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The uniqueness and wellposedness of vector equilibrium problems with a representation theorem for the solution set
Fixed Point Theory and Applicationsvolume 2014, Article number: 115 (2014)
Abstract
This paper aims to present some uniqueness and wellposedness results for vector equilibrium problems (for short, VEPs). We first construct a complete metric space M consisting of VEPs satisfying some conditions. Using the method of setvalued analysis, we prove that there exists a dense everywhere residual subset Q of M such that each VEP in Q has a unique solution. Moreover, we introduce and obtain the generalized Hadamard wellposedness and generic Hadamard wellposedness of VEPs by considering the perturbations of both vectorvalued functions and feasible sets. As an application, we provide a representation theorem for the solution set to each VEP in M.
MSC:49K40, 90C31, 46B40, 47H04.
1 Introduction
The vector equilibrium problem (for short, VEP) is a natural generalization of the equilibrium problem for the vectorvalued function. It is well known that the vector equilibrium problem is a unified model of several fundamental mathematical problems, namely, the vector optimization problem, the vector variational inequality, the vector complementarity problem, the multiobjective game, the vector network equilibrium problem etc. Since the VEP was proposed at about 1997 lots of peoples have made many contributions to this problem and hundreds of papers have been published; see, e.g., the collection [1] and the monograph [2]. However, works on the uniqueness of solutions to VEPs were hardly seen. The only work we can find about the uniqueness of solutions to VEPs is [3], in which Khanh and Tung established sufficient conditions for the local uniqueness of solutions to VEPs by using approximations as generalized derivatives under the assumption that the functions have first and second Fréchet derivatives. The reason why results on uniqueness are so few is due to the fact that except for a few types of mathematical problems, most of the mathematical problems cannot guarantee the uniqueness of the solution. Therefore, to consider the generic uniqueness of the solutions may be more suitable, which will answer the question how many problems there are in a class of problems having a unique solution. The generic uniqueness of solutions to VEPs is our main motivation in this paper.
As we know, several works have been achieved about the generic uniqueness of solutions to some optimizationrelated problems such as optimization problems [4, 5], twoperson zerosum continuous games [6], saddle point problems [7], large crowding games [8]. Recently, some new results were obtained. Yu et al. [9] obtained the generic uniqueness of equilibrium points for a class of equilibrium problems. The results in [9] showed that most of the monotone equilibrium problems (in the sense of Baire category) have a unique equilibrium point and that each monotone equilibrium problem can be arbitrarily approached by a sequence of such equilibrium problems that each of them has a unique equilibrium point. Moreover, Peng et al. [10] provided a unified approach to the generic uniqueness and applied it to several nonlinear problems. However, the study of the generic uniqueness of solutions to VEPs has an essential difficulty: that the values of different vectorvalued functions are incomparable. To overcome such a difficulty is one of the main tasks in this paper.
The stability of solutions to nonlinear problems is also an important topic. The notion of wellposedness is just one of the approaches to the stability. There have been several notions of wellposedness about optimizationrelated problems. We refer to [11–15] for more details. For wellposedness of equilibrium problems or vector equilibrium problems, there are some results. Fang et al. [16] investigated the wellposedness of equilibrium problems; Kimura et al. [17] studied the parametric wellposedness for vector equilibrium problems; Bianchi et al. [18] introduced and studied two types of wellposedness for vector equilibrium problems; Li and Li [19] investigated the LevitinPolyak wellposedness of vector equilibrium problems with variable domination structures; Salamon [20] analyzed the Hadamard wellposedness of parametric vector equilibrium problems; Peng et al. [21] investigated several types of LevitinPolyak wellposedness of generalized vector equilibrium problems. Most of these works considered the perturbation of the parameters in the vectorvalued functions. Different from these works, we will not only consider the perturbation of objective functions but also consider the perturbation of feasible sets.
This paper aims to present some generic uniqueness and wellposedness results for VEPs. We consider both the perturbation of vectorvalued functions and the perturbation of feasible sets. The paper is organized as follows. In Section 2, we recall some definitions and preliminaries. In Section 3, we investigate the uniqueness of solutions to VEPs. We first construct a complete metric space M consisting of VEPs satisfying some conditions. Then we prove that most of the VEPs (in the sense of Baire category) in M have a unique solution. In Section 4, the Hadamard wellposedness of VEPs is introduced and studied. The generalized Hadamard wellposedness and generic Hadamard wellposedness of VEPs are derived. In Section 5, applying the above results we provide an interesting representation theorem for the solution set of each VEP in M. Finally, we briefly conclude our results in Section 6.
