The uniqueness and well-posedness of vector equilibrium problems with a representation theorem for the solution set

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R E S E A R C H

Open Access

The uniqueness and well-posedness of vector

equilibrium problems with a representation

theorem for the solution set

Ding-Tao Peng

1*

_{, Jian Yu}

1

_{and Nai-Hua Xiu}

2

*_{Correspondence:}

[email protected] 1_{College of Science, Guizhou}

University, Guiyang, Guizhou 550025, China

Full list of author information is available at the end of the article

Abstract

This paper aims to present some uniqueness and well-posedness results for vector equilibrium problems (for short, VEPs). We ﬁrst construct a complete metric spaceM consisting of VEPs satisfying some conditions. Using the method of set-valued analysis, we prove that there exists a dense everywhere residual subsetQofMsuch that each VEP inQhas a unique solution. Moreover, we introduce and obtain the generalized Hadamard well-posedness and generic Hadamard well-posedness of VEPs by considering the perturbations of both vector-valued functions and feasible sets. As an application, we provide a representation theorem for the solution set to each VEP inM.

MSC: 49K40; 90C31; 46B40; 47H04

Keywords: vector equilibrium problem; set-valued mapping; dense everywhere

residual set; uniqueness; well-posedness

1 Introduction

The vector equilibrium problem (for short, VEP) is a natural generalization of the equilib-rium problem for the vector-valued function. It is well known that the vector equilibequilib-rium problem is a unified model of several fundamental mathematical problems, namely, the vector optimization problem, the vector variational inequality, the vector complementar-ity problem, the multiobjective game, the vector network equilibrium problemetc.Since the VEP was proposed at about  lots of peoples have made many contributions to this problem and hundreds of papers have been published; see,e.g., the collection [] and the monograph []. 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 [], 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 optimization-related problems such as optimization problems [, ], two-person

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zero-sum continuous games [], saddle point problems [], large crowding games []. Re-cently, some new results were obtained. Yuet al.[] obtained the generic uniqueness of equilibrium points for a class of equilibrium problems. The results in [] showed that most of the monotone equilibrium problems (in the sense of Baire category) have a unique equi-librium point and that each monotone equiequi-librium problem can be arbitrarily approached by a sequence of such equilibrium problems that each of them has a unique equilibrium point. Moreover, Penget al.[] 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 vector-valued 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 well-posedness is just one of the approaches to the stability. There have been several notions of well-posedness about optimization-related problems. We refer to [–] for more details. For well-posedness of equilibrium problems or vector equilibrium prob-lems, there are some results. Fanget al.[] investigated the well-posedness of equilibrium problems; Kimuraet al.[] studied the parametric well-posedness for vector equilibrium problems; Bianchiet al.[] introduced and studied two types of well-posedness for vec-tor equilibrium problems; Li and Li [] investigated the Levitin-Polyak well-posedness of vector equilibrium problems with variable domination structures; Salamon [] ana-lyzed the Hadamard well-posedness of parametric vector equilibrium problems; Penget al.[] investigated several types of Levitin-Polyak well-posedness of generalized vector equilibrium problems. Most of these works considered the perturbation of the parameters in the vector-valued 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 well-posedness results for VEPs. We consider both the perturbation of vector-valued functions and the perturbation of feasible sets. The paper is organized as follows. In Section , we recall some definitions and preliminaries. In Section , we investigate the uniqueness of solutions to VEPs. We first construct a complete metric spaceMconsisting of VEPs satisfying some conditions. Then we prove that most of the VEPs (in the sense of Baire category) inMhave a unique so-lution. In Section , the Hadamard well-posedness of VEPs is introduced and studied. The generalized Hadamard well-posedness and generic Hadamard well-posedness of VEPs are derived. In Section , applying the above results we provide an interesting representation theorem for the solution set of each VEP inM. Finally, we briefly conclude our results in Section .

2 Preliminaries

Throughout this section, let H be a Hausdorﬀ topological vector space and C be a nonempty closed, convex and pointed cone inHwithintC=∅, whereintCdenotes the topological interior ofC. We note thatintC+C⊂intC(see []).

