1 Introduction

Let R = (−∞,+∞) and A be a nonempty subset of a topological vector space E. Assume that f : A × A → R is a function such that f(x,x) > 0 for all x ∈ A. The scalar equilibrium problem considered by Blum and Oettli^[1] is to find x0 ∈ A such that f(x0,y) > 0 for all y ∈ A. This scalar equilibrium problem includes some fundamental mathematical problems such as the optimization problems,the Nash equilibrium problems,the fixed points,the complementary problems,and the variational inequalities^[1].

Recently,the vector equilibrium problem has received much more attention than the scalar equilibrium problem. Various vector equilibrium problems with moving cones C(·) have been studied^{[2, 3, 4, 5, 6, 7, 8, 9]},where C(·) is a set-valued mapping from a topological space X to a topological vector space V such that C(x) is a closed convex cone of V for each given x ∈ X. When dealing with such moving cones,some authors used the assumption of the upper semicontinuity of C(·) or W(·) = V \ intC(·). In fact,we know that this assumption is a very strong one,while the closedness of C(·) or W(·) is a reasonable assumption^{[7, 10]}.

More recently,by a general coincidence theorem,Balaj^[2] showed some existence theorems of solutions for a class of generalized vector equilibrium problems in topological vector spaces. Using the generalization of Fan-Browder fixed point theorem,Balaj and Lin^[3] further showed some existence theorems of solutions concerned with the generalized vector equilibrium problems in G-convex spaces.

Moveover,Park^[11] first introduced the concepts of abstract convex spaces,class and class . As pointed out by Park^[11],the abstract convex spaces include convex subsets of topo- logical vector spaces,convex spaces,H-spaces,and G-convex spaces as special cases. Therefore, it seems quite natural and interesting for us to study some generalized vector equilibrium prob- lems with moving cones in abstract convex spaces under some suitable conditions.

Assume that X is a topological space,Z is a nonempty set,and V is a topological vector space. Let F : X × Z ⊸ V and C : X ⊸ V be two maps. In this paper,we consider the following four types of generalized vector equilibrium problems:

Furthermore,suppose that h : X ⊸ L is a map,where L is a real topological vector space ordered by a proper closed convex cone D. Obviously,the existence of solutions to the generalized vector equilibrium problems (i)-(iv) is closely related to the existence of feasible solutions to the following four types of generalized semi-infinite programs with generalized vector equilibrium constraints:

Let L = V = R,X = Z,and C(x) = [0,+∞) for all x ∈ X. If h and F are single-valued, then it is easy to see that the generalized semi-infinite programs (v)-(viii) are reduced to the following semi-infinite program:

where h : X → R and F : X × X → R are two maps. It is well known that the semi-infinite program represents an important class of optimization problems (see,for example,Refs. ^[12]- ^[15] and the references therein).

In this paper,we first show some existence theorems of solutions to the generalized vector equilibrium problems (i)-(iv) under suitable conditions. Then,we give some existence theorems of solutions concerned with the generalized semi-infinite programs (v)-(viii).

2 Preliminaries

In this section,we recall some known concepts,definitions,and lemmas which can be found in Refs. ^[6],^[11],and ^[16]-^[20].

For a given set X,let hXi denote the family of all nonempty finite subsets of X. Suppose that A is a subset of a topological space. Let int A and A denote the interior and closure of A, respectively.

A multimap (or simply a map) T : X ⊸Y is a function from a given set X into the power set 2^Y of Y such that the value T (x) is a subset Y for all x ∈ X. Let T : X ⊸ Y be a given map. We define a map T− : Y ⊸ X by T−(y) = {x ∈ X : y ∈ T (x)} for all y ∈ Y and call it the (lower) inverse of T . Assume that T : X ⊸ Y is a map defined by T(x) = T (x) for all x ∈ X. For any given subset A ⊂ X,define . Let B ⊂ Y be a given subset, and

.Moreover,the graph of T is defined as

.

Let X and Y be two topological spaces. We say that a map T : X ⊸Y is

(i) closed if its graph Graph(T ) is a closed subset of X × Y ;

(ii) upper semicontinuous (in short,u.s.c.) if,for any x ∈ X and any open set V in Y with T (x) ⊂ V ,there exists a neighborhood U of x such that T (x′) ⊂ V for all x′ ∈ U;

(iii) lower semicontinuous (in short,l.s.c.) if,for any x ∈ X and any open set V in Y with T (x) ∩ V ≠ ∅,there exists a neighborhood U of x such that T (x′) ∩ V ≠ ∅ for all x′ ∈ U;

(iv) continuous if T is both u.s.c. and l.s.c.;

(v) compact if T (X) is contained in a compact subset of Y .

