditions of Corollary.are satisfied.Moreover, as-sume that F(·,·)and S are l.s.c.Then there is a solution to the problem
wMinHh(K),
where
K=x∈E:x∈S(x),F(x,y)⊆C(x),∀y∈S(x).
Remark . WhenS(x) =Efor allx∈E, Corollary . was given by Theorem of Yang and Huang [] under some different conditions.
Theorem . Suppose that all the conditions of Theorem.are satisfied.Moreover, as-sume that F(·,·,·)and S are l.s.c.Then there is a solution to the problem
wMinHh(K),
where
K=x∈E:x∈S(x),∃z∈B(x),F(x,y,z)⊆C(x),∀y∈S(x).
Proof Obviously, Theorem . shows thatK=∅. By Lemma ., it is sufficient to prove thath(K) is compact. Sincehis u.s.c. andK⊆E, from Lemma ., we only need to show thatK is closed. Let{xα} ⊆K be a net withxα→x. Thenxα∈S(xα) and there exists
zα∈B(xα) such that
Fxα,y,zα
⊆C(xα), ∀y∈S(xα).
SinceBis a u.s.c. mapping with compact values, it follows from Lemma . that there exists a subnet of{zα}, denoted again by{zα}, such thatzα→z∈B(x). For anyy∈S(x), the
lower semi-continuity ofStogether with Lemma . implies that there existsyα∈S(xα) such thatyα→y. Forv∈F(x,y,z), by the fact thatFis l.s.c., it follows from Lemma .
that there existsvα∈F(xα,yα,zα) such thatvα→v. Now the closedness ofCwithvα∈
C(xα) shows thatv∈C(x) and so there existsz∈B(x) such thatF(x,y,z)⊆C(x) for
ally∈S(x). Moreover, the closedness ofE\Gshows thatx∈S(x). Thus,Kis closed.
This completes the proof.
Theorem . Suppose that all the conditions of Theorem.are satisfied.Moreover, as-sume that F(·,·,·)and S are l.s.c.Then there is a solution to the problem
where
K=x∈E:x∈S(x),F(x,y,z)∩–intC(x) =∅,∀y∈S(x),∀z∈B(x).
Proof Theorem . shows thatK=∅. From Lemma ., it is sufficient to show thath(K) is compact. Sincehis u.s.c. andK⊆E, by Lemma ., we only need to show thatKis closed. Let{xα} ⊆Kbe a net withxα→x. Thenxα∈S(xα),
Fxα,y,z∩–intC(x) =∅, ∀y∈S(xα),∀z∈B(xα)
and so
Fxα,y,z⊆W(xα), ∀y∈S(xα),∀z∈B(xα).
Similar to the proof of Theorem ., we havex∈S(x),
F(x,y,z)⊆W(x), ∀y∈S(x),∀z∈B(x)
and so
F(x,y,z)∩–intC(x) =∅, ∀y∈S(x),∀z∈B(x).
Thus,Kis closed. This completes the proof.
Corollary . Suppose that all the conditions of Corollary.are satisfied.Moreover,
as-sume that F(·,·)and S are l.s.c.Then there is a solution to the problem
wMinHh(K),
where
K=x∈E:x∈S(x),F(x,y)∩C(x) =∅,∀y∈S(x).
Remark . WhenS(x) =Efor allx∈E, Corollary . was given by Theorem of Yang
and Huang [] under some different conditions.
Theorem . Suppose that all the conditions of Theorem.are satisfied.Moreover,
as-sume that F(·,·,·)and S are l.s.c.Then there is a solution to the problem
wMinHh(K),
where
Proof It follows from Theorem . thatK=∅. From Lemma ., it is sufficient to show thath(K) is compact. Sincehis u.s.c. andK⊆E, by Lemma ., we only need to showK
is closed. Let{xα} ⊆Kbe a net withxα→x. Thenxα∈S(xα) and there existszα∈B(xα) such that
Fxα,y,zα
∩–intC(xα) =∅, ∀y∈S(xα).
Thus, there existszα∈B(xα) such that
Fxα,y,zα
⊆C(xα), ∀y∈S(xα).
Similar to the proof of Theorem ., we know thatx∈S(x) and there existsz∈B(x)
such that
F(x,y,z)⊆W(x), ∀y∈S(x).
Thus,x∈S(x) and there existsz∈B(x) such that
F(x,y,z)∩–intC(x) =∅, ∀y∈S(x).
It follows thatKis closed. This completes the proof.
Theorem . Suppose that all the conditions of Theorem.are satisfied.Moreover, as-sume that F(·,·,·)is a u.s.c.mapping with compact values and S is l.s.c.Then there is a solution to the problem
wMinHh(K),
where
K=x∈E:x∈S(x),F(x,y,z)–intC(x),∀y∈S(x),∀z∈B(x).
Proof Theorem . shows thatK=∅. From Lemma ., it is sufficient to show thath(K) is compact. Sincehis u.s.c. andK⊆E, by Lemma ., we only need to prove thatK is closed. Let{xα} ⊆Kbe a net withxα→x. Thenxα∈S(xα),
Fxα,y,z–intC(xα), ∀y∈S(xα),∀z∈B(xα)
and so there existsvα∈Vsuch that
vα∈F
xα,y,z\–intC(xα).
By the lower semi-continuity ofSandB, for anyy∈S(x) andz∈B(x), it follows from
Lemma . that there existyα∈S(xα) andzα∈B(xα) such thatyα→yandzα→z. Since
thatvα∈/ –intC(xα) shows thatvα∈W(xα). Now the closedness ofW shows that v∈ W(x) and sov∈/–intC(x). Moreover, the closedness ofE\Gshows thatx∈S(x).
