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Wellposed generalized vector equilibrium problems
Journal of Inequalities and Applications volume 2014, Article number: 127 (2014)
Abstract
In this paper, we establish the bounded rationality model M for generalized vector equilibrium problems by using a nonlinear scalarization technique. By using the model M, we introduce a new wellposedness concept for generalized vector equilibrium problems, which unifies its Hadamard and LevitinPolyak wellposedness. Furthermore, sufficient conditions for the wellposedness for generalized vector equilibrium problems are given. As an application, sufficient conditions on the wellposedness for generalized equilibrium problems are obtained.
MSC: 49K40, 90C31.
1 Introduction
As is well known, the notion of wellposedness can be divided into two different groups: Hadamard type and Tykhonov type [1]. Roughly speaking, Hadamard types of wellposedness for a problem means the continuous dependence of the optimal solution from the data of the problem. Tykhonov types of wellposedness for a problem such as Tykhonov and LevitinPolyak wellposedness are based on the convergence of approximating solution sequences of the problem. Some researchers have investigated the relations between them for different problems (see [1–4]). The notion of extended wellposedness has been proposed by Zolezzi [5] in the context of scalar optimization. In some sense this notion unifies the ideas of Tykhonov and Hadamard wellposedness. Moreover, the notion of extended wellposedness has been generalized to vector optimization problems by Huang [6–8].
On the other hand, the vector equilibrium problem provides a very general model for a wide range of problems, for example, the vector optimization problem, the vector variational inequality problem, the vector complementarity problem and the vector saddle point problem. In the literature, existence results for various types of vector equilibrium problems have been investigated intensively; see, e.g., [9, 10] and the references therein. The study of wellposedness for vector equilibrium problems is another important topic in vector optimization theory. Recently, Tykhonov types wellposedness for vector optimization problems, vector variational inequality problems and vector equilibrium problems have been intensively studied in the literature, such as [11–17]. Among those papers, we observe that the scalarization technique is an efficient approach to deal with Tykhonov types wellposedness for vector optimization problems. As noted in [12, 16], the notions of wellposedness in the scalar case can be extended to the vector case and, for this end, one needs an appropriate scalarizarion technique. Such a technique is supposed to preserve some wellposedness properties when one passes from the vectorial to the scalar case, and simple examples show that linear scalarization is not useful from this point of view even in the convex case. An effort in this direction was made in the papers (see [11, 12, 16, 18]). Miglierina et al. [11] investigated several types of wellposedness concepts for vectorial optimization problems by using a nonlinear scalarization procedure. Some equivalences between wellposedness of vectorial optimization problems and wellposedness of corresponding scalar optimization problems are given. By virtue of a nonlinear scalarization function, Durea [12] proved the Tykhonov wellposedness of the scalar optimization problems are equivalent to the Tykhonov wellposedness of the original vectorial optimization problems. Very recently, Li and Xia [16] investigated LevitinPolyak wellposedness for vectorial optimization problems by using a nonlinear scalarization function. They also showed the equivalence relations between the LevitinPolyak wellposedness of scalar optimization problems and the vectorial optimization problems.
Motivated and inspired by the research work mentioned above, we introduce a new wellposedness concept for generalized vector equilibrium problems (in short (GVEP)), which unifies its Hadamard and LevitinPolyak wellposedness. The concept of wellposedness for (GVEP) is investigated by using a new method which is different from the ones used in [5–8]. Our method is based on a nonlinear scalarization technique and the bounded rationality model M (see [19–22]). Furthermore, we give some sufficient conditions on various types of wellposedness for (GVEP). Finally, we apply these results to generalized equilibrium problems (in short (GEP)).
2 Preliminaries
Let X be a nonempty subset of the Hausdorff topological space H and Y be a Hausdorff topological vector space. Assume that C denotes a nonempty, closed, convex, and pointed cone in Y with apex at the origin and intC\ne \mathrm{\varnothing}, where intC denotes the topological interior of C.
Let G:X\rightrightarrows X be a setvalued mapping and \phi :X\times X\to Y be a vectorvalued mapping, the problem of interest, called generalized vector equilibrium problems (in short (GVEP)), which consist of finding an element x\in X such that x\in G(x) and
When Y=\mathbb{R} and C=[0,+\mathrm{\infty}[, the generalized vector equilibrium problem becomes the generalized equilibrium problem (in short (GEP)): finding an element x\in X such that x\in G(x) and
Now we introduce the notion of LevitinPolyak approximating solution sequence for (GVEP).
Definition 2.1 A sequence \{{x}_{n}\}\subset X is called a LevitinPolyak approximating solution sequence (in short LP sequence) for (GVEP), if there exists \{{\u03f5}_{n}\}\subset {\mathbb{R}}_{+} with {\u03f5}_{n}\to 0 such that
and
Next, we introduce a nonlinear scalarization function and their related properties.
For fixed e\in intC, the nonlinear scalarization function is defined by
The nonlinear scalarization function {\xi}_{e} has the following properties:

(i)
{\xi}_{e}(y)\ge r\u27fay\notin reintC;

(ii)
{\xi}_{e}(re)=r;

(iii)
{\xi}_{e}({y}_{1}+{y}_{2})\le {\xi}_{e}({y}_{1})+{\xi}_{e}({y}_{2}), for all {y}_{1},{y}_{2}\in Y.
Definition 2.2 Let \phi :X\to Y be a vectorvalued mapping.

(i)
φ is said to be Cupper semicontinuous at x if for any open neighborhood V of zero element in Y, there is an open neighborhood U at x in X such that for any {x}^{\prime}\in U, \phi ({x}^{\prime})\in \phi (x)+VC;

(ii)
φ is said to be Cupper semicontinuous on X if φ is Cupper semicontinuous on each x\in X;

(iii)
φ is said to be Clower semicontinuous at x if for any open neighborhood V of zero element in Y, there is an open neighborhood U at x in X such that for any {x}^{\prime}\in U, \phi ({x}^{\prime})\in \phi (x)+V+C;

(iv)
φ is said to be Clower semicontinuous on X if φ is Clower semicontinuous on each x\in X.
Remark 2.1 In Definition 2.2, when Y=\mathbb{R}, C=[0,+\mathrm{\infty}[, being Cupper semicontinuous reduces to being upper semicontinuous and being Clower semicontinuous reduces to being lower semicontinuous.
Lemma 2.2 If \phi :X\times X\to Y is Cupper semicontinuous on X\times X, then {\xi}_{e}\circ \phi :X\times X\to \mathrm{\Re} is upper semicontinuous on X\times X.
Proof In order to show that {\xi}_{e}\circ \phi :X\times X\to \mathrm{\Re} is upper semicontinuous on X\times X, we must check, for any r\in \mathrm{\Re}, the set L=\{(x,y)\in X\times X:{\xi}_{e}(\phi (x,y))\ge r\} is closed.
Let ({x}_{n},{y}_{n})\in L and ({x}_{n},{y}_{n})\to ({x}_{0},{y}_{0}), we have {\xi}_{e}(\phi ({x}_{n},{y}_{n}))\ge r, that is to say, by Lemma 2.1(i), \phi ({x}_{n},{y}_{n})\notin reintC. Next, we only need to prove that {\xi}_{e}(\phi ({x}_{0},{y}_{0}))\ge r, that is, \phi ({x}_{0},{y}_{0})\notin reintC. By way of contradiction, assume that \phi ({x}_{0},{y}_{0})\in reintC, then there exists an open neighborhood V of zero element in Y such that
Since φ is Cupper semicontinuous at ({x}_{0},{y}_{0})\in X\times X, we have
It contradicts \phi ({x}_{n},{y}_{n})\notin reintC. So L is closed. It shows {\xi}_{e}\circ \phi :X\times X\to \mathrm{\Re} is upper semicontinuous on X\times X. □
Finally, we recall some useful definitions and lemmas.
Let F:X\rightrightarrows Y be a setvalued mapping. F is said to be upper semicontinuous at x\in X if for any open set U\supset F(x), there is an open neighborhood O(x) of x such that U\supset F({x}^{\prime}) for each {x}^{\prime}\in O(x); F is said to be lower semicontinuous at x if for any open set U\cap F(x)\ne \mathrm{\varnothing}, there is an open neighborhood O(x) of x such that U\cap F({x}^{\prime})\ne \mathrm{\varnothing}, for each {x}^{\prime}\in O(x); F is said to be an usco mapping if F is upper semicontinuous and F(x) is nonempty compact for each x\in X; F is said to be closed if Graph(F) is closed, where Graph(F)=\{(x,y)\in X\times Y:x\in X,y\in F(x)\} is the graph of F.
Lemma 2.3 [22]
Let X and Y be two metric spaces. Suppose that F:Y\rightrightarrows X is a usco mapping. Then for any {y}_{n}\to y and any {x}_{n}\in F({y}_{n}), there is a subsequence \{{x}_{{n}_{k}}\}\subset \{{x}_{n}\} such that {x}_{{n}_{k}}\to x\in F(y).
Lemma 2.4 [24]
If F:Y\rightrightarrows X is closed and X is compact, then F is upper semicontinuous on Y.
Let (X,d) be a metric space. Denote by K(X) all nonempty compact subsets of X. For arbitrary {C}_{1},{C}_{2}\subset X, define
where
and
It is obvious that h is a Hausdorff metric on K(X).
Lemma 2.5 [25]
Let (X,d) be a metric space and h be Hausdorff metric on X. Then (K(X),h) is complete if and only if (X,d) is complete.
3 Bounded rationality model and definition of wellposedness for (GVEP)
Let (X,d) be a metric space. The problem space Λ of (GVEP) is given by
For any {\lambda}_{1}=({\phi}_{1},{G}_{1}), {\lambda}_{2}=({\phi}_{2},{G}_{2})\in \mathrm{\Lambda}, define
where h denotes a Hausdorff distance on X. Clearly, (\mathrm{\Lambda},\rho ) is a metric space.
Next, the bounded rationality model M=\{\mathrm{\Lambda},X,f,\mathrm{\Phi}\} for (GVEP) is defined as follows:

(i)
Λ and X are two metric spaces;

(ii)
the feasible set of the problem \lambda \in \mathrm{\Lambda} is defined by
f(\lambda ):=\{x\in X:x\in G(x)\}; 
(iii)
the solution set of the problem \lambda \in \mathrm{\Lambda} is defined by
E(\lambda ):=\{x\in G(x):\phi (x,y)\notin intC,\mathrm{\forall}y\in G(x)\}; 
(iv)
the rationality function of the problem \lambda \in \mathrm{\Lambda} is defined by
\mathrm{\Phi}(\lambda ,x):=\underset{y\in G(x)}{sup}\{{\xi}_{e}(\phi (x,y))\}.
Lemma 3.1

(1)
x\in f(\lambda ), \mathrm{\Phi}(\lambda ,x)\ge 0.

(2)
For all \lambda \in \mathrm{\Lambda}, E(\lambda )\ne \mathrm{\varnothing}.

(3)
For all \lambda \in \mathrm{\Lambda} and \u03f5\ge 0, \mathrm{\Phi}(\lambda ,x)={sup}_{y\in G(x)}\{{\xi}_{e}(\phi (x,y))\}\le \u03f5 if and only if \phi (x,y)+\u03f5e\notin intC, \mathrm{\forall}y\in G(x). Particularly, x\in E(\lambda ) if and only if \mathrm{\Phi}(\lambda ,x)=0.
Proof (1) If x\in f(\lambda ), then x\in G(x). By Lemma 2.1(i), we have

(2)
Obvious.

