Zhang et al. Fixed Point Theory and Applications (2015) 2015:29 DOI 10.1186/s13663-015-0277-6

0 Fixed Point Theory and Applications

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Existence of solutions for generalized vector quasi-equilibrium problems in abstract convex spaces with applications

Wei-bing Zhang,,Shu-qiang Shan1 and Nan-jing Huang1,2*

"Correspondence: nanjinghuang@hotmail.com 1 Department 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 T-convex and KC-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 generalization of the vector equilibrium problem which provides a unified model for vector quasi-variational inequalities, vector quasi-complementarity problems, vector optimization problems and vector saddle point problems. In 2000, Fu [1] established the existence theorems for the generalized vector quasi-equilibrium problems and the set-valued vector equilibrium problems. In 2003, Ansari and Fabián [2] considered a generalized vector quasi-equilibrium problem with or without involving $-condensing mappings and proved the existence of its solution in real topological vector spaces. In 2005, Li et al. [3] studied the existence of solutions for two classes of generalized vector quasi-equilibrium problems. Recently, Lin et al. [4] introduced and studied a class of generalized vector quasi-equilibrium problems involving pseudomonotonicity hemicontinuity mappings under different conditions in topological vector spaces. Lin etal. [5] proved the existence of equilibria for generalized abstract economy with a lower semicontinuous constraint correspondence and a fuzzy constraint correspondence defined on a noncompact/nonparacompact strategy set. They also considered a systems of generalized vector quasi-equilibrium problems in topological vector spaces. Very recently, Yang and Pu [6] studied the existence and essential components in connection with the set of solutions for the system of strong vector quasi-equilibrium problems. Fu and Wang [7] considered the generalized strong vector quasi-equilibrium problems with domination structure. On the other hand, Ding [8] studied the existence of solutions for generalized vector quasi-equilibrium problems in locally

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G-convex spaces. Balaj and Lin [9] investigated existence of solutions for the generalized equilibrium problems in G-convex spaces.

The abstract convex space, introduced by Park [10] in 2006, includes the convex subset of a topological vector space, the convex space, the W-space, and the G-convex space as special cases. Moreover, Park [11] investigated the property of the abstract convex spaces and showed some applications. Recently, several authors have focused on the studies concerned with the set-valued maps and optimization problems in abstract convex spaces with applications. For instance, Cho et al. [12] studied some coincidence theorems and minimax inequalities in abstract convex spaces. Yang et al. [13] proved some maximal element theorems for set-valued maps in abstract convex spaces with applications. Yang and Huang [14] gave some coincidence theorems for compact and noncompact KC-maps in abstract convex spaces with applications. Lu and Hu [15] established a new collectively fixed point theorem in noncompact abstract convex spaces with applications to equilibria for generalized abstract economies. Park [16] gave some comments on fixed points, maximal elements, and equilibria of economies in abstract convex spaces. Yang and Huang [17] studied the existence of solutions for the generalized vector equilibrium problems in abstract convex spaces. At the end of the paper [17], 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 knowledge, 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 interesting 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 [18] initiated semi-infinite programming precisely to deduce important 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 ellipsoid, many researchers have been investigated the theory, applications and methods for the semi-infinite programming (see, for example, [19-22]). 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 applications to numerous real-life problems such as Chebyshev approximation, design centering, robust optimization, optimal layout of an assembly line, time minimal control, and disjunctive optimization (see [23] and the references therein). Therefore, it is important and interesting to study the existence of solutions concerned with some generalized semiinfinite 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 semiinfinite programs. We give some sufficient conditions to guarantee the existence of solutions for these generalized vector quasi-equilibrium problems in abstract convex spaces. As applications, we give some existence theorems of solutions for the generalized semiinfinite programs under suitable conditions.

2 Preliminaries

Let X, Y be two nonempty sets. A set-valued mapping T : X ^ Y is a mapping from X into the power set 2Y. The inverse T-1 of T is the set-valued mapping from Y to X defined

T-1(y) = {x e X: y e T(x)}.

An abstract convex space (X, D, r) consists of a nonempty set X, a nonempty set D, and a set-valued mapping F : (D) ^ X with nonempty values, where (D) denotes the set of all nonempty finite subset of a set D. If for each A e (D) with the cardinality |A | = n + 1, there exists a continuous function 4>a : An ^ r(A) such that J e (A) implies $a(Aj) c F J), where An is the standard n-simplex and Aj the face of An corresponding to J e (A), then the abstract convex space reduces to the G-convex space. Let rA := r(A) for A e (D). When D c X, the space is defined by (X 2 D, r). In this case, a subset M of X is said to be r-convex if, for any A e (M n D), we have Fa C M. In the caseX = D, let (X, r) := (X, X, r).

It is easy to see that any vector space Y is an abstract convex space with r := co, where co denotes the convex hull in the vector space Y. Next we give more examples as follows.

Example 2.1 ([10]) Let E be a topological vector space with a neighborhood system V of its origin. A subset X of E is said to be almost convex (see [24] for more details) if for any V e V and for any finite subset A = (xi,x2, ...,xn} of X, there exists a subset B = {yi, y2,...,yn} of X such that yi - xi e V for all i = 1,2,...,n and co B c X. Let FA = co B for any A e (X). Then (X, F) is a G-convex space and hence an abstract convex space.

Example 2.2 ([10]) Usually, a convex space (E,C) in the classical sense consists of a nonempty set E and a family C of subsets of E such that E itself is an element of C and C is closed under arbitrary intersection. For any given subset X c E, the C-convex hull of X is defined as by

CocX = p|(Y e C : X c Y}.

We say that X is C-convex if X = Coc X. Consider the mapping F : (E) ^ E defined by FA = Coc A. Then (E, F) is an abstract convex space.

Example 2.3 Let (M, d) be a pseudo-metric space, that is, d: M x M ^ [0, such that, for every x, y, z e M,

(i) d(x, x) = 0;

(ii) d(x, y)=d(y, x);

(iii) d(x, z) < d(x, y) + d(y, z).

For any A e (M), define a set-valued mapping F : (M) ^ M by

Fa = F(A) = p|(B: B is a closed ball containing A}. Then it is easy to see that (M, F) is an abstract convex space.

