Lower semicontinuity of approximate solution mappings for parametric generalized vector equilibrium problems

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R E S E A R C H

Open Access

Lower semicontinuity of approximate

solution mappings for parametric generalized

vector equilibrium problems

Rabian Wangkeeree

1*

_{, Panatda Boonman}

1

_{and Pakkapon Preechasilp}

2

*_{Correspondence:} [email protected]

1_{Department of Mathematics,}

Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand

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

Abstract

In this paper, we obtain suﬃcient conditions for the lower semicontinuity of an approximate solution mapping for a parametric generalized vector equilibrium problem involving set-valued mappings. By using a scalarization method, we obtain the lower semicontinuity of an approximate solution mapping for such a problem without the assumptions of monotonicity and compactness.

Keywords: lower semicontinuity; approximate solution mapping; parametric generalized vector equilibrium problems; scalarization method

1 Introduction

The vector equilibrium problem is a uniﬁed model of several problems, for example, the vector optimization problem, the vector variational inequality problem, the vector com-plementarity problem and the vector saddle point problem. In the literature, existence re-sults for various types of vector equilibrium problems have been investigated intensively,

e.g., see [–] and the references therein. The stability analysis of the solution mappings for VEP is an important topic in vector equilibrium theory. Recently, the semicontinuity, especially the lower semicontinuity, of solution mappings to parametric vector equilib-rium problems has been studied in the literature, see [–]. In the mentioned results, the lower semicontinuity of solution mappings to parametric generalized strong vector equi-librium problems is established under the assumptions of monotonicity and compactness. Very recently, Han and Gong [] studied the lower semicontinuity of solution mappings to parametric generalized strong vector equilibrium problems without the assumptions of monotonicity and compactness.

On the other hand, exact solutions of the problems may not exist in many practical prob-lems because the data of the probprob-lems are not sufficiently ‘regular’. Moreover, these math-ematical models are solved usually by numerical methods which produce approximations to the exact solutions. So it is impossible to obtain an exact solution of many practical problems. Naturally, investigating approximate solutions of parametric equilibrium prob-lems is of interest in both practical applications and computations. Anh and Khanh [] considered two kinds of approximate solution mappings to parametric generalized vector quasiequilibrium problems and established the sufficient conditions for their Hausdorff semicontinuity (or Berge semicontinuity). Among many approaches for dealing with the

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lower semicontinuity and continuity of solution mappings for parametric vector varia-tional inequalities and parametric vector equilibrium problems, the scalarization method is of considerable interest. By using a scalarization method, Li and Li [] discussed the Berge lower semicontinuity and Berge continuity of an approximate solution mapping for a parametric vector equilibrium problem.

Motivated by the work reported in [–], in this paper we aim to establish eﬃcient conditions for the lower semicontinuity of an approximate solution mapping for a para-metric generalized vector equilibrium problem involving set-valued mappings. By using a scalarization method, we obtain the lower semicontinuity of an approximate solution mapping for such a problem without the assumptions of monotonicity and compactness.

2 Preliminaries

Throughout this paper, letXandYbe real Hausdorﬀ topological vector spaces, and letZ

be a real topological space. We also assume thatCis a pointed closed convex cone inYwith its interiorintC=∅. LetY∗be the topological dual space ofY. LetC∗:={ξ∈Y∗:ξ,y ≥

,∀y∈C}be the dual cone ofC, whereξ,ydenotes the value ofξaty. SinceintC=∅, the dual coneC∗ofChas a weak∗compact base. Lete∈intC. ThenB_e∗:={ξ∈C∗:ξ,e= } is a weak∗compact base ofC∗.

Suppose thatKis a nonempty subset ofXandF:K×K→Y_\{∅}_{is a set-valued}

map-ping. We consider the following generalized vector equilibrium problem (GVEP) of ﬁnd-ingx∈Ksuch that

F(x,y)⊂Y\–intC, ∀y∈K. (.)