Lemma 1^[16] Assume that T is a map from a topological space X to a topological space Y . It is known that T is l.s.c. at x ∈ X if and only if,for any given point y ∈ T (x) and any given net {x_α} in X converging to x,there exists a net {y_α} ∈ T (x_α) such that y_α converges to y.

Lemma 2^[17] Suppose that X and Y are two Hausdorff topological spaces and T : X ⊸Y is a map. Then,the following statements hold.

(i) If T is an u.s.c. map with closed values,then T is closed;

(ii) If X is a compact space and T is an u.s.c. map with compact values,then T (X) is a compact subset of Y .

Definition 1^[18] Let X be a nonempty set and Y be a topological space. A map T : X ⊸Y is said to be transfer open valued if,for any (x,y) ∈ X×Y with y ∈ T (x),there exists an x′ ∈ X such that y ∈ int T (x′).

It is easy to see that a map with open values is transfer open valued. However,the converse is not true in general.

Lemma 3^[6] Assume that X is a topological space and Z is a nonempty set. Let P : X ⊸Z be a map. Then,the following statements are equivalent.

(i) The map P− is transfer open valued,and the map P has nonempty values;

(ii) .

Definition 2^[11] A triple (E,D; Γ) is said to be an abstract convex space,if E and D are two nonempty sets and Γ : hDi ⊸E is a map with nonempty values. An abstract convex space (E,D; Γ) with any topology on E is said to be an abstract convex topological space.

For any given A ∈ hDi,let Γ_A := Γ(A).

When E = D,we let (E; Γ) := (E,E; Γ). It is easy to see that any vector space E is an abstract convex space with Γ=co,where co denotes the convex hull in the vector space E. Obviously,we know that (R;co) is an abstract convex space.

Assume that (E,D; Γ) is an abstract convex space. For any D′ ⊂ D,we define the Γ-convex hull of D′ as follows:

We say that a subset X of E is a Γ-convex subset of (E,D; Γ) relative to D′ if,for any given N ∈ hD′i,one has ΓN ⊂ X,i.e.,coΓD′ ⊂ X. This shows that (X,D′; Γ|hD′i) itself is an abstract convex space called a subspace of (E,D; Γ). If D is a subset of E,then the space is denoted by (E ⊃ D; Γ). In this case,a subset X of E is called to be Γ-convex if coΓ(X ∩D) ⊂ X,i.e.,X is Γ-convex relative to D′ = X ∩ D. It is easy to see that,if (E; Γ) = (R;co),then the Γ-convex subset reduces to the ordinary convex subset.

Suppose that (E,D; Γ) is an abstract convex space and Z is a set. Let F : E ⊸ Z be a map with nonempty values. A map G : D ⊸ Z is called a Knaster-Kuratowski-Mazurkiewicz (KKM) map with respect to F if

It is well known that a KKM map G : D ⊸E is a KKM map with respect to the identity map 1E. A map F : E ⊸Z is called to have the KKM property (called a R-map) if,for any KKM map G : D ⊸ Z with respect to F,the family {G(y)}y∈D has the finite intersection property. In the sequal,let

.

In the case that Z is a topological space,we define the -map for a closed-valued map G and the -map for a open-valued map G,respectively. Obviously,if Z is discrete,then three classes R,RC,and are identical. We would like to mention that some authors use the notation KKM(E,Z) instead of RC(E,Z).

For a G-convex space (E,D; Γ),it is easy to see that the identity map 1E ∈ (E,E) ∩ (E,E). Furthermore,if F : E → Z is a continuous single-valued map or if F : E ⊸ Z has a continuous selection,then it is easy to see that F ∈ (E,Z) ∩ (E,Z). For more details about classes R,RC,and RO,we refer the reader to Ref. ^[11] and the references therein.

