Abstract
In this paper, several kinds of generalized vector quasiequilibrium problems are introduced and studied in abstract convex spaces. Using the properties of Γconvex and \(\mathfrak{KC}\)maps, some sufficient conditions are given to guarantee the existence of solutions in connection with these generalized vector quasiequilibrium problems. As applications, some existence theorems of solutions for the generalized semiinfinite programs with vector quasiequilibrium constraints are also given.
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1 Introduction
It is well known that the vector quasiequilibrium problem is an important generalization of the vector equilibrium problem which provides a unified model for vector quasivariational inequalities, vector quasicomplementarity problems, vector optimization problems and vector saddle point problems. In 2000, Fu [1] established the existence theorems for the generalized vector quasiequilibrium problems and the setvalued vector equilibrium problems. In 2003, Ansari and Fabián [2] considered a generalized vector quasiequilibrium problem with or without involving Φcondensing 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 quasiequilibrium problems. Recently, Lin et al. [4] introduced and studied a class of generalized vector quasiequilibrium problems involving pseudomonotonicity hemicontinuity mappings under different conditions in topological vector spaces. Lin et al. [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 quasiequilibrium 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 quasiequilibrium problems. Fu and Wang [7] considered the generalized strong vector quasiequilibrium problems with domination structure. On the other hand, Ding [8] studied the existence of solutions for generalized vector quasiequilibrium problems in locally Gconvex spaces. Balaj and Lin [9] investigated existence of solutions for the generalized equilibrium problems in Gconvex spaces.
The abstract convex space, introduced by Park [10] in 2006, includes the convex subset of a topological vector space, the convex space, the Hspace, and the Gconvex 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 setvalued 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 setvalued maps in abstract convex spaces with applications. Yang and Huang [14] gave some coincidence theorems for compact and noncompact \(\mathfrak{KC}\)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 quasiequilibrium 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 quasiequilibrium problems in abstract convex spaces. Therefore, it is natural and interesting to study some generalized vector quasiequilibrium problems in abstract convex spaces under some suitable conditions.
On the other hand, we know that semiinfinite 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 semiinfinite programming precisely to deduce important results about two such geometric problems: the problems of covering a compact body in finite dimensional spaces by the minimumvolume disk and the minimumvolume ellipsoid, many researchers have been investigated the theory, applications and methods for the semiinfinite programming (see, for example, [19–22]). As a generalization of semiinfinite programming, the generalized semiinfinite programming has been become a vivid field of active research in mathematical programming in recent years due to its important applications to numerous reallife 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 quasiequilibrium constraints in abstract convex spaces.
The main purpose of this paper is to study several classes of generalized vector quasiequilibrium 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 quasiequilibrium 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 setvalued mapping \(T:X\rightrightarrows Y\) is a mapping from X into the power set \(2^{Y}\). The inverse \(T^{1}\) of T is the setvalued mapping from Y to X defined by
An abstract convex space \((X,D,\Gamma)\) consists of a nonempty set X, a nonempty set D, and a setvalued mapping \(\Gamma:\langle D\rangle\rightrightarrows X\) with nonempty values, where \(\langle D\rangle\) denotes the set of all nonempty finite subset of a set D. If for each \(A\in\langle D\rangle\) with the cardinality \(A=n+1\), there exists a continuous function \(\phi_{A}:\triangle _{n}\rightarrow\Gamma(A)\) such that \(J\in\langle A\rangle\) implies \(\phi_{A}(\triangle_{J})\subseteq\Gamma(J)\), where \(\triangle_{n}\) is the standard nsimplex and \(\triangle_{J}\) the face of \(\triangle _{n}\) corresponding to \(J\in\langle A\rangle\), then the abstract convex space reduces to the Gconvex space. Let \(\Gamma_{A}:=\Gamma (A)\) for \(A\in \langle D\rangle\). When \(D\subset X\), the space is defined by \((X\supseteq D,\Gamma)\). In this case, a subset M of X is said to be Γconvex if, for any \(A\in\langle M\cap D \rangle\), we have \(\Gamma_{A}\subseteq M\). In the case \(X=D\), let \((X,\Gamma):=(X,X,\Gamma)\).
It is easy to see that any vector space Y is an abstract convex space with \(\Gamma:=\operatorname{co}\), 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 \(\mathcal{V}\) of its origin. A subset X of E is said to be almost convex (see [24] for more details) if for any \(V\in\mathcal{V}\) and for any finite subset \(A=\{x_{1}, x_{2}, \ldots, x_{n}\}\) of X, there exists a subset \(B=\{ y_{1}, y_{2}, \ldots,y_{n}\}\) of X such that \(y_{i}  x_{i} \in V\) for all \(i = 1, 2, \ldots, n\) and \(\operatorname{co} B \subset X\). Let \(\Gamma_{A} = \operatorname{co} B\) for any \(A \in\langle X \rangle\). Then \((X, \Gamma)\) is a Gconvex space and hence an abstract convex space.
Example 2.2
([10])
Usually, a convex space \((E, \mathcal{C})\) in the classical sense consists of a nonempty set E and a family \(\mathcal{C}\) of subsets of E such that E itself is an element of \(\mathcal{C}\) and \(\mathcal{C}\) is closed under arbitrary intersection. For any given subset \(X\subset E\), the \(\mathcal{C}\)convex hull of X is defined as by
We say that X is \(\mathcal{C}\)convex if \(X=\operatorname{Co}_{\mathcal{C}}X\). Consider the mapping \(\Gamma: \langle E\rangle\rightrightarrows E\) defined by \(\Gamma_{A}=\operatorname{Co}_{\mathcal{C}}A\). Then \((E, \Gamma)\) is an abstract convex space.
Example 2.3
Let \((M,d)\) be a pseudometric space, that is, \(d:M\times M\to [0,+\infty)\) such that, for every \(x,y,z \in M\),