2 Preliminaries
Throughout this section, let H be a Hausdorff topological vector space and C be a nonempty closed, convex and pointed cone in H with $intC\ne \mathrm{\varnothing}$, where intC denotes the topological interior of C. We note that $intC+C\subset intC$ (see [22]).
Let X be a nonempty set and $\varphi :X\times X\to H$ be a vectorvalued function. The socalled vector equilibrium problem (for short, VEP) [2] is to find ${x}^{\ast}\in X$ such that
We call ${x}^{\ast}$ a solution of $VEP(\varphi )$. If $H=\mathbb{R}$, $C=(\mathrm{\infty},0]$, the VEP becomes the Ky Fan inequality [23, 24]. Similarly, if $H=\mathbb{R}$, $C=[0,+\mathrm{\infty})$, the VEP becomes the equilibrium problem [25].
Definition 2.1 (see [2])
Let X be a nonempty subset of a Hausdorff topological vector space E and $f:X\to H$ be a vectorvalued function. f is said to be Cupper semicontinuous at $x\in X$ iff for any open neighborhood V of 0 in H, there exists an open neighborhood U of x in X such that, for any ${x}^{\prime}\in U$,
f is said to be Cupper semicontinuous on X iff f is Cupper semicontinuous at each $x\in X$; and f is said to Clower semicontinuous on X iff −f is Cupper semicontinuous on X.
Definition 2.2 Let X be a nonempty subset of a Hausdorff topological vector space E and $\varphi :X\times X\to H$ be a vectorvalued function. ϕ is said to be Cstrictlyquasimonotone on $X\times X$ iff for any $x,y\in X$ with $x\ne y$,
Example 2.3 Let $E=\mathbb{R}$, $X=[1,1]\subset E$, $H={\mathbb{R}}^{2}$ and $C={\mathbb{R}}_{+}^{2}\subset H$. Define
One can easily check that ${f}_{1}$ is Cupper semicontinuous on X but not Clower semicontinuous at $x=0$; ${f}_{2}$ is both Cupper semicontinuous and Clower semicontinuous on X; ϕ is both Cupper semicontinuous and Clower semicontinuous on $X\times X$; ϕ is Cstrictlyquasimonotone on $X\times X$; and that $x=\pm 1\in X$ are the only two solutions to $VEP(\varphi )$.
To investigate the uniqueness of solutions to VEPs, we will use the way of setvalued analysis. So let us recall some definitions and lemmas about setvalued mappings; for more details see [26].
Definition 2.4 Let X, M be two topological spaces. Denote by ${2}^{X}$ the space of all nonempty subsets of X. Let $S:M\to {2}^{X}$ be a setvalued mapping. Then (i) S is said to be upper (respectively, lower) semicontinuous at $u\in M$ iff for each open set G in X with $G\supset S(u)$ (respectively, $G\cap S(u)\ne \mathrm{\varnothing}$), there exists an open neighborhood O of u such that $G\supset S({u}^{\prime})$ (respectively, $G\cap S({u}^{\prime})\ne \mathrm{\varnothing}$) for each ${u}^{\prime}\in O$; (ii) S is said to be continuous at $u\in M$ iff S is both upper semicontinuous and lower semicontinuous at u; (iii) S is said to be a usco mapping iff S is upper semicontinuous on M and $S(u)$ is compact for each $u\in M$; (iv) A subset Q of M is called residual iff it contains the intersection of countably many dense everywhere open subsets of M.
Let M be a Baire space, X be a metric space and $S:M\to {2}^{X}$ be a usco mapping, then there exists a dense everywhere residual subset Q of M such that S is lower semicontinuous at each $x\in Q$.
Remark 2.6 If there exists a dense everywhere residual subset Q of M such that, for each $u\in Q$, a certain property P depending on u holds, then we say that the property P is generic on M. Since Q is a second category set, we may say that the property P holds for most of the points (in the sense of Baire category) in M. The research on generic properties (including generic existence, generic uniqueness, generic stability, generic wellposedness and so on) has attracted much attention; see, e.g., [4–7, 9, 22, 24, 28, 29] and the references therein.
Lemma 2.7 (see [28])
Let A and ${A}_{n}$ ($n=1,2,\dots $) all be nonempty compact subsets of a metric space X with ${A}_{n}\to A$ in the Hausdorff distance topology, then the following statements hold:

(i)
${\bigcup}_{n=1}^{+\mathrm{\infty}}{A}_{n}\cup A$ is also nonempty compact subset of X;

(ii)
If ${x}_{n}\in {A}_{n}$, ${x}_{n}\to x$, then $x\in A$.