As pointed out by Park [25], the abstract convex space includes many generalized convex spaces as special cases such as L-spaces, spaces having property (H), pseudo H-spaces, M-spaces, G-H-spaces, another L-spaces, FC-spaces and others. Some more examples of the abstract convex space and comments on it can be found in the literature [10, 25, 26] and the references therein.

Let (X, Г) be an abstract convex space and V be a real topological vector space. Let E be a nonempty subset of X. Assume that S: E ^ E and B: E ^ E are two set-valued mappings. Suppose that F: X x X x X ^ V and C: X ^ V are two set-valued mappings such that for each x e X, C(x) is a closed convex cone with int C(x) = 0, here int C(x) denotes the interior of C(x). In this paper, we will consider the following generalized vector quasi-equilibrium problems in abstract convex spaces.

• (GVQEP1) Find x e E such that

x e S(x) and F (x, y, z) Я C(x), Vy e S(x), Vz e B(x).

We would like to mention that (GVQEP1) was considered by Lin et al. [4] in topological vector spaces. When S(x) = B(x) = E for all x e E, (GVQEP1) was considered by Yang and Huang [17] in abstract convex spaces and by Balaj and Lin [9] in G-convex spaces, respectively.

• (GVQEP2) Find x e E and Z e B(x) such that

xe S(x) and F(x,y,Z) Я C(x), Vy e S(x).

When C(x) was replaced by -C(x), (GVQEP2) was considered by Li and Li [27] in topological vector spaces. If S(x) = E for all x e E, then (GVQEP2) was investigated by Fu and Wang [7] in topological vector spaces.

• (GVQEP3) Find x e E such that

x e S(x) and F(x,y, z) П - int C(x) = 0, Vy e S(x), Vz e B(x).

We note that (GVQEP3) was considered by Lin et al. [4] in topological vector spaces. When S(x) = B(x) = E for all x e E, (GVQEP3) was studied by Yang and Huang [17] in abstract convex spaces and by Balaj and Lin [9] in G-convex spaces, respectively.

• (GVQEP4) Find xe E and z e B(x) such that

x e S(x) and F(x,y, z) П - int C(x) = 0, Vy e S(x).

We note that (GVQEP4) was considered by Li and Li [27] in topological vector spaces.

• (GVQEP5) Find x e E such that

x e S(x) and F(x,y, z) ^ - int C(x), Vy e S(x), Vz e B(x).

When S(x) = B(x)= E for all x e E, (GVQEP5) was investigated by Yang and Huang [17] in abstract convex spaces and by Balaj and Lin [9] in G-convex spaces, respectively.

• (GVQEP6) Find xe E and z e B(x) such that

x e S(x) and F(x,y, z) ^ - int C(x), Vy e S(x).

It is worth mentioning that (GVQEP6) was considered by Lin etal. [4], andLiandLi [27] in topological vector spaces, respectively. Moreover, some special cases of (GVQEP6) were considered by Ansari and Fabián [2] in topological vector spaces.

• (GVQEP7) Find x e E such that

x e S(x) and F(x,y,z) n C(x) = 0, Vy e S(x), Vz e B(x).

When S(x) = B(x) = E for all x e E, (GVQEP7) was studied by Yang and Huang [17] in abstract convex spaces and by Balaj and Lin [9] in G-convex spaces, respectively.

• (GVQEP8) Find x e E and z e B(x) such that

xe S(x) and F(x, y, z) n C(x) = 0, Vy e S(x).

When S(x) = B(x) = E for all x e E, (GVQEP8) was considered by Lin [28] in topological vector spaces.

We would like to point out that, for a suitable choice of the spaces E, X, V and the mappings S, B, F, C, one can obtain a number of well-known insights into the generalized vector quasi-equilibrium problem [2, 4, 5, 7, 8, 27], the generalized vector equilibrium problem [9,17, 28], the vector equilibrium problem, and the vector variational inequality problem [29, 30] as special cases of the problems (GVQEP1)-(GVQEP8).

Furthermore, assume that h: X ^ L is a set-valued mapping, where L is a real topological vector space ordered by a closed convex pointed cone H c L with intH = 0. It is clear that the existence of solutions for problems (GVQEP1)-(GVQEP8) is closely analogous to the existence of solutions in connection with the following generalized semi-infinite programs with generalized vector quasi-equilibrium constraints:

• (GSIP1) Generalized semi-infinite program with constraint (GVQEP1):

wMinHh(K), where

K = {x e E: x e S(x), F(x, y, z) n - int C(x) = 0, Vy e S(x), Vz e B(x)}.

When S(x) = B(x) = E for all x e E, (GSIP1) was considered by Yang and Huang [17] in abstract convex spaces.

• (GSIP2) Generalized semi-infinite program with constraint (GVQEP2):

wMinHh(K), where

K = {x e E: x e S(x), 3z e B(x), F(x,y, z) C C(x), Vy e S(x)}.

Some special cases of (GSIP2) were considered by Lin [28] in topological vector spaces.

• (GSIP3) Generalized semi-infinite program with constraint (GVQEP3):

wMinHh(K), where

K = {x e E: x e S(x), F(x, y, z) n - int C(x) = 0, Vy e S(x), Vz e B(x)}.

When S(x) =B(x) =E for all x e E, (GSIP3) was studiedbyYangandHuang [17] inabstract convex spaces.

• (GSIP4) Generalized semi-infinite program with constraint (GVQEP4):

wMinHh(K), where

K = {x e E: x e S(x), 3z e B(x), F(x,y, z) n - int C(x) = 0, Vy e S(x)}.

We would like to mention that some special cases of (GSIP4) were studied by Lin [28] in topological vector spaces.

• (GSIP5) Generalized semi-infinite program with constraint (GVQEP5):

wMinHh(K), where

K = {x e E: x e S(x), F(x, y, z) ^ - int C(x), Vy e S(x), Vz e B(x)}.

When S(x) = B(x) = E for all x e E, (GSIP5) was investigated by Yang and Huang [17] in abstract convex spaces.