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

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

(iii)
\(d(x,z) \leq d(x,y) + d(y,z)\).
For any \(A\in\langle M\rangle\), define a setvalued mapping \(\Gamma: \langle M\rangle\rightrightarrows M\) by
Then it is easy to see that \((M,\Gamma)\) 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 Lspaces, spaces having property (H), pseudo Hspaces, Mspaces, GHspaces, another Lspaces, FCspaces 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 \((X,\Gamma)\) be an abstract convex space and V be a real topological vector space. Let E be a nonempty subset of X. Assume that \(S:E\rightrightarrows E\) and \(B:E\rightrightarrows E\) are two setvalued mappings. Suppose that \(F:X\times X\times X\rightrightarrows V\) and \(C:X \rightrightarrows V\) are two setvalued mappings such that for each \(x\in X\), \(C(x)\) is a closed convex cone with \(\operatorname{int} C(x)\neq\emptyset\), here \(\operatorname{int} C(x)\) denotes the interior of \(C(x)\). In this paper, we will consider the following generalized vector quasiequilibrium problems in abstract convex spaces.
• (GVQEP1) Find \(\tilde{x}\in E\) such that
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 \(x\in E\), (GVQEP1) was considered by Yang and Huang [17] in abstract convex spaces and by Balaj and Lin [9] in Gconvex spaces, respectively.
• (GVQEP2) Find \(\tilde{x}\in E\) and \(\tilde{z}\in B(\tilde{x})\) such that
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 \(x\in E\), then (GVQEP2) was investigated by Fu and Wang [7] in topological vector spaces.
• (GVQEP3) Find \(\tilde{x}\in E\) such that
We note that (GVQEP3) was considered by Lin et al. [4] in topological vector spaces. When \(S(x)=B(x)=E\) for all \(x\in E\), (GVQEP3) was studied by Yang and Huang [17] in abstract convex spaces and by Balaj and Lin [9] in Gconvex spaces, respectively.
• (GVQEP4) Find \(\tilde{x}\in E\) and \(\tilde{z}\in B(\tilde{x})\) such that
We note that (GVQEP4) was considered by Li and Li [27] in topological vector spaces.
• (GVQEP5) Find \(\tilde{x}\in E\) such that
When \(S(x)=B(x)=E\) for all \(x\in E\), (GVQEP5) was investigated by Yang and Huang [17] in abstract convex spaces and by Balaj and Lin [9] in Gconvex spaces, respectively.
• (GVQEP6) Find \(\tilde{x}\in E\) and \(\tilde{z}\in B(\tilde{x})\) such that
It is worth mentioning that (GVQEP6) was considered by Lin et al. [4], and Li and Li [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 \(\tilde{x}\in E\) such that
When \(S(x)=B(x)=E\) for all \(x\in E\), (GVQEP7) was studied by Yang and Huang [17] in abstract convex spaces and by Balaj and Lin [9] in Gconvex spaces, respectively.
• (GVQEP8) Find \(\tilde{x}\in E\) and \(\tilde{z}\in B(\tilde{x})\) such that
When \(S(x)=B(x)=E\) for all \(x\in E\), (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 wellknown insights into the generalized vector quasiequilibrium 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 \(h: X\rightrightarrows L\) is a setvalued mapping, where L is a real topological vector space ordered by a closed convex pointed cone \(H\subseteq L\) with \(\operatorname{int} H\neq\emptyset\). 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 semiinfinite programs with generalized vector quasiequilibrium constraints:
• (GSIP1) Generalized semiinfinite program with constraint (GVQEP1):
where
When \(S(x)=B(x)=E\) for all \(x\in E\), (GSIP1) was considered by Yang and Huang [17] in abstract convex spaces.
• (GSIP2) Generalized semiinfinite program with constraint (GVQEP2):
where
Some special cases of (GSIP2) were considered by Lin [28] in topological vector spaces.
• (GSIP3) Generalized semiinfinite program with constraint (GVQEP3):
where
When \(S(x)=B(x)=E\) for all \(x\in E\), (GSIP3) was studied by Yang and Huang [17] in abstract convex spaces.
• (GSIP4) Generalized semiinfinite program with constraint (GVQEP4):
where
We would like to mention that some special cases of (GSIP4) were studied by Lin [28] in topological vector spaces.
• (GSIP5) Generalized semiinfinite program with constraint (GVQEP5):
where
When \(S(x)=B(x)=E\) for all \(x\in E\), (GSIP5) was investigated by Yang and Huang [17] in abstract convex spaces.
• (GSIP6) Generalized semiinfinite program with constraint (GVQEP6):
where
We note that some special cases of (GSIP6) were considered by Lin [28] in topological vector spaces.
• (GSIP7) Generalized semiinfinite program with constraint (GVQEP7):
where
When \(S(x)=B(x)=E\) for all \(x\in E\), (GSIP7) was studied by Yang and Huang [17] in abstract convex spaces.
• (GSIP8) Generalized semiinfinite program with constraint (GVQEP8):
where
It is worth mentioning that (GSIP8) can be considered as a generalization of the generalized vector semiinfinite 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 semiinfinite program [17], the mathematical program with equilibrium constraint [19], the generalized semiinfinite program [23], the generalized vector semiinfinite 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 \(K\subseteq V\) be a nonempty set and \(C\subseteq V\) be the closed convex pointed cone with \(\operatorname{int} C\neq\emptyset\). The set of all weak minimal points of K with respect to the ordering cone C is defined as
Definition 2.2
Let \((X,D,\Gamma)\) be an abstract convex space and Z be a set. For a setvalued mapping \(T:X\rightrightarrows Z\) with nonempty values, if a setvalued mapping \(G:D\rightrightarrows Z\) satisfies
then G is called a KKM mapping with respect to T. A KKM mapping \(G:D\rightrightarrows X\) is a KKM mapping with respect to the identity mapping \(I_{X}\).
A setvalued mapping \(F:X\rightrightarrows Z\) is called to be a \(\mathfrak{KC}\)map if, for any closedvalued KKM mapping \(G:D\rightrightarrows Z\) with respect to F, the family \(\{G(y)\} _{y\in D}\) has the finite intersection property. We denote
Definition 2.3
([33])
Let X and Y be two topological spaces. A setvalued mapping \(F:X\rightrightarrows Y\) is said to be

(i)
upper semicontinuous (u.s.c.) at \(x_{0}\) if for any open set \(V \supseteq F(x_{0})\), there is an open neighborhood \(O_{x_{0}}\) of \(x_{0}\) such that \(F(x')\subseteq V\) for each \(x'\in O_{x_{0}}\),

(ii)
lower semicontinuous (l.s.c.) at \(x_{0}\) if for any open set \(V\cap F(x_{0})\neq\emptyset\), there is an open neighborhood \(O_{x_{0}}\) of \(x_{0}\) such that \(F(x')\cap V\neq\emptyset\) for each \(x'\in O_{x_{0}}\),

(iii)
continuous at \(x_{0}\) if it is both upper and lower semicontinuous at \(x_{0}\),

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

(v)
closed if and only if its graph \(\operatorname{Graph}(F):=\{(x,y)\in X\times Y:y\in F(x)\}\) is closed.
Lemma 2.1
([34])
Let X and Y be two topological spaces and \(F:X\rightrightarrows Y\) a setvalued mapping.