When the setKand the mappingFare perturbed by a parameterμwhich varies over a setMofZ, we consider the following parametric generalized vector equilibrium problem (PGVEP) of ﬁndingx∈K(μ) such that

F(x,y,μ)⊂Y\–intC, ∀y∈K(μ), (.) whereK:M→X_\{∅}_{is a set-valued mapping,}_F_:_B_×_B_×_M_⊂_X_×_X_×_Z_→_Y_\{∅}

is a set-valued mapping withK(M) =μ∈MK(μ)⊂B. For each ε>  and μ∈M, the

approximate solution set of (PGVEP) is deﬁned by

S(ε,μ) :=x∈K(μ) :F(x,y,μ) +εe⊂Y\–intC,∀y∈K(μ),

where e∈intC. For eachξ ∈B∗_e and (ε,μ)∈R+_×_M_{, by}_S

ξ(ε,μ) we denote the ξ -ap-proximate solution set of (PGVEP),i.e.,

Sξ(ε,μ) :=

x∈K(μ) : inf

z∈F(x,y,μ)ξ(z) +ε≥,∀y∈K(μ)

.

Deﬁnition . LetDbe a nonempty convex subset ofX. A set-valued mappingG:X→

Y _{is said to be:}

(i) C-convexonDif, for anyx,x∈Dand for anyt∈[, ], we have

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(ii) C-concaveonDif, for anyx,x∈Dand for anyt∈[, ], we have

Gtx+ ( –t)x ⊆tG(x) + ( –t)G(x) +C.

Deﬁnition .[] LetMandM be topological vector spaces. LetDbe a nonempty

subset ofM. A set-valued mappingG:M→M_{is said to be}_{uniformly continuous}_on_D

if, for any neighborhoodVof ∈M, there exists a neighborhoodUof ∈Msuch that

G(x)⊆G(x) +Vfor anyx,x∈Dwithx–x∈U.

Deﬁnition .[] LetMandM be topological vector spaces. A set-valued mapping

G:M→M_{is said to be:}

(i) Hausdorﬀ upper semicontinuous(H-u.s.c.) atu∈Mif, for any neighborhoodV of ∈M, there exists a neighborhoodU(u)ofusuch that

G(u)⊆G(u) +V for everyu∈U(u).

(ii) Lower semicontinuous(l.s.c.) atu∈Mif, for anyx∈G(u)and any neighborhood

Vofx, there exists a neighborhoodU(u)ofusuch that

G(u)∩V=∅ for everyu∈U(u).

The following lemma plays an important role in the proof of the lower semicontinuity of the solution mappingS(·,·).

Lemma .[, Theorem ] The union=_i_∈_Iiof a family of l.s.c.set-valued mappings

ifrom a topological space X into a topological space Y is also an l.s.c.set-valued mapping from X into Y,where I is an index set.

3 Lower semicontinuity of the approximate solution mapping for (PGVEP) In this section, we establish the lower semicontinuity of the approximate solution mapping for (PGVEP) at the considered point (ε,μ)∈R+_×_M_with_ε

> .

Firstly, using the same argument as in the proof given in [, Lemma .], we can prove the following useful result.

Lemma . For eachε> ,μ∈M,if for each x∈K(μ),F(x,K(μ),μ) +C is a convex set,

then

S(ε,μ) = ξ∈C∗\{}

Sξ(ε,μ) =

ξ∈B∗e

Sξ(ε,μ).

Proof For anyx∈_ξ_∈_C∗_\{}Sξ(ε,μ), there existsξ∈C∗\{}such thatx∈Sξ(ε,μ). Thus, we can obtain thatx∈K(μ) andinfz∈F(x,y,μ)ξ(z) +ε≥,∀y∈K(μ). Then, for eachy∈

K(μ) andz∈F(x,y,μ),ξ(z) +ε≥, which arrives atz∈/ –intC. It then follows that, for eachz∈F(x,y,μ),

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which gives thatx∈S(ε,μ). Hence,ξ∈C∗\{}Sξ(ε,μ)⊆S(ε,μ). Conversely, letx∈S(ε,μ) be arbitrary. Thenx∈K(μ) andF(x,y,μ) +εe⊆Y\–intC,∀y∈K(μ). Thus, we have

Fx,K(μ),μ ∩(–intC) =∅, and hence

Fx,K(μ),μ +C ∩(–intC) =∅.