Lemma 4 (see Theorem 3.4 in Ref. ^[19]) Assume that X is a topological space,(Y ; Γ) is an abstract convex topological space,and Z is a nonempty set. Let P : X ⊸ Z,Q : Y ⊸ Z, and T : Y ⊸X be three maps such that

there exists a nonempty compact subset D of X such that, for each given subset N ∈ hY i, there exists a compact Γ-convex subset LN of Y containing N and satisfying

Then,there exists (x₀,y₀) ∈ X × Y with x₀ ∈ T (y₀) and P(x₀) ∩ Q(y₀) 6≠ ∅

Remark 1 When T is compact or X is compact,it is easy to see that the conditions (iv) and (v) of Lemma 4 become superfluous.

Suppose that Z is a real topological vector space,C is a convex cone in Z with int C ≠ ?, and A is a nonempty subset of Z. For any given points z1,z2 ∈ A,we define z₁ < z₂,if and only if z₂ − z₁ ∈ int C. A point ¯y ∈ A is said to be a weakly vector minimal point of A if,for any given point y ∈ A,one has y − ¯y /∈ −int C. Let wMinC A denote the set of all weakly vector minimal points of A.

Lemma 5^[20] Assume that A is a nonempty compact subset of a real topological vector space Z,and D is a closed convex cone in Z with D ≠ Z. Then,one has wMinDA ≠ ∅.

3 Existence theorems of solutions to generalized vector equilibrium prob- lems

In this section,we prove the following existence theorems of solutions concerned with the generalized vector equilibrium problems (i)-(iv),which unify and generalize corresponding re- sults presented in Refs. ^[2] and ^[3].

Theorem 1 Assume that X and V are two topological spaces. Let (Y ; Γ) be an abstract convex topological space,and let Z be a nonempty set. Suppose that F : X×Z ⊸V ,C : X ⊸ V , Q : Y ⊸ Z,and T : Y ⊸X are four maps such that

(i) Q(Y ) = Z;

(ii) for each given point x ∈ X, {z ∈ Z : F(x, z) 6⊂ C(x)} ≠ ? implies that there is an open neighborhood N(x) of x with z′ ∈ Z such that F(x′, z′) 6⊂ C(x′) for all x′ ∈ N(x);

(iii) for each given point x ∈ X, the set {y ∈ Y : F(x, z) 6⊂ C(x) for some z ∈ Q(y)} is 􀀀-convex;

(iv) for each given point y ∈ Y with x ∈ T (y) and z ∈ Q(y), one has F(x, z) ⊂ C(x);

(v) T ∈ (Y,X);

(vi) for each given compact subset A of Y, T (A) is a compact sunset of X;

(vii) there exists a nonempty compact subset D of X such that, for each N ∈ hY i, there is a compact 􀀀-convex subset LN of Y containing N and satisfying

Then, there exists a point x0 ∈ X with F(x0, z) ⊂ C(x0) for all z ∈ Z.

Proof Let P : X ⊸Z be a map defined by

If the conclusion of Theorem 1 is not true,then for each x ∈ X,there exists a point z ∈ Z such that F(x,z) 6⊂ C(x) and so P(x) ≠ ? for all x ∈ X.

Obviously,the condition (ii) of Theorem 1 is equivalent to the fact that the map P− is transfer open valued. By the condition (ii) and the definition of P,it is easy to see that,for each x ∈ X,P(x) ≠ ? shows that there is an open neighborhood N(x) of x and a point z′ ∈ Z with z′ ∈ P(x′) for all x′ ∈ N(x). Therefore,for each given x ∈ X and z ∈ P(x),we know that there is an open neighborhood N(x) of x and a point z′ ∈ Z such that N(x) ⊂ P−(z′). This implies that P− is transfer open valued. By Lemma 3 and the condition (i) in Theorem 1,it follows that

Thus,the condition (i) of Lemma 4 is satisfied. From the conditions (iii) and (v)-(vii) of Theorem 1,we know that the conditions (ii)-(v) of Lemma 4 are satisfied. It follows from Lemma 4 that there exists (x0,y0) ∈ X × Y such that x0 ∈ T (y0) and P(x0) ∩ Q(y0) ≠ ?. Thus,there exists z0 ∈ Q(y0) with F(x0,z0) 6⊂ C(x0),which contradicts with the condition (iv) of Theorem 1. Therefore,the conclusion of Theorem 1 is true. The proof is complete.

Remark 2 It is easy to see that Theorem 1 includes Theorem 11 of Ref. ^[2] and Theorem 8 of Ref. ^[3] as special cases. If T is compact or X is compact,then we know that the conditions (vi) and (vii) of Theorem 1 become superfluous.