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

(ii)
if X 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 \(F:X\rightrightarrows Y\) be upper semicontinuous and \(F(x)\) is compact. Then for any net \(\{ x_{\alpha}\}\subset X\) with \(x_{\alpha}\rightarrow x\) and \(y_{\alpha}\in F(x_{\alpha})\), there exists a subnet \(\{y_{\beta}\}\subset y_{\alpha}\) such that \(y_{\beta}\rightarrow y\in F(x)\).
Lemma 2.3
([36])
Let X and Y be two topological spaces and \(F:X\rightrightarrows Y\) be lower semicontinuous at \(x\in X\) if and only if for any \(y\in F(x)\) and any net \(\{x_{\alpha}\}\) with \(x_{\alpha}\rightarrow x\), there is a net \(\{y_{\alpha}\}\) such that \(y_{\alpha}\in F(x_{\alpha})\) and \(y_{\alpha}\rightarrow y\).
Lemma 2.4
([10])
Let \((X,D,\Gamma)\) be an abstract convex space, Z a set, and \(T:X\rightrightarrows Z\) a setvalued mapping. Then \(F\in\mathfrak{KC}(X,Z)\) if and only for any \(G:D\rightrightarrows Z\) satisfying

(i)
G is closedvalues;

(ii)
\(F(\Gamma_{N})\subseteq G(N)\) for any \(N\in\langle D\rangle\),
we have
for each \(N\in\langle D\rangle\).
Lemma 2.5
([32])
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 \(D\neq V\). Then one has \(\operatorname{wMin}_{D} A\neq\emptyset\).
An abstract convex space with any topology is called an abstract convex topological space. In the rest of this paper, let \((X,\Gamma)\) 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 \(T:X\rightrightarrows X\), \(B:E\rightrightarrows E\), \(S:E\rightrightarrows E\), \(F:E\times E\times E\rightrightarrows V\) and \(Q:E\rightrightarrows V\) are five setvalued mappings. Let ρ be a binary relation on \(2^{V}\) and \(\rho^{c}\) be the complementary relation of ρ. Let α be any of the quantifiers ∀, ∃, and \(\bar {\alpha}\) be the other of the quantifiers ∀, ∃.
3 Main results
In order to show the existence of solutions for the vector quasiequilibrium problems (GVQEP1)(GVQEP8), we first give the following general result.
Theorem 3.1
Suppose that the following conditions are satisfied:

(i)
\(T\in\mathfrak{K}\mathfrak{C}(X,X)\);

(ii)
for each \(y\in E\), the set \(\{x\in E:(\bar{\alpha})z\in B(x), \rho^{c}(F(x,y,z), Q(x))\}\) is open in E;

(iii)
\(G_{0}=\{x\in E:x\notin S(x)\}\) is open in E;

(iv)
for each \(x\in E\), \(S(x)\) is nonempty Γconvex, \(S^{1}(y)\) is open for all \(y\in E\);

(v)
for each \((x_{0}, y_{0})\in E\times E\) with \(x_{0}\in T(y_{0})\) such that \(y_{0}\notin S(x_{0})\).
Then there exists \(\tilde{x}\in S(\tilde{x})\) such that \((\alpha) z\in B(\tilde{x})\), \(\rho(F(\tilde{x},y,z), Q(\tilde{x}))\) for any \(y\in S(\tilde{x})\).
Proof
For any \(x \in E\), define \(A:E\rightrightarrows E\) by
From the definition of \(A(x)\), one has
Define \(P:E\rightrightarrows E\) by
Let \(M(y)=E\setminus P^{1}(y)\). We show that \(M(y)\) is closed for all \(y\in E\). In fact, it follows from (1) that
Since \(S^{1}(y)\), \(A^{1}(y)\), and \(G_{0}\) 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 \(x_{0}\in E\) such that \(x_{0}\in T(\Gamma _{N})\setminus M(N)\). This shows that there exists a point \(y_{0}\in \Gamma_{N}\) such that \(x_{0}\in T(y_{0})\), \(x_{0}\in P^{1}(y)\) for any \(y\in N\), and so \(N\subset P(x_{0})\subset S(x_{0})\). Since \(S(x_{0})\) is Γconvex and \(N\in\langle S(x_{0})\rangle\), we know that \(y_{0}\in\Gamma_{N}\subset S(x_{0})\), 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 \(M(y)\subset E\) is closed and E is compact, we know that \(M(y)\) is compact for any \(y\in E\) and so
Thus, there exists a point \(\tilde{x}\in E\) such that
This implies that \(\tilde{x}\notin P^{1}(y)\) for all \(y\in E\) and so \(P(\tilde{x})=\emptyset\).
If \(\tilde{x}\in G_{0}\), then it is easy to see that \(S(\tilde {x})=P(\tilde{x})=\emptyset\), which is a contradiction. Therefore, we have
and so
that is, \(\tilde{x}\in S(\tilde{x})\), \((\alpha)z\in B(\tilde{x})\), \(\rho (F(\tilde{x},y,z), C(\tilde{x}))\) for all \(y\in S(\tilde{x})\). 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)′:

\(S:E\rightrightarrows E\) is a u.s.c. setvalued mapping.
Next we give some existence theorems in connection with the solution of the vector quasiequilibrium 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 \(y\in E\), \(F(\cdot,y,\cdot)\) is l.s.c. and C is closed;