• (GSIP6) Generalized semi-infinite program with constraint (GVQEP6):

wMinHh(K), where

K = {x e E: x e S(x), 3z e B(x), F(x,y, z) ^ - int C(x), Vy e S(x)}.

We note that some special cases of (GSIP6) were considered by Lin [28] in topological vector spaces.

• (GSIP7) Generalized semi-infinite program with constraint (GVQEP7):

wMinHh(K), where

K = {x e E: x e S(x), F(x, y, z) n C(x) = 0, Vy e S(x), Vz e B(x)}.

When S(x) =B(x) =E for all x e E, (GSIP7) was studiedbyYangandHuang [17] inabstract convex spaces.

• (GSIP8) Generalized semi-infinite program with constraint (GVQEP8):

wMin^(K),

K = {x e E: x e S(x), 3z e B(x),F(x,y,z) n C(x) = 0, Vy e S(x)}.

It is worth mentioning that (GSIP8) can be considered as a generalization of the generalized vector semi-infinite programming introduced and studied by Lin [28] in topological vector spaces.

In brief, for suitable choice of the spaces L, V, X, E and the mappings S, B, F, C, h,one can obtain a number of known the generalized semi-infinite program [17], the mathematical program with equilibrium constraint [19], the generalized semi-infinite program [23], the generalized vector semi-infinite programming [28], and the vector optimization problem [30-32] as special cases from the problems (GSIP1)-(GSIP8). Now, we recall some useful definitions and lemmas as follows.

Definition 2.1 Let K C V be a nonempty set and C c V be the closed convex pointed cone with int C = 0. The set of all weak minimal points of K with respect to the ordering cone C is defined as

wMinC(K) = {x e K: (x - K) n int C = 0}.

Definition 2.2 Let (X,D, F) be an abstract convex space and Z be a set. For a set-valued mapping T: X ^ Z with nonempty values, if a set-valued mapping G: D ^ Z satisfies

T(Fn) c G(N) := U G(y) for all N e (D),

then G is called a KKM mapping with respect to T. A KKM mapping G: D ^ X is a KKM mapping with respect to the identity mapping IX .

A set-valued mapping F: X ^ Z is called to be a KC-map if, for any closed-valued KKM mapping G: D ^ Z with respect to F, the family (G(y)}yeD has the finite intersection property. We denote

KC(X, Z) := (F: F is KC-map}.

Definition 2.3 ([33]) Let X and Y be two topological spaces. A set-valued mapping F: X ^ Y is said to be

(i) upper semicontinuous (u.s.c.) at x0 if for any open set V 2 F(x0), there is an open neighborhood Ox0 of x0 such that F(x') c V for each x' e Ox0,

(ii) lower semicontinuous (l.s.c.) at x0 if for any open set V n F(x0) = 0, there is an open neighborhood Ox0 of x0 such that F(x') n V = 0 for each x' e Ox0,

(iii) continuous at x0 if it is both upper and lower semicontinuous at x0,

(iv) upper semicontinuous (lower semicontinuous or continuous) on X if it is upper semicontinuous (lower semicontinuous or continuous) at every x e X,

(v) closed if and only if its graph Graph(F) := ((x,y) e X x Y: y e F(x)} is closed.

Lemma 2.1 ([34]) Let X and Y be two topological spaces and F : X ^ Y a set-valued mapping.

(i) If Y is compact, then F is closed if and only if it is upper semicontinuous,

(ii) ifX is a compact space and F is a u.s.c. mapping with compact values, then F(X) is a compact subset of Y.

Lemma 2.2 ([35]) 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 {xa} с X with xa ^ x andya e F(xa), there exists a subnet {yp} с ya such thatyp ^ y e F(x).

Lemma 2.3 ([36]) Let X and Y be two topological spaces and F: X ^ Y be lower semi-continuous atx e X if and only if for any y e F(x) and any net {xa} with xa ^ x, there is a net {ya} such thatya e F(xa) and ya ^ y.

Lemma 2.4 ([10]) Let (X, D, Г) be an abstract convex space, Z a set, and T : X ^ Z a set-valued mapping. Then F e KC(X, Z) if and only for any G: D ^ Z satisfying

(i) G is closed-values;

(ii) F(rw) с G(N) for any N e {D), we have

F (E)f| {G(y):y e N} = 0

for each N e {D).

Lemma2.5 ([32]) Assume that A isanonempty compact subset ofa real topological vector space V andD is a closed convex cone in V with D = V. Then one has wMinDA = 0.

An abstract convex space with any topology is called an abstract convex topological space. In the rest of this paper, let (X, Г) be an abstract convex Hausdorff topological space and E be a nonempty compact subset of X. Let V be a topological vector spaces. Assume that T : X ^ X, B : E ^ E, S: E ^ E, F: E x E x E ^ V and Q : E ^ V are five set-valued mappings. Let p be a binary relation on 2V and pc be the complementary relation of p. Let a be any of the quantifiers V, 3, and a be the other of the quantifiers V, 3.

3 Main results

In order to show the existence of solutions for the vector quasi-equilibrium problems (GVQEP1)-(GVQEP8), we first give the following general result.

Theorem 3.1 Suppose that the following conditions are satisfied:

(i) T e KC(X, X);

(ii) for each y e E, the set {x e E: (a)z e B(x), pc(F(x,y, z), Q(x))} is open in E;

(iii) G0 = {x e E: xe S(x)} is open in E;

(iv) for each x e E, S(x) is nonempty Г-convex, S-1(y) is open for all y e E;

(v) for each (x0,y0) e E x E with x0 e T(y0) such that y0 e S(x0).

Then there exists x e S(x) such that (a)z e B(x), p(F(x,y, z), Q(x)) for anyy e S(x).

Proof For any x e E, define A : E ^ E by

A(x) = {y e E: (a)z e B(x), pc(F(x,y, z), Q(x))}.

From the definition of A(x), one has

A-1(y) = {x e E: (a)z e B(x), pc(F(x,y,z), Q(x))}. Define P: E ^ E by

P(x)=( S(x) n A(x), x e E \ Gq; (1)

IS(x), x e Gq.