LetXbe a nonempty set andφ:X×X→Hbe a vector-valued function. The so-called vector equilibrium problem (for short, VEP) [] is to ﬁndx∗∈Xsuch that

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We callx∗ a solution ofVEP(φ). IfH=R,C= (–∞, ], the VEP becomes the Ky Fan inequality [, ]. Similarly, ifH=R,C= [, +∞), the VEP becomes the equilibrium problem [].

Definition . (see []) Let X be a nonempty subset of a Hausdorff topological vec-tor spaceE andf :X→H be a vector-valued function.f is said to beC-upper semi-continuous at x∈X iff for any open neighborhoodV of  inH, there exists an open neighborhoodUofxinXsuch that, for anyx∈U,

fx∈f(x) +V–C or equivalently,f(x)∈fx+V+C;

f is said to beC-upper semi-continuous onXiﬀf isC-upper semi-continuous at each x∈X; andf is said toC-lower semi-continuous onXiﬀ –f isC-upper semi-continuous onX.

Deﬁnition . LetXbe a nonempty subset of a Hausdorﬀ topological vector spaceEand

φ:X×X→Hbe a vector-valued function.φis said to beC-strictly-quasi-monotone on X×Xiﬀ for anyx,y∈Xwithx=y,

φ(x,y) /∈–intC ⇒ φ(y,x)∈–C.

Example . LetE=R,X= [–, ]⊂E,H=R_and_C₌_R

+⊂H. Deﬁne

f(x) :=

(, ), –≤x< , (, ), ≤x≤; f(x) := (x,x), –≤x≤;

φ(x,y) :=|x|–|y|,|x|–|y|, –≤x,y≤.

One can easily check that f isC-upper semi-continuous on X but not C-lower

semi-continuous at x= ;f is bothC-upper semi-continuous andC-lower semi-continuous

onX;φis bothC-upper semi-continuous andC-lower semi-continuous onX×X;φis C-strictly-quasi-monotone onX×X; and thatx=±∈Xare the only two solutions to

VEP(φ).

To investigate the uniqueness of solutions to VEPs, we will use the way of set-valued analysis. So let us recall some deﬁnitions and lemmas about set-valued mappings; for more details see [].

Deﬁnition . Let X, M be two topological spaces. Denote by X _{the space of all}

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Lemma .(see [, ]) Let M be a Baire space,X be a metric space and S:M→X_be

a usco mapping,then there exists a dense everywhere residual subset Q of M such that S is lower semi-continuous at each x∈Q.

Remark . If there exists a dense everywhere residual subsetQofMsuch that, for each

u∈Q, a certain propertyPdepending onuholds, then we say that the propertyPis generic onM. SinceQis a second category set, we may say that the propertyPholds for most of the points (in the sense of Baire category) inM. The research on generic properties (including generic existence, generic uniqueness, generic stability, generic well-posedness and so on) has attracted much attention; see,e.g., [–, , , , , ] and the references therein.

Lemma .(see []) Let A and An(n= , , . . .)all be nonempty compact subsets of a

met-ric space X with An→A in the Hausdorﬀ distance topology,then the following statements

hold:

(i) +_n∞_=An∪Ais also nonempty compact subset ofX;

(ii) Ifxn∈An,xn→x,thenx∈A.

3 Uniqueness of solutions to VEPs

In the rest of this paper, letXbe a nonempty and closed subset of a complete metric space E, (H, · ) be a Banach space, andCbe a nonempty, closed, convex, and pointed cone in H withintC=∅. For any> , denote byB() :={z∈H:z ≤}andB◦() :={z∈H:

z<}. We emphasize that the open neighborhoodV in Deﬁnition . can be replaced byB◦() in the case thatHis a normed space.

Let a spaceMof VEPs be deﬁned by

M=

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

u= (φ,A) :

φ:X×X→HisC-upper semi-continuous onX×X;

φisC-strictly-quasi-monotone onX×X;

sup₍_x_,_y₎_∈_X_×_Xφ(x,y)< +∞;

Ais a nonempty compact subset ofX; and∃x∈Asuch thatφ(x,y) /∈–intC,∀y∈A

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

.