(b)
B is l.s.c.
Then there exists \(\tilde{x}\in E\) such that \(\tilde{x}\in S(\tilde {x})\) and \(F(\tilde{x},y,z)\subset C(\tilde{x})\) for all \(y\in S(\tilde{x})\) and \(z\in B(\tilde{x})\).
Proof
Let
We show that
is open. Let \(\{x_{\alpha}\}\subseteq E\setminus A^{1}(y)\) be a net with \(x_{\alpha}\rightarrow x_{0}\). Then
Since B and \(F(\cdot,y,\cdot)\) are l.s.c., by Lemma 2.3, for any \(z\in B(x_{0})\) and \(v\in F(x_{0},y,z)\), there exist \(z_{\alpha}\in B(x_{\alpha})\) and \(v_{\alpha}\in F(x_{\alpha},y,z_{\alpha})\) such that \(z_{\alpha}\rightarrow z\) and \(v_{\alpha}\rightarrow v\). Now the closedness of C with \(v_{\alpha}\in C(x_{\alpha})\) shows that \(v\in C(x)\) and so \(F(x_{0},y,z)\subseteq C(x)\) for any \(z\in B(x_{0})\). This shows that \(x_{0}\in E\setminus A^{1}(y)\) and so \(E\setminus A^{1}(y)\) is closed. Thus, \(A^{1}(y)\) is open. It follows from Theorem 3.1 that there exists \(\tilde{x}\in E\) such that \(\tilde{x}\in S(\tilde{x})\) and \(F(\tilde{x},y,z)\subseteq C(\tilde{x})\) for all \(y\in S(\tilde{x})\) and \(z\in B(\tilde{x})\). 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 \(y\in E\), \(F(\cdot,y)\) is l.s.c. and C is closed. Then there exists \(\tilde {x}\in E\) such that \(\tilde{x}\in S(\tilde{x})\) and \(F(\tilde {x},y)\subseteq C(\tilde{x})\) for all \(y\in S(\tilde{x})\).
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 \(x\in E\), 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 \(y\in E\), \(F(\cdot,y,\cdot)\) is l.s.c. and C is closed;

(b)
B is u.s.c. and \(B(x)\) is compact for each \(x\in E\).
Then there exist \(\tilde{x}\in E\) and \(\tilde{z}\in B(\tilde{x})\) such that \(\tilde{x}\in S(\tilde{x})\) and \(F(\tilde{x},y,\tilde{z})\subseteq C(\tilde{x})\) for all \(y\in S(\tilde{x})\).
Proof
Let
We first show that
is open. Let \(\{x_{\alpha}\}\subseteq E\setminus A^{1}(y)\) be a net with \(x_{\alpha}\rightarrow x_{0}\). Then there exists \(z_{\alpha}\in B(x_{\alpha})\) such that \(F(x_{\alpha},y,z_{\alpha })\subseteq C(x_{\alpha})\). Since B is u.s.c. with compact values, by Lemma 2.2, there exists a subset net of \(\{z_{\alpha}\}\), denoted again by \(\{z_{\alpha}\}\), such that \(z_{\alpha}\rightarrow z_{0}\in B(x_{0})\). The fact that \(F(\cdot,y,\cdot)\) is l.s.c. together with Lemma 2.3 shows that, for any \(v\in F(x_{0},y,z_{0})\), there exists \(v_{\alpha}\in F(x_{\alpha},y,z_{\alpha})\) such that \(v_{\alpha}\rightarrow v\). Since \(v_{\alpha}\in C(x_{\alpha})\) and C is closed, we know that \(v\in C(x_{0})\) and so \(F(x_{0},y,z_{0})\subseteq C(x_{0})\) for some \(z_{0}\in B(x_{0})\). This implies that \(x_{0}\in E\setminus A^{1}(y)\) and so \(E\setminus A^{1}(y)\) is closed. Thus, \(A^{1}(y)\) is open. It follows from Theorem 3.1 that there exist \(\tilde{x}\in E\) and \(\tilde{z}\in B(\tilde{x})\) such that \(\tilde{x}\in S(\tilde {x})\) and \(F(\tilde{x},y,\tilde{z})\subseteq C(\tilde{x})\) for any \(y\in S(\tilde{x})\). This completes the proof. □
Remark 3.4
When \(S(x)=E\) for all \(x\in E\), the existence of the solutions for generalized vector quasiequilibrium 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 \(y\in E\), \(F(\cdot,y,\cdot)\) is l.s.c., \(C(x)\) is a setvalued mapping with a nonempty interior for each \(x\in E\), the mapping \(W: E\rightrightarrows V\), defined by \(W(x)=V\setminus ({}\operatorname{int} C(x))\), is closed;

(b)
B is l.s.c.
Then there exists \(\tilde{x}\in E\) such that \(\tilde{x}\in S(\tilde {x})\) and \(F(\tilde{x},y,z)\cap({}\operatorname{int} C(\tilde{x}))=\emptyset\) for all \(y\in S(\tilde{x})\) and \(z\in B(\tilde{x})\).
Proof
Let
We prove that
is open. Let \(\{x_{\alpha}\}\subseteq E\setminus A^{1}(y)\) be a net with \(x_{\alpha}\rightarrow x_{0}\). Then
and so
Similar to the proof of Theorem 3.2, we get
This shows that \(x_{0}\in E\setminus A^{1}(y)\) and so \(E\setminus A^{1}(y)\) is closed. Thus, \(A^{1}(y)\) is open. It follows from Theorem 3.1 that there exists \(\tilde{x}\in E\) such that \(\tilde{x}\in S(\tilde{x})\) and
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 \(y\in E\), \(F(\cdot,y)\) is l.s.c., \(C(x)\) has a nonempty interior for each \(x\in E\), the map \(W: E\rightrightarrows V\), defined by \(W(x)=V\setminus({}\operatorname{int} C(x))\), is closed.
Then there exists \(\tilde{x}\in E\) such that \(\tilde{x}\in S(\tilde {x})\) and \(F(\tilde{x},y)\cap({}\operatorname{int} C(\tilde{x}))=\emptyset\) for all \(y\in S(\tilde{x})\).
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 \(x\in E\), 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 \(y\in E\), \(F(\cdot,y,\cdot)\) is l.s.c., \(C(x)\) has a nonempty interior for each \(x\in E\), and the mapping \(W: E\rightrightarrows V\), defined by \(W(x)=V\setminus({}\operatorname{int} C(x))\), is closed;

