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Existence and Hadamard wellposedness of a system of simultaneous generalized vector quasiequilibrium problems
Journal of Inequalities and Applications volume 2017, Article number: 58 (2017)
Abstract
An existence result for the solution set of a system of simultaneous generalized vector quasiequilibrium problems (for short, (SSGVQEP)) is obtained, which improves Theorem 3.1 of the work of Ansari et al. (J. Optim. Theory Appl. 127:2744, 2005). Moreover, a definition of Hadamardtype wellposedness for (SSGVQEP) is introduced and sufficient conditions for Hadamard wellposedness of (SSGVQEP) are established.
Introduction
Recently, a vector equilibrium problem has received lots of attention because it unifies several classes of problems, for instance, vector variational inequality problems, vector optimization problems, vector saddle point problems and vector complementarity problems, for details, see [2] and the references therein. Moreover, many authors further investigated several general types of it, for instance, see [3–8].
Let I be a finite index set and \(i\in I\). Assume that \(E_{i}\), \(F_{i}\) and \(Z_{i}\) are locally convex Hausdorff spaces, \(X_{i}\subset E_{i}\) and \(Y_{i}\subset F_{i}\) are two nonempty convex subsets. Let \(X=\prod _{i\in I}X_{i}\) and \(Y=\prod _{i\in I}Y_{i}\). Assume that \(C_{i}:X\rightarrow2^{Z_{i}}\) is a setvalued mapping, the values of which are closed convex cones with apex at the origin, \(C_{i}(x)\subsetneqq Z_{i}\) and \(\operatorname {int}C_{i}(x)\neq\emptyset\). Let \(Z_{i}^{*}\) be the dual of \(Z_{i}\), \(S_{i}: X\rightarrow2^{X_{i}}\) and \(T_{i}: X\rightarrow2^{Y_{i}}\) be setvalued mappings with nonempty values. Assume that \(f_{i}:X\times Y\times X_{i}\rightarrow Z_{i}\), \(g_{i}: X\times Y\times Y_{i}\rightarrow Z_{i}\) are two trifunctions.
One of the general types, a system of simultaneous generalized vector quasiequilibrium problems (for short, (SSGVQEP)), as follows, is considered: find \((\bar{x},\bar{y})\in X\times Y\) such that \(\forall i\in I\), \(\bar{x}_{i}\in S_{i}(\bar{x})\), \(\bar{y}_{i}\in T_{i}(\bar{x})\),
The problem (SSGVQEP) was introduced by Ansari in [1]. By suitable choices of \(f_{i}\), \(g_{i}\), \(S_{i}\) and \(T_{i}\), (SSGVQEP) reduces to several classical systems of (quasi)equilibrium problems and systems of variational inequalities, which are studied in the literatures (see [9–13] and the references therein). Furthermore, by suitable conditions and suitable choices of i, (SSGVQEP) contains vector equilibrium problems as special cases. A solution of (SSGVQEP) is an ideal solution. It is better than other solutions such as weak efficient solutions, efficient solutions and proper efficient solutions (see [2, 14–16] and the references therein). Therefore, it is meaningful to study the existence result for the solution set of (SSGVQEP).
The classical concept of Hadamard wellposedness requires not only the existence and uniqueness of the optimal solution but also the continuous dependence of the optimal solution on the problem data. Recently, the classical concept together with its generalized types has been studied in other more complicated situations such as scalar optimization problems, vector optimization problems, nonlinear optimal control problems, and so on, see [4, 17–29] and the references therein. However, as far as we know, there are few results about Hadamard wellposedness of (SSGVQEP). Therefore, it is necessary to study Hadamard wellposedness of (SSGVQEP).
In this paper, by using demicontinuity and natural quasiconvexity, we obtain an existence theorem of solutions for (SSGVQEP). Moreover, we introduce the definition of Hadamard wellposedness for (SSGVQEP) and discuss sufficient conditions for Hadamard wellposedness of (SSGVQEP). The rest of the paper goes as follows. In Section 2, we recall some necessary notations and definitions. In Section 3, we obtain the existence theorem of solutions for (SSGVQEP). In Section 4, we investigate Hadamard wellposedness of (SSGVQEP).
Preliminaries and notations
Let us recall some notations and definitions of vectorvalued mappings and setvalued mappings together with their properties.
Let X, Y be two topological spaces and \(F:X\rightarrow2^{Y}\) be a setvalued mapping. Assume that \(x\in X\). If for any open set V with \(F(x)\subset V\), there exists a neighborhood N of x such that
F is called upper semicontinuous (\(\mathit{u.s.c.}\) for short) at x. If F is \(\mathit{u.s.c.}\) at each point of X, F is called \(\mathit{u.s.c.}\) If for any \(z\in F(x)\) and any neighborhood N of z, there exists a neighborhood U of x such that \(\forall y\in U\), we have
F is called lower semicontinuous (\(\mathit{l.s.c.}\) for short) at x. If F is \(\mathit{l.s.c.}\) at every point of X, F is called \(\mathit{l.s.c.}\) In addition, F is called continuous if F is both \(\mathit{l.s.c.}\) and \(\mathit{u.s.c.}\) If the set \(\operatorname{Graph}(F)\), i.e., \(\operatorname{Graph}(F)=\{(x,y):x\in X,y\in F(x)\}\), is a closed set in \(X\times Y\), F is called a closed mapping. F is called compact if the closure of \(F(X)\), i.e., \(\overline {F(X)}\), is compact, where \(F(X)=\bigcup_{x\in X}F(x)\).
Definition 1
[30]
Let Y, Z be topological vector spaces. A vectorvalued mapping \(f:Y\rightarrow Z\) is called demicontinuous if for each closed half space \(M\subset Z\),
is closed in Y.
Definition 2
Let \((Z,P)\) be an ordered topological vector space, E be a nonempty convex subset of a vector space X, and \(f: E\rightarrow Z\) be a vectorvalued mapping.