Corollary 1 Assume that X is a compact topological space. Suppose that (Y ; Γ) is an abstract convex space,Z is a nonempty set,and V is a topological space. Let F : X × Z ⊸ V, C : X ⊸V,Q : Y ⊸ Z,and T : Y ⊸X be four maps such that

(i) Q(Y ) = Z;

(ii) for each given point z ∈ Z, the map x ⊸F(x, z) is l.s.c., and C is closed;

(iii) for each given point x ∈ X, the set {y ∈ Y : F(x, z) 6⊂ C(x) for some z ∈ Q(y)} is 􀀀-convex;

(iv) for each given point y ∈ Y with x ∈ T (y) and z ∈ Q(y), one has F(x, z) ⊂ C(x);

(v) T ∈ (Y,X).

Then, there exists a point x0 ∈ X with F(x0, z) ⊂ C(x0) for all z ∈ Z.

Theorem 2 Assume that X is a topological space,(Y ; Γ) is an abstract convex topological space,Z is a nonempty set,and V is a topological vector space. Let F : X×Z ⊸ V,C : X ⊸V, Q : Y ⊸ Z,and T : Y ⊸X be four maps such that

(i) Q(Y ) = Z;

(ii) for each given point x ∈ X, the condition {z ∈ Z : F(x, z) ∩ (−int C(x)) ≠ ?} ≠ ? implies that there is an open neighborhood N(x) of x with z′ ∈ Z such that F(x′, z′) ∩ (−int C(x′)) ≠ ? for all x′ ∈ N(x);

(iii) for each given point x ∈ X, the set {y ∈ Y : F(x, z) ∩ (−int C(x)) ≠ ? for some z ∈ Q(y)} is 􀀀-convex;

(iv) for each point y ∈ Y with x ∈ T (y) and z ∈ Q(y), one has F(x, z) ∩ (−int C(x)) = ?;

(v) T ∈ (Y,X);

(vi) for each given compact subset A of Y, T (A) is a compact subset of X;

(vii) there is a nonempty compact subset D of X such that, for each N ∈ hY i, there exists a compact 􀀀-convex subset LN of Y containing N with

Then, there is a point x0 ∈ X with F(x0, z) ∩ (−int C(x0)) = ? for all z ∈ Z.

Proof Let P : X ⊸Z be the map defined by

Then,we know that the proof of Theorem 2 is similar to the proof of Theorem 1. Therefore, we omit it here. The proof is complete.

Remark 3 Theorem 2 includes Theorem 12 of Ref. ^[2] and Theorem 9 of Ref. ^[3] as special cases. If T is compact or X is compact,then we know that the conditions (vi) and (vii) of Theorem 2 become superfluous.

Corollary 2 Assume that X is a compact topological space,(Y ; Γ) is an abstract convex space,Z is a nonempty set,and V is a topological vector space. Let F : X×Z ⊸ V,C : X ⊸V, Q : Y ⊸ Z,and T : Y ⊸X be four maps such that

(i) Q(Y ) = Z;

(ii) for each given point z ∈ Z, the map x ⊸ F(x, z) is l.s.c., C(x) has nonempty interior for each x ∈ X, the map W : X ⊸ V defined by W(x) = V \ (−int C(x)) is closed;

(iii) for each given point x ∈ X, the set {y ∈ Y : F(x, z) ∩ (−int C(x)) ≠ ∅ for some z ∈ Q(y)} is 􀀀-convex;

(iv) for each given point y ∈ Y with x ∈ T (y) and z ∈ Q(y), one has F(x, z)∩(−int C(x)) = ∅;

(v) T ∈ (Y,X). Then, there is x0 ∈ X with F(x0, z) ∩ (−int C(x0)) = ∅for all z ∈ Z.