(b)
B is u.s.c. and \(B(x)\) is compact for each \(x\in E\).
Then there exist \(\tilde{x}\in E\) and \(\tilde{z}\in B(\tilde{x})\) such that \(\tilde{x}\in S(\tilde{x})\) and \(F(\tilde{x},y,\tilde{z})\cap ({}\operatorname{int} C(\tilde{x}))=\emptyset\) for all \(y\in S(\tilde{x})\).
Proof
Let
We show that
is open. Let \(\{x_{\alpha}\}\subseteq E\setminus A^{1}(y)\) be a net with \(x_{\alpha}\rightarrow x_{0}\). Then
for some \(z_{\alpha}\in B(x_{\alpha})\), that is,
Using similar arguments to the proof of Theorem 3.3, we have
for some \(z_{0}\in B(x_{0})\). This shows that \(x_{0}\in E\setminus A^{1}(y)\) and so \(E\setminus A^{1}(y)\) is closed. Thus, \(A^{1}(y)\) is open. It follows from Theorem 3.1 that there exist \(\tilde{x}\in E\) and \(\tilde{z}\in B(\tilde{x})\) such that \(\tilde{x}\in S(\tilde {x})\) and
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 \(y\in E\), \(F(\cdot,y,\cdot)\) is u.s.c. with compact valued on \(E\times E\times E\) and \(C(x)\) has a nonempty interior for each \(x\in E\), the mapping \(W: E\rightrightarrows V\), defined by \(W(x)=V\setminus({}\operatorname{int} C(x))\), is closed;

(b)
B is l.s.c.
Then there exists \(\tilde{x}\in E\) such that \(\tilde{x}\in S(\tilde {x})\) and \(F(\tilde{x},y,z)\nsubseteq{}\operatorname{int} C(\tilde{x})\) for all \(y\in S(\tilde{x})\) and \(z\in B(\tilde{x})\).
Proof
Let
We prove that
is open. Let \(x_{\alpha}\in E\setminus A^{1}(y)\) be a net with \(x_{\alpha}\rightarrow x_{0}\). Then
for any \(z'\in B(x_{\alpha})\) and so there exists \(v_{\alpha}\in V\) such that
Since B is l.s.c., by Lemma 2.3, for any \(z\in B(x_{0})\), there exists \(z_{\alpha}\in B(x_{\alpha})\) such that \(z_{\alpha }\rightarrow z\). Since \(F(\cdot,y,\cdot)\) is u.s.c. with compact valued, by Lemma 2.2, there exists a subset net of \(\{v_{\alpha }\}\), denoted again by \(\{v_{\alpha}\}\), such that \(v_{\alpha }\rightarrow v_{0}\in F(x_{0},y,z)\). On the other hand, the fact that \(v_{\alpha}\notin{}\operatorname{int} C(x_{\alpha})\) shows that \(v_{\alpha}\in W(x_{\alpha})\). Now the closedness of W shows that \(v_{0}\in W(x_{0})\) and so \(v_{0}\notin{}\operatorname{int} C(x_{0})\). Thus \(F(x_{0},y,z)\nsubseteq{}\operatorname{int} C(x_{0})\) for any \(z\in B(x_{0})\). This implies that \(x_{0}\in E\setminus A^{1}(y)\) and so \(E\setminus A^{1}(y)\) is closed. Thus, \(A^{1}(y)\) is open. It follows from Theorem 3.1 that there exists \(\tilde{x}\in E\) such that \(\tilde{x}\in S(\tilde{x})\) and \(F(\tilde {x},y,z)\nsubseteq{}\operatorname{int} C(\tilde{x})\) for all \(y\in S(\tilde{x})\) and \(z\in B(\tilde{x})\). 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 \(y\in E\), \(F(\cdot,y)\) is u.s.c. with compact valued on \(E\times E\) and \(C(x)\) has a nonempty interior for each \(x\in E\), the mapping \(W: E\rightrightarrows V\), defined by \(W(x)=V\setminus ({}\operatorname{int} C(x))\), is closed.
Then there exists \(\tilde{x}\in E\) such that \(\tilde{x}\in S(\tilde {x})\) and \(F(\tilde{x},y)\nsubseteq{}\operatorname{int} C(\tilde{x})\) for all \(y\in S(\tilde{x})\).
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 \(x\in E\), 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 \(y\in E\), \(F(\cdot,y,\cdot)\) is u.s.c. with compact valued on \(E\times E\times E\) and \(C(x)\) has a nonempty interior for each \(x\in E\), the mapping \(W: E\rightrightarrows V\), defined by \(W(x)=V\setminus({}\operatorname{int} C(x))\), is closed.

(b)
B is u.s.c. and \(B(x)\) is compact for each \(x\in E\).
Then there exist \(\tilde{x}\in E\) and \(\tilde{z}\in B(\tilde{x})\) such that \(\tilde{x}\in S(\tilde{x})\) and \(F(\tilde{x},y,\tilde {z})\nsubseteq{}\operatorname{int} C(\tilde{x})\) for all \(y\in S(\tilde{x})\).
Proof
Let
We prove that
is open. Let \(x_{\alpha}\in E\setminus A^{1}(y)\) be a net with \(x_{\alpha}\rightarrow x_{0}\). Then \(F(x_{\alpha },y,z_{\alpha})\nsubseteq{}\operatorname{int} C(x_{\alpha})\) for some \(z_{\alpha}\in B(x_{\alpha})\) and so there exists \(v_{\alpha}\in V\) such that
Since B is u.s.c. with compact valued, by Lemma 2.2, there exists a subnet of \(\{z_{\alpha}\}\), denoted again by \(\{z_{\alpha}\}\), such that \(z_{\alpha}\rightarrow z_{0}\in B(x_{0})\). The fact that \(F(\cdot,y,\cdot)\) is u.s.c. with compact valued together with Lemma 2.2 shows that there exists a subset net of \(\{ v_{\alpha}\}\), denoted again by \(\{v_{\alpha}\}\), such that \(v_{\alpha }\rightarrow v_{0}\in F(x_{0},y,z_{0})\). On the other hand, it is easy to see that \(v_{\alpha}\in W(x_{\alpha})\). Since W is closed, we know that \(v_{0}\in W(x_{0})\) and so \(v_{0}\notin{}\operatorname{int} C(x_{0})\). Thus \(F(x_{0},y,z_{0})\nsubseteq{}\operatorname{int} C(x_{0})\) for some \(z_{0}\in B(x_{0})\) and so \(x_{0}\in E\setminus A^{1}(y)\). This implies that \(E\setminus A^{1}(y)\) is closed and so \(A^{1}(y)\) is open. It follows from Theorem 3.1 that there exist \(\tilde{x}\in E\) and \(\tilde{z}\in B(\tilde{x})\) such that \(\tilde{x}\in S(\tilde {x})\) and \(F(\tilde{x},y,\tilde{z})\nsubseteq{}\operatorname{int} C(\tilde{x})\) for all \(y\in S(\tilde{x})\). 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 \(y\in E\), \(F(\cdot,y,\cdot)\) is u.s.c. with compact valued on \(E\times E\times E\) and C is closed;