Let M(y) =E \ P-1(y). We show that M(y) is closed for all y e E. In fact, it follows from (1) that

P-1(y) = {x e E \ Gq :y e S(x) n A(x)} U {x e Gq :y e S(x)}

= {x e E \ Gq : x e S-1(y) n A-1(y)} U {x e Gq : x e S-1(y)} = {(E \ Gq) n S-1(y) n A-1(y)} U {Gq n S-1(y)} = S-1(y) n (Gq U A-1(y)).

Since S-1(y), A-1(y), and Gq are open, we know that P-1(y) is open and so M(y) is closed.

We show that M is a KKM mapping with respect to T. Suppose that M is not a KKM mapping with respect to T. Then there exist a finite subset N and a point xQ e E such that xQ e T(Fn) \ M(N). This shows that there exists a point yQ e FN such that xQ e T(yQ), xQ e P-1(y) for any y e N, and so N c P(xQ) c S(xQ). Since S(xQ) is F-convex and N e (S(xQ)),we know that yQ e FN c S(xQ), which is a contradiction. It follows that M is a KKM mapping with respect to T.

It follows from Lemma 2.4 that M has finite intersection property. From the facts that M(y) c E is closed and E is compact, we know that M(y) is compact for any y e E and so

M(y) = 0.

Thus, there exists a point x e E such that xef| M(y) = E \U P-1(y).

yeE yeE

This implies that x e P-1(y) for all y e E and so P(x) = 0.

If x e Gq, then it is easy to see that S(x) = P(x) = 0, which is a contradiction. Therefore, we have

x e E \ Gq with S(x) n A(x)= P(x) = 0 and so

x e S(x) and y e A(x), Vy e S(x),

that is, x e S(x), (a)z e B(x), p(F(x,y, z), C(x)) for all y e S(x). This completes the proof.

Remark 3.1 By Lemma 2.1, it is easy to see that the condition (iii) can be replaced by the following condition:

(iii)' S : E ^ E is a u.s.c. set-valued mapping.

Next we give some existence theorems in connection with the solution of the vector quasi-equilibrium problems (GVQEP1)-(GVQEP8).

Theorem 3.2 Assume that the conditions (i), (iii), (iv), and (v) in Theorem 3.1 are satisfied. Moreover, suppose that

(a) for each y e E, F(■,y, ■) is l.s.c. and C is closed;

(b) B is l.s.c.

Then there exists x e Esuch thatx e S(x) and F(x,y, z) С C(x) for ally e S(x) andz e B(x). Proof Let

A(x) = {y e E : 3z e B(x), F (x, y, z) £ C(x)}.

We show that

A-1(y) = {x e E : 3z e B(x), F(x,y, z) £ C(x)}

is open. Let {xa} Я E \ A-1(y) be a net with xa ^ xo. Then

F(xa,y, z') Я C(xœ), Vz! e B(xa).

Since B and F(-,y, ■) are l.s.c., by Lemma 2.3, for any z e B(xo) and v e F(xo,y, z), there exist za e B(xa ) and va e F(xa, y, za ) suchthat za z and va ^ v. Now the closedness of C with va e C(xa) shows that v e C(x) and so F(xo,y, z) Я C(x) for any z e B(xo). This shows that xo e E \ A-1(y) andsoE \ A-1(y) is closed. Thus, A-1(y) is open. It follows from Theorem 3.1 that there exists x e E such that x e S(x) and F (x, y, z) Я C(x) for all y e S(x) and z e B(x). This completes the proof. □

Remark 3.2 Theorem 3.2 can be considered as a generalization of Theorem 3.3 in [4] under different conditions from the topological vector space to the abstract convex space.

Corollary 3.1 Assume that the conditions (i), (iii), (iv), and (v) in Theorem 3.1 are satisfied with B = S. Suppose that, for each y e E, F(■,y) is l.s.c. and C is closed. Then there exists xe E such thatx e S(x) and F (x, y) Я C(x) for ally e S(x).

Proof The proof is similar to that of Theorem 3.2 and so we omit it here. □

Remark 3.3 When S(x) = E for all x e E, Corollary 3.1 was given by Theorem 1 of Yang and Huang [17] under quite different conditions.

Theorem 3.3 Assume that the conditions (i), (iii), (iv), and (v) in Theorem 3.1 are satisfied. Moreover, suppose that (a) for each y e E, F(■,y, ■) is l.s.c. and C is closed;

(b) B is u.s.c. and B(x) is compact for each x e E. Then there existx e E andze B(x) such thatx e S(x) andF(x,y,z) C C(x) for ally e S(x).

Proof Let

A(x) = {y e E: Vz e B(x),F(x,y,z) £ C(x)}.

We first show that

A-1(y) = {x e E: Vz e B(x), F(x,y, z) £ C(x)}

is open. Let {xa} C E \ A-1(y) be a net with xa ^ xQ. Then there exists za e B(xa) such that F(xa,y, za) c C(xa). Since B is u.s.c. with compact values, by Lemma 2.2, there exists a subset net of {za}, denoted again by {za}, such that za ^ zQ e B(xQ). The fact that F(■,y, ■) is l.s.c. together with Lemma 2.3 shows that, for any v e F(xQ,y, zQ), there exists va e F(xa,y, za) such that va ^ v. Since va e C(xa) and C is closed, we know that v e C(xQ) and so F(xQ,y, zQ) C C(xQ) for some zQ e B(xQ). This implies that xQ e E \ A-1(y) and so E \ A-1(y) is closed. Thus, A-1(y) is open. It follows from Theorem 3.1 that there exist x e E and z e B(x) such that x e S(x) and F(x,y,z) C C(x) for any y e S(x). This completes the proof. □

Remark 3.4 When S(x) = E for all x e E, the existence of the solutions for generalized vector quasi-equilibrium was studied in Theorem 3.1 of [7] in real Hausdorff topological vector spaces.

Theorem 3.4 Assume that the conditions (i), (iii), (iv), and (v) in Theorem 3.1 are satisfied. Moreover, suppose that

(a) for each y e E, F(-,y, ■) is l.s.c., C(x) is a set-valued mapping with a nonempty interior for each x e E, the mapping W: E ^ V, defined by W(x) = V \ (- int C(x)), is closed;

(b) B is l.s.c.