(i)
f is called convex if for every \(x_{1},x_{2}\in E\) and for every \(\lambda\in[0,1]\), one has
$$f\bigl(\lambda x_{1}+(1\lambda)x_{2}\bigr)\in\lambda f(x_{1})+(1\lambda)f(x_{2})P. $$ 
(ii)
f is called properly quasiconvex if for every \(x_{1},x_{2}\in E\) and \(\lambda\in[0,1]\), one has either \(f(\lambda x_{1}+(1\lambda)x_{2})\in f(x_{1})P\) or \(f(\lambda x_{1}+(1\lambda )x_{2})\in f(x_{2})P\).

(iii)
f is said to be naturally quasiconvex if for every \(x_{1}, x_{2}\in E\), \(\lambda\in[0, 1]\), there exists \(\mu\in[0, 1]\) such that
$$f\bigl(\lambda x_{1}+(1\lambda)x_{2}\bigr)\in\mu f(x_{1})+ (1\mu)f(x_{2})P. $$
It is clear that every properly quasiconvex or convex mapping is naturally quasiconvex, but a naturally quasiconvex mapping may not be convex or properly quasiconvex.
Results and discussion
In this section, we will consider the existence results of (SSGVQEP) and give an example to show that our existence theorem extends the corresponding result in [1]. Moreover, we will introduce Hadamardtype wellposedness for (SSGVQEP) and establish sufficient conditions of Hadamardtype wellposedness for (SSGVQEP).
Existence of solutions for (SSGVQEP)
In this subsection, we will consider the existence results of (SSGVQEP) and give example to show that our existence theorem extends the corresponding result in [1].
Let Z be a locally convex Hausdorff space, \(P\subset Z\) be a closed convex and pointed cone, and \(\operatorname {int}P\neq\emptyset\). We denote
We can deduce from [31], p.165, Theorem 2, that \(T\neq \emptyset\).
Lemma 1
For arbitrary \(x\in Z\), if \((x^{*},x)\geq0\) for all \(x^{*}\in T\), then \(x\in P\).
Proof
If we assume that \((x^{*},x)\geq0\) for all \(x^{*}\in T\), but \(x\notin P\). Let \(A=\{\lambda x+(1\lambda) p: \lambda\in (0,1), p\in\operatorname {int}P\}\), then we have A is an open convex set,
If not, there exist \(y\in P\), \(\lambda\in(0,1)\) and \(p\in \operatorname {int}P\) such that \(y= \lambda x+(1\lambda)p\). Thus,
It is a contradiction. Thus, (1) holds. By [31], p.165, Theorem 2, there exists \(x^{*'}\in Z^{*}\) such that for all \(y\in P\),
and for all \(y\in A\),
Then \(x^{*'}(x)< 0\) and \(x^{*'}\in T\). However, this contradicts the fact that \((x^{*},x)\geq0\) for all \({x^{*}\in T}\). □
The following wellknown KakutaniFanGlicksberg theorem is our main tool.
Lemma 2
[32]
Let X be a locally convex Hausdorff space, \(E\subset X\) be a nonempty, convex compact subset. Let \(F: E\rightarrow2^{E}\) be u.s.c. with nonempty, closed and convex set \(F(x)\), \(\forall x\in E\). Then F has a fixed point in E.
Lemma 3
[33], Theorems 6, 7
Assume that X and Y are two locally convex Hausdorff spaces and X is also compact. The setvalued mapping \(F: X\rightarrow2^{Y}\) is u.s.c. with compact values if and only if it is a closed mapping.
Theorem 1
Let \(i\in I\). Assume that \(E_{i}\), \(F_{i}\) and \(Z_{i}\) are locally convex Hausdorff spaces, \(X_{i}\) and \(Y_{i}\) are nonempty and convex subsets of \(E_{i}\) and \(F_{i}\), respectively. Let \(X=\prod _{i\in I}X_{i}\) and \(Y=\prod _{i\in I}Y_{i}\). The setvalued mappings \(S_{i}:X \rightarrow2^{X_{i}}\) and \(T_{i}:Y\rightarrow2^{Y_{i}}\) are compact closed mappings with nonempty and convex values. Assume that the following conditions hold:

(i)
\(C_{i}:X\rightarrow2^{Z_{i}}\) is a closed setvalued mapping. For arbitrary \(x\in X\), \(C_{i}(x)\) is a convex closed cone with apex at the origin. Assume that \(P_{i}=\bigcap_{x\in X}C_{i}(x)\),

(ii)
\(P_{i}^{*}\) has a weak ^{∗} compact convex base \(B_{i}^{*}\) and \(Z_{i}\) is ordered by \(P_{i}\),

(iii)
\(f_{i}:X\times Y\times X_{i}\rightarrow Z_{i}\) is a demicontinuous function such that for arbitrary \((x,y)\in X\times Y\),

(a)
\(0\leq_{P_{i}}f_{i}(x,y,x_{i})\),

(b)
the map \(u_{i}\mapsto f_{i}(x,y,u_{i})\) is naturally quasiconvex,

(a)

(iv)
\(g_{i}:X\times Y\times Y_{i}\rightarrow Z_{i}\) is a demicontinuous function such that for arbitrary \((x,y)\in X\times Y\),

(a)
\(0\leq_{P_{i}}g_{i}(x,y,y_{i})\),

(b)
the map \(v_{i}\mapsto g_{i}(x,y,v_{i})\) is naturally quasiconvex.

(a)
Then (SSGVQEP) has a solution \((\bar{x},\bar{y})\in X\times Y\).
Proof
We denote the setvalued mapping \(T_{i}: X\rightarrow 2^{Z^{*}_{i}}\) by
By (iii), (iv) and Lemma 2.2 of [34], for every \(x_{i}^{*}\in T_{i}\), the composite functions \(x_{i}^{*}\circ f_{i}\) and \(x_{i}^{*}\circ g_{i}\) are continuous. For each \(i\in I\), \(\forall(x,y)\in X\times Y\), define:
Firstly, we show that for arbitrary \((x,y)\in X\times Y\), \(A_{i}(x,y)\) and \(B_{i}(x,y)\) are nonempty. In fact, \(x_{i}^{*}\circ f_{i}\) and \(x_{i}^{*}\circ g_{i}\) are respectively continuous on compact sets \(S_{i}(x)\) and \(T_{i}(x)\). Secondly, we show that \(A_{i}\) is a closed mapping (similar to \(B_{i}\)). In fact, let \((x_{n},y_{n},u_{n})\in \operatorname{Graph}(A_{i})\) and \((x_{n},y_{n},u_{n})\rightarrow(x,y,u)\in X\times Y\times X_{i}\). Then
which means \((x_{i}^{*}\circ f_{i})(x,y,u)=F_{i}(x,y)\). Since \(\operatorname{Graph}(S_{i})\) is closed in \(X\times X_{i}\), \(u_{n}\in S_{i}(x_{n})\), we obtain that \(u\in S_{i}(x)\). Hence, \((x,y,u)\in \operatorname{Graph}(A_{i})\). By Lemma 3, \(A_{i}\) is u.s.c. Thirdly, we show that the set \(A_{i}(x,y)\) is convex. For this, let \(u_{i,1}, u_{i,2}\in A_{i}(x,y)\). According to the definition of \(A_{i}(x,y)\), we have \(u_{i,1}, u_{i,2}\in S_{i}(x,y)\), and
Let \(\lambda\in(0,1)\), since \(S_{i}:X\times Y\rightarrow2^{X_{i}}\) has convex values, we have \((1\lambda)u_{i,1}+\lambda u_{i,2}\in S_{i}(x,y)\). Since the map \(f_{i}(x,y,\cdot)\) is naturally quasiconvex, there exists \(t\in(0,1)\) such that
That is, \(x_{i}^{*}\circ f_{i}(x,y,(1\lambda)u_{i,1}+\lambda u_{i,2})=F_{i}(x,y)\), which means \((1\lambda)u_{i,1}+\lambda u_{i,2}\in A_{i}(x,y)\).
Assume that \(L_{i}=T_{i}(X)\), \(i\in I\). Since \(T_{i}:X\rightarrow 2^{Y_{i}}\) is nonempty convexvalued, \(L_{i}\) are nonempty convex subsets of \(F_{i}\) and \(L=\prod_{i\in I}L_{i}\) is a nonempty convex subset of \(F=\prod_{i\in I}F_{i}\). Since \(E_{i}\) is a locally convex topological vector space, \(X_{i}\) is a nonempty convex subset of \(E_{i}\). It is similar to knowing that \(X=\prod _{i\in I}X_{i}\) is a nonempty convex subset of \(E=\prod_{i\in I}E_{i}\).
Define setvalued mappings \(H_{i}:X\times L\rightarrow 2^{X_{i}\times L_{i}}\), \(i\in I\) as
According to the proof above, we obtain that X and L are nonempty convex. Define \(H:X\times L\rightarrow2^{X\times L}\) as \(H(x,y)=\prod_{i\in I}H_{i}(x,y)\). Obviously, H is a u.s.c. setvalued mapping with convex and compact values. By Lemma 2, there exists \((\bar{x},\bar{y})\in X\times L\) such that \((\bar{x},\bar{y})\in H(\bar{x},\bar{y})\). Thus, \(\bar{x}_{i}\in S_{i}(\bar{x})\), \(\bar{y}_{i}\in T_{i}(\bar{x})\) with \(\bar{x}_{i}\in A_{i}(\bar{x},\bar{y})\) and \(\bar{y}_{i}\in B_{i}(\bar{x},\bar{y})\). According to (4) and (5), it means that for each \(i\in I\), \(\bar{x}_{i}\in S_{i}(\bar{x})\), \(\bar{y}_{i}\in T_{i}(\bar{x})\) such that