3 Uniqueness of solutions to VEPs
In the rest of this paper, let X be a nonempty and closed subset of a complete metric space E, $(H,\parallel \cdot \parallel )$ be a Banach space, and C be a nonempty, closed, convex, and pointed cone in H with $intC\ne \mathrm{\varnothing}$. For any $\u03f5>0$, denote by $B(\u03f5):=\{z\in H:\parallel z\parallel \le \u03f5\}$ and ${B}^{\circ}(\u03f5):=\{z\in H:\parallel z\parallel <\u03f5\}$. We emphasize that the open neighborhood V in Definition 2.1 can be replaced by ${B}^{\circ}(\u03f5)$ in the case that H is a normed space.
Let a space M of VEPs be defined by
For any ${u}_{1}=({\varphi}_{1},{A}_{1}),{u}_{2}=({\varphi}_{2},{A}_{2})\in M$, define
where h is the Hausdorff distance on X.
Lemma 3.1 $(M,\rho )$ is a complete metric space.
Proof Clearly, ρ is a metric on M. We only need to show that $(M,\rho )$ is complete. Let $\{{u}_{n}=({\varphi}_{n},{A}_{n})\}$ be a Cauchy sequence of M, then for any $\u03f5>0$, there exists a positive integer $N(\u03f5)$ such that
Hence
Since H is a Banach space, for any $x,y\in X$, there exists $\varphi (x,y)\in H$ such that ${lim}_{m\to \mathrm{\infty}}{\varphi}_{m}(x,y)=\varphi (x,y)$ and
Since X is complete, $K(X)$ is also complete, where $K(X)$ denotes the space of all nonempty compact subsets of X and is endowed with the Hausdorff distance h induced by the metric on X. Consequently, by $h({A}_{m},{A}_{n})<\u03f5$, there exists $A\in K(X)$ such that ${A}_{n}\to A$. Next, we will prove $u:=(\varphi ,A)\in M$.

(i)
Fix $n\ge N(\u03f5)$. Since ${\varphi}_{n}$ is Cupper semicontinuous on $X\times X$, there exists a neighborhood $U(x,y)\subset X\times X$ of $(x,y)$ such that
$${\varphi}_{n}({x}^{\prime},{y}^{\prime})\in {\varphi}_{n}(x,y)+{B}^{\circ}(\u03f5)C,\phantom{\rule{1em}{0ex}}\mathrm{\forall}({x}^{\prime},{y}^{\prime})\in U(x,y).$$(3.2)
Thus, by (3.1) and (3.2), for any $({x}^{\prime},{y}^{\prime})\in U(x,y)$, we have
It follows that ϕ is Cupper semicontinuous on $X\times X$.
(ii) For any $x,y\in X$ with $x\ne y$, suppose $\varphi (x,y)\notin intC$. Since intC is open and ${lim}_{n\to \mathrm{\infty}}{\varphi}_{n}(x,y)=\varphi (x,y)$, we have ${\varphi}_{n}(x,y)\notin intC$ when n is big enough. It follows from the Cstrictlyquasimonotonicity of ${\varphi}_{n}$ that ${\varphi}_{n}(y,x)\in C$. Since C is closed and ${lim}_{n\to \mathrm{\infty}}{\varphi}_{n}(y,x)=\varphi (y,x)$, we obtain $\varphi (y,x)\in C$. Therefore, ϕ is Cstrictlyquasimonotone on $X\times X$.
(iii) For each $n\ge N(\u03f5)$, we have
Hence ${sup}_{(x,y)\in X\times X}\parallel \varphi (x,y)\parallel \le {sup}_{(x,y)\in X\times X}\parallel {\varphi}_{n}(x,y)\parallel +\u03f5<+\mathrm{\infty}$.