For anyu= (φ,A),u= (φ,A)∈M, deﬁne

ρ(u,u) = sup (x,y)∈X×X

φ(x,y) –φ(x,y)+h(A,A),

wherehis the Hausdorﬀ distance onX.

Lemma . (M,ρ)is a complete metric space.

Proof Clearly,ρ is a metric on M. We only need to show that (M,ρ) is complete. Let

{un= (φn,An)}be a Cauchy sequence ofM, then for any> , there exists a positive integer

N() such that

ρ(um,un) = sup (x,y)∈X×X

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Hence

sup (x,y)∈X×X

φm(x,y) –φn(x,y)< and h(Am,An) <, ∀m,n≥N().

SinceHis a Banach space, for anyx,y∈X, there existsφ(x,y)∈Hsuch thatlimm→∞φm(x,

y) =φ(x,y) and

sup (x,y)∈X×X

φn(x,y) –φ(x,y)≤, ∀n≥N(). (.)

SinceXis complete,K(X) is also complete, whereK(X) denotes the space of all nonempty compact subsets ofXand is endowed with the Hausdorﬀ distancehinduced by the metric onX. Consequently, byh(Am,An) <, there existsA∈K(X) such thatAn→A. Next, we

will proveu:= (φ,A)∈M.

(i) Fixn≥N(). SinceφnisC-upper semi-continuous onX×X, there exists a

neigh-borhoodU(x,y)⊂X×Xof (x,y) such that

φn

x,y∈φn(x,y) +B◦() –C, ∀

x,y∈U(x,y). (.)

Thus, by (.) and (.), for any (x,y)∈U(x,y), we have

φx,y∈φn

x,y+B()⊂φn(x,y) +B◦() –C⊂φ(x,y) +B◦() –C.

It follows thatφisC-upper semi-continuous onX×X.

(ii) For any x,y∈ X with x =y, suppose φ(x,y) /∈ –intC. Since intC is open and

lim_n_→∞φn(x,y) =φ(x,y), we have φn(x,y) /∈–intC when n is big enough. It follows

from the C-strictly-quasi-monotonicity of φn that φn(y,x) ∈ –C. Since C is closed

and limn→∞φn(y,x) =φ(y,x), we obtainφ(y,x)∈–C. Therefore, φ is

C-strictly-quasi-monotone onX×X.

(iii) For eachn≥N(), we have

sup (x,y)∈X×X

φn(x,y) –φ(x,y)≤ and sup (x,y)∈X×X

φn(x,y)< +∞.

Hencesup₍_x_,_y₎_∈_X_×_Xφ(x,y) ≤sup₍_x_,_y₎_∈_X_×_Xφn(x,y)+< +∞.

(iv) Sinceun= (φn,An)∈Mfor eachn= , , . . . , there existsxn∈Ansuch that

φn(xn,y) /∈–intC, ∀y∈An. (.)

SinceAandAn(n= , , . . .) are all compact andAn→A, by Lemma .(i), +∞

n=An∪Ais

also compact. Note that{xn} ⊂ +∞

n=An∪A. Without loss of generality, supposexn→x∗.

By Lemma .(ii),x∗∈A. We shall show thatx∗fulﬁlls

φx∗,y∈/–intC, ∀y∈A.

Assume, by contradiction, that there existsy∈Asuch thatφ(x∗,y)∈–intC. SinceintC

is open, there exists>  such that

φx∗,y

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Fory∈A, by virtue ofAn→A, there existyn∈An(n= , , . . .) such thatyn→y. By

(.), it follows fromyn∈Anthat

φn(xn,yn) /∈–intC. (.)

SinceφisC-upper semi-continuous onX×Xandxn→x∗,yn→y, there existsN≥

N() such that

φ(xn,yn) –φ

x∗,y

∈B◦() –C, ∀n≥N. (.)