Thus,
F(x,y,z)–intC(x)
for ally∈S(x) andz∈B(x) and soKis closed. This completes the proof.
Corollary . Suppose that all the conditions of Corollary.are satisfied.Moreover, as-sume that F(·,·)is u.s.c.and S is l.s.c.Then there is a solution to the problem
wMinHh(K),
where
K=x∈E:x∈S(x),F(x,y)C(x),∀y∈S(x).
Remark . WhenS(x) =Efor allx∈E, Corollary . was given by Theorem of Yang and Huang [] under some different conditions.
Theorem . Suppose that all the conditions of Theorem.are satisfied.Moreover, as-sume that F(·,·,·)is a u.s.c.mapping with compact values and S is l.s.c.Then there is a solution to the problem
wMinHh(K),
where
K=x∈E:x∈S(x),∃z∈B(x),F(x,y,z)–intC(x),∀y∈S(x).
Proof Theorem . shows thatK=∅. By Lemma ., it is sufficient to prove thath(K) is compact. Sincehis u.s.c. andK⊆E, from Lemma ., we only need to show thatK is closed. Let{xα} ⊆K be a net withxα→x. Thenxα∈S(xα) and there existszα∈B(xα) such that
Fxα,y,zα
–intC(xα), ∀y∈S(xα).
Thus, there existsvα∈V such that
vα∈F
xα,y,zα
\–intC(xα).
SinceBis a u.s.c. mapping with compact values, it follows from Lemma . that there exists a subnet of{zα}, denoted again by{zα}, such thatzα→z∈B(x). By the lower
semi-continuity ofS, for anyy∈S(x), Lemma . shows that there existsyα∈S(xα) such thatyα→y. SinceF(·,·,·) is a u.s.c. mapping with compact values, Lemma . implies that there exists a subnet of{vα}, denoted again by{vα}, such thatvα→v∈F(x,y,z). Similar
Theorem . Suppose that all the conditions of Theorem.are satisfied.Moreover, as-sume that F(·,·,·)is a u.s.c.mapping with compact values and S is l.s.c.Then there is a solution to the problem
wMinHh(K),
where
K=x∈E:x∈S(x),F(x,y,z)∩C(x)=∅,∀y∈S(x),∀z∈B(x).
Proof Theorem . shows thatK=∅. From Lemma ., it is sufficient to show thath(K) is compact. Sincehis u.s.c. andK⊆E, by Lemma ., we only need to showKis closed. Let{xα} ⊆Kbe a net withxα→x. Thenxα∈S(xα),
Fxα,y,z∩C(xα)= ∅, ∀y∈S(xα),z∈B(xα)
and so there existsvα∈Vsuch that
vα∈F
xα,y,z∩C(xα).
By the lower semi-continuity ofSandB, for anyy∈S(x) andz∈B(x), Lemma . shows
that there existyα∈S(xα) andzα∈B(xα) such thatyα→yandzα→z. SinceF(·,·,·) is u.s.c. with compact values, by Lemma ., there exists a subnet of{vα}, denoted again by
{vα}, such thatvα→v∈F(x,y,z). Now the closedness ofCwithvα∈C(xα) shows that
v∈C(x) and so
F(x,y,z)∩C(x)=∅, ∀y∈S(x),∀z∈B(x).
Moreover, the closedness ofE\Gshows thatx∈S(x). Thus,Kis closed. This completes
the proof.
Corollary . Suppose that all the conditions of Corollary.are satisfied.Moreover, as-sume that F(·,·)and S are l.s.c.Then there is a solution to the problem
wMinHh(K),
where
K=x∈E:x∈S(x),F(x,y)∩C(x)=∅,∀y∈S(x).
Remark . WhenS(x) =Efor allx∈E, Corollary . was given by Theorem of Yang and Huang [] under some different conditions.
Theorem . Suppose that all the conditions of Theorem.are satisfied.Moreover, as-sume that F(·,·,·)is a u.s.c.mapping with compact values and S is l.s.c.Then there is a solution to the problem
where
K=x∈E:x∈S(x),∃z∈B(x),F(x,y,z)∩C(x)=∅,∀y∈S(x).
Proof Theorem . shows thatK=∅. From Lemma ., it is sufficient to prove thath(K) is compact. Sincehis u.s.c. andK⊆E, by Lemma ., we only need to showKis closed. Let{xα} ⊆Kbe a net withxα→x. Thenxα∈S(xα) and there existszα∈B(xα) such that
Fxα,y,zα
∩C(xα)=∅, ∀y∈S(xα).
Thus, there existsvα∈V such that
vα∈F
xα,y,zα
∩C(xα).
SinceBis a u.s.c. mapping with compact values, it follows from Lemma . that there exists a subnet of{zα}, denoted again by{zα}, such thatzα→z∈B(x). By the lower
semi-continuity ofS, for anyy∈S(x), Lemma . implies that there existsyα∈S(xα) such thatyα→y. SinceF(·,·,·) is a u.s.c. mapping with compact values, by Lemma ., there exists a subnet of{vα}, denoted again by{vα}, such thatvα→v∈F(x,y,z). Now the
closedness ofCwithvα∈C(xα) shows thatv∈C(x) and so there existsz∈B(x) such
that
F(x,y,z)∩C(x)=∅, ∀y∈S(x).
Moreover, the closedness ofE\G shows thatx∈S(x). Therefore,K is closed. This
completes the proof.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank Professor Ravi P Agarwal for his valuable suggestions and comments. This work was supported the National Natural Science Foundation of China (11171237, 11471230).
Received: 6 November 2014 Accepted: 30 January 2015 References
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