(iv)
Since ${u}_{n}=({\varphi}_{n},{A}_{n})\in M$ for each $n=1,2,\dots $ , there exists ${x}_{n}\in {A}_{n}$ such that
$${\varphi}_{n}({x}_{n},y)\notin intC,\phantom{\rule{1em}{0ex}}\mathrm{\forall}y\in {A}_{n}.$$(3.3)
Since A and ${A}_{n}$ ($n=1,2,\dots $) are all compact and ${A}_{n}\to A$, by Lemma 2.7(i), ${\bigcup}_{n=1}^{+\mathrm{\infty}}{A}_{n}\cup A$ is also compact. Note that $\{{x}_{n}\}\subset {\bigcup}_{n=1}^{+\mathrm{\infty}}{A}_{n}\cup A$. Without loss of generality, suppose ${x}_{n}\to {x}^{\ast}$. By Lemma 2.7(ii), ${x}^{\ast}\in A$. We shall show that ${x}^{\ast}$ fulfills
Assume, by contradiction, that there exists ${y}_{0}\in A$ such that $\varphi ({x}^{\ast},{y}_{0})\in intC$. Since intC is open, there exists ${\u03f5}_{0}>0$ such that
For ${y}_{0}\in A$, by virtue of ${A}_{n}\to A$, there exist ${y}_{n}\in {A}_{n}$ ($n=1,2,\dots $) such that ${y}_{n}\to {y}_{0}$. By (3.3), it follows from ${y}_{n}\in {A}_{n}$ that
Since ϕ is Cupper semicontinuous on $X\times X$ and ${x}_{n}\to {x}^{\ast}$, ${y}_{n}\to {y}_{0}$, there exists ${N}_{1}\ge N({\u03f5}_{0})$ such that
From (3.1), we derive
By (3.4), (3.6), and (3.7), we obtain for all $n\ge {N}_{1}$,
which contradicts (3.5). Hence $\varphi ({x}^{\ast},y)\notin intC$ for all $y\in A$. Thus we have shown $u:=(\varphi ,A)\in M$. Consequently, the inequality (3.1) and $h({A}_{n},A)\to 0$ imply that ${lim}_{n\to \mathrm{\infty}}\rho ({u}_{n},u)=0$. Therefore, $(M,\rho )$ is a complete metric space. □
Lemma 3.2 Let $f:X\to H$ be Cupper semicontinuous on X, then the set $L:=\{x\in X:f(x)\notin intC\}$ is closed in X.
Proof Let ${x}_{n}\in L$ with ${x}_{n}\to x\in X$. We only need to prove $x\in L$. Assume, by contradiction, that $x\notin L$, then $f(x)\in intC$. Note that $intC$ is open, there exists $\u03f5>0$ such that $f(x)+{B}^{\circ}(\u03f5)\subset intC$. Since $f:X\to H$ is Cupper semicontinuous at x and ${x}_{n}\to x$, there exists $N>0$ such that, for any $n>N$, we have $f({x}_{n})\in f(x)+{B}^{\circ}(\u03f5)C\subset intCC\subset intC$. But it follows from ${x}_{n}\in L$ that $f({x}_{n})\notin intC$, which is a contradiction. The proof is complete. □
For each $u=(\varphi ,A)\in M$, by the definition of M, $VEP(\varphi )$ must have at least one solution in A, i.e., $\mathrm{\exists}{x}^{\ast}\in A$ such that $\varphi ({x}^{\ast},y)\notin intC$ for all $y\in A$. Denote by $S(u)$ the set of all solutions to $VEP(\varphi )$ in A. Then the correspondence $u\to S(u)$ yields a setvalued mapping $S:M\to {2}^{X}$.
Lemma 3.3 $S:M\to {2}^{X}$ is a usco mapping.
Proof For each $u=(\varphi ,A)\in M$, note that
Since ϕ is Cupper semicontinuous on $X\times X$, it is also Cupper semicontinuous on $A\times A$. Moreover, $x\to \varphi (x,y)$ is also Cupper semicontinuous on A. By Lemma 3.2, for each $y\in A$, the set $\{x\in A:\varphi (x,y)\notin intC\}$ is closed in A. Thus $S(u)$ is closed in A. Furthermore, $S(u)$ is compact since A is compact.
Next, we will prove that S is upper semicontinuous on M. We assume, by contradiction, that there exists $u=(\varphi ,A)\in M$ such that S is not upper semicontinuous at u, then there exists an open neighborhood G in X with $G\supset S(u)$ such that, for each $n=1,2,\dots $ and each open neighborhood ${U}_{n}=\{{u}^{\prime}=({\varphi}^{\prime},{A}^{\prime})\in M:\rho ({u}^{\prime},u)<\frac{1}{n}\}$ of u, there exist ${u}_{n}=({\varphi}_{n},{A}_{n})\in {U}_{n}$ and ${x}_{n}\in S({u}_{n})$ but ${x}_{n}\notin G$.