From (.), we derive

φn(xn,yn) –φ(xn,yn)∈B(), ∀n≥N. (.)

By (.), (.), and (.), we obtain for alln≥N,

φn(xn,yn) = φ

x∗,y

+φn(xn,yn) –φ(xn,yn)

+φ(xn,yn) –φ

x∗,y

∈φx∗,y

+B() +B◦() –C

⊂φx∗,y

+B◦() –C

⊂–intC–C⊂–intC,

which contradicts (.). Hence φ(x∗,y) /∈–intCfor ally∈A. Thus we have shownu:= (φ,A)∈M. Consequently, the inequality (.) andh(An,A)→ imply thatlimn→∞ρ(un,

u) = . Therefore, (M,ρ) is a complete metric space.

Lemma . Let f :X→H be C-upper semi-continuous on X,then the set L:={x∈X: f(x) /∈–intC}is closed in X.

Proof Letxn∈Lwithxn→x∈X. We only need to provex∈L. Assume, by contradiction,

thatx∈/L, thenf(x)∈–intC. Note that –intCis open, there exists>  such thatf(x) + B◦()⊂–intC. Sincef :X→HisC-upper semi-continuous atxandxn→x, there exists

N>  such that, for anyn>N, we havef(xn)∈f(x) +B◦() –C⊂–intC–C⊂–intC. But

it follows fromxn∈Lthatf(xn) /∈–intC, which is a contradiction. The proof is complete.

For eachu= (φ,A)∈M, by the deﬁnition ofM,VEP(φ) must have at least one solution inA,i.e.,∃x∗∈Asuch thatφ(x∗,y) /∈–intCfor ally∈A. Denote byS(u) the set of all solutions toVEP(φ) inA. Then the correspondenceu→S(u) yields a set-valued mapping S:M→X_.

Lemma . S:M→X_{is a usco mapping.}

Proof For eachu= (φ,A)∈M, note that

S(u) =x∈A:φ(x,y) /∈–intC,∀y∈A=

y∈A

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Sinceφ isC-upper semi-continuous onX×X, it is alsoC-upper semi-continuous on A×A. Moreover,x→φ(x,y) is alsoC-upper semi-continuous onA. By Lemma ., for eachy∈A, the set{x∈A:φ(x,y) /∈–intC}is closed inA. ThusS(u) is closed inA. Fur-thermore,S(u) is compact sinceAis compact.

Next, we will prove thatSis upper semi-continuous onM. We assume, by contradiction, that there existsu= (φ,A)∈Msuch thatSis not upper semi-continuous atu, then there exists an open neighborhoodGinXwithG⊃S(u) such that, for eachn= , , . . . and each open neighborhoodUn={u= (φ,A)∈M:ρ(u,u) < _n}ofu, there existun= (φn,An)∈

Unandxn∈S(un) butxn∈/G.

Sinceun= (φn,An)∈Unfor eachn= , , . . . , we haveρ(un,u) <_n→. Then

φn→φ and An→A. (.)

It follows fromxn∈S(un) thatxn∈Anand

φn(xn,y) /∈–intC, ∀y∈An. (.)

By Lemma .(i),+_n_=∞An∪Ais compact due to the compactness ofAnandA. Note that

{xn}+n=∞⊂ +∞

n=An∪A. Without loss of generality, we suppose that{xn}+n=∞is convergent.

Moreover, by Lemma .(ii), the limitx∗of{xn}n+=∞belongs toA,i.e.,xn→x∗∈A.

Mean-while,xn∈/GandGis open, thusx∗∈/G. SinceS(u)⊂G, we havex∗∈/S(u). Consequently,

there existsy∈Asuch thatφ(x∗,y)∈–intC. Note that –intCis open; then there exists

>  such that

φx∗,y+B◦()⊂–intC. (.)

SinceAn→Aandy∈A, there exists a sequence{yn}+n=∞ such thatyn∈Anandyn→y.

Sinceφn→φ, there existsN>  such that, for anyn≥N,

φn(xn,yn) –φ(xn,yn)∈B◦



. (.)