BecauseF(x,K(μ),μ) +Cis a convex set, by the well-known Edidelheit separation theorem (see [], Theorem .), there exist a continuous linear functionalξ∈Y∗\{}and a real numberγ such that

ξ(ˆc) <γ≤ξ(z+c)

for allz∈F(x,K(μ),μ),c∈Candcˆ∈–intC. SinceCis a cone, we haveξ(cˆ)≤ for all

ˆ

c∈–intC. Thus,ξ(ˆc)≥ for allcˆ∈C, that is,ξ∈C∗. Moreover, it follows fromc∈C,

ˆ

c∈–intCand the continuity ofξ thatξ(z) +ε≥ for allz∈F(x,K(μ),μ). Thus, for all

y∈K(μ), we haveinfz∈F(x,y,μ)ξ(z) +ε≥,i.e.,x∈Sξ(ε,μ)⊆

ξ∈C∗\{}Sξ(ε,μ).

Theorem . We assume that for any givenξ∈B∗_e,there existsδ> such that theξ -ap-proximate solution setSξ(·,·)exists in[ε,δ)×N(μ),where N(μ)is a neighborhood ofμ.

Assume further that the following conditions are satisﬁed: (i) K(μ)is nonempty convex;

(ii) Kis H-u.s.c.atμand l.s.c.atμ;

(iii) for anyy∈K(μ),F(·,y,μ)isC-concave onK(μ); (iv) F(·,·,·)is uniformly continuous onK(M)×K(M)×N(μ).

Then theξ-approximate solution mappingSξ : [ε,δ)×N(μ)→Xis l.s.c.at(ε,μ).

Proof Suppose to the contrary thatSξ(·,·) is not l.s.c. at (ε,μ), then there exist x∈

Sξ(ε,μ) and a neighborhoodWof X∈X. For any neighborhoodsJ(ε) andU(μ) of

εandμ, respectively, there existε∈J(ε)∩[ε,δ) andμ∈U(μ) such that (x+W)∩

Sξ(ε,μ) =∅. In particular, there exist sequences{εn} ↓εand{μn} →μsuch that

(x+W)∩Sξ(εn,μn) =∅, ∀n∈N. (.)

For the aboveW, there exists a neighborhoodWof X∈Xsuch that

W+W⊆W. (.)

We deﬁne aξ-set-valued mappingHξ: [,δ)→Xby

Hξ(ε) =

x∈K(μ) : inf

z∈F(x,y,μ)

ξ(z) +ε+ε≥,∀y∈K(μ)

, ε∈[,δ).

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X∈X. For any neighborhoodUof , there existsε∈Usuch that (x¯+O)∩Hξ(ε) =∅. In particular, there exists a nonnegative sequence{ε_n} ↓ such that

(x¯+O)∩Hξ

ε_n =∅, ∀n∈N. (.)

SinceHξ()=∅, we choosex∗∈Hξ(). Sinceεn →, there existsεnsuch that

ε ε+εn

¯

x+ ε

n

ε+εn

x∗=x¯+ ε

n

ε+εn

x∗–x¯ ∈ ¯x+O. (.)

We claim that ε

ε+εnx¯+

ε_n



ε+εnx

∗_∈_H

ξ(ε_n_). In fact, sincex¯∈Hξ() andx∗∈Hξ(), for any

y∈K(μ), we haveinft∈F(¯x,y,μ)ξ(t) +ε≥ andinfk∈F(x∗,y,μ)ξ(k) +ε≥. Then, for any

u∈F(x¯,y,μ), ε ε+εn

ξ(u) + ε ε+εn

ε≥, (.)

and for anyv∈F(x∗,y,μ), ε_n_

ε+εn

ξ(v) + ε

n

ε+εn

ε≥. (.)

By theC-concavity ofF(·,y,μ), we have that

F

ε ε+εn

¯

x+ ε

n

ε+εn

x∗,y,μ

⊆ ε ε+εn

F(x¯,y,μ) + ε_n_ ε+εn

Fx∗,y,μ +C.