Theorem 3 Assume that X and V are two topological spaces,(Y ; Γ) is an abstract convex topological space,and Z is a nonempty set. Let F : X × Z ⊸ V,C : X ⊸ V,Q : Y ⊸ Z,and T : Y ⊸X be four maps such that

(i) Q(Y ) = Z;

(ii) for each given point x ∈ X, the condition {z ∈ Z : F(x, z) ∩ C(x) = ∅} ≠ ∅ implies that there is an open neighborhood N(x) of x with z′ ∈ Z such that F(x′, z′) ∩ C(x′) = ∅ for all x′ ∈ N(x);

(iii) for each given point x ∈ X, the set {y ∈ Y : F(x, z) ∩ C(x) = ∅ for some z ∈ Q(y)} is 􀀀-convex;

(iv) for each given point y ∈ Y with x ∈ T (y) and z ∈ Q(y), one has F(x, z) ∩ C(x) ≠ ∅;

(v) T ∈ (Y,X);

(vi) for each given compact subset A of Y, T (A) is a compact subset of X;

(vii) there is a nonempty compact subset D of X such that, for each N ∈ hY i, there exists a compact 􀀀-convex subset LN of Y containing N with

Then, there is x0 ∈ X with F(x0, z) ∩ C(x0) ≠ ∅ for all z ∈ Z.

Proof Let P : X ⊸Z be the map defined by P(x) = {z ∈ Z : F(x,z) ∩ C(x) = ∅},∀x ∈ X. We know that the proof of Theorem 3 is similar to the proof of Theorem 1. Therefore,we omit it here. The proof is complete.

Remark 4 Theorem 3 includes Theorem 13 of Ref. ^[2] and Theorem 10 of Ref. ^[3] as special cases. If T is compact or X is compact,then we know that the conditions (vi) and (vii) of Theorem 3 become superfluous.

Corollary 3 Assume that X is a compact topological space,(Y ; Γ) is an abstract convex space,Z is a nonempty set,and V is a topological space. Let F : X × Z ⊸ V,C : X ⊸ V, Q : Y ⊸ Z,and T : Y ⊸X be four maps such that

(i) Q(Y ) = Z;

(ii) the map F is u.s.c. with compact values, and the map C is closed;

(iii) for each given point x ∈ X, the set {y ∈ Y : F(x, z) ∩ C(x) = ∅ for some z ∈ Q(y)} is 􀀀-convex;

(iv) for each given point y ∈ Y with x ∈ T (y) and z ∈ Q(y), one has F(x, z) ∩ C(x) 6= ∅;

(v) T ∈ (Y,X).

Then, there is x0 ∈ X with F(x0, z) ∩ C(x0) 6= ∅ for all z ∈ Z.

Theorem 4 Assume that X is a topological space, (Y ; 􀀀) is an abstract convex topological space, Z is a nonempty set, and V is a topological vector space. Let F : X×Z ⊸ V, C : X ⊸V, Q : Y ⊸ Z, and T : Y ⊸X be four maps such that

(i) Q(Y ) = Z;

(ii) for each given point x ∈ X, the condition {z ∈ Z : F(x, z) ⊂ −int C(x)} 6= ? implies that there is an open neighborhood N(x) of x with z′ ∈ Z such that F(x′, z′) ⊂ −int C(x′) for all x′ ∈ N(x);

(iii) for each given point x ∈ X, the set {y ∈ Y : F(x, z) ⊂ −int C(x) for some z ∈ Q(y)} is 􀀀-convex;

(iii) for each given point x ∈ X, the set {y ∈ Y : F(x, z) ⊂ −int C(x) for some z ∈ Q(y)} is 􀀀-convex;

(iv) for each given point y ∈ Y with x ∈ T (y) and z ∈ Q(y), one has F(x, z) 6⊂ −int C(x);

(v) T ∈ (Y,X);

(vi) for each given compact subset A of Y, T (A) is a compact subset of X;

(vii) there is a nonempty compact subset D of X such that, for each N ∈ hY i, there exists a compact 􀀀-convex subset LN of Y containing N with

Then, there exists x0 ∈ X such that F(x0, z) 6⊂ −int C(x0) for all z ∈ Z.

Proof Let P : X ⊸Z be the map defined by

We know that the proof of Theorem 4 is similar to the proof of Theorem 1. Therefore,we omit it here. The proof is complete.

Remark 5 Theorem 4 includes Theorem 14 of Ref. ^[2] and Theorem 11 of Ref. ^[3] as special cases. If T is compact or X is compact,then we know that the conditions (vi) and (vii) of Theorem 4 become superfluous.