Moreover,φisC-upper semi-continuous onX×Xas well asxn→x∗,yn→y, hence there

existsN>Nsuch that, for anyn≥N,

φ(xn,yn) –φ

x∗,y∈B◦



–C. (.)

By (.)-(.), we have for anyn≥N, φn(xn,yn) =φ

x∗,y+φn(xn,yn) –φ(xn,yn)

+φ(xn,yn) –φ

x∗,y

∈φx∗,y+B◦



+B◦



–C

=φx∗,y+B◦() –C⊂–intC–C⊂–intC,

which is in contradiction with (.). Therefore,Smust be upper semi-continuous onM.

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Theorem . There exists a dense everywhere residual subset Q of M such that S(u)is a singleton for each u= (φ,A)∈Q,that is,VEP(φ)has a unique solution in A.

Proof By Lemma .,Mis a complete metric space, so it is a Baire space. SinceS:M→X

is a usco mapping (Lemma .) andXis a metric subspace, by Lemma ., there exists a dense everywhere residual subset QofMsuch thatSis lower semi-continuous at each u= (φ,A)∈Q.

Assume, by contradiction, thatS(u) is not a singleton for someu= (φ,A)∈Q. Then

there exist at least two pointsx,x∈S(u)⊂Awithx=x. Consequently, there exist

two open subsetsUandVinXsuch thatx∈U,x∈VandU∩V=∅.

Deﬁne a functiong:X→Ras follows:

g(x) = d(x,x) d(x,x) +d(x,X\U)

, ∀x∈X,

wheredis the metric onX. Note thatg is continuous onX; ≤g(x)≤ for allx∈X; g(x) =  if and only ifx=x;g(x) =  for allx∈V.

Takez∈–intC. For eachn= , , . . . , letφn:X×X→Hbe deﬁned by

φn(x,y) =φ(x,y) +

 ng(x)

z, ∀x,y∈X.

Furthermore, deﬁne

un= (φn,A).

For eachn= , , . . . , we will proveun∈M.

(i) It is easy to check thatφnisC-upper semi-continuous onX×X;

(ii) For any x,y∈X with x=y, suppose φn(x,y) /∈–intC. Then we can claim that

φ(x,y) /∈–intC. Otherwiseφ(x,y)∈–intC. Note that [_ng(x)]z∈–C, then

φn(x,y) =φ(x,y) +

 ng(x)

z∈–intC–C⊂–intC,

which is a contradiction. By theC-strictly-quasi-monotonicity ofφandφ(x,y) /∈–intC,

we getφ(y,x)∈–C. Hence

φn(y,x) =φ(y,x) +

 ng(y)

z∈–C–C⊂–C.

That is,φnisC-strictly-quasi-monotone onX×X.

(iii)sup₍_x_,_y₎_∈_X_×_Xφn(x,y) ≤sup(x,y)∈X×Xφ(x,y)+_n· z< +∞.

(iv) Fromx∈S(u) andg(x) = , we derive

φn(x,y) =φ(x,y) +

 ng(x)

z=φ(x,y) /∈–intC, ∀y∈A,

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Thus we have shownun∈Mfor eachn= , , . . . . Consequently,ρ(un,u)≤_n· z →

asn→ ∞.

Note thatx∈V∩S(u), thenV∩S(u)=∅. SinceSis lower semi-continuous atuand

un→u, there exists a positive integern>  big enough such thatV∩S(un)=∅. Take

xn∈V∩S(un), then we havexn∈V∩A,g(xn) =  andφn(xn,y) /∈–intCfor any

y∈A. Takey=x(∈A), then we get

φ(xn,x) +

 n

z=φ(xn,x) +

 n

g(xn)

z=φn(xn,x) /∈–intC. (.)

Note that _nz∈–intC. Ifφ(xn,x)∈–C, thenφn(xn,x)∈–C–intC⊂–intC, which

contradicts (.). Hence we have

φ(xn,x) /∈–C. (.)

