R E S E A R C H
Open Access
Well-posedness for parametric generalized
vector quasivariational inequality problems
of the Minty type
Nguyen Van Hung
**Correspondence:
[email protected] Department of Mathematics, Dong Thap University, 783 Pham Huu Lau Street, Ward 6, Cao Lanh City, Vietnam
Abstract
In this paper, we introduce the concepts of well-posedness, and of well-posedness in the generalized sense for parametric generalized vector quasivariational inequality problems of the Minty type. The necessary and sufficient conditions for the various kinds of well-posedness of these problems are obtained. Our results are different from some main results in the literature and extend them.
MSC: 90C31; 49J53; 49J40; 49J45
Keywords: parametric generalized vector quasivariational inequality problems of the Minty type; well-posed; well-posedness in the generalized sense; upper semicontinuity
1 Introduction and preliminaries
A vector variational inequality in a finite-dimensional Euclidean space was introduced first by Giannessi []. Later, this problem has been extended and studied by many authors in abstract spaces; see [–]. Moreover, vector variational inequality problems have many important applications in vector optimization problems [–], vector equilibria problems [, ], and variational relation problems [, ].
The concept of well-posedness for unconstrained scalar optimization problems was first introduced and studied by Tykhonov [], which has become known as Tykhonov well-posedness. In , Levitin and Polyak [] introduced the concept of well-posedness for constrained scalar optimization problems. With the development of the theory about opti-mization problems, the concept of well-posedness has been generalized to several related problems, as vector optimization problems, see [–], variational inequality problems, see [, –], equilibria problems, see [–] and the references therein. Recently, Fang and Huang [] studied the well-posedness for a vector variational inequality of the Minty type and the Stampacchia type. Very recently, Lalitha and Bhatia [] also studied a quasi-variational inequality problem of the Minty type, and the well-posedness for this problem was obtained.
Motivated and inspired by the work mentioned, in this paper, we also study the paramet-ric generalized vector quasivariational inequality problems. However, we only study the well-posedness for generalized vector quasivariational inequality problems of the Minty type. The well-posedness for generalized vector quasivariational inequality problems of the Stampacchia type is the same as the Minty type. LetX,Y,,be metric spaces and
C⊂Y be a closed, convex, and pointed cone withintC=∅. The coneCinduces a partial ordering inYdefined by
y<x ⇔ y–x∈–intC, ∀x,y∈Y,
y≮x ⇔ y–x∈/–intC, ∀x,y∈Y,
whereintCdenotes the interior ofC.
LetL(X,Y) be the space of all linear continuous operators fromXintoY, andA⊂X
be a nonempty subset. LetK:A×→A,K:A×→A, andT:A×→L(X,Y)be
set-valued mappings. LetQ:L(X,Y)→L(X,Y),η:A×A×→Abe continuous single-valued mappings. We denote by z,xthe value of a linear operatorz∈L(X;Y) atx∈X, and we always assume that ·,·is continuous.
Now we adopt the following notations (see [, , ]). For subsetsMandN under consideration we adopt the notations
(u,v) wM×N means∀u∈M,∃v∈N,
(u,v) mM×N means∃v∈N,∀u∈M,
(u,v) sM×N means∀u∈M,∀v∈N,
(u,v)w M¯ ×N means∃u∈M,∀v∈Nand similarly form¯,¯s.
where w, m, and s are used for weak, middle, and strong, respectively, kinds of considered problems. Letα∈ {w, m, s},α¯∈ { ¯w,m¯,¯s}, and, forγ ∈,λ∈. We consider the following parametric generalized vector quasivariational inequality problems of the Minty type (in short: (MQVIPγ λ)).
(MQVIPγ λ) Findx¯∈K(x¯,γ)such that(y,z)αK(x¯,γ)×T(y,γ)satisfies
Q(z),η(y,x¯,λ)≮.
Denote by (MQVIP) the family{(MQVIPγ λ) : (γ,λ)∈×}. For eachγ ∈,λ∈, and letE(γ) :={x∈A:x∈K(x,γ)}. We denote byα(γ,λ) the solution sets of (MQVIPγ λ).