Since ${u}_{n}=({\varphi}_{n},{A}_{n})\in {U}_{n}$ for each $n=1,2,\dots $ , we have $\rho ({u}_{n},u)<\frac{1}{n}\to 0$. Then
It follows from ${x}_{n}\in S({u}_{n})$ that ${x}_{n}\in {A}_{n}$ and
By Lemma 2.7(i), ${\bigcup}_{n=1}^{+\mathrm{\infty}}{A}_{n}\cup A$ is compact due to the compactness of ${A}_{n}$ and A. Note that ${\{{x}_{n}\}}_{n=1}^{+\mathrm{\infty}}\subset {\bigcup}_{n=1}^{+\mathrm{\infty}}{A}_{n}\cup A$. Without loss of generality, we suppose that ${\{{x}_{n}\}}_{n=1}^{+\mathrm{\infty}}$ is convergent. Moreover, by Lemma 2.7(ii), the limit ${x}^{\ast}$ of ${\{{x}_{n}\}}_{n=1}^{+\mathrm{\infty}}$ belongs to A, i.e., ${x}_{n}\to {x}^{\ast}\in A$. Meanwhile, ${x}_{n}\notin G$ and G is open, thus ${x}^{\ast}\notin G$. Since $S(u)\subset G$, we have ${x}^{\ast}\notin S(u)$. Consequently, there exists $y\in A$ such that $\varphi ({x}^{\ast},y)\in intC$. Note that $intC$ is open; then there exists $\u03f5>0$ such that
Since ${A}_{n}\to A$ and $y\in A$, there exists a sequence ${\{{y}_{n}\}}_{n=1}^{+\mathrm{\infty}}$ such that ${y}_{n}\in {A}_{n}$ and ${y}_{n}\to y$. Since ${\varphi}_{n}\to \varphi $, there exists ${N}_{1}>0$ such that, for any $n\ge {N}_{1}$,
Moreover, ϕ is Cupper semicontinuous on $X\times X$ as well as ${x}_{n}\to {x}^{\ast}$, ${y}_{n}\to y$, hence there exists ${N}_{2}>{N}_{1}$ such that, for any $n\ge {N}_{2}$,
By (3.10)(3.12), we have for any $n\ge {N}_{2}$,
which is in contradiction with (3.9). Therefore, S must be upper semicontinuous on M. The proof is thus complete. □
Theorem 3.4 There exists a dense everywhere residual subset Q of M such that $S(u)$ is a singleton for each $u=(\varphi ,A)\in Q$, that is, $VEP(\varphi )$ has a unique solution in A.
Proof By Lemma 3.1, M is a complete metric space, so it is a Baire space. Since $S:M\to {2}^{X}$ is a usco mapping (Lemma 3.3) and X is a metric subspace, by Lemma 2.5, there exists a dense everywhere residual subset Q of M such that S is lower semicontinuous at each $u=(\varphi ,A)\in Q$.
Assume, by contradiction, that $S({u}_{0})$ is not a singleton for some ${u}_{0}=({\varphi}_{0},{A}_{0})\in Q$. Then there exist at least two points ${x}_{1},{x}_{2}\in S({u}_{0})\subset {A}_{0}$ with ${x}_{1}\ne {x}_{2}$. Consequently, there exist two open subsets U and V in X such that ${x}_{1}\in U$, ${x}_{2}\in V$ and $U\cap V=\mathrm{\varnothing}$.
Define a function $g:X\to \mathbb{R}$ as follows:
where d is the metric on X. Note that g is continuous on X; $0\le g(x)\le 1$ for all $x\in X$; $g(x)=0$ if and only if $x={x}_{1}$; $g(x)=1$ for all $x\in V$.
Take $z\in intC$. For each $n=1,2,\dots $ , let ${\varphi}_{n}:X\times X\to H$ be defined by
Furthermore, define
For each $n=1,2,\dots $ , we will prove ${u}_{n}\in M$.

(i)
It is easy to check that ${\varphi}_{n}$ is Cupper semicontinuous on $X\times X$;

(ii)
For any $x,y\in X$ with $x\ne y$, suppose ${\varphi}_{n}(x,y)\notin intC$. Then we can claim that ${\varphi}_{0}(x,y)\notin intC$. Otherwise ${\varphi}_{0}(x,y)\in intC$. Note that $[\frac{1}{n}g(x)]z\in C$, then
$${\varphi}_{n}(x,y)={\varphi}_{0}(x,y)+[\frac{1}{n}g(x)]z\in intCC\subset intC,$$
which is a contradiction. By the Cstrictlyquasimonotonicity of ${\varphi}_{0}$ and ${\varphi}_{0}(x,y)\notin intC$, we get ${\varphi}_{0}(y,x)\in C$. Hence
That is, ${\varphi}_{n}$ is Cstrictlyquasimonotone on $X\times X$.