Corollary 4 Assume that X is a compact topological space,(Y ; Γ) is an abstract convex space,Z is a nonempty set,and V is a topological vector space. Let F : X×Z ⊸ V,C : X ⊸V, Q : Y ⊸ Z,and T : Y ⊸X be four maps such that

(i) Q(Y ) = Z;

(ii) the map F is u.s.c. with compact values, C(x) has a nonempty interior for each x ∈ X, and the map W : X ⊸V defined by W(x) = V \ (−int C(x)) is closed;

(iii) for each given point x ∈ X, the set {y ∈ Y : F(x, z) ⊂ −int C(x) for some z ∈ Q(y)} is Γ-convex;

(iv) for each given point y ∈ Y with x ∈ T (y) and z ∈ Q(y), one has F(x, z) ∉ −int C(x);

(v) T ∈ (Y,X).

Then, there is x0 ∈ X with F(x0, z) ∉ −int C(x0) for all z ∈ Z.

4 Applications to 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 (v)-(viii).

Assume that X is a compact topological space,L is a real topological vector space ordered by a proper closed convex cone D,and h : X ⊸L is an u.s.c. map with compact values.

Theorem 5 Assume that all conditions of Corollary 1 are satisfied. Then,there is a solution to the problem

where

.

Proof By Corollary 1,we know that K ≠ ?. Next,we show that the set K is closed. In fact,if x ∈ K,then there is a net {x_α}α∈Λ in K with x_α → x. For each α ∈ Λ,one has

(2) For any given point z ∈ Z,let v ∈ F(x,z). Since the map x ⊸ F(x,z) is l.s.c.,it follows from Lemma 1 that there exists a net {vα}α∈Λ in V converging to v with vα ∈ F(x_α,z) for all α ∈ Λ. It follows from (2) that vα ∈ C(x_α) for all α ∈ Λ. By the closedness of C,one has v ∈ C(x) and F(x,z) ⊂ C(x). It follows that x ∈ K and K is a closed subset of X. Moreover,since X is compact,we know that K is compact. Now,Lemma 2 (ii) implies that h(K) is a nonempty compact subset of L. It follows from Lemma 5 that wMinD h(K) ≠ ?. The proof is complete.

Remark 6 The special case of the problem (1) was considered by Lin^[5] as Theorem 5.1 under different assumptions.

Theorem 6 Assume that all conditions of Corollary 2 are satisfied. Then,there is a solution to the problem

.

where

.

.

Proof By Corollary 2,one has K ≠ ∅. We now show that K is closed. In fact,if x ∈ K, then there is a net {x_α}α∈Λ in K with x_α → x. For each α ∈ Λ,one has

Let z ∈ Z be a given point with v ∈ F(x,z). Since the map x ⊸ F(x,z) is l.s.c.,it follows from Lemma 1 that there exists a net {vα}α∈Λ in V converging to v with vα ∈ F(x_α,z) for all α ∈ Λ. By (3),we know that vα /∈ −int C(x_α) for all α ∈ Λ and vα ∈ W(x_α) for all α ∈ Λ. The closedness of W shows that v ∈ W(x) and v /∈ −int C(x). It follows that F(x,z) ∩ (−int C(x)) = ∅ and x ∈ K,i.e.,K is a closed subset of X. The compactness of X implies that K is compact. From Lemma 2 (ii),we know that h(K) is a nonempty compact subset of L. It follows from Lemma 5 that wMinD h(K) ≠ ∅. The proof is complete.

Theorem 7 Assume that all conditions of Corollary 3 are satisfied. Then,there is a solution to the problem

, where

.

Proof By Corollary 3,one has K ≠ ∅. We show that K is closed. In fact,if x ∈ K,then there exists a net {x_α}α∈Λ in K with x_α → x. For each α ∈ Λ,one has F(x_α,z) ∩ C(x_α) ≠ ∅ for all z ∈ Z. For any given point z ∈ Z,let vα ∈ F(x_α,z)∩C(x_α) and A = {x_α : α ∈ Λ}∪{x}. Then,it is easy to see that A is a compact set. Since the map F is u.s.c. with compact values,from Lemma 2 (ii),we know that F(A × {z}) is a compact set and {vα}α∈Λ has a subnet {vαλ}αλ∈Λ with vαλ → v. Moreover,it follows from Lemma 2 (i) that F is closed and v ∈ F(x,z). Now,the closedness of C implies that v ∈ C(x) and F(x,z) ∩ C(x) ≠ ∅. This means that x ∈ K and K is a closed subset of X. Thus,we know that K is compact. From Lemma 2 (ii),it is easy to see that h(K) is a nonempty compact subset of L and Lemma 5 yields that wMinD h(K) ≠ ∅. The proof is complete.