Throughout the article, we assume thatα(γ,λ)=∅for each (γ,λ) in the neighborhoods (γ,λ)∈×.
Next, we recall some basic definitions and some of their properties.
Definition .([, ]) LetXandZbe two topological vector spaces and letG:X→Z
be a multifunction.
(i) Gis said to belower semicontinuous(lsc)atxifG(x)∩U=∅for each open set
U⊆Zimplies the existence of a neighborhoodVofxsuch thatG(x)∩U=∅,
∀x∈V.
(ii) Gis said to beupper semicontinuous(usc)atxif for each open setU⊇G(x), there is a neighborhoodVofxsuch thatU⊇G(x),∀x∈V.
(iii) Gis said to beclosed atxif for each net{(xn,yn)} ∈graphG:={(x,y)|y∈G(x)},
Lemma .([, ]) Let X and Z be two topological vector spaces and G:X→Zbe a multifunction.
(i) IfZis compact andGis closed atx,thenGis usc atx.
(ii) IfGis usc atxandG(x)is closed,thenGis closed atx.
The structure of this article is as follows. In the remaining part of this section, we re-call definitions for later use. In Section , we introduce concepts of well-posedness, and well-posedness in the generalized sense for parametric generalized vector quasivariational inequality problems of the Minty type. Moreover, the necessary and sufficient conditions for the various kinds of well-posedness of these problems are obtained.
2 Main results
Definition . Let{(γn,λn)} ⊆×converges to (γ,λ). A sequence{xn} ⊆Ais said to
be an approximating sequence for (MQVIP) corresponding to{(γn,λn)}, if
(i) xn∈K(xn,λn),∀n;
(ii) there exists a sequence{εn} ∈intCthat converges tosuch that
(y,z)αK(xn,γn)×T(y,γn) satisfies
Q(z),η(y,xn,λn)
+εn≮.
Definition . The problem (MQVIP) is said to be well-posed at (γ,λ) if
(i) the problem (MQVIP) has a unique solutionx,i.e.,α(γ,λ) ={x};
(ii) for any sequence{(γn,λn)} ⊆×converges to(γ,λ), every approximating
sequence{xn}for (MQVIP) corresponding to{(γn,λn)}converges tox.
Definition . The problem (MQVIP) is said to be well-posed in the generalized sense
at (γ,λ) if
(i) the solution setα(γ,λ)of (MQVIP) is nonempty;
(ii) for any sequence{(γn,λn)} ⊆×that converges to(γ,λ), every approximating
sequence{xn}for (MQVIP) corresponding to{(γn,λn)}has a subsequence which
converges to some point ofα(γ,λ).
Forγ∈,λ∈, andε∈intC, we denote the approximate solution set of (MQVIP) by (γ,λ,ε):
(γ,λ,ε) :=x∈K(x,γ)|(y,z)αK(x,γ)×T(y,γ) :Q(z),η(y,x,λ)+ε≮.
Remark .
(i) In the special case, whereA=B,X=Y,=,K(x,γ) =K(x,γ) =A, η(y,x,λ) =y–x, andQis an identity map, letT:A×→L(X,Y)be a single-valued mapping, then the problem (MQVIPγ λ) reduces to the problem (MVVIλ) studied in [].
(ii) In the special case as in Remark .(i), then Definitions ., ., and . reduce to Definitions ., ., and ., respectively, of Fang and Huang in [].
(iii) Well-posedness for vector problems has been defined in different ways. In this paper, we denoteε∈intCinstead ofe, withbeing positive numbers and
Remark .([]) LetXandZbe two metric spaces andG:X→Zbe a multifunction.
IfG(x) is compact, thenGis usc atxif and only if for any sequence{xn}that converges
toxand for any sequence{yn} ⊆G(xn), there is a subsequence{ynk}of{yn}converging to somey∈G(x). If, in addition,G(x) ={y}is a singleton, then the above limit pointy
must beyand the whole{yn}converges toy.
The following theorem gives sufficient conditions for the posedness and the well-posedness in the generalized sense for (MQVIP).
Theorem . Assume for problem(MQVIP)that
(i) Eis usc atγandE(γ)is a compact set;
(ii) inK(A,)× {γ},Kis lsc;
(iii) inK(K(A,),)× {γ},Tis usc and compact-valued ifα= w(orα= m),and lsc if
α= s.