(iii)
${sup}_{(x,y)\in X\times X}\parallel {\varphi}_{n}(x,y)\parallel \le {sup}_{(x,y)\in X\times X}\parallel {\varphi}_{0}(x,y)\parallel +\frac{1}{n}\cdot \parallel z\parallel <+\mathrm{\infty}$.

(iv)
From ${x}_{1}\in S({u}_{0})$ and $g({x}_{1})=0$, we derive
$${\varphi}_{n}({x}_{1},y)={\varphi}_{0}({x}_{1},y)+[\frac{1}{n}g({x}_{1})]z={\varphi}_{0}({x}_{1},y)\notin intC,\phantom{\rule{1em}{0ex}}\mathrm{\forall}y\in {A}_{0},$$
which implies ${x}_{1}\in S({u}_{n})\ne \mathrm{\varnothing}$.
Thus we have shown ${u}_{n}\in M$ for each $n=1,2,\dots $ . Consequently, $\rho ({u}_{n},{u}_{0})\le \frac{1}{n}\cdot \parallel z\parallel \to 0$ as $n\to \mathrm{\infty}$.
Note that ${x}_{2}\in V\cap S({u}_{0})$, then $V\cap S({u}_{0})\ne \mathrm{\varnothing}$. Since S is lower semicontinuous at ${u}_{0}$ and ${u}_{n}\to {u}_{0}$, there exists a positive integer ${n}_{0}>0$ big enough such that $V\cap S({u}_{{n}_{0}})\ne \mathrm{\varnothing}$. Take ${x}_{{n}_{0}}\in V\cap S({u}_{{n}_{0}})$, then we have ${x}_{{n}_{0}}\in V\cap {A}_{0}$, $g({x}_{{n}_{0}})=1$ and ${\varphi}_{{n}_{0}}({x}_{{n}_{0}},y)\notin intC$ for any $y\in {A}_{0}$. Take $y={x}_{1}$ ($\in {A}_{0}$), then we get
Note that $\frac{1}{{n}_{0}}z\in intC$. If ${\varphi}_{0}({x}_{{n}_{0}},{x}_{1})\in C$, then ${\varphi}_{{n}_{0}}({x}_{{n}_{0}},{x}_{1})\in CintC\subset intC$, which contradicts (3.13). Hence we have
Since ${x}_{1}\in S({u}_{0})$, we have ${\varphi}_{0}({x}_{1},y)\notin intC$ for any $y\in {A}_{0}$. Taking $y={x}_{{n}_{0}}$ ($\in {A}_{0}$), we get ${\varphi}_{0}({x}_{1},{x}_{{n}_{0}})\notin intC$. It follows from the Cstrictlyquasimonotonicity of ${\varphi}_{0}$ that ${\varphi}_{0}({x}_{{n}_{0}},{x}_{1})\in C$, which is in contradiction with (3.14). Therefore, $S(u)$ must be a singleton for each $u\in Q$. □
When $H=\mathbb{R}$, $C=[0,+\mathrm{\infty})$, we get Corollary 3.5 as follows.
Corollary 3.5 Let
where f is called pseudomonotone (see [30]) on $X\times X$ iff for any $x,y\in X$ with $x\ne y$,
Then there exists a dense everywhere residual subset ${Q}^{\prime}$ of ${M}^{\prime}$ such that, for each $u=(f,A)\in {Q}^{\prime}$, f has a unique equilibrium point in A.
Remark 3.6 Corollary 3.5 generalized Theorem 3.2 of [9], one of main results of [9], as regards the following four aspects:

(i)
we do not require the convexity of function $f\in {M}^{\prime}$;

(ii)
we do not require the convexity and linear structure of the set X;

(iii)
we omit the requirement that $f(x,x)\ge 0$ for all $x\in X$;

(iv)
we replace the monotonicity of f by pseudomonotonicity which is weaker than the former.
4 Wellposedness of VEPs
As is well known, the notions of wellposedness can be mainly divided into three groups, namely, Hadamard type, Tykhonov type and LevitinPolyak type. Generally speaking, to consider Tykhonov wellposedness of a problem, one introduces the notion of ‘approximating sequence’ for the solution and requires some convergence of such sequences to a solution of the problem; while, Hadamard wellposedness of a problem means the continuous dependence of the solutions on the data or the parameter of the problem; as for LevitinPolyak wellposedness, we mean the convergence of the approximating solution sequence to a solution of the problem with some constraints; for more details to see [11–15]. In this section, we will investigate the Hadamard wellposedness of VEPs.