It follows that, for any w∈F( ε

ε+εn_x¯ + εn

ε+εn_x

∗_,_y_,_μ_{), there exist} _z_¯_∈_F₍_x_¯_,_y_,_μ_), _z∗ _∈

F(x∗,y,μ) andc∈Csuch thatw= ε

ε+εn_z¯+ ε_n



ε+εn_z

∗₊_c_{. It follows from the linearity}

ofξ thatξ(w) – ε

ε+εn

ξ(z¯) – εn

ε+εn

ξ(z∗) =ξ(c)≥, which gives thatξ(w)≥ ε

ε+εn

ξ(¯z) + ε_n



ε+εn

ξ(z∗). For allw∈F( ε

ε+εnx¯+

ε_n



ε+εnx

∗_,_y_,_μ_{), by (.) and (.), we have}

ξ(w)≥– ε ε+εn

ε– ε_n_ ε+εn

ε= – ε ε+εn

ε_n

+ε ≥–

ε_n

+ε .

This implies thatinf z∈F( ε

ε+εn_x¯+

εn

ε+εn_x

∗_,_y_,μ)

ξ(z) +ε_n

+ε≥, that is,

ε

ε+εnx¯+

ε_n



ε+εnx

∗_∈

Hξ(ε_n_). By (.), we get that _ε ε

+εnx¯

+ εn

ε+εnx

∗ _∈ ₍_x_¯ ₊_O

)∩Hξ(ε_n_), which contra-dicts (.). Therefore,Hξ is l.s.c. at . SinceHξ is l.s.c. at , for abovex∈Sξ(ε,μ) =

Hξ() and for aboveW, there exists a balanced neighborhoodV of  such that (x+

W)∩Hξ(ε)=∅, ∀ε∈V. In particular, from {εn} ↓ε, there exits N∈N such that (x+W)∩Hξ(εN–ε)=∅. Letx∈(x+W)∩Hξ(εN–ε).

For anyε¯> , sincee∈intC, there existsδ>  such that

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SinceF(·,·,·) is uniformly continuous onK(M)×K(M)×N(μ), for aboveδBY, there

exists a neighborhoodVof ∈B, a neighborhoodUof ∈Band a neighborhoodNof ∈M, for any (x,y,μ), (x,y,μ)∈K(M)×K(M)×N(μ) withx–x∈V,y–y∈U andμ–μ∈N, we have

F(x,y,μ)⊆δBY+F(x,y,μ). (.)

SinceKis H-u.s.c. atμ, for aboveU, there exists a neighborhoodU(μ) ofμsuch that

K(μ)⊆K(μ) +U, ∀μ∈U(μ). (.)

We see thatx∈K(μ). SinceKis l.s.c. atμ, forV∩W, there exists a neighborhood

U(μ) ofμsuch that

x+V∩W ∩K(μ)=∅, ∀μ∈U(μ). (.) It follows from μn→μ that there exists a positive integerN ≥Nsuch thatμN_ ∈ U(μ)∩U(μ)∩U(μ)∩(μ+N). Noting that (.) and (.), we obtain

K(μ_N

)⊆K(μ) +U (.)

and

x+V∩W ∩K(μN_)=∅. (.)

By (.), we choose

x∈x+V∩W ∩K(μN_). (.)

Next, we prove thatx∈Sξ(εN_,μN_). For anyy∈K(μN_), by (.), there existsy∈K(μ) such thaty–y∈U. It follows from (.) thatx–x∈V. Noting thatμN_ ∈U(μ)∩

(μ+N) and (.), we have

Fx,y,μ_N

 ⊆δBY+F

x,y,μ .

By (.), we have

Fx,y,μ_N

 ⊆C–ε¯e+F

x,y,μ . (.)

Hence, for anyy∈K(μ_N

) andz

_∈_F₍_x_,_y_,_μ

N_), there existc∈Candz∈F(x,y,μ)

such that

z=c–ε¯e+z.

It follows from the linearity ofξthatξ(z)+ε¯≥ξ(z) for allε¯> . This leads toξ(z)≥ξ(z). Thus

ξz +ε_N

≥ξ

z +ε_N

=ξ

z + (ε_N

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Hencex∈Sξ(εN_,μN_). Also, sincex∈(x+W) and by (.) and (.), we have

x∈x+V∩W⊆x+W+W⊆x+W.

This means that (x+W)∩Sξ(εN_,μN_)=∅, which contradicts (.). This completes the

proof.