Theorem 8 Assume that all conditions of Corollary 4 are satisfied. Then,there is a solution to the problem

where K = {x ∈ X : F(x,z) 6⊂ −int C(x) for all z ∈ Z}.

Proof By Corollary 4,one has K ≠ ∅. Now,we prove that K is closed. In fact,if x ∈ K,then there is a net {x_α}α∈Λ in K with x_α → x. For each α ∈ Λ,one has F(x_α,z) 6⊂ −int C(x_α) for all z ∈ Z. For any given point z ∈ Z,let vα ∈ F(x_α,z) \ (−int C(x_α)) and A = {x_α : α ∈ Λ} ∪ {x}. Then,it is easy to see that A is compact. Since the map F is u.s.c. with compact values,by Lemma 2 (ii),we know that F(A × {z}) is compact and {vα}α∈Λ has a subnet {vαλ}αλ∈Λ with v_αλ → v. From Lemma 2 (i),we have that F is closed and v ∈ F(x,z). Since v_α ∈ W(x_α) for all α ∈ Λ,the closedness of W implies that v ∈ W(x) and v ∈ F(x,z) \ (−int C(x)). Thus,x ∈ K and K is a closed subset of X. Now,the compactness of X shows that K is compact. From Lemma 2 (ii),we know that h(K) is a nonempty compact subset of L. Thus,Lemma 5 yields that wMinD h(K) ≠ ∅. The proof is complete.

Remark 7 The special case of the problem (4) was considered by Lin^[5] as Theorem 5.3 under different assumptions.

Finally,we give the following example to illustrate that all the conditions of Theorems 5,6, 7,and 8 can be satisfied.

Example 1 Let X = [0 1] be endowed with the Euclidean topology. Let (Y ; Γ) = ([0 1];co), Z = [0 1],and V = R. Let F : X × Z ⊸ V ,C : X ⊸ V ,Q : Y ⊸ Z,and T : Y ⊸ X be four maps defined by

Obviously,Q(Y )=Z and T ∈RC(Y,X). Moreover,it is easy to have the following conclusions.

(i) For each z ∈ Z, the map x ⊸ F(x, z) is l.s.c., and the map F is u.s.c. with compact values;

(ii) C is closed, and for each x ∈ X, int C(x) = (0,+∞) 6= ?, and W(x) = V \(−int C(x)) = C(x). Hence, W is closed.

(iii) For each x ∈ X, the set

is convex.

(iv) For each y ∈ Y with x ∈ T (y) and z ∈ Q(y), one has F(x, z) = F(x, x) = 0. Thus, we have

Therefore,all conditions of Corollaries 1,2,3,and 4 are satisfied. Further,let L=R, D=[0,+∞),and h : X ⊸L be defined by

Then,it is easy to see that h is an u.s.c. map with compact values. Thus,all conditions of Theorems 5,6,7,and 8 are satisfied.

5 Conclusions

In this paper,we introduce and study four types of the generalized vector equilibrium problems (i)-(iv) with moving cones in topological spaces without linear structures. By a coincidence theorem of Yang and Huang^[19],we obtain some existence theorems of solutions tothe generalized vector equilibrium problems under suitable conditions. We also give applications to the generalized semi-infinite programs with generalized vector equilibrium constraints.

We would like to point out that the problems considered in this paper are quite different from the ones studied by Balaj^[2],Yang et al.^{[9, 21]},and Ding and Ding^[22]. In fact,Yang et al.^[9] studied the system of generalized equilibrium problems in abstract convex spaces which extended the generalized equilibrium problems of Balaj^[2]. Yang et al.^[21] considered the system of generalized quasivariational inclusion problems in the LΓ-spaces,and Ding and Ding^[22] studied the generalized vector equilibrium problems in the noncompact FC-spaces. Since the abstract convex space includes the LΓ-space and the FC-space as special cases,by Remarks 2-7,we know that the results presented in this paper can be considered as generalization and improvement of the corresponding results of Balaj^[2],Balaj and Lin^[3],Lin^[5],Yang et al.^{[9, 21]}, and Ding and Ding^[22].

It is well known that the quasi-equilibrium problem is an important generalization of the equilibrium problem. Therefore,it is interesting and important to study some types of gener- alized vector quasi-equilibrium problems with moving cones in topological spaces.

Acknowledgements The authors are grateful to the editor and the referees for their valuable comments and suggestions.