Then there existsxxe Esuch thatx e S(x) andF(x,y,z) n (- int C(x)) = 0for ally e S(x) and z e B(x).

Proof Let

A(x) = {y e E: 3z e B(x),F(x,y,z) n (-int C(x)) = 0}. We prove that

A-1(y) = {x e E: 3z e B(x), F(x,y, z) n (- int C(x)) = 0} is open. Let {xa} C E \ A-1(y) be a net with xa ^ xQ. Then

F(xa,y, z') n (- int C(xa)) = 0, Vz' e B(xa) and so

F(xa,y,z') C W(xa) = V \ (-int C(xa

Similar to the proof ofTheorem 3.2, we get

F(x0,y,z) ç W(x0) = V \ (-int C(xo)).

This shows that x0 e E \ A-1(y) and so £ \ A-1(y) is closed. Thus, A-1(y) is open. It follows from Theorem 3.1 that there exists X e E such that X e S(X) and

F(X, y, z) n (- int C(X)) = 0, Vy e S(X), Vz e B(X).

This completes the proof. □

Remark 3.5 Theorem 3.4 can be considered as a generalization of Theorem 3.2 in [4] under different conditions from the topological vector space to the abstract convex space.

Corollary 3.2 Assume that the conditions (i), (iii), (iv), and (v) in Theorem 3.1 are satisfied with S = B. Moreover, suppose that (a) for each y e E, F (■, y) is l.s.c., C(x) has a nonempty interior for each x e E, the map W : E ^ V, defined by W(x) = V \ (- int C(x)), is closed. Then there exists X e E such that X e S(X) and F(X, y) n (- int C(X)) = 0 for all y e S(X).

Proof The proof is similar to that of Theorem 3.4 and so we omit it here. □

Remark 3.6 When S(x) = E for all x e E, Corollary 3.2 was given by Theorem 2 of Yang and Huang [17] under quite different conditions.

Theorem 3.5 Suppose the conditions (i), (iii), (iv), and (v) in Theorem 3.1 are satisfied. Moreover, suppose that

(a) for each y e E, F (■, y, ■) is l.s.c., C(x) has a nonempty interior for each x e E, and the mapping W : E ^ V, defined by W (x) = V \ (- int C(x)), is closed;

(b) B is u.s.c. and B(x) is compact for each x e E.

Then there exist X e Eandze B(X) such thatX e S(X) and F(X,y, z) n (- int C(X)) = 0 for all y e S(X).

Proof Let

A(x) = {y e E : Vz e B(x), F (x, y, z) n (-int C(x)) = 0}. We show that

A-1(y) = {x e E : Vz e B(x), F (x, y, z) n (- int C(x)) = 0} is open. Let {xa} Ç E \ A-1(y) be a net with xa ^ x0. Then

F(xa, y, za ) n (-int C(xa )) = 0 for some za e B(xa ), that is,

F (Xa, y, za ) Ç V \ (- int C(xa )).

Using similar arguments to the proof of Theorem 3.3, we have

F(x0,y, z0) C W(x0) = V \ (- int C(x0})

for some z0 e B(x0). This shows that x0 e E\ A-1(y) andsoE\ A-1(y) is closed. Thus, A-1(y) is open. It follows from Theorem 3.1 that there exist x e E and z e B(x) such that x e S(x) and

F(x, y, z) n (- int C(x)) = 0, Vye S(x). This completes the proof. □

Remark 3.7 When E is a nonempty convex compact of a topological vector space, Li and Li [27] studied the existence of solutions for (GVQEP4).

Theorem 3.6 Assume that the conditions (i), (iii), (iv), and (v) are satisfied in Theorem 3.1. Moreover, suppose that

(a) for each y e E, F(■,y, ■) is u.s.c. with compact valued on E x E x E and C(x) has a nonempty interior for each xeE, the mapping W: E ^ V, defined by

W(x) = V \ (- int C(x)), is closed;

(b) B is l.s.c.

Then there exists x e E such that x e S(x) and F(x, y, z) £ - int C(x) for all y e S(x) and z e B(x).

Proof Let

A(x) = {yeE: 3z e B(x), F (x, y, z) c - int C(x)}.

We prove that

A-1(y) = {x e E: 3z e B(x), F(x,y, z) c - int C(x)}

is open. Let xa eE \ A-1(y) be a net with xa ^ x0. Then

F(xa, y, z') £ - int C(xa)

for any z e B(xa) and so there exists va e V such that

va e F(xa,y,z') \ (- int C(xa

Since B is l.s.c., by Lemma 2.3, for any z e B(x0), there exists za e B(xa) such that za ^ z. Since F(■,y, ■) is u.s.c. with compact valued, by Lemma 2.2, there exists a subset net of {va}, denoted again by {va}, such that va ^ v0 e F(x0,y,z). On the other hand, the fact that va e - int C(xa) shows that va e W(xa). Now the closedness of W shows that v0 e W(x0) and so v0 e - int C(x0). Thus F(x0,y, z) £ - int C(x0) for any z e B(x0). This implies that x0 e E \ A-1(y) andsoE \ A-1(y) is closed. Thus, A-1(y) is open. It follows from Theorem 3.1 that there exists x e E such that x e S(x) and F(x, y, z) £ - int C(x) for all y e S(x) and z e B(x). This completes the proof. □

Corollary 3.3 Assume that the conditions (i), (iii), (iv), and (v) in Theorem 3.1 are satisfied with S = B. Moreover, suppose that (a) for each yeE, F (■, y) is u.s.c. with compact valued on E x E and C(x) has a nonempty interior for each xeE, the mapping W: E ^ V, defined by W(x) = V \ (- int C(x)), is closed. Then there exists X e E such that X e S(X) and F(X,y) £ - int C(X) for all y e S(X).

Proof The proof is similar to that of Theorem 3.6 and so we omit it here. □

Remark 3.8 When S(x) = E for all x e E, Corollary 3.3 was given by Theorem 4 of Yang and Huang [17] under quite different conditions.

Theorem 3.7 Assume that the conditions (i), (iii), (iv), and (v) are satisfied in Theorem 3.1. Moreover, suppose that

(a) for each yeE, F (■, y, ■) is u.s.c. with compact valued on E x E x E and C(x) has a nonempty interior for each xeE, the mapping W: E ^ V, defined by

W(x) = V \ (- int C(x)), is closed.