By conditions (iii)(a), (iv)(a), we have
By Lemma 1, we obtain that
Then the (SSGVQEP) has a solution. □
Remark 1
The following example is given to show that Theorem 1 improves [1], Theorem 3.1.
Example 1
For each \(i\in I\), \(E_{i}=F_{i}=\mathbb{R}\) and \(Z_{i}=\mathbb{R}^{2}\), \(X_{i}=Y_{i}=[0,1]\). Let \(X=\prod _{i\in I}X_{i}\) and \(Y=\prod _{i\in I}Y_{i}\). For each \(i\in I\), the setvalued mappings \(S_{i}:X \rightarrow2^{X_{i}}\) and \(T_{i}:Y\rightarrow2^{Y_{i}}\) are defined as \(S_{i}(x)=T_{i}(x)=[0,1]\). For all \((x, y, u_{i})\in X\times Y\times X_{i}\), let
and for all \((x,y,v_{i})\in X\times Y\times Y_{i}\),
Then the assumptions of Theorem 1 hold. But the vectorvalued mapping \(f_{i}\) is not a properly quasiconvex mapping, and thus this example does not satisfy all the conditions of Theorem 3.1 in [1].
Hadamard wellposedness of (SSGVQEP)
In this subsection, we will introduce Hadamardtype wellposedness for (SSGVQEP) and establish sufficient conditions of Hadamardtype wellposedness for (SSGVQEP). Broadly speaking, we say that a problem is Hadamard wellposed if it is possible to obtain ‘small’ changes in the solutions in correspondence to ‘small’ changes in the data. More precisely, let us recall the notions of Hadamard wellposedness and generalized Hadamard wellposedness.
Assume that Z is a metric space, the excess of the set \(A\subset Z\) to the set \(B\subset Z\) is defined by
and the Hausdorff distance between A and B is defined as
For convenience, in what follows, assume that \(P_{0}\) is a set of problems of (SSGVQEP) and \(p_{n}\) (\(n=1,2,\ldots\)) means a sequence of problems of (SSGVQEP) which belong to \(P_{0}\). We show that the formula of \(p_{n}\) is as follows: find \((x^{n},y^{n})\in X\times Y\) such that \(\forall i\in I\), \(x^{n}_{i}\in S^{n}_{i}(x^{n})\), \(y^{n}_{i}\in T^{n}_{i}(x^{n})\),
Meanwhile, for any problem \(p\in P_{0}\), the formula of p is showed as follows: find \((x,y)\in X\times Y\) such that \(\forall i\in I\), \(x_{i}\in S_{i}(x)\), \(y_{i}\in T_{i}(y)\),
Given a set \(P_{0}\) of (SSGVQEP), let us define the distance function \(d_{P_{0}}\) as follows:
where \(p_{1}=(f_{1}^{1},f_{2}^{1},\ldots,f_{N}^{1},g_{1}^{1},g_{2}^{1},\ldots ,g_{N}^{1},S_{1}^{1},S_{2}^{1},\ldots ,S_{N}^{1},T_{1}^{1},T_{2}^{1},\ldots ,T_{N}^{1})\), \(p_{2}=(f_{1}^{2},f_{2}^{2},\ldots, f_{N}^{2},g_{1}^{2},g_{2}^{2}, \ldots ,g_{N}^{2},S_{1}^{2},S_{2}^{2},\ldots ,S_{N}^{2},T_{1}^{2},T_{2}^{2},\ldots,T_{N}^{2})\in P_{0}\). Let
Clearly, \((P_{0}, d_{P_{0}})\) is a metric space.
We say that \(p_{n}\rightarrow p\) if \(d_{P_{0}}(p_{n},p)\rightarrow0\). Moveover, let \(\Gamma(p)\) be the set of solutions of \(p\in P_{0}\). Γ is a setvalued mapping from \(P_{0}\) to \(2^{X\times Y}\), and it is called the solution mapping of p.
Definition 3
Let \((P_{0}, d_{P_{0}})\) be the metric space of data of (SSGVQEP) problems mentioned above, let \((X\times Y, d_{X\times Y})\) be the metric space for the solutions of a problem p in \((P_{0}, d_{P_{0}})\) and Γ be the solution mapping from the space \((P_{0}, d_{P_{0}})\) of problems to the space \(2^{X\times Y}\) of all nonempty solution subsets in \((X\times Y, d_{X\times Y})\).