Then(MQVIP)is well-posed in the generalized sense at(γ,λ).Moreover,ifα(γ,λ)is a singleton,then this problem is well-posed at(γ,λ).
Proof Sinceα={w, m, s}, we have in fact three cases. However, the proof techniques are similar. We consider only the caseα= s. We first prove thatsis upper semicontinuous
at (γ,λ, ). Indeed, we suppose to the contrary the existence of an open subset V of
s(γ,λ, ) such that for all{(γn,λn,εn)} ⊆××Cit converges to{(γ,λ, )}, that
is, xn∈s(γn,λn,εn), xn∈/ V, for alln. SinceE is usc and is compact-valued atγ, we
can assume thatxntends toxfor somex∈E(γ). Ifx∈/s(γ,λ, ) =(γ,λ),∃y∈ K(x,γ),∃z∈T(y,γ) such that
Q(z),η(y,x,λ)
< .
By the lower semicontinuity of K, T at (x,γ) and (y,γ), ∀y∈ K(x,γ), ∀z ∈ T(y,γ) there exists yn∈K(xn,γn), zn∈T(yn,γn) such that yn→y, zn→z. Since xn∈(γn,λn,εn), we have
Q(zn),η(yn,xn,λn)
+εn≮. (.)
Let id :C→Cbe an identity map, by the continuity ofη,Q, and ·,·, it follows that ·,·+id is continuous (where id is continuous). So (.) implies
Q(z),η(y,x,λ)
≮,
which is impossible. Hence,xbelongs tos(γ,λ, )⊆V, which is again a contradiction,
sincexn∈/V, for alln. Therefore,sis usc at (γ,λ, ).
Now we prove thats(γ,λ, ) is compact, by checking its closedness. Indeed, letxn∈
s(γ,λ, ),xn→x. This proof is similar to above and so we havex∈s(γ,λ, ) and
hences(γ,λ, ) is compact. By Remark ., we complete the proof.
Example . LetA=B=X=Y=R,== [, ],C=R+,γ= ,Qbe an identity map, K,K:A×→A,T:A×→L(X,Y), andη:A×A×→Abe defined by
K(x,γ) =
–γ – ,γ
,
η(y,x,γ) =γ+γ + +ε,
T(y,γ) =
γ+ ,
K(x,γ) =, γ+.
Then we haveE() = (–, ] andE(γ) = (–γ–,γ],∀γ ∈(, ]. We show that assumptions (ii) and (iii) of Theorem . are fulfilled. But the family{(MQVIPγ λ) : (γ,λ)∈×}is not well-posed in the generalized sense at (, ). The reason is thatEis not usc at and
E() is not compact. In fact
α(γ,λ,ε) = ⎧ ⎨ ⎩
(–, ], ifγ = ,
(–γ–
,γ], ifγ ∈(, ].
The following example shows that the lower semicontinuity ofKis essential.
Example . LetA=B= [–, ],X=Y=R,== [, ],C=R+,γ= ,Qbe an identity
map,K,K:A×→A,T:A×→L(X,Y), andη:A×A×→Abe defined by
K(x,γ) = [, ],
η(y,x,γ) ={x+y–ε},
T(y,γ) ={},
K(x,γ) =
⎧ ⎨ ⎩
{–, , }, ifγ= ,
{, }, otherwise.
We have E(γ) = [, ],∀γ ∈[, ]. Hence E is usc at andE() is compact and the conditions (ii) and (iii) of Theorem . are easily seen to be fulfilled. But the family
{(MQVIPγ λ) : (γ,λ)∈×}is not well-posed in the generalized sense at (, ). The rea-son is thatKis not lower semicontinuous at (x, ). In fact
α(γ,λ,ε) = ⎧ ⎨ ⎩
{}, ifγ = , [, ], ifγ∈(, ].
Theorem . Assume for problem(MQVIP)the assumptions(ii) and(iii)as in
Theo-rem.and replace(i)by(i):
(i) Ais compact,Kis closed inA× {γ}.
Proof We omit the proof since the technique is similar to that for Theorem . with
suit-able modifications.