(b) B is u.s.c. and B(x) is compact for each xeE.

Then there exist X e E and z e B(X) such that X e S(X) and F(X, y, z) £ - int C(X) for all y e S(X).

Proof Let

A(x) = {yeE: VzeB(x), F(x, y, z) c - int C(x)}.

We prove that

A-1(y) = {x e E: Vz e B(x), F(x,y, z) c - int C(x)}

is open. Let xa e E \ A-1(y) be a netwith xa ^ xo. ThenF(xa,y, za) £ - int C(xa) for some za e B(xa) and so there exists va e V such that

va e F(Xa,y, za) \ (- int C(xa)).

Since B is u.s.c. with compact valued, by Lemma 2.2, there exists a subnet of {za}, denoted again by {za}, suchthat za ^ zo e B(xo). The fact that F (■, y, ■) is u.s.c. with compact valued together with Lemma 2.2 shows that there exists a subset net of {va}, denoted again by {va}, such that va ^ vo e F(xo,y, zo). On the other hand, it is easy to see that va e W(xa). Since W is closed, we know that vo e W(xo)andso vo e - int C(xo).Thus F(xo, y, zo) £ - int C(xo) for some zo e B(xo) and so xo e E \ A-1(y). This implies that E \ A-1(y) is closed and so A-1(y) is open. It follows from Theorem 3.1 that there exist X e E and z e B(X) such that X e S(X) and F(X,y, z) £ - int C(X) for all y e S(X). This completes the proof. □

Remark 3.9 Theorem 3.7 can be considered as a generalization of Theorem 3.1 in [2, 4] under different conditions from the topological vector space to the abstract convex space.

Remark 3.10 When E is a nonempty convex compact of topological vector space, Li and Li [27] studied the existence of solutions for (GVQEP6).

Theorem 3.8 Suppose the conditions (i), (iii), (iv), and (v) in Theorem 3.1 are satisfied. Moreover, assume that

(a) for each y e E, F(■,y, ■) is u.s.c. with compact valued on E x E x E and C is closed;

(b) B is l.s.c.

There exists xe Esuch thatx e S(x) andF (x, y, z) n C(x) = 0 for allye S(x) andze B(x). Proof Let

A(x) = {yeE: 3z e B(x),F(x,y,z) n C(x) = 0}.

We show that

A-1(y) = {x e E: 3z e B(x),F(x,y,z) n C(x) = 0}

is open. Let {xa} Q E \ A-1(y) be a net with xa ^ xo. Then

F(xa,y, z') n C(xa) = 0, Vz e B(xa).

It follows that there exists va e F(xa,y,z!) n C(xa). Since B is l.s.c., by Lemma 2.3, there exists za e B(xa) such that za ^ z for any z e B(xo). By the fact that F(■,y, ■) is u.s.c. with compact valued, there exists a subset of {va}, denoted again by {va}, such that va ^ vo e F(xo, y, z).Since va e C(xa )andC is closed, we know that vo e C(xo)andso vo eF(xo, y, z) n C(xo). Thus,

F(xo,y, z) n C(xo) = 0, Vze B(xo).

This shows that xo eE \ A-1(y) and so E \ A-1(y) is closed. Thus, A-1 (y) is open. By Theorem 3.1, there exists xe E such that x e S(x) and

F (x, y, z) n C(x) = 0, Vye S(x), Vz e B(x).

This completes the proof. □

Corollary 3.4 Assume that the conditions (i), (iii), (iv), and (v) in Theorem 3.1 are satisfied with S = B. Moreover, suppose that, for each y eE, F (■, y) is u.s.c. with compact valued on E x E and C is closed. Then there exists xe E such that x e S(x) and F (x, y) n C(x) = 0 for ally e S(x).

Proof The proof is similar to that of Theorem 3.8 and so we omit it here. □

Remark 3.11 When S(x) = E for all xeE, Corollary 3.4 was given by Theorem 3 of Yang and Huang [17] under some different conditions.

Theorem 3.9 Suppose the conditions (i), (iii), (iv), and (v) are satisfied in Theorem 3.1. Moreover, assume that

(a) for each y eE, F (■, y, ■) is u.s.c. with compact valued on E x E x E and C is closed;

(b) B is u.s.c. and B(x) is compact for each xeE.

There exist x e E andze B(x) such thatx e S(x) andF (x, y, z) n C(x) = 0 for ally e S(x).

Proof Let

A(x) = {y e E: Vz e B(x),F(x,y,z) n C(x) = 0}.

We prove that

A-1(y) = {x e E: Vz e B(x),F(x,y,z) n C(x) = 0}

is open. Let {xa} C E \ A-1(y) be a net with xa ^ xQ. Then

F(xa, y, za ) n C(xa ) = 0

for some za e B(xa), that is, there exists va e F(xa,y,za) n C(xa). Since B is u.s.c. and B(x) is compact, it follows from Lemma 2.2 that there exists a subset of {za}, denoted again by {za}, such that za ^ zQ e B(xQ). Similar to the proof of Theorem 3.8, we can prove that F(xQ,y,zQ) n C(xQ) = 0 for some zQ e B(xQ). This shows that xQ e E \ A-1(y) and so E \ A-1(y) is closed. Thus, A-1(y) is open. It follows from Theorem 3.1 that there exist x e E and z e B(x) such that x e S(x) and

F(x,y, z) n C(x) = 0, Vy e S(x).

This completes the proof. □

4 Applications to the generalized semi-infinite programs

In this section, by the results presented in Section 3, we give some existence theorems of solutions to the generalized semi-infinite programs. Let L be a real topological vector space ordered by a closed convex pointed cone H C L with intH = 0 and h : X ^ L be a u.s.c. mapping with compact values.

Theorem 4.1 Suppose that all the conditions of Theorem 3.2 are satisfied. Moreover, assume that F(-, ■, ■) and S are l.s.c. Then there is a solution to the problem

wMinHh(K),

K = {xeE:xeS(x),F(x,y,z) C C(x),Vy eS(x),VzeB(x)}.