(1)
Let \(p_{n}\rightarrow p\). A problem \(p\in P\) is called Hadamard wellposed (in short, Hwp) with respect to \((P_{0}, d_{P_{0}})\) and \((X\times Y, d_{X\times Y})\) if the set \(\Gamma(p)\) of solutions of p is a singleton and any sequence \(x_{n}\in\Gamma(p_{n})\) converges to the unique solution of p.

(2)
Let \(p_{n}\rightarrow p\). A problem \(p\in P\) is called generalized Hadamard wellposed (in short, gHwp) with respect to \((P_{0}, d_{P_{0}})\) and \((X\times Y, d_{X\times Y})\) if the set \(\Gamma(p)\) of solutions of p is nonempty, and any sequence \(x_{n}\in\Gamma(p_{n})\) has a subsequence converging to some solution in \(\Gamma(p)\).
Example 2
Let \(I=\{ 1, 2\} \) for each \(i\in I\), \(E_{i}=F_{i}=\mathbb{R}\) and \(Z_{i}=\mathbb{R}\), \(X_{i}=Y_{i}=[0,1]\). Assume that the problem p is defined by \(S_{i}(x)=(1,1)\), \(T_{i}(x)=\{0\}\), \(C_{i}(x)=\mathbb{R}_{+}\), \(f_{i}(x,y,u_{i})=x_{i}u_{i}\) and \(g_{i}(x,y,v_{i})=0\) for every \(i\in I\). Define a sequence of problems \(\{p_{n}\}\) by \(S^{n}_{i}(x)=[1+\frac{1}{n},1\frac{1}{n}]\), \(T^{n}_{i}(x)=\{0\}\), \(C^{n}_{i}(x)=\mathbb{R}_{+}\), \(f^{n}_{i}(x,y,u_{i})=x_{i}u_{i}+\frac{1}{n}\) and \(g^{n}_{i}(x,y,v_{i})=0\) for every \(i\in I\). It is clear that \(d(p,p_{n})\rightarrow0\), the solution set \(\Gamma(p_{n})\) of \(p_{n}\) is \([1\frac{1}{2n},1\frac{1}{n}]\times[1\frac{1}{2n},1\frac {1}{n}]\times\{0\}\times\{0\}\), but the problem p has not any solution. Therefore, the problem p is not Hadamard wellposed.
Lemma 4
Let \(I=\{1,2,\ldots,n\}\) be a finite set. For each \(i\in I\), \(E_{i}\), \(F_{i}\) and \(Z_{i}\) are metric spaces. Let \(X_{i}\subseteq E_{i}\) and \(Y_{i}\subseteq F_{i}\) be compact convex subsets and \(X=\prod _{i\in I}X_{i}\) and \(Y=\prod _{i\in I}Y_{i}\). Assume that the set \(\Gamma(p)\) of solutions of \(p\in P_{0}\) is nonempty and the following conditions are satisfied: for each \(i\in I\),