Theorem . We assume that for any givenξ ∈B∗_e,there existsδ> such that the ap-proximate solution setSξ(·,·)exists in[ε,δ)×N(μ).Suppose that conditions(i)-(iv)as

in Theorem.are satisﬁed.Assume further that for each x∈K(μ),F(x,K(μ),μ) +C is a convex set.Then the approximate solution mappingS: [ε,δ)×N(μ)→Xis l.s.c.at (ε,μ).

Proof SinceF(x,K(μ),μ) +Cis a convex set for eachx∈K(μ), by virtue of Lemma ., it holds thatS(ε,μ) =ξ∈B∗eSξ(ε,μ). It follows from Theorem . that for eachξ ∈

B∗_e,Sξ(·,·) is l.s.c. at (ε,μ). Thus, in view of Lemma ., we obtain thatS(·,·) is l.s.c. at

(ε,μ).

The following example illustrates all of the assumptions in Theorem ..

Example . LetY =R, C=R₊:={(x,x)∈R :x≥,x≥} andZ=X=R. Let

B(,_) be the closed ball of radius / inR_{. Let}_B_{= [–, ],}_M_{= [–, ] and the set-valued} mappingF:B×B×M→Y_{be deﬁned by}

F(x,y,μ) =w(x,y,μ),v(x,y,μ) +B(, /),

wherew(x,y,μ) :=y_(μ_{– ) +}_x₍_y_–_x_{+ ) – }_y_{+  and}_v₍_x_,_y_,_μ_{) :=}_y_(μ_{– ) –}_x_{+ }_xy_{+ .} Deﬁne a set-valued mappingK:M→X_{for all}_μ_∈_M_{, by}_K₍_μ_{) := [– +}_μ_{,  +}_μ_]_∩_{[–, ].}

We choose e= (, )∈intC, ε= ., μ=  and ξ = (, ). We can see that B∗(,)=

{(x,x) :x+x= ,x,x≥}and ∈S(,)(ε, ). Further, for anyμ∈(–, ), there ex-istsε∈[., .) such that ∈S(,)(ε,μ). Hence,S(,)(·,·) exists in [., .)×[–, ]. It is easy to observe that for anyy∈K(),F(·,y, ) isC-concave onK(). Clearly, condi-tion (ii) is true. It is obvious thatK(M) = [–, ]. LetN(μ) = [–, ], we can see thatF(·,·,·) is uniformly continuous onK(M)×K(M)×N(μ). Finally, we can check that for each

x∈[–, ],F(x, [–, ], ) +Cis a convex set. Applying Theorem ., we obtain thatSis l.s.c. at (., ).

The following example illustrates that the concavity ofFcannot be dropped.

Example . LetY=R_,_C₌_R

+andZ=X=R. LetB= [–, ],M= [–, ] and the set-valued mappingF:B×B×M→Y _{be deﬁned by}

F(x,y,μ) =μx(x–y) – ., ×x(x–y) – ..

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except (iii). Indeed, takingy= ,x= ,x=  andt= ., we have

(–., –.) = (–., –.) – .(, –.) – .(, –.)

∈[–., ]× {–.}– .[–., ]× {–.} – .[–., ]× {–.}

∈F.() + .(), ,  – .F(, , ) – .F(, , ) =F(., , ) – .F(, , ) – .F(, , ),

but (–., –.) /∈C. The direct computation shows that

˜

S(ε,μ) =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

{, } ifμ∈(, ], [, ] ifμ= ,

{} ifμ∈[–, ).

(.)

Clearly, we see thatS˜(·,·) is even not l.s.c. at (ε,μ) sinceF(·,y,μ) is notC-concave on

K(μ).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand.}2_{Program in}

Mathematics, Faculty of Education, Pibulsongkram Rajabhat University, Phitsanulok, 65000, Thailand.

Acknowledgements

The authors were partially supported by the Thailand Research Fund, Grant No. PHD/0078/2554 and Grant

No. RSA5780003. The authors would like to thank the referees for their remarks and suggestions, which helped to prove the paper.

Received: 3 July 2014 Accepted: 7 October 2014 Published:21 Oct 2014

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