Proof Theorem 3.2 shows that K = 0. From Lemma 2.5, it is sufficient to show that h(K) is compact. Since h is u.s.c. and K C E, by Lemma 2.1, we only need to prove that K is closed. Let {xa} C K be a net with xa ^ xQ. Then xa e S(xa) and

F(xa,y,z') C C(xa), Vy' e S(xa), Vz' e B(xa).

Since S and B are l.s.c., for any y e S(xQ) and z e B(xQ), it follows from Lemma 2.3 that there exist ya e S(xa) and za e B(xa) such that ya ^ y and za ^ z. By the lower semi-continuity

of F and Lemma 2.3, for any v e F(xo,y, z), there exists va e F(xa,ya, za) such that va ^ v. Now the closedness of C with va e C(xa) shows that v e C(xo) and so F(xo,y, z) Q C(xo) for all y e S(xo) and z e B(xo). Moreover, the closedness of E \ Go shows that xo e S(xo). Thus, K is closed. This completes the proof. □

Corollary 4.1 Suppose that all the conditions of Corollary 3.1 are satisfied. Moreover, assume that F(■, ■) and S are l.s.c. Then there is a solution to the problem

wMinHh(K),

K = {xeE: xe S(x), F(x, y) Q C(x), Vy e S(x)}.

Remark 4.1 When S(x) = E for all xeE, Corollary 4.1 was given by Theorem 5 of Yang and Huang [17] under some different conditions.

Theorem 4.2 Suppose that all the conditions of Theorem 3.3 are satisfied. Moreover, assume that F( ■, ■, ■) and S are l.s.c. Then there is a solution to the problem

wMinHh(K),

K = {xeE: xe S(x), 3z e B(x), F (x, y, z) Q C(x), Vy e S(x)}.

Proof Obviously, Theorem 3.3 shows that K = 0. By Lemma 2.5, it is sufficient to prove that h(K) is compact. Since h is u.s.c. and K Q E, from Lemma 2.1, we only need to show that K is closed. Let {xa} QK be a net with xa ^ xo. Then xa e S(xa) and there exists za eB(xa) such that

F(xa,y,za) Q C(xa), Vy' e S(xa).

Since B is a u.s.c. mapping with compact values, it follows from Lemma 2.2 that there exists a subnet of {za}, denoted again by {za}, such that za ^ zo e B(xo). For any y e S(xo), the lower semi-continuity of S together with Lemma 2.3 implies that there exists ya e S(xa) such that ya ^ y. For v e F(xo,y, zo), by the fact that F is l.s.c., it follows from Lemma 2.3 that there exists va e F(xa, ya, za) such that va ^ v. Now the closedness of C with va e C(xa) shows that v e C(xo) and so there exists zo e B(xo) such that F(xo,y, z) Q C(xo) for all y e S(xo). Moreover, the closedness of E \ Go shows that xo e S(xo). Thus, K is closed. This completes the proof. □

Theorem 4.3 Suppose that all the conditions of Theorem 3.4 are satisfied. Moreover, assume that F(^, ■, ■) and S are l.s.c. Then there is a solution to the problem

wMinHh(K ),

K = {x e E : x e S(x), F(x, y, z) n - int C(x) = 0, Vy e S(x), Vz e B(x)}.

Proof Theorem 3.4 shows that K = 0. From Lemma 2.5, it is sufficient to show that h(K) is compact. Since h is u.s.c. and K ç E, by Lemma 2.1, we only need to show that K is closed. Let {xa} ç K be a net with xa ^ x0. Then xa e S(xa),

F(xa,y, z') n - int C(x) = 0, Vy' e S(xa), Vz' e B(xa)

and so

F(xa,y, z') ç W(xa), Vy' e S(xa), Vz' e B(xa). Similar to the proof ofTheorem 4.1, we have x0 e S(x0),

F(X0,y, z) ç W(X0), Vy e S(x0), Vz e B(x0) and so

F(x0,y, z) n - int C(x0) = 0, Vy e S(x0), Vz e B(x0). Thus, K is closed. This completes the proof. □

Corollary 4.2 Suppose that all the conditions of Corollary 3.2 are satisfied. Moreover, assume that F(■, ■) and S are l.s.c. Then there is a solution to the problem

wMinHh(K ),

K = {x e E : x e S(x), F (x, y) n C(x) = 0, Vy e S(x)}.

Remark 4.2 When S(x) = E for all x e E, Corollary 4.2 was given by Theorem 6 of Yang and Huang [17] under some different conditions.

Theorem 4.4 Suppose that all the conditions ofTheorem 3.5 are satisfied. Moreover, assume that F(-, ■, ■) and S are l.s.c. Then there is a solution to the problem

wMinHh(K ),

K = {x e E : x e S(x), 3z e B(x), F (x, y, z) n - int C(x) = 0, Vy e S(x)}.

Proof It follows from Theorem 3.5 that K = 0. From Lemma 2.5, it is sufficient to show that h(K) is compact. Since h is u.s.c. and K Q E, by Lemma 2.1, we only need to show K is closed. Let {xa} QK be a net with xa ^ xo. Then xa e S(xa) and there exists za e B(xa) such that

F(xa,y,za) n -int C(xa) = 0, vy e S(xa).

Thus, there exists za e B(xa) such that

F(xa,y',za) C C(xa), vy e S(xa).

Similar to the proof of Theorem 4.2, we know that x0 e S(x0) and there exists z0 e B(x0) such that

F(xo,y, zo) Q W(xo), Vy e S(xo). Thus, xo e S(xo) and there exists zo e B(xo) such that

F(xo,y, zo) n - int C(xo) = 0, Vy e S(xo). It follows that K is closed. This completes the proof. □

Theorem 4.5 Suppose that all the conditions of Theorem 3.6 are satisfied. Moreover, assume 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 ),

K = {xeE: xe S(x), F (x, y, z) £ - int C(x), Vy e S(x), Vz e B(x)}.