(i)
the setvalued mappings \(S_{i}:X \rightarrow2^{X_{i}}\) and \(T_{i}:X\rightarrow2^{Y_{i}}\) are compact closed continuous mappings with nonempty convex values,

(ii)
the vectorvalued mappings \(f_{i}:X\times Y\times X_{i}\rightarrow Z_{i}\) and \(g_{i}:X\times Y\times Y_{i}\rightarrow Z_{i}\) are continuous.
Then \(\Gamma(p):P_{0}\rightarrow2^{X\times Y}\) is u.s.c.
Proof
Since \(X\times Y\) is compact, by Lemma 3, we need only to show that Γ is a closed mapping, i.e., to show that for any \(p_{n}\in P\), \(m=1,2,3,\ldots\) with \(p_{n}\rightarrow p\), and for any \((x^{n},y^{n})\in\Gamma(p_{n})\) with \((x^{n},y^{n})\rightarrow(x,y)\), we have \((x,y)\in\Gamma(p)\). Since \((x^{n},y^{n})\in\Gamma(p_{n})\), we obtain \(x^{n}_{i}\in S^{n}_{i}(x^{n})\) and \(y^{n}_{i}\in T^{n}_{i}(y^{n})\). For any \(i\in I\), by the continuity of \(S_{i}\), \(T_{i}\) and \(p_{n}\rightarrow p\), we have \(x_{i}\in S_{i}(x)\) and \(y_{i}\in T_{i}(y)\). Therefore, to prove \((x,y)\in\Gamma(p)\), we only need to prove
Suppose that (9) is not true, we have
Without loss of generality, we assume that \(\exists u_{i}\in S_{i}(x)\), s.t. \(f_{i}(x,y,u_{i})\notin C(x_{i})\). Thus, there exists some open neighborhood V of the zero element of \(Z_{i}\) such that
Since \(p_{n}\rightarrow p\), there exists \(n_{1}\in Z^{+}\) (\(Z^{+}\) is a set of positive integers) such that when \(n\geq n_{1}\), we have
Since \(u_{i}\in S_{i}(x)\), \((x^{n},y^{n})\rightarrow(x,y)\) and \(S_{i}\) is a compact continuous mapping, we have that there exists \(u^{n}_{i}\in S^{n}_{i}(\bar{x}^{n})\) such that \(u^{n}_{i}\rightarrow u_{i}\). Since \(f_{i}\) is continuous at \((x,y,u_{i})\), there exists \(n_{2}\in Z^{+}\) such that for any \(n\geq n_{2}\), we have
Let \(N=\max\{n_{1},n_{2}\}\). By (10) and (11), we obtain that for any \(n\geq N\),
Since \((f_{i}(x,y,u_{i})+V)\cap C_{i}(x)=\emptyset\), we have \(f^{n}_{i}(x^{n},y^{n},u^{n}_{i})\notin C_{i}(x)\), which contradicts \((x^{n},y^{n})\in\Gamma(p_{n})\). Therefore, Γ is a closed mapping. □
Now we establish the sufficient condition of Hadamardtype wellposedness for (SSGVQEP).
Theorem 2
Let \(I=\{1,2,\ldots,n\}\) be a finite set, for each \(i\in I\), let \(E_{i}\), \(F_{i}\) and \(Z_{i}\) be metric spaces, and \(X_{i}\subseteq E_{i}\) and \(Y_{i}\subseteq F_{i}\) be compact convex subsets. Let \(X=\prod _{i\in I}X_{i}\) and \(Y=\prod _{i\in I}Y_{i}\). Assume that the set \(\Gamma(p)\) of solutions of \(p\in P_{0}\) is nonempty and the following conditions are satisfied: for each \(i\in I\),