(b)
B is l.s.c.
There exists \(\tilde{x}\in E\) such that \(\tilde{x}\in S(\tilde{x})\) and \(F(\tilde{x},y,z)\cap C(\tilde{x})\neq\emptyset\) for all \(y\in S(\tilde{x})\) and \(z\in B(x)\).
Proof
Let
We show that
is open. Let \(\{x_{\alpha}\}\subseteq E\setminus A^{1}(y)\) be a net with \(x_{\alpha}\rightarrow x_{0}\). Then
It follows that there exists \(v_{\alpha}\in F(x_{\alpha},y,z')\cap C(x_{\alpha})\). Since B is l.s.c., by Lemma 2.3, there exists \(z_{\alpha}\in B(x_{\alpha})\) such that \(z_{\alpha}\rightarrow z\) for any \(z\in B(x_{0})\). By the fact that \(F(\cdot,y,\cdot)\) is u.s.c. with compact valued, there exists a subset of \(\{v_{\alpha}\}\), denoted again by \(\{v_{\alpha}\}\), such that \(v_{\alpha}\rightarrow v_{0} \in F(x_{0},y,z)\). Since \(v_{\alpha}\in C(x_{\alpha})\) and C is closed, we know that \(v_{0}\in C(x_{0})\) and so \(v_{0}\in F(x_{0},y,z)\cap C(x_{0})\). Thus,
This shows that \(x_{0}\in E\setminus A^{1}(y)\) and so \(E\setminus A^{1}(y)\) is closed. Thus, \(A^{1}(y)\) is open. By Theorem 3.1, there exists \(\tilde{x}\in E\) such that \(\tilde{x}\in S(\tilde{x})\) and
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 \(y\in E\), \(F(\cdot,y)\) is u.s.c. with compact valued on \(E\times E\) and C is closed. Then there exists \(\tilde{x}\in E\) such that \(\tilde {x}\in S(\tilde{x})\) and \(F(\tilde{x},y)\cap C(\tilde{x})\neq \emptyset\) for all \(y\in S(\tilde{x})\).
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 \(x\in E\), 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 \(y\in E\), \(F(\cdot,y,\cdot)\) is u.s.c. with compact valued on \(E\times E\times E\) and C is closed;