Definition 4.1 Let $u\in M$. (1) The VEP associated with u is said to be generalized Hadamard wellposed iff for any ${u}_{n}\in M$ and any ${x}_{n}\in S({u}_{n})$, ${u}_{n}\to u$ implies that $\{{x}_{n}\}$ has a subsequence converging to an element of $S(u)$; (2) The VEP associated with u is said to be Hadamard wellposed iff $S(u)=\{x\}$ (a singleton) and for any ${u}_{n}\in M$ and any ${x}_{n}\in S({u}_{n})$, ${u}_{n}\to u$ implies that $\{{x}_{n}\}$ converges to x.
Theorem 4.2 For each $u=(\varphi ,A)\in M$, the VEP associated with u is generalized Hadamard wellposed.
Proof Let ${u}_{n}=({\varphi}_{n},{A}_{n})\in M$, ${x}_{n}\in S({u}_{n})$, and ${u}_{n}\to u$. Note that ${A}_{n}\to A$ because ${u}_{n}\to u$. According to Lemma 2.7(i), it follows from the compactness of ${A}_{n}$ and A that ${\bigcup}_{n=1}^{+\mathrm{\infty}}{A}_{n}\cup A$ is compact. Since ${x}_{n}\in S({u}_{n})\subset {\bigcup}_{n=1}^{+\mathrm{\infty}}{A}_{n}\cup A$, there exists a convergent subsequence $\{{x}_{{n}_{k}}\}$ of $\{{x}_{n}\}$. Moreover, by Lemma 2.7(ii), the limit point ${x}^{\ast}$ of $\{{x}_{{n}_{k}}\}$ belongs to A, i.e., ${x}_{{n}_{k}}\to {x}^{\ast}\in A$. By Lemma 3.3, S is upper semicontinuous at u and $S(u)$ is compact.
If ${x}^{\ast}\notin S(u)$, then there exists an open set O in X such that $O\supset S(u)$ and ${x}^{\ast}\notin \overline{O}$. Since S is upper semicontinuous at u and ${u}_{n}\to u$, there is a positive integer N such that $O\supset S({u}_{n})$ for all $n\ge N$. From ${x}_{{n}_{k}}\in S({u}_{{n}_{k}})\subset O$ for all ${n}_{k}\ge N$ and ${x}_{{n}_{k}}\to {x}^{\ast}$, it follows that ${x}^{\ast}\in \overline{O}$, which is a contradiction. Hence ${x}^{\ast}\in S(u)$. The proof is complete. □
Theorem 4.3 There exists a dense everywhere residual subset Q of M such that, for each $u=(\varphi ,A)\in Q$, the VEP associated with u is Hadamard wellposed, that is, VEPs in M are generic Hadamard wellposed.
Proof By Theorem 3.4, there exists a dense everywhere residual subset Q of M such that, for each $u\in Q$, $S(u)$ is a singleton.
Let $u=(\varphi ,A)\in Q$ and $S(u)=\{x\}$. Suppose ${u}_{n}\in M$, ${x}_{n}\in S({u}_{n})$, and ${u}_{n}\to u$. We shall prove ${x}_{n}\to x$. If it is not true, then there exist an open neighborhood O of x and a subsequence $\{{x}_{{n}_{k}}\}$ such that ${x}_{{n}_{k}}\notin O$. By Theorem 4.2, the VEP associated u is generalized Hadamard wellposed. Since $S(u)=\{x\}$, $\{{x}_{{n}_{k}}\}$ has a subsequence converging to x, which contradicts ${x}_{{n}_{k}}\notin O$. □
5 A representation theorem of the solution set to VEPs
In this section, we use the limits of the solutions to VEPs, each of which has a unique solution, to provide an interesting representation of the solution set of each VEP in M, which in forms is very similar with the Clarke subdifferentials of the local Lipschitz functions.
Denote by
By Theorem 3.4, $P\ne \mathrm{\varnothing}$. It is clear that P is the largest dense everywhere residual subset (ordered by the set inclusion) of M such that, for each $u\in P$, $S(u)$ is a singleton and the VEP associated with u is Hadamard wellposed.
Theorem 5.1 For each $u=(\varphi ,A)\in M$,
Proof The righthand side of (5.1) means that we only consider such sequences $\{{x}_{n}\}$ and $\{{u}_{n}\}$ satisfying that $\{{u}_{n}\}\subset P$; $\{{u}_{n}\}$ converges to u; ${x}_{n}$ is the unique point of $S({u}_{n})$; and that $\{{x}_{n}\}$ is convergent.