(i)
the setvalued mappings \(S_{i}:X \rightarrow2^{X_{i}}\) and \(T_{i}:X\rightarrow2^{Y_{i}}\) are compact closed continuous mappings with nonempty convex values,

(ii)
the vectorvalued mappings \(f_{i}:X\times Y\times X_{i}\rightarrow Z_{i}\) and \(g_{i}:X\times Y\times Y_{i}\rightarrow Z_{i}\) are continuous.
Then the problem (SSGVQEP) is generalized Hadamard wellposed.
Proof
By Lemma 4 and Theorem 2.1 of [35], the conclusion naturally holds. □
Remark 2
It is easy to verify that if (SSGVQEP) has a unique solution, then the fact that (SSGVQEP) is generalized Hadamard wellposed implies that (SSGVQEP) is Hadamard wellposed.
Conclusions
Under some weaker conditions, we have established an existence result for the solution set of a system of simultaneous generalized vector quasiequilibrium problems, and it improved the relevant Theorem 3.1 in the work of Ansari et al. [1]. We have defined a new concept of Hadamardtype wellposedness for (SSGVQEP) and established sufficient conditions for Hadamard wellposedness of (SSGVQEP).
References
Ansari, QH, Lin, LJ, Su, LB: Systems of simultaneous generalized vector quasiequilibrium problems and their applications. J. Optim. Theory Appl. 127, 2744 (2005)
Giannessi, F: Vector Variational Inequalities and Vector Equilibria: Mathematical Theories. Kluwer Academic, Dordrecht (2000)
Chen, CR, Li, SJ, Teo, KL: Solution semicontinuity of parametric generalized vector equilibrium problems. J. Glob. Optim. 45, 309318 (2009)
Li, SJ, Zhang, WY: Hadamard wellposed vector optimization problems. J. Glob. Optim. 46, 383393 (2010)
Long, XJ, Huang, YQ, Peng, ZY: Optimality conditions for the Henig efficient solution of vector equilibrium problems with constraints. Optim. Lett. 5, 717728 (2010)
Peng, ZY, Li, Z, Yu, KZ, Wang, DC: A note on solution lower semicontinuity for parametric generalized vector equilibrium problems. J. Inequal. Appl. 2014, 325 (2014)
Wang, QL, Lin, Z, Li, XB: Semicontinuity of the solution set to a parametric generalized strong vector equilibrium problem. Positivity 18, 733748 (2014)
Xu, YD, Li, SJ: On the lower semicontinuity of the solution mappings to a parametric generalized strong vector equilibrium problem. Positivity 17, 341353 (2013)
Fang, YP, Huang, NJ, Kim, JK: Existence results for systems of vector equilibrium problems. J. Glob. Optim. 35, 7183 (2006)
Fu, JY: Simultaneous vector variational inequalities and vector implicit complementarity problems. J. Optim. Theory Appl. 93, 141151 (1997)
Husain, T, Tarafdar, E: Simultaneous variational inequalities, minimization problems, and related results. Math. Jpn. 39, 221231 (1994)
Li, J, Huang, NJ: An extension of gap functions for a system of vector equilibrium problems with applications to optimization. J. Glob. Optim. 39, 247260 (2007)
Lin, Z: The study of the system of generalized vector quasiequilibrium problems. J. Glob. Optim. 36, 627635 (2006)
Gong, XH: Symmetric strong vector quasiequilibrium problems. Math. Methods Oper. Res. 65, 305314 (2007)
Hou, SH, Gong, XH, Yang, XM: Existence and stability of solutions for generalized Ky Fan inequality problems with trifunctions. J. Optim. Theory Appl. 146, 387398 (2010)
Tan, NX: On the existence of solutions of quasivariational inclusion problems. J. Optim. Theory Appl. 123, 619638 (2004)
Ceng, LC, Hadjisavvas, N, Schaible, S, Yao, JC: Wellposedness for mixed quasivariationallike inequalities. J. Optim. Theory Appl. 139, 109125 (2008)
Ceng, LC, Yao, JC: Wellposedness of generalized mixed variational inequalities, inclusion problems and fixedpoint problems. Nonlinear Anal. TMA 69, 45854603 (2008)