(b)
B is u.s.c. and \(B(x)\) is compact for each \(x\in E\).
There exist \(\tilde{x}\in E\) and \(\tilde{z}\in B(\tilde{x})\) such that \(\tilde{x}\in S(\tilde{x})\) and \(F(\tilde{x},y,\tilde{z})\cap C(\tilde{x})\neq\emptyset\) for all \(y\in S(\tilde{x})\).
Proof
Let
We prove that
is open. Let \(\{x_{\alpha}\}\subseteq E\setminus A^{1}(y)\) be a net with \(x_{\alpha}\rightarrow x_{0}\). Then
for some \(z_{\alpha}\in B(x_{\alpha})\), that is, there exists \(v_{\alpha }\in F(x_{\alpha},y,z_{\alpha})\cap C(x_{\alpha})\). Since B is u.s.c. and \(B(x)\) is compact, it follows from Lemma 2.2 that there exists a subset of \(\{z_{\alpha}\}\), denoted again by \(\{z_{\alpha }\}\), such that \(z_{\alpha}\rightarrow z_{0}\in B(x_{0})\). Similar to the proof of Theorem 3.8, we can prove that \(F(x_{0},y,z_{0})\cap C(x_{0})\neq\emptyset\) for some \(z_{0}\in B(x_{0})\). This shows that \(x_{0}\in E\setminus A^{1}(y)\) and so \(E\setminus A^{1}(y)\) is closed. Thus, \(A^{1}(y)\) is open. It follows from Theorem 3.1 that there exist \(\tilde{x}\in E\) and \(\tilde{z}\in B(\tilde{x})\) such that \(\tilde{x}\in S(\tilde {x})\) and
This completes the proof. □
4 Applications to the generalized semiinfinite programs
In this section, by the results presented in Section 3, we give some existence theorems of solutions to the generalized semiinfinite programs. Let L be a real topological vector space ordered by a closed convex pointed cone \(H\subseteq L\) with \(\operatorname{int} H\neq\emptyset\) and \(h: X\rightrightarrows L\) 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 \(F(\cdot,\cdot,\cdot)\) and S are l.s.c. Then there is a solution to the problem
where
Proof
Theorem 3.2 shows that \(K\neq\emptyset\). From Lemma 2.5, it is sufficient to show that \(h(K)\) is compact. Since h is u.s.c. and \(K\subseteq E\), by Lemma 2.1, we only need to prove that K is closed. Let \(\{x_{\alpha}\}\subseteq K\) be a net with \(x_{\alpha}\rightarrow x_{0}\). Then \(x_{\alpha}\in S(x_{\alpha})\) and
Since S and B are l.s.c., for any \(y\in S(x_{0})\) and \(z\in B(x_{0})\), it follows from Lemma 2.3 that there exist \(y_{\alpha }\in S(x_{\alpha})\) and \(z_{\alpha}\in B(x_{\alpha})\) such that \(y_{\alpha} \rightarrow y\) and \(z_{\alpha}\rightarrow z\). By the lower semicontinuity of F and Lemma 2.3, for any \(v\in F(x_{0},y,z)\), there exists \(v_{\alpha}\in F(x_{\alpha},y_{\alpha },z_{\alpha})\) such that \(v_{\alpha}\rightarrow v\). Now the closedness of C with \(v_{\alpha}\in C(x_{\alpha})\) shows that \(v\in C(x_{0})\) and so \(F(x_{0},y,z)\subseteq C(x_{0})\) for all \(y\in S(x_{0})\) and \(z\in B(x_{0})\). Moreover, the closedness of \(E\setminus G_{0}\) shows that \(x_{0}\in S(x_{0})\). 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 \(F(\cdot,\cdot)\) and S are l.s.c. Then there is a solution to the problem
where
Remark 4.1
When \(S(x)=E\) for all \(x\in E\), 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 \(F(\cdot,\cdot,\cdot)\) and S are l.s.c. Then there is a solution to the problem
where
Proof
Obviously, Theorem 3.3 shows that \(K\neq\emptyset\). By Lemma 2.5, it is sufficient to prove that \(h(K)\) is compact. Since h is u.s.c. and \(K\subseteq E\), from Lemma 2.1, we only need to show that K is closed. Let \(\{x_{\alpha}\}\subseteq K\) be a net with \(x_{\alpha}\rightarrow x_{0}\). Then \(x_{\alpha}\in S(x_{\alpha})\) and there exists \(z_{\alpha}\in B(x_{\alpha})\) such that
Since B is a u.s.c. mapping with compact values, it follows from Lemma 2.2 that there exists a subnet of \(\{z_{\alpha}\}\), denoted again by \(\{z_{\alpha}\}\), such that \(z_{\alpha}\rightarrow z_{0}\in B(x_{0})\). For any \(y\in S(x_{0})\), the lower semicontinuity of S together with Lemma 2.3 implies that there exists \(y_{\alpha}\in S(x_{\alpha})\) such that \(y_{\alpha} \rightarrow y\). For \(v\in F(x_{0},y,z_{0})\), by the fact that F is l.s.c., it follows from Lemma 2.3 that there exists \(v_{\alpha}\in F(x_{\alpha},y_{\alpha},z_{\alpha})\) such that \(v_{\alpha}\rightarrow v\). Now the closedness of C with \(v_{\alpha }\in C(x_{\alpha})\) shows that \(v\in C(x_{0})\) and so there exists \(z_{0}\in B(x_{0})\) such that \(F(x_{0},y,z)\subseteq C(x_{0})\) for all \(y\in S(x_{0})\). Moreover, the closedness of \(E\setminus G_{0}\) shows that \(x_{0}\in S(x_{0})\). 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 \(F(\cdot,\cdot,\cdot)\) and S are l.s.c. Then there is a solution to the problem
where
Proof
Theorem 3.4 shows that \(K\neq\emptyset\). From Lemma 2.5, it is sufficient to show that \(h(K)\) is compact. Since h is u.s.c. and \(K\subseteq E\), by Lemma 2.1, we only need to show that K is closed. Let \(\{x_{\alpha}\}\subseteq K\) be a net with \(x_{\alpha}\rightarrow x_{0}\). Then \(x_{\alpha}\in S(x_{\alpha})\),
and so
Similar to the proof of Theorem 4.1, we have \(x_{0}\in S(x_{0})\),
and so
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 \(F(\cdot,\cdot)\) and S are l.s.c. Then there is a solution to the problem
where
Remark 4.2
When \(S(x)=E\) for all \(x\in E\), 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 of Theorem 3.5 are satisfied. Moreover, assume that \(F(\cdot,\cdot,\cdot)\) and S are l.s.c. Then there is a solution to the problem
where
Proof
It follows from Theorem 3.5 that \(K\neq\emptyset\). From Lemma 2.5, it is sufficient to show that \(h(K)\) is compact. Since h is u.s.c. and \(K\subseteq E\), by Lemma 2.1, we only need to show K is closed. Let \(\{x_{\alpha}\}\subseteq K\) be a net with \(x_{\alpha}\rightarrow x_{0}\). Then \(x_{\alpha}\in S(x_{\alpha})\) and there exists \(z_{\alpha}\in B(x_{\alpha})\) such that
Thus, there exists \(z_{\alpha}\in B(x_{\alpha})\) such that
Similar to the proof of Theorem 4.2, we know that \(x_{0}\in S(x_{0})\) and there exists \(z_{0}\in B(x_{0})\) such that
Thus, \(x_{0}\in S(x_{0})\) and there exists \(z_{0}\in B(x_{0})\) such that
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 \(F(\cdot,\cdot,\cdot)\) is a u.s.c. mapping with compact values and S is l.s.c. Then there is a solution to the problem
where
Proof