The following example shows that the compactness ofAcannot be dropped.
Example . LetA=B=X=Y= (–∞, +∞),== [, +∞),C= [, +∞),γ= ,Hbe
the identity map,K,K:A×→A,T:A×→B, andη:A×A×→Abe defined
by
K(x,γ) = ⎧ ⎨ ⎩
{}, ifγ = ,
{+γ}, ifγ = ,
K(x,γ) = [, ],
η(y,x,λ) =
γ+γ –ε
cos(γ)+γ+ ,
T(y,γ) =
cos(γ)+γ+ .
We see thatKis closed at (x, ), the assumptions (ii) and (iii) of Theorem . are satisfied.
But the family{(MQVIPγ λ) : (γ,λ)∈×}is not well-posed in the generalized sense at (, ). The reason is thatAis not compact. In fact,α(, , ) ={}andα(γ,λ,ε) ={+γ},
∀γ∈(, +∞).
The following example shows that the closedness ofKis essential.
Example . LetA=B=X=Y= [–, ],== [, ],C=R+,γ= ,Hbe an identity
map,K,K:A×→A,T:A×→B, andη:A×A×→Abe defined by
K(x,γ) = (–γ, ],
K(x,γ) = [, ],
η(y,x,γ) =x–yx–ε,
T(y,γ) ={}.
We show thatAis compact and the conditions (ii), (iii) of Theorem . are easily seen to be fulfilled. But the family{(MQVIPγ λ) : (γ,λ)∈×}is not well-posed in the generalized sense at (, ). The reason is thatKis not closed at (x, ). In fact,
α(γ,λ,ε) = ⎧ ⎨ ⎩
{}, ifγ= ,
{, }, ifγ= .
The following example shows that all assumptions of Theorem . are satisfied.
Example . LetX=Y=R,A=B= [, ],== [, ],C=R+,γ= ,Hbe an identity
by
K(x,γ) =K(x,γ) = [, ],
η(y,x,γ) =γ+ γ + –ε,
T(y,γ) ={}.
ThenE(γ) = [, ],∀γ ∈[, ]. We see that all assumptions of Theorem . are satisfied. So, the family{(MQVIPγ λ) : (γ,λ)∈×}is well-posed in the generalized sense at (, ). In fact,(γ,λ,ε) = [, ],∀γ ∈[, ].
For (γ,λ)∈×,ε∈intC, and positiveξ, we define the following sets of approximate solutions of the family{(MQVIPγ λ) : (γ,λ)∈×}:
γλ
α (ξ,ε)
=
γ∈B(γ,ξ),λ∈B(λ,ξ)
α(γ,λ,ε)
=
γ∈B(γ,ξ),λ∈B(λ,ξ)
x∈K(x,γ)|(y,z)αK(x,γ)×T(y,γ) :
Q(z),η(y,x,λ)+ε≮,
whereB(γ,ξ) andB(λ,ξ) are the closed balls centered atγandλwith radiusξ.
Observe that, for every (γ,λ)∈×, (i) γλ
α (, ) =α(γ,λ, ) =α(γ,λ);
(ii) α(γ,λ)⊆α(γ,λ,ε)⊆γαλ(ξ,ε).
Theorem . Assume X is complete and the following conditions hold:
(i) Kis closed inA× {γ},and inK(A,)× {γ},Kis lsc;
(ii) inK(K(A,),)× {γ},Tis usc and compact-valued ifα= w(orα= m),and lsc if
α= s.
Then(MQVIP)is well-posed at(γ,λ)if and only if
γλ
α (ξ,ε)=∅, ∀ξ> ,ε∈intC and diam γλ
α (ξ,ε)→ as(ξ,ε)→(, ).
Proof Similar arguments can be applied to the three cases. We present only the proof for the case whereα= s. If (MQVIP) is well-posed at (γ,λ), then (MQVIP) has a unique
solu-tionx∈s(γ,λ) and hence
γλ
s (ξ,ε)=∅,∀ξ> ,ε∈intCass(γ,λ)⊆
γλ s (ξ,ε).
Ifdiamγλ
s (ξ,ε) as (ξ,ε)→(, ), then there existq> andξn> ,εn∈intC, such
thatεn→,ξn→, and
diamγλ
s (ξn,εn) >q> , ∀n∈N.
Then there existxn,xn∈γλ
s (ξn,εn) such thatd(xn,xn) > q
> . Hence there existγ n,γn∈ B(γ,ξn), andλn,λn∈B(λ,ξn) such that∀y∈K(xn,γn),∀z∈T(y,γn) satisfy
Q(z),ηy,xn,λn+εn≮,
and∀y∈K(xn,γn),∀z∈T(y,γn) satisfy
i.e.,{x
n}and{xn}are approximating sequences for (MQVIP) corresponding to{(γn,λn)}
and{(γ
n,λn)}, respectively. Hence, the sequences{xn}and{xn}converges to the unique
solutionxof (MQVIPγλ), contradicting the fact thatd(x n,xn) >
q
> ,∀n∈N.
Conversely, let {γn} →γ and{λn} →λ, and {xn} be approximating sequences for
(MQVIP) corresponding to{γn}and{λn}. Then there is{εn} → such that∀y∈K(xn,γn), ∀z∈T(y,γn) satisfying
Q(z),η(y,xn,λn)
+εn≮, ∀n∈N.
This yieldsxn∈sγλ(ξn,εn) with{ξn}=max{d(γn,γ),d(λn,λ)} →, asn→+∞. Since
diamγλ
s (ξn,εn)→ as (ξn,εn)→(, ), it follows that{xn}is Cauchy and converges to
a pointx. By the closedness ofKat (x,γ),x∈K(x,γ).
Next, we verify thatx∈s(γ,λ). Using the same argument as for Theorem ., we
deduce thatx∈s(γ,λ).
Now we prove that (MQVIPγλ) has a unique solution. If
s(γ,λ) has two distinct
solutionsxandx, it is not hard to see thatx,x∈γλ
s (ξ,ε),∀ξ> ,ε∈intC. It follows
that
<d(x,x)≤γλ
s (ξ,ε)→,
which is impossible. Hence, (MQVIP) is well-posed at (γ,λ).
The following example shows that the uniqueness of well-posed is essential.
Example . LetX=Y =R,A=B= [–, ],== [, ],C=R+, γ= ,H be an
identity map, and letK,K:A×→A,T:A×→L(X,Y), andη:A×A×→Abe
defined by
K(x,γ) =K(x,γ) = [, ],
η(y,x,γ) =γ+ –ε,
T(y,γ) ={}.
We show that the conditions (i) and (ii) of Theorem . are easily seen to be fulfilled and the family{(MQVIPγ λ) : (γ,λ)∈×}is well-posed at (, ). Butdiamγλ
α (ξ,ε) =
[, ] as (ξ,ε)→(, ).
The following example shows that all assumptions of Theorem . are satisfied.
Example . LetA=B=X=Y=R,== [, ],C=R+,γ= ,Hbe an identity map,
and letK,K:A×→A,T:A×→L(X,Y), andη:A×A×→Abe defined by
K(x,γ) = [, +∞),
η(y,x,γ) =y–x+γ,
T(y,γ) ={},
We show that the conditions (i) and (ii) of Theorem . are easily seen to be fulfilled and α(, ) ={}and
α(γ,λ,ε) = ⎧ ⎨ ⎩
[,ε], ifγ = ,
[,γ +ε], ifγ ∈(, ]
andγλ
α (ξ,ε) = [,ε] and the family{(MQVIPγ λ) : (γ,λ)∈×}is well-posed at (, ), anddiamγλ
α (ξ,ε)→ as (ξ,ε)→(, ).
Next, we consider the following notions of measures of noncompactness.
Definition .([, ]) LetXis complete. The Kuratowski measure of the setA⊆Xis
defined by
ζ(A) =inf
ϑ> A⊆ n
i=
Li,diamLi<ϑ,i= , , . . . ,n, for somen∈N
.
Definition .([, ]) A,Bbe nonempty subsets ofX. The Hausdorff metricH(·,·)
betweenAandBis defined by
H(A,B) =maxH∗(A,B),H∗(B,A),
whereH∗(A,B) =supa∈Ad(a,B) withd(a,B) =infb∈Ba–b.
By the definitions ofζ andH, we have
ζ(A)≤ζ(B) + H(A,B),
for every all bounded setsAandB.
Remark .([, ]) The functionζ is a regular measure of noncompactness defined
byζ : X→[, +∞] that satisfies the following conditions:
(i) ζ(D) = +∞if and only if the setDis unbounded; (ii) ζ(D) =ζ(cl(D));
(iii) fromζ(D) = it follows thatDis a totally bounded set; (iv) fromP⊆Qit follows thatζ(P)≤ζ(Q);
(v) ifXis a complete space, and if{Bn}is a sequence of closed subsets ofXsuch that Bn+⊆Bnfor eachn∈Nandlimn→+∞ζ(Bn) = , thenM=
n∈NBnis a nonempty
compact set andlimn→+∞H(Bn,M) = , whereHis a Hausdorff metric.
Lemma . Assume we have problem(MQVIP).Let,be finite dimensional and the
following conditions hold:
(i) Kis closed inA× {γ},and inK(A,)× {γ},Kis lsc;
(ii) inK(K(A,),)× {γ},Tis usc and compact-valued ifα= w(orα= m),and lsc if
α= s.
Thenγλ
α (ξ,ε)is closed,for allξ> ,ε∈intC.
existγn∈B(γ,ξ) andλn∈B(λ,ξ) and∀y∈K(xn,γn),∀z∈T(y,γn) such that
Q(z),η(y,xn,λn)
+ε≮, ∀n∈N.
Since B(γ,ξ) and B(λ,ξ) are compact, we can assume that {γn} →γ ∈B(γ,ξ) and {λn} →λ∈B(λ,ξ). By the closedness ofK at (x,γ), we find thatx∈K(x,γ). We show
that∀y∈K(x,γ),∀z∈T(y,γ) such that
Q(z),η(y,x,λ)+ε≮,
i.e.,x∈γλ
s (ξ,ε). Indeed, ifx∈/sγλ(ξ,ε), then∃y∈K(x,γ),∃z∈T(y,γ) such that
Q(z),η(y,x,λ)+ε< .
By the lower semicontinuity ofKandT, there existyn∈K(xn,γn),zn∈T(yn,γn) such
that{yn} →y,{zn} →z, for alln. Asxn∈sγλ(ξ,ε) we have
Q(zn),η(yn,xn,λn)
+ε≮.
By the continuity ofQ,η, ·,·and id, it follows that ·,·+ id is continuous. So we have
Q(z),η(y,x,λ)+ε≮,
and we see a contradiction. Hencex∈γλ
s (ξ,ε). Thus,sγλ(ξ,ε) is closed.
Next, we provide sufficient conditions for the two sets to coincide.
Lemma . Assume for problem(MQVIP)the following conditions to hold:
(i) K(x,·)is closed atγ,andK(x,·)is lsc atγ;
(ii) inK(K(A,),)× {γ},Tis usc and compact-valued ifα= w(orα= m),and lsc if
α= s.
Thenα(γ,λ) =
ε∈intC,ξ>
γλ
α (ξ,ε),for every(γ,λ)∈×.
Proof We present only the proof for the case where α = s. We first prove that
ε∈intC
γλ
s (ξ,ε) =s(γ,λ,ε). It is easy to see that
ε∈intC
γλ
s (ξ,ε)⊇s(γ,λ,ε).
Thus, we only need to show that ε∈intC
γλ
s (ξ,ε) ⊆ s(γ,λ,ε). Indeed, let x ∈
ε∈intC
γλ
s (ξ,ε), there are γn∈B(γ,ξ) and λn ∈B(λ,ξ) such that ∀y∈K(x,γn), ∀z∈T(y,γn) satisfying
Q(z),η(y,x,λn)
+ε≮.
Sincex∈K(x,γn),γn→γandKis closed, we havex∈K(x,γ). Now we verify that x∈(γ,λ,ε). Indeed, for eachy∈K(x,γ), by the semicontinuity ofK(x,·) atγand
the semicontinuity ofT at (y,γ), there existyn∈K(x,γn) andzn∈T(yn,γn) such that {yn} →y,{zn} →z. Asx∈s(γn,λn,ε), we have
Q(zn),η(yn,x,λn)
By the continuity ofQ,η, ·,·, ·,·+ id, and (yn,zn,γn,λn)→(y,z,γ,λ). We have
Q(z),η(y,x,λ)
+ε≮,
i.e.,
ε∈intC
γλ
s (ξ,ε)⊆s(γ,λ,ε).
Hence
ε∈intC
γλ
s (ξ,ε) =s(γ,λ,ε).
It is clear that
s(γ,λ) =
ξ>
s(γ,λ,ε) =
ξ>,ε∈intC
γλ
s (ξ,ε).
The following theorem shows the well-posedness in the generalized sense at (γ,λ) for
(MQVIP) by using the Kuratowski measureζ.
Theorem . Let X be complete,,be finite dimensional and the following conditions
hold:
(i) Kis closed inA× {γ},and inK(A,)× {γ},Kis lsc;
(ii) inK(K(A,),)× {γ},Tis usc and compact-valued ifα= w(orα= m),and lsc if
α= s.
Then(MQVIP)is well-posed in the generalized sense at(γ,λ)if and only if
γλ
α (ξ,ε)=∅, ∀ξ> ,ε∈intC and ζ
γλ
α (ξ,ε)
→ as(ξ,ε)→(, ).
Proof Similar arguments can be applied in the three cases. We present only the proof for the case whereα= s. Now we suppose that (MQVIP) is well-posed in the generalized sense at (γ,λ). Letsbe a solution set of (MQVIPγ λ) for all (γ,λ)∈×. Then, from
The-orem ., we see thats(γ,λ) is a nonempty compact. Clearlys(γ,λ)⊆sγλ(ξ,ε), ∀ξ> ,ε∈intC. Now we show that
ζγλ s (ξ,ε)
→ as (ξ,ε)→(, ).
Indeed, sinces(γ,λ)⊆
γλ
s (ξ,ε),∀ξ > ,ε∈intC. Using the concept of Hausdorff
metric, we have
Hγλ
s (ξ,ε),s(γ,λ)
=maxH∗γλ
s (ξ,ε),s(γ,λ)
,H∗s(γ,λ),γsλ(ξ,ε)
=H∗γλ
s (ξ,ε),s(γ,λ)
.
Suppose thats(γ,λ)⊆
n
We seti={t∈A|d(t,Li)≤H(sγλ(ξ,ε),s(γ,λ))}.
We claim thatγλ s (ξ,ε)⊆
n
i=i. Indeed, letx∈γsλ(ξ,ε). Thend(x,s(γ,λ))≤ H(γλ
s (ξ,ε),s(γ,λ)). Sinces(γ,λ)⊆
n
i=Li, we see that
d x, n i= Li
≤Hγλ
s (ξ,ε),s(γ,λ)
.
Hence, there isksuch thatd(x,Lk)≤H(γsλ(ξ,ε),s(γ,λ)),i.e.,x∈k. So
γλ s (ξ,ε)⊆
n
i=
i.
Note further that
diami=diamLi+ H
γλ
s (ξ,ε),s(γ,λ)
≤ϑ+ Hγλ
s (ξ,ε),s(γ,λ)
.
Hence,
ζγλ s (ξ,ε)
≤Hγλ
s (ξ,ε),s(γ,λ)
+ζs(γ,λ)
.
Sinces(γ,λ) is compact,ζ(s(γ,λ)) = , so we have
ζγλ s (ξ,ε)
≤H∗γλ
s (ξ,ε),s(γ,λ)
.
Now we prove that
H∗γλ s (ξ,ε)
→ as (ξ,ε)→(, ).
Suppose to the contrary that
H∗γλ s (ξ,ε)
as (ξ,ε)→(, ).
There areθ> , (ξn,εn)→(, ), andxn∈
γλ
s (ξn,εn) such that
dxn,s(γ,λ)
≥θ> , ∀n∈N.
{xn}is an approximating sequence of (MQVIP). By the well-posedness in the generalized
sense of (MQVIP) at (γ,λ), there is a subsequence{xk}of{xn}converging to some point
ofs(γ,λ), which is impossible asd(xn,s(γ,λ))≥θ> ,∀n∈N. Hence
ζγλ s (ξ,ε)
→ as (ξ,ε)→(, ).
Conversely,ζ(γλ
s (ξ,ε))→ as (ξ,ε)→(, ). By Lemma ., we see thatγsλ(ξ,ε)
is closed, for allξ> ,ε∈intC. By Lemma ., we have
s(γ,λ) =
ε∈intC,ξ>
Sinceζ(γλ
s (ξ,ε))→ as (ξ,ε)→(, ), the regular measure properties ofζ imply that
s(γ,λ) is compact and
Hγλ
s (ξ,ε),s(γ,λ)
→ as (ξ,ε)→(, ).
Let{xn}be an approximating sequence for (MQVIP) corresponding to{(γn,λn)}, where {γn} →γand{λn} →λ. There is{εn} → such that∀y∈K(xn,γn),∀z∈T(y,γn)
satis-fying
Q(z),η(y,xn,λn)
+εn≮, ∀n∈N.
This means thatxn∈
γλ
s (ξn,εn) withξn:=max{d(γ,γn),d(λ,λn)}. We see that
dxn,s(γ,λ)
≤Hγλ
s (ξn,εn),s(γ,λ)
→ asn→+∞.
Hence, there isx¯n∈s(γ,λ) such that
d(xn,x¯n)→ asn→+∞.
By the compactness ofs(γ,λ), there is a subsequence{¯xnk}of{¯xn}convergent to some pointxofs(γ,λ). Therefore, the corresponding subsequence{xnk}of{xn}tends tox. Hence, (MQVIP) is well-posed in the generalized sense at (γ,λ).
Remark . In cases as in Remark .(i), Theorems ., ., and .-. in [] are
par-ticular cases of Theorems ., ., and ., respectively. However, the assumptions and our proof methods are very different from Theorems ., ., and .-. in [].
The following example shows that the closedness of K in Theorem . cannot be dropped.
Example . LetX=Y =R, A=B= [–, ],== [, ],C=R+,γ= , H be an
identity map, and letK,K:A×→A,T:A×→L(X,Y), andη:A×A×→Abe
defined by
K(x,γ) = [–γ, ],
η(y,x,γ) =x(x–y),
T(y,γ) ={},
K(x,γ) = [, ].
We show thatKis lsc inK(A,)×and the condition (ii) of Theorem . is easily seen
to be fulfilled andγλ
α (ξ,ε)⊆[–, ]. Hence,ζ(αγλ(ξ,ε))→ as (ξ,ε)→(, ). But the family{(MQVIPγ λ) : (γ,λ)∈×}is not well-posed in the generalized sense at (, ). The reason is thatKis not closed at (A, ). Indeed, we letγn=xn=n→, asn→ ∞and tn=n∈K(xn,γn) = (–n, ],∀n∈N. It is clear that{tn}is convergent to /∈K(, ) = (, ].
The following example shows that the lower semicontinuity ofK in Theorem . is essential.
Example . LetX=Y=R,A=B= [–, ],== [, ], C=R+,ε∈intC,ξ > ,
γ= ,Hbe an identity map, and letK,K:A×→A,T:A×→L(X,Y), andη: A×A×→Abe defined by
K(x,γ) = [, ],
η(y,x,γ) =x+y,
T(y,γ) ={},
K(x,γ) = ⎧ ⎨ ⎩
{–, , }, ifγ = ,
{, }, otherwise.
We show thatK is lsc inK(A,)× and the condition (ii) of Theorem . is easily seen to be fulfilled andγλ
α (ξ,ε)⊆[–, ]. Hence,ζ( γλ
α (ξ,ε))→ as (ξ,ε)→(, ). But the family{(MQVIPγ λ) : (γ,λ)∈×}is not well-posed in the generalized sense at (, ). The reason is thatKis not lower semicontinuous. In fact
α(γ,λ,ε) =γ λα (ξ,ε) = ⎧ ⎨ ⎩
[ –ε, ]∩[, ], ifγ = ,
[, ], ifγ ∈(, ].
The following example shows that all assumptions of Theorem . are fulfilled.
Example . LetX