Ceng, LC, Wong, NC, Yao, JC: Wellposedness for a class of strongly mixed variationalhemivariational inequalities with perturbations. J. Appl. Math. 2012, Article ID 712306 (2012)
Ceng, LC, Gupta, H, Wen, CF: Wellposedness by perturbations of variationalhemivariational inequalities with perturbations. Filomat 26(5), 881895 (2012)
Ceng, LC, Lin, YC: Metric characterizations of αwellposedness for a system of mixed quasivariationallike inequalities in Banach spaces. J. Appl. Math. 2012, Article ID 264721 (2012)
Ceng, LC, Wen, CF: Wellposedness by perturbations of generalized mixed variational inequalities in Banach spaces. J. Appl. Math. 2012, Article ID 194509 (2012)
Ceng, LC, Liou, YC, Wen, CF: On the wellposedness of generalized hemivariational inequalities and inclusion problems in Banach spaces. J. Nonlinear Sci. Appl. 9, 38793891 (2016)
Ceng, LC, Liou, YC, Wen, CF: Some equivalence results for wellposedness of generalized hemivariational inequalities with Clarke’s generalized directional derivative. J. Nonlinear Sci. Appl. 9, 27982812 (2016)
Chen, JW, Wan, ZP, Cho, J: LevitinPolyak wellposedness by perturbations for systems of setvalued vector quasiequilibrium problems. Math. Methods Oper. Res. 77, 3364 (2013)
Dontchev, AL, Zolezzi, T: WellPosed Optimization Problems. Lecture Notes in Mathematics, vol. 1543. Springer, Berlin (1993)
Li, SJ, Li, MH: LevitinPolyak wellposedness of vector equilibrium problems. Math. Methods Oper. Res. 69, 125140 (2009)
Luccchetti, R, Revaliski, J (eds.): Recent Developments in WellPosed Variational Problems. Kluwer Academic, Dordrecht (1995)
Peng, ZY, Yang, XM: PainlevéKuratowski convergences of the solution sets for perturbed vector equilibrium problems without monotonicity. Acta Math. Appl. Sinica (Engl. Ser.) 30, 845858 (2014)
Farajzadeh, AP: On the symmetric vector quasiequilibrium problems. J. Math. Anal. Appl. 322, 10991110 (2006)
Swartz, C: An Introduction to Functional Analysis. Dekker, New York (1992)
Glicksberg, I: A further generalization of the Kakutani fixed point theorem with application to Nash equilibrium points. Proc. Am. Math. Soc. 3, 170174 (1952)
Berge, C: Espaces Topologiques. Dunod, Paris (1959)
Tanaka, T: Generalized quasiconvexities, cone saddle points and minimax theorems for vector valued functions. J. Optim. Theory Appl. 81, 355377 (1994)
Zhou, YH, Yu, JY, Yang, H, Xiang, SW: Hadamard types of wellposedness of nonself setvalued mappings for coincide points. Nonlinear Anal. 63, 24272436 (2005)
Acknowledgements
This research was partially supported by the National Natural Science Foundation of China (Grant number 11401058), by the Basic and Advanced Research Project of Chongqing (Grant numbers cstc2016jcyjA0219, cstc2014jcyjA00033), by the Education Committee Project Research Foundation of Chongqing (Grant number KJ1400630), by the Scientific Research Fund of Sichuan Provincial Science and Technology Department (Grant number 2015JY0237), by the Program for University Innovation Team of Chongqing (Grant number CXTDX201601026) and by the Young Doctor Fund Project of Chongqing Technology and Business University (Grant number 1352014).
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Zhang, W., Zeng, J. Existence and Hadamard wellposedness of a system of simultaneous generalized vector quasiequilibrium problems. J Inequal Appl 2017, 58 (2017). https://doi.org/10.1186/s1366001713302
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DOI: https://doi.org/10.1186/s1366001713302
MSC
 49J53
 49K40
 90C33
 90C46
Keywords
 demicontinuity
 natural quasiconvexity
 existence theorem
 Hadamard wellposedness