Theorem 3.6 shows that \(K\neq\emptyset\). From Lemma 2.5, it is sufficient to show that \(h(K)\) is compact. Since h is u.s.c. and \(K\subseteq E\), by Lemma 2.1, we only need to prove that K is closed. Let \(\{x_{\alpha}\}\subseteq K\) be a net with \(x_{\alpha}\rightarrow x_{0}\). Then \(x_{\alpha}\in S(x_{\alpha})\),
and so there exists \(v_{\alpha}\in V\) such that
By the lower semicontinuity of S and B, for any \(y\in S(x_{0})\) and \(z\in B(x_{0})\), it follows from Lemma 2.3 that there exist \(y_{\alpha}\in S(x_{\alpha})\) and \(z_{\alpha}\in B(x_{\alpha})\) such that \(y_{\alpha} \rightarrow y\) and \(z_{\alpha}\rightarrow z\). Since \(F(\cdot,\cdot,\cdot)\) is a u.s.c. mapping with compact values, Lemma 2.2 shows that there exists a subnet of \(\{v_{\alpha}\}\), denoted again by \(\{v_{\alpha}\}\), such that \(v_{\alpha}\rightarrow v_{0}\in F(x_{0},y,z)\). On the other hand, the fact that \(v_{\alpha }\notin{}\operatorname{int} C(x_{\alpha})\) shows that \(v_{\alpha}\in W(x_{\alpha})\). Now the closedness of W shows that \(v_{0}\in W(x_{0})\) and so \(v_{0}\notin{}\operatorname{int} C(x_{0})\). Moreover, the closedness of \(E\setminus G_{0}\) shows that \(x_{0}\in S(x_{0})\). Thus,
for all \(y\in S(x_{0})\) and \(z\in B(x_{0})\) 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 \(F(\cdot,\cdot)\) is u.s.c. and S is l.s.c. Then there is a solution to the problem
where
Remark 4.3
When \(S(x)=E\) for all \(x\in E\), 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 \(F(\cdot,\cdot,\cdot)\) is a u.s.c. mapping with compact values and S is l.s.c. Then there is a solution to the problem
where
Proof
Theorem 3.7 shows that \(K\neq\emptyset\). By Lemma 2.5, it is sufficient to prove that \(h(K)\) is compact. Since h is u.s.c. and \(K\subseteq E\), from Lemma 2.1, we only need to show that K is closed. Let \(\{x_{\alpha}\}\subseteq K\) be a net with \(x_{\alpha}\rightarrow x_{0}\). Then \(x_{\alpha}\in S(x_{\alpha})\) and there exists \(z_{\alpha}\in B(x_{\alpha})\) such that
Thus, there exists \(v_{\alpha}\in V\) such that
Since B is a u.s.c. mapping with compact values, it follows from Lemma 2.2 that there exists a subnet of \(\{z_{\alpha}\}\), denoted again by \(\{z_{\alpha}\}\), such that \(z_{\alpha}\rightarrow z_{0}\in B(x_{0})\). By the lower semicontinuity of S, for any \(y\in S(x_{0})\), Lemma 2.3 shows that there exists \(y_{\alpha}\in S(x_{\alpha})\) such that \(y_{\alpha} \rightarrow y\). Since \(F(\cdot ,\cdot,\cdot)\) is a u.s.c. mapping with compact values, Lemma 2.2 implies that there exists a subnet of \(\{v_{\alpha}\}\), denoted again by \(\{v_{\alpha}\}\), such that \(v_{\alpha}\rightarrow v_{0}\in F(x_{0},y,z_{0})\). 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 \(F(\cdot,\cdot,\cdot)\) is a u.s.c. mapping with compact values and S is l.s.c. Then there is a solution to the problem
where
Proof
Theorem 3.8 shows that \(K\neq\emptyset\). From Lemma 2.5, it is sufficient to show that \(h(K)\) is compact. Since h is u.s.c. and \(K\subseteq E\), by Lemma 2.1, we only need to show K is closed. Let \(\{x_{\alpha}\}\subseteq K\) be a net with \(x_{\alpha}\rightarrow x_{0}\). Then \(x_{\alpha}\in S(x_{\alpha})\),
and so there exists \(v_{\alpha}\in V\) such that
By the lower semicontinuity of S and B, for any \(y\in S(x_{0})\) and \(z\in B(x_{0})\), Lemma 2.3 shows that there exist \(y_{\alpha }\in S(x_{\alpha})\) and \(z_{\alpha}\in B(x_{\alpha})\) such that \(y_{\alpha} \rightarrow y\) and \(z_{\alpha}\rightarrow z\). Since \(F(\cdot ,\cdot,\cdot)\) is u.s.c. with compact values, by Lemma 2.2, there exists a subnet of \(\{v_{\alpha}\}\), denoted again by \(\{v_{\alpha }\}\), such that \(v_{\alpha}\rightarrow v_{0}\in F(x_{0},y,z)\). Now the closedness of C with \(v_{\alpha}\in C(x_{\alpha})\) shows that \(v_{0}\in C(x_{0})\) and so
Moreover, the closedness of \(E\setminus G_{0}\) shows that \(x_{0}\in S(x_{0})\). 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 \(F(\cdot,\cdot)\) and S are l.s.c. Then there is a solution to the problem
where
Remark 4.4
When \(S(x)=E\) for all \(x\in E\), 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 \(F(\cdot,\cdot,\cdot)\) is a u.s.c. mapping with compact values and S is l.s.c. Then there is a solution to the problem
where
Proof
Theorem 3.8 shows that \(K\neq\emptyset\). From Lemma 2.5, it is sufficient to prove that \(h(K)\) is compact. Since h is u.s.c. and \(K\subseteq E\), by Lemma 2.1, we only need to show K is closed. Let \(\{x_{\alpha}\}\subseteq K\) be a net with \(x_{\alpha}\rightarrow x_{0}\). Then \(x_{\alpha}\in S(x_{\alpha})\) and there exists \(z_{\alpha}\in B(x_{\alpha})\) such that
Thus, there exists \(v_{\alpha}\in V\) such that
Since B is a u.s.c. mapping with compact values, it follows from Lemma 2.2 that there exists a subnet of \(\{z_{\alpha}\}\), denoted again by \(\{z_{\alpha}\}\), such that \(z_{\alpha}\rightarrow z_{0}\in B(x_{0})\). By the lower semicontinuity of S, for any \(y\in S(x_{0})\), Lemma 2.3 implies that there exists \(y_{\alpha}\in S(x_{\alpha})\) such that \(y_{\alpha} \rightarrow y\). Since \(F(\cdot,\cdot,\cdot)\) is a u.s.c. mapping with compact values, by Lemma 2.2, there exists a subnet of \(\{v_{\alpha}\}\), denoted again by \(\{v_{\alpha}\}\), such that \(v_{\alpha}\rightarrow v_{0}\in F(x_{0},y,z_{0})\). Now the closedness of C with \(v_{\alpha}\in C(x_{\alpha})\) shows that \(v_{0}\in C(x_{0})\) and so there exists \(z_{0}\in B(x_{0})\) such that
Moreover, the closedness of \(E\setminus G_{0}\) shows that \(x_{0}\in S(x_{0})\). Therefore, K is closed. This completes the proof. □
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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).
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Zhang, Wb., Shan, Sq. & Huang, Nj. Existence of solutions for generalized vector quasiequilibrium problems in abstract convex spaces with applications. Fixed Point Theory Appl 2015, 29 (2015). https://doi.org/10.1186/s1366301502776
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DOI: https://doi.org/10.1186/s1366301502776