Proof Theorem 3.6 shows that K = 0. From Lemma 2.5, it is sufficient to show that h(K) is compact. Since h is u.s.c. and K Q E, by Lemma 2.1, we only need to prove that K is closed. Let {xa} QK be a net with xa ^ xo. Then xa e S(xa),

F(xa,y, z') £ - int C(xa), vy e S(xa), Vz' e B(xa) and so there exists va e V such that va e F(xa,y,z') \ (-int C(xa)).

By the lower semi-continuity of S and B, for any y e S(xo) and z e B(xo), it follows from Lemma 2.3 that there exist ya e S(xa) and za e B(xa) such that ya ^ y and za ^ z. Since F(■, ■, ■) is a u.s.c. mapping with compact values, Lemma 2.2 shows that there exists a subnet of {va}, denoted again by {va}, such that va ^ vo e F(xo,y,z). On the other hand, the fact

that va e -int C(xa) shows that va e W(xa). Now the closedness of W shows that vQ e W(xQ) and so vQ e - int C(xQ). Moreover, the closedness of E \ GQ shows that xQ e S(xQ). Thus,

F(xo,y, z) £ - int C(xq)

for all y e S(xQ) and z e B(xQ) and so K is closed. This completes the proof. □

Corollary 4.3 Suppose that all the conditions of Corollary 3.3 are satisfied. Moreover, assume that F(■, ■) is u.s.c. and S is l.s.c. Then there is a solution to the problem

wMinHh(K),

K = {x e E: x e S(x), F(x, y) £ C(x), Vy e S(x)}.

Remark 4.3 When S(x) = E for all x e E, Corollary 4.3 was given by Theorem 8 of Yang and Huang [17] under some different conditions.

Theorem 4.6 Suppose that all the conditions of Theorem 3.7 are satisfied. Moreover, assume 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),

K = {x e E: x e S(x), 3z e B(x), F(x,y, z) £ - int C(x), Vy e S(x)}.

Proof Theorem 3.7 shows that K = 0. By Lemma 2.5, it is sufficient to prove that h(K) is compact. Since h is u.s.c. and K C E, from Lemma 2.1, we only need to show that K is closed. Let {xa} C K be a net with xa ^ xQ. Then xa e S(xa) and there exists za e B(xa) such that

F(xa,y, za) £ - int C(xa), Vy' e S(xa).

Thus, there exists va e V such that

va e F(xa,y, za) \ (- int C(xa)).

Since B is a u.s.c. mapping with compact values, it follows from Lemma 2.2 that there exists a subnet of {za}, denoted again by {za}, such that za ^ zQ e B(xQ). By the lower semi-continuity of S, for any y e S(xQ), Lemma 2.3 shows that there exists ya e S(xa) such that ya ^ y. Since F(-, ■, ■) is a u.s.c. mapping with compact values, Lemma 2.2 implies that there exists a subnet of {va}, denoted again by {va},suchthat va ^ vQ eF(x0,y, zQ). Similar

to the proof of Theorem 4.5, we can prove that K is closed. This completes the proof.

Theorem 4.7 Suppose that all the conditions of Theorem 3.8 are satisfied. Moreover, assume 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 ),

K = {x e E : x e S(x), F (x, y, z) n C(x) = 0, Vy e S(x), Vz e B(x)}.

Proof Theorem 3.8 shows that K = 0. From Lemma 2.5, it is sufficient to show that h(K) is compact. Since h is u.s.c. and K ç E, by Lemma 2.1, we only need to show K is closed. Let {xa} çK be a net with xa ^ xo. Then xa e S(xa),

F(xa,y, z') n C(xa) = 0, Vy' e S(xa),z e B(xa) and so there exists va e V such that va e F(xa,y, z)n C(xa).

By the lower semi-continuity of S and B, for any y e S(x0) andz e B(x0), Lemma 2.3 shows that there exist ya e S(xa) and za e B(xa) such that ya ^ y and za ^ z. Since F(^, ■, ■) is u.s.c. with compact values, by Lemma 2.2, there exists a subnet of {va}, denoted again by {va}, such that va ^ v0 e F(x0,y,z). Now the closedness of C with va e C(xa) shows that v0 e C(x0) and so

F(x0,y, z) n C(x0) = 0, Vy e S(x0), Vz e B(x0).

Moreover, the closedness of E \ G0 shows that x0 e S(x0). Thus, K is closed. This completes the proof. □

Corollary 4.4 Suppose that all the conditions of Corollary 3.4 are satisfied. Moreover, assume that F(■, ■) and S are l.s.c. Then there is a solution to the problem

wMinHh(K ),

K = {x e E : x e S(x), F (x, y) n C(x) = 0, Vy e S(x)}.

Remark 4.4 When S(x) = E for all x e E, Corollary 4.4 was given by Theorem 7 of Yang and Huang [17] under some different conditions.

Theorem 4.8 Suppose that all the conditions of Theorem 3.9 are satisfied. Moreover, assume 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 ),

K = {x e E: xe S(x), 3z e B(x), F (x, y, z) n C (x) = 0, Vy e S(x)}.

Proof Theorem 3.8 shows that K = 0. From Lemma 2.5, it is sufficient to prove that h(K) is compact. Since h is u.s.c. and K c E, by Lemma 2.1, we only need to show K is closed. Let {xa} c K be a net with xa ^ x0. Then xa e S(xa) and there exists za e B(xa) such that

F(xa,y, za) n C(xa) = 0, Vyr e S(xa).

Thus, there exists va e V such that

Va eF(xa,y', z^j n C(xa).

Since B is a u.s.c. mapping with compact values, it follows from Lemma 2.2 that there exists a subnet of {za}, denoted again by {za}, such that za ^ z0 e B(x0). By the lower semi-continuity of S, for any y e S(x0), Lemma 2.3 implies that there exists ya e S(xa) such that ya ^ y. Since F(•, •, •) is a u.s.c. mapping with compact values, by Lemma 2.2, there exists a subnet of {va}, denoted again by {va}, such that va ^ v0 e F(x0,y,z0). Now the closedness of C with va e C(xa) shows that v0 e C(x0) and so there exists z0 e B(x0) such that

F(x0,y,z0) n C(x0) = 0, Vy e S(x0).

Moreover, the closedness of E \ G0 shows that x0 e S(x0). 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 Published online: 24 February 2015

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