(3)
If \phi (x,y)+\u03f5e\notin intC, \mathrm{\forall}y\in G(x), by Lemma 2.1(i), {\xi}_{e}(\phi (x,y))\ge \u03f5, \mathrm{\forall}y\in G(x). Thus, we have \mathrm{\Phi}(\lambda ,x)={sup}_{y\in G(x)}\{{\xi}_{e}(\phi (x,y))\}\le \u03f5.
Conversely, if \mathrm{\Phi}(\lambda ,x)={sup}_{y\in G(x)}\{{\xi}_{e}(\phi (x,y))\}\le \u03f5, then {\xi}_{e}(\phi (x,y))\ge \u03f5, \mathrm{\forall}y\in G(x). By Lemma 2.1(i), we get \phi (x,y)+\u03f5e\notin intC, \mathrm{\forall}y\in G(x). □
Remark 3.1 By Definition 2.1 and Lemma 3.1, for all {\u03f5}_{n}>0 with {\u03f5}_{n}\to 0, the set of LP approximating solution for the problem λ is defined as
the set of solutions for the problem λ is defined as
Hence, LevitinPolyak wellposedness for (GVEP) is defined as follows.
Definition 3.1

(i)
If \mathrm{\forall}{x}_{n}\in E(\lambda ,{\u03f5}_{n}), {\u03f5}_{n}>0 with {\u03f5}_{n}\to 0, there must exist a subsequence \{{x}_{{n}_{k}}\}\subset \{{x}_{n}\} such that {x}_{{n}_{k}}\to x\in E(\lambda ), then the problem \lambda \in \mathrm{\Lambda} is said to be generalized LevitinPolyak wellposed (in short GLPwp);

(ii)
If E(\lambda )=\{x\} (a singleton), \mathrm{\forall}{x}_{n}\in E(\lambda ,{\u03f5}_{n}), {\u03f5}_{n}>0 with {\u03f5}_{n}\to 0, there must have {x}_{n}\to x, then the problem \lambda \in \mathrm{\Lambda} is said to be LevitinPolyak wellposed (in short LPwp).
Referring to [3], Hadamard wellposedness for (GVEP) is defined as follows.
Definition 3.2

(i)
If \mathrm{\forall}{\lambda}_{n}\in \mathrm{\Lambda}, {\lambda}_{n}\to \lambda, \mathrm{\forall}{x}_{n}\in E({\lambda}_{n}), there must exist a subsequence \{{x}_{{n}_{k}}\}\subset \{{x}_{n}\} such that {x}_{{n}_{k}}\to x\in E(\lambda ), then the problem \lambda \in \mathrm{\Lambda} is said to be generalized Hadamard wellposed (in short GHwp);

(ii)
If E(\lambda )=\{x\} (a singleton), \mathrm{\forall}{\lambda}_{n}\in \mathrm{\Lambda}, {\lambda}_{n}\to \lambda, \mathrm{\forall}{x}_{n}\in E({\lambda}_{n}), we must have {x}_{n}\to x, then the problem \lambda \in \mathrm{\Lambda} is said to be Hadamard wellposed (in short Hwp).
Next, we establish a new wellposedness concept for (GVEP), which unifies its Hadamard and LevitinPolyak wellposedness.
Definition 3.3

(i)
If \mathrm{\forall}{\lambda}_{n}\in \mathrm{\Lambda}, {\lambda}_{n}\to \lambda, \mathrm{\forall}{x}_{n}\in E({\lambda}_{n},{\u03f5}_{n}), {\u03f5}_{n}>0 with {\u03f5}_{n}\to 0, there must exist a subsequence \{{x}_{{n}_{k}}\}\subset \{{x}_{n}\} such that {x}_{{n}_{k}}\to x\in E(\lambda ), then the problem \lambda \in \mathrm{\Lambda} is said to be generalized wellposed (in short Gwp);

(ii)
If E(\lambda )=\{x\} (a singleton), \mathrm{\forall}{\lambda}_{n}\in \mathrm{\Lambda}, {\lambda}_{n}\to \lambda, \mathrm{\forall}{x}_{n}\in E({\lambda}_{n},{\u03f5}_{n}), {\u03f5}_{n}>0 with {\u03f5}_{n}\to 0, there must have {x}_{n}\to x, then \lambda \in \mathrm{\Lambda} is said to be wellposed (in short wp).
By Definitions 3.1, 3.2 and 3.3, it is easy to check the following.
Lemma 3.2

(1)
If the problem \lambda \in \mathrm{\Lambda} is Gwp, then λ must be GLPwp.

