Existence of solutions for generalized vector quasiequilibrium problems in abstract convex spaces with applications
 Weibing Zhang^{1},
 Shuqiang Shan^{1} and
 Nanjing Huang^{1, 2}Email author
https://doi.org/10.1186/s1366301502776
© Zhang et al.; licensee Springer. 2015
Received: 6 November 2014
Accepted: 30 January 2015
Published: 24 February 2015
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.
Keywords
abstract convex space generalized vector quasiequilibrium problem generalized semiinfinite program setvalued mapping KKM mapping1 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
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])
Example 2.3
 (i)
\(d(x,x) = 0\);
 (ii)
\(d(x,y) = d(y,x)\);
 (iii)
\(d(x,z) \leq d(x,y) + d(y,z)\).
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.
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:
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
Definition 2.2
Definition 2.3
([33])
 (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])
 (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])
 (i)
G is closedvalues;
 (ii)
\(F(\Gamma_{N})\subseteq G(N)\) for any \(N\in\langle D\rangle\),
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
 (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 \(y_{0}\notin S(x_{0})\).
Proof
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.
Remark 3.1
 (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
 (a)
for each \(y\in E\), \(F(\cdot,y,\cdot)\) is l.s.c. and C is closed;
 (b)
B is l.s.c.
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
 (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 \(x\in E\).
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
 (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.
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
 (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.
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
 (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 \(x\in E\).
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
 (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.
Proof
Corollary 3.3
 (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.
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
 (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 \(x\in E\).
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
 (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.
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
 (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 \(x\in E\).
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
Proof
Corollary 4.1
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
Proof
Theorem 4.3
Proof
Corollary 4.2
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
Proof
Theorem 4.5
Proof
Corollary 4.3
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
Proof
Theorem 4.7
Proof
Corollary 4.4
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
Proof
Declarations
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).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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