Existence and Hadamard well-posedness of a system of simultaneous generalized vector quasi-equilibrium problems
- Wenyan Zhang^{1} and
- Jing Zeng^{2}Email author
DOI: 10.1186/s13660-017-1330-2
© The Author(s) 2017
Received: 23 November 2016
Accepted: 24 February 2017
Published: 7 March 2017
Abstract
An existence result for the solution set of a system of simultaneous generalized vector quasi-equilibrium problems (for short, (SSGVQEP)) is obtained, which improves Theorem 3.1 of the work of Ansari et al. (J. Optim. Theory Appl. 127:27-44, 2005). Moreover, a definition of Hadamard-type well-posedness for (SSGVQEP) is introduced and sufficient conditions for Hadamard well-posedness of (SSGVQEP) are established.
Keywords
demicontinuity natural quasi-convexity existence theorem Hadamard well-posednessMSC
49J53 49K40 90C33 90C461 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 set-valued 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 set-valued 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.
The classical concept of Hadamard well-posedness 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 well-posedness of (SSGVQEP). Therefore, it is necessary to study Hadamard well-posedness of (SSGVQEP).
In this paper, by using demicontinuity and natural quasi-convexity, we obtain an existence theorem of solutions for (SSGVQEP). Moreover, we introduce the definition of Hadamard well-posedness for (SSGVQEP) and discuss sufficient conditions for Hadamard well-posedness 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 well-posedness of (SSGVQEP).
2 Preliminaries and notations
Let us recall some notations and definitions of vector-valued mappings and set-valued mappings together with their properties.
Definition 1
[30]
Definition 2
- (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 quasi-convex 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 quasi-convex 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 quasi-convex or convex mapping is naturally quasi-convex, but a naturally quasi-convex mapping may not be convex or properly quasi-convex.
3 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 Hadamard-type well-posedness for (SSGVQEP) and establish sufficient conditions of Hadamard-type well-posedness for (SSGVQEP).
3.1 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].
Lemma 1
For arbitrary \(x\in Z\), if \((x^{*},x)\geq0\) for all \(x^{*}\in T\), then \(x\in P\).
Proof
The following well-known Kakutani-Fan-Glicksberg 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 set-valued mapping \(F: X\rightarrow2^{Y}\) is u.s.c. with compact values if and only if it is a closed mapping.
Theorem 1
- (i)
\(C_{i}:X\rightarrow2^{Z_{i}}\) is a closed set-valued 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 quasi-convex,
- (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 quasi-convex.
- (a)
Proof
Assume that \(L_{i}=T_{i}(X)\), \(i\in I\). Since \(T_{i}:X\rightarrow 2^{Y_{i}}\) is nonempty convex-valued, \(L_{i}\) 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 \(E=\prod_{i\in I}E_{i}\).
Example 1
3.2 Hadamard well-posedness of (SSGVQEP)
In this subsection, we will introduce Hadamard-type well-posedness for (SSGVQEP) and establish sufficient conditions of Hadamard-type well-posedness for (SSGVQEP). Broadly speaking, we say that a problem is Hadamard well-posed 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 well-posedness and generalized Hadamard well-posedness.
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 set-valued mapping from \(P_{0}\) to \(2^{X\times Y}\), and it is called the solution mapping of p.
Definition 3
- (1)
Let \(p_{n}\rightarrow p\). A problem \(p\in P\) is called Hadamard well-posed (in short, H-wp) 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 well-posed (in short, gH-wp) 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 well-posed.
Lemma 4
- (i)
the set-valued 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 vector-valued 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.
Proof
Now we establish the sufficient condition of Hadamard-type well-posedness for (SSGVQEP).
Theorem 2
- (i)
the set-valued 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 vector-valued 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.
Remark 2
It is easy to verify that if (SSGVQEP) has a unique solution, then the fact that (SSGVQEP) is generalized Hadamard well-posed implies that (SSGVQEP) is Hadamard well-posed.
4 Conclusions
Under some weaker conditions, we have established an existence result for the solution set of a system of simultaneous generalized vector quasi-equilibrium problems, and it improved the relevant Theorem 3.1 in the work of Ansari et al. [1]. We have defined a new concept of Hadamard-type well-posedness for (SSGVQEP) and established sufficient conditions for Hadamard well-posedness of (SSGVQEP).
Declarations
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).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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