(2)
If the problem \lambda \in \mathrm{\Lambda} is wp, then λ must be LPwp.
Lemma 3.3

(1)
If the problem \lambda \in \mathrm{\Lambda} is Gwp, then λ must be GHwp.

(2)
If the problem \lambda \in \mathrm{\Lambda} is wp, then λ must be Hwp.
4 Sufficient conditions for wellposedness of (GVEP)
Assume that the bounded rationality model M=\{\mathrm{\Lambda},X,f,\mathrm{\Phi}\} for (GVEP) is given. Now, let (X,d) be a compact metric space, (Y,\parallel \cdot \parallel ) be a Banach space, and C be a nonempty, closed, convex, and pointed cone in Y with apex at the origin and intC\ne \mathrm{\varnothing}.
In order to show sufficient conditions for wellposedness of (VEP), we first give the following lemmas.
Lemma 4.1 (\mathrm{\Lambda},\rho ) is a complete metric space.
Proof Let \{{\lambda}_{n}=({\phi}_{n},{G}_{n})\} be any Cauchy sequence in Λ, then for any \u03f5>0, there is a positive integer N such that for any n,m\ge N,
Then, for any fixed (x,y)\in X\times Y, \{{\phi}_{n}(x,y)\} is a Cauchy sequence in Y, and \{{G}_{n}(x)\} is a Cauchy sequence in K(X). By Lemma 2.5, (K(X),h) is a complete spaces and (Y,\parallel \cdot \parallel ) is also complete spaces. It follows that there exist \phi (x,y)\in Y and G(x)\in K(X) such that {lim}_{m\to \mathrm{\infty}}{\phi}_{m}(x,y)=\phi (x,y) and {lim}_{m\to \mathrm{\infty}}{G}_{m}(x)=G(x). Thus, for all n\ge N, we have
Next, we will prove that \lambda =(\phi ,G)\in \mathrm{\Lambda}, thus (\mathrm{\Lambda},\rho ) is a complete metric space.

(i)
For any open convex neighborhood V of zero element in Y, there is a positive integer {n}_{0} such that for all x,y\in X,
\phi (x,y)\in {\phi}_{{n}_{0}}(x,y)+\frac{V}{3},(1)
and
Since {\lambda}_{{n}_{0}}=({\phi}_{{n}_{0}},{G}_{{n}_{0}})\in \mathrm{\Lambda}, {\phi}_{{n}_{0}} is Cupper semicontinuous on X\times X, thus there are an open neighborhood of {U}_{1} at x and an open neighborhood of {U}_{2} at y such that
By (1), (2), and (3), we have
It shows \phi :X\times X\to Y is Cupper semicontinuous on X\times X.

(ii)
It is easy to check that \phi (x,x)=\mathbf{0}, \mathrm{\forall}x\in X, {sup}_{(x,y)\in X\times X}\parallel \phi (x,y)\parallel <+\mathrm{\infty}, G:X\rightrightarrows X is continuous on X and \mathrm{\forall}x\in X, G(x) is a nonempty compact set.