Since P is dense in M and $u=(\varphi ,A)\in M$, there exists $\{{u}_{n}=({\varphi}_{n},{A}_{n})\}\subset P$ such that $\{{u}_{n}\}$ converges to u. By the definition of P, $S({u}_{n})$ has a unique point, denoted by ${x}_{n}$. Note that $\{{x}_{n}\}\subset {\bigcup}_{n=1}^{+\mathrm{\infty}}{A}_{n}\cup A$, as well as ${A}_{n}$ and A are compact and ${A}_{n}\to A$, due to Lemma 2.7, $\{{x}_{n}\}$ or its subsequence converges to a point of A. Hence the righthand side set of (5.1) is welldefined and nonempty.
First, suppose ${u}_{n}\in P$, ${x}_{n}\in S({u}_{n})$, ${u}_{n}\to u$ and ${x}_{n}\to x$. By Theorem 4.2, the VEP associated with u is generalized Hadamard wellposed. It follows from ${x}_{n}\to x$ that $x\in S(u)$. Hence $S(u)\supset \{{lim}_{n}{x}_{n}:{x}_{n}\in S({u}_{n}),{u}_{n}\in P,{u}_{n}\to u\}$.
Next, let ${x}^{\ast}\in S(u)$. Define a function $g:X\to \mathbb{R}$ as follows:
where d is the metric on X. Note that g is continuous on X; $0\le g(x)<1$ for all $x\in X$; $g(x)=0$ if and only if $x={x}^{\ast}$.
Take $z\in intC$. For each $n=1,2,\dots $ , define
Same as in the proof of Theorem 3.4, one can check that ${u}_{n}\in M$ and ${x}^{\ast}\in S({u}_{n})$ for each $n=1,2,\dots $ , and that ${u}_{n}\to u$ as $n\to \mathrm{\infty}$. Moreover, we shall show that $S({u}_{n})$ is a singleton, and hence $S({u}_{n})=\{{x}^{\ast}\}$ for each $n=1,2,\dots $ . By way of contradiction, assume that $S({u}_{{n}_{0}})$ is not a singleton for some ${u}_{{n}_{0}}=({\varphi}_{{n}_{0}},A)$. Then there exists ${x}^{\prime}\in S({u}_{{n}_{0}})\subset A$ with ${x}^{\prime}\ne {x}^{\ast}$. It follows from ${x}^{\ast}\in S(u)$ that $\varphi ({x}^{\ast},{x}^{\prime})\notin intC$. By the Cstrictlyquasimonotonicity of ϕ, we get $\varphi ({x}^{\prime},{x}^{\ast})\in C$. Note that $g({x}^{\prime})>0$ and $[\frac{1}{{n}_{0}}g({x}^{\prime})]z\in intC$. Then
But it follows from ${x}^{\prime}\in S({u}_{{n}_{0}})$ that ${\varphi}_{{n}_{0}}({x}^{\prime},{x}^{\ast})\notin intC$, which is a contradiction. Thus $S({u}_{n})=\{{x}^{\ast}\}$ for each $n=1,2,\dots $ .
Take ${x}_{n}={x}^{\ast}$ for each $n=1,2,\dots $ , then we have ${u}_{n}\in P$, ${u}_{n}\to u$, ${x}_{n}\in S({u}_{n})$, and ${x}^{\ast}={lim}_{n}{x}_{n}$. From the arbitrariness of ${x}^{\ast}\in S(u)$, we derive
Combining the above two parts, we get the conclusion. □
6 Conclusions
In this paper, we considered a class of vector equilibrium problems. By considering the perturbations of vectorvalued functions and feasible sets, we proved that each of the problems is generalized Hadamard wellposed, and that in the sense of Baire category, most of the problems have unique solution and are Hadamard wellposed. As an application, an interesting representation theorem for the solution set to each of the problems was provided.
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Acknowledgements
The first author was supported by the NSFC (11271098), the Science and Technology Foundation of Guizhou Province (20102133), and the Scientific Research Projects for the Introduced Talents of Guizhou University (201343). The third author was supported by the National Basic Research Program of China (2010CB732501) and the NSFC (71271021). The authors, therefore, acknowledge these supports.
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Keywords
 vector equilibrium problem
 setvalued mapping
 dense everywhere residual set
 uniqueness
 wellposedness