Since x∈S(u), we have φ(x,y) /∈–intC for any y∈A. Taking y=xn (∈A), we

get φ(x,xn) /∈ –intC. It follows from the C-strictly-quasi-monotonicity of φ that φ(xn,x)∈–C, which is in contradiction with (.). Therefore,S(u) must be a singleton

for eachu∈Q.

WhenH=R,C= [, +∞), we get Corollary . as follows. Corollary . Let

M=

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

u= (f,A) :

f:X×X→Ris upper semi-continuous on X×X; f is pseudo-monotone on X×X;

sup₍_x_,_y₎_∈_X_×_X|f(x,y)|< +∞;

A is a nonempty compact subset of X; and∃x∈A such that f(x,y)≥,∀y∈A

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

,

where f is called pseudo-monotone(see[])on X×X iﬀ for any x,y∈X with x=y, f(x,y)≥ ⇒ f(y,x)≤.

Then there exists a dense everywhere residual subset Qof Msuch that,for each u= (f,A)∈ Q,f has a unique equilibrium point in A.

Remark . Corollary . generalized Theorem . of [], one of main results of [], as regards the following four aspects:

(i) we do not require the convexity of functionf ∈M;

(ii) we do not require the convexity and linear structure of the setX; (iii) we omit the requirement thatf(x,x)≥for allx∈X;

(iv) we replace the monotonicity off by pseudo-monotonicity which is weaker than the former.

4 Well-posedness of VEPs

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consider Tykhonov well-posedness of a problem, one introduces the notion of ‘approxi-mating sequence’ for the solution and requires some convergence of such sequences to a solution of the problem; while, Hadamard well-posedness of a problem means the con-tinuous dependence of the solutions on the data or the parameter of the problem; as for Levitin-Polyak well-posedness, we mean the convergence of the approximating solution sequence to a solution of the problem with some constraints; for more details to see [– ]. In this section, we will investigate the Hadamard well-posedness of VEPs.

Deﬁnition . Let u ∈M. () The VEP associated with u is said to be generalized Hadamard well-posed iﬀ for anyun∈Mand anyxn∈S(un),un→uimplies that{xn}has

a subsequence converging to an element ofS(u); () The VEP associated withuis said to be Hadamard well-posed iﬀS(u) ={x}(a singleton) and for anyun∈Mand anyxn∈S(un),

un→uimplies that{xn}converges tox.

Theorem . For each u= (φ,A)∈M,the VEP associated with u is generalized Hadamard well-posed.

Proof Letun= (φn,An)∈M,xn∈S(un), andun→u. Note thatAn→Abecauseun→u.

According to Lemma .(i), it follows from the compactness ofAnandAthat +∞

n=An∪A

is compact. Sincexn∈S(un)⊂+n=∞An∪A, there exists a convergent subsequence{xnk}of

{xn}. Moreover, by Lemma .(ii), the limit pointx∗of{xnk}belongs toA,i.e.,xnk→x∗∈A. By Lemma .,Sis upper semi-continuous atuandS(u) is compact.

Ifx∗∈/S(u), then there exists an open setOinXsuch thatO⊃S(u) andx∗∈ ¯/O. SinceSis upper semi-continuous atuandun→u, there is a positive integerNsuch thatO⊃S(un)

for alln≥N. Fromxnk ∈S(unk)⊂Ofor allnk≥N andxnk →x

∗_{, it follows that}_x∗_{∈ ¯}_O, which is a contradiction. Hencex∗∈S(u). The proof is complete. Theorem . There exists a dense everywhere residual subset Q of M such that,for each u= (φ,A)∈Q,the VEP associated with u is Hadamard well-posed,that is,VEPs in M are generic Hadamard well-posed.

Proof By Theorem ., there exists a dense everywhere residual subsetQofMsuch that, for eachu∈Q,S(u) is a singleton.

Letu= (φ,A)∈QandS(u) ={x}. Supposeun∈M,xn∈S(un), andun→u. We shall

provexn→x. If it is not true, then there exist an open neighborhoodOofxand a

sub-sequence{xnk} such thatxnk ∈/ O. By Theorem ., the VEP associateduis generalized Hadamard well-posed. SinceS(u) ={x},{xnk}has a subsequence converging tox, which

contradictsxnk∈/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 solu-tion, to provide an interesting representation of the solution set of each VEP inM, which in forms is very similar with the Clarke subdiﬀerentials of the local Lipschitz functions.

Denote by

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By Theorem .,P=∅. It is clear thatPis the largest dense everywhere residual subset (ordered by the set inclusion) ofMsuch that, for eachu∈P,S(u) is a singleton and the VEP associated withuis Hadamard well-posed.

Theorem . For each u= (φ,A)∈M, S(u) =

lim

n xn:xn∈S(un),un∈P,un→u

. (.)

Proof The right-hand side of (.) means that we only consider such sequences{xn}and

{un}satisfying that{un} ⊂P;{un}converges tou;xnis the unique point ofS(un); and that

{xn}is convergent.

SincePis dense inMandu= (φ,A)∈M, there exists{un= (φn,An)} ⊂Psuch that{un}

converges tou. By the deﬁnition ofP,S(un) has a unique point, denoted byxn. Note that

{xn} ⊂

+∞

n=An∪A, as well asAnandAare compact andAn→A, due to Lemma .,

{xn}or its subsequence converges to a point ofA. Hence the right-hand side set of (.) is

well-deﬁned and nonempty.

First, supposeun∈P,xn∈S(un),un→uandxn→x. By Theorem ., the VEP

asso-ciated withuis generalized Hadamard well-posed. It follows fromxn→xthatx∈S(u).

HenceS(u)⊃ {limnxn:xn∈S(un),un∈P,un→u}.

Next, letx∗∈S(u). Deﬁne a functiong:X→Ras follows:

g(x) = d(x,x ∗₎

 +d(x,x∗), ∀x∈X,

where dis the metric onX. Note thatg is continuous onX; ≤g(x) <  for allx∈X; g(x) =  if and only ifx=x∗.

Takez∈–intC. For eachn= , , . . . , deﬁne

φn(x,y) :=φ(x,y) +

 ng(x)

z, ∀x,y∈X and un:= (φn,A).

Same as in the proof of Theorem ., one can check thatun∈Mandx∗∈S(un) for each

n= , , . . . , and thatun→uasn→ ∞. Moreover, we shall show thatS(un) is a singleton,

and henceS(un) ={x∗}for eachn= , , . . . . By way of contradiction, assume thatS(un) is

not a singleton for someun= (φn,A). Then there existsx∈S(un)⊂Awithx=x∗. It

follows fromx∗∈S(u) thatφ(x∗,x) /∈–intC. By theC-strictly-quasi-monotonicity ofφ, we getφ(x,x∗)∈–C. Note thatg(x) >  and [_n g(x)]z∈–intC. Then

φn

x,x∗:=φx,x∗+

 n

gxz∈–C–intC⊂–intC.

But it follows from x∈S(un) that φn(x,x∗) /∈–intC, which is a contradiction. Thus

S(un) ={x∗}for eachn= , , . . . .

Takexn=x∗ for eachn= , , . . . , then we haveun∈P,un→u, xn∈S(un), andx∗= limnxn. From the arbitrariness ofx∗∈S(u), we derive

S(u)⊂

lim

n xn:xn∈S(un),un∈P,un→u

.

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6 Conclusions

In this paper, we considered a class of vector equilibrium problems. By considering the perturbations of vector-valued functions and feasible sets, we proved that each of the problems is generalized Hadamard well-posed, and that in the sense of Baire category, most of the problems have unique solution and are Hadamard well-posed. As an appli-cation, an interesting representation theorem for the solution set to each of the problems was provided.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the ﬁnal manuscript.

Author details

1_{College of Science, Guizhou University, Guiyang, Guizhou 550025, China.}2_{School of Science, Beijing Jiaotong University,}

Beijing, 100044, China.

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.

Received: 1 December 2013 Accepted: 22 April 2014 Published:09 May 2014 References

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