(iii)
Since {\lambda}_{n}=({\phi}_{n},{G}_{n})\in \mathrm{\Lambda}, there exists {x}_{n}\in X such that {x}_{n}\in {G}_{n}({x}_{n}) and {\phi}_{n}({x}_{n},y)\notin intC, \mathrm{\forall}y\in {G}_{n}({x}_{n}). Firstly, we may suppose that {x}_{n}\to x, since X is a compact metric space. For all n\ge N,
\begin{array}{rl}h({G}_{n}({x}_{n}),G(x))& \le h({G}_{n}({x}_{n}),G({x}_{n}))+h(G({x}_{n}),G(x))\\ \le \u03f5+h(G({x}_{n}),G(x)).\end{array}(4)
Note that G is continuous on X, we have
By (4) and (5), we get
Hence, x\in G(x).
Finally, we only need to prove that \phi (x,y)\notin intC, \mathrm{\forall}y\in G(x). By way of contradiction, assume that there exists {y}_{0}\in G(x) such that \phi (x,{y}_{0})\in intC. It shows that there exists an open convex neighborhood V of zero element in Y such that \phi (x,{y}_{0})+V\subset intC.
Since {sup}_{(x,y)\in X\times X}\parallel {\phi}_{n}(x,y)\phi (x,y)\parallel \to 0, there is a positive integer {N}_{1} such that \mathrm{\forall}n\ge {N}_{1},
By virtue of h({G}_{n}({x}_{n}),G(x))\to 0, there exists {y}_{n}\in {G}_{n}({x}_{n}) such that {y}_{n}\to {y}_{0}. Note that \phi :X\times X\to Y is Cupper semicontinuous on X\times X, then there exists a positive integer {N}_{2} such that \mathrm{\forall}n\ge {N}_{2},
Let N=max\{{N}_{1},{N}_{2}\}, \mathrm{\forall}n\ge N, by (6) and (7), we have
This is a contradiction to {\phi}_{n}({x}_{n},y)\notin intC, \mathrm{\forall}y\in {G}_{n}({x}_{n}). □
Lemma 4.2 f:\mathrm{\Lambda}\rightrightarrows X is an usco mapping.
Proof Since X is a compact metric space, by Lemma 2.4, it suffices to show that Graph(f) is closed, where Graph(f)=\{(\lambda ,x)\in \mathrm{\Lambda}\times X:x\in f(\lambda )\}. That is to say, \mathrm{\forall}{\lambda}_{n}\in \mathrm{\Lambda}, {\lambda}_{n}\to \lambda, \mathrm{\forall}{x}_{n}\in f({\lambda}_{n}), {x}_{n}\to x, we need to show that x\in f(\lambda ).
In fact, for each n=1,2,3,\dots , since {x}_{n}\in f({\lambda}_{n}), then there exists {x}_{n}\in X such that {x}_{n}\in {G}_{n}({x}_{n}). Let {sup}_{x\in X}h({G}_{n}(x),G(x))={\u03f5}_{n} with {\u03f5}_{n}\to 0, there must be h({G}_{n}({x}_{n}),G({x}_{n}))\le {\u03f5}_{n}. Since {x}_{n}\in {G}_{n}({x}_{n}), there exists {x}_{n}^{\prime}\in G({x}_{n}) such that d({x}_{n},{x}_{n}^{\prime})\le {\u03f5}_{n}. By
we get {x}_{n}^{\prime}\to x. Note that setvalue mapping G is continuous on X, then
By (8) and (9), we get
Since G(x) is a nonempty compact subset of X, by (10), we have x\in G(x). It shows that x\in f(\lambda ). □
Lemma 4.3 For all (\lambda ,x)\in \mathrm{\Lambda}\times X, \mathrm{\Phi}(\lambda ,x) is lower semicontinuous at (\lambda ,x).
Proof By Lemma 4.1, it is only need to show that \mathrm{\forall}\u03f5>0, \mathrm{\forall}{\lambda}_{n}=({\phi}_{n},{G}_{n})\in \mathrm{\Lambda}, {\lambda}_{n}\to \lambda =(\phi ,G)\in \mathrm{\Lambda}, \mathrm{\forall}{x}_{n}\in X, {x}_{n}\to x\in X, there exists a positive integer N such that \mathrm{\forall}n\ge N,
By definition of the least upper bound, there exists {y}_{0}\in G(x) such that
Note that {sup}_{x\in X}h({G}_{n}(x),G(x))\to 0 and h(G({x}_{n}),G(x))\to 0, we have
From (13), there exists {y}_{n}\in {G}_{n}({x}_{n}) such that d({y}_{n},{y}_{0})\to 0.
Since {sup}_{(x,y)\in X\times X}\parallel {\phi}_{n}(x,y)\phi (x,y)\parallel \to 0, that is to say, there exists a positive integer {N}_{1} such that \mathrm{\forall}n\ge {N}_{1},
where r\in \phantom{\rule{0.2em}{0ex}}]\frac{\u03f5}{4},\frac{\u03f5}{4}[, e\in intC, re is an open neighborhood of zero element in Y. Thus, by (14) and Lemma 2.1(ii), we get
By (15) and Lemma 2.1(iii), we get
By Lemma 2.2, {\xi}_{e}\circ \phi is upper semicontinuous on X\times X. Then there exists a positive integer {N}_{2} such that \mathrm{\forall}n\ge {N}_{2},
Let N=max\{{N}_{1},{N}_{2}\}, \mathrm{\forall}n\ge N, by (16), (17), and (12), we get the inequality (11):
□
Next, we give sufficient conditions for Gwp and wp of (VEP).
Theorem 4.1

(1)
For all \lambda \in \mathrm{\Lambda}, the problems λ is Gwp.

(2)
For all \lambda \in \mathrm{\Lambda}, if E(\lambda )=\{x\} (a singleton), then the problem λ is wp.
Proof (1) \mathrm{\forall}{\lambda}_{n}\in \mathrm{\Lambda}, {\lambda}_{n}\to \lambda, \mathrm{\forall}{x}_{n}\in E({\lambda}_{n},{\u03f5}_{n}), {\u03f5}_{n}>0 with {\u03f5}_{n}\to 0, then
and
From (18), there exists {u}_{n}\in f({\lambda}_{n}) such that d({u}_{n},{x}_{n})\to 0 as n\to \mathrm{\infty}. It follows by Lemma 4.2 and Lemma 2.3 that there exists \{{u}_{{n}_{k}}\}\subset \{{u}_{n}\} such that {u}_{{n}_{k}}\to x\in f(\lambda ). By
we get
By Lemma 4.3 and (19), we have
That is,
By (20) and (21), we have x\in E(\lambda ). It shows that λ is Gwp.

(2)
By way of contradiction. If the sequence \{{x}_{n}\} does not converge x, then there exists an open neighborhood O at x and a subsequence \{{x}_{{n}_{k}}\} of \{{x}_{n}\} such that {x}_{{n}_{k}}\notin O. Since E(\lambda )=\{x\} (a singleton), by the proof of (1), we get {x}_{{n}_{k}}\to x. This is a contradiction to {x}_{{n}_{k}}\notin O. □
Similarly, by Lemmas 3.2 and 3.3, it is easy to check the following.
Theorem 4.2

(1)
For all \lambda \in \mathrm{\Lambda}, the problems λ must be GLPwp and GHwp.

(2)
For all \lambda \in \mathrm{\Lambda}, if E(\lambda )=\{x\} (a singleton), then the problem λ must be LPwp and Hwp.
Finally, we apply these results to (GEP). Let Y=\mathbb{R}, C=[0,+\mathrm{\infty}[, the problem space of (GEP) is defined as
For any {\lambda}_{1}=({\phi}_{1},{G}_{1}),{\lambda}_{2}=({\phi}_{2},{G}_{2})\in {\mathrm{\Lambda}}^{\prime}, define
It is easy to check that ({\mathrm{\Lambda}}^{\prime},\varrho ) is a complete metric space. Hence, we have the following.
Corollary 4.1

(1)
For all \lambda \in {\mathrm{\Lambda}}^{\prime}, the problem λ is Gwp.

(2)
For all \lambda \in {\mathrm{\Lambda}}^{\prime}, if E(\lambda )=\{x\} (a singleton), then the problem λ is wp.
Corollary 4.2

(1)
For all \lambda \in {\mathrm{\Lambda}}^{\prime}, the problem λ must be GLPwp and GHwp.

(2)
For all \lambda \in {\mathrm{\Lambda}}^{\prime}, if E(\lambda )=\{x\} (a singleton), then the problem λ must be LPwp and Hwp.
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Acknowledgements
This research was supported by NSFC (Grant Number: 11161008), the Guizhou Provincial Science and Technology Foundation (20132235), (20122289). The authors thank the anonymous referees for their constructive comments which help us to revise the paper.
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Deng, X., Xiang, S. Wellposed generalized vector equilibrium problems. J Inequal Appl 2014, 127 (2014). https://doi.org/10.1186/1029242X2014127
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DOI: https://doi.org/10.1186/1029242X2014127
Keywords
 wellposedness
 generalized vector equilibrium problems
 bounded rationality model
 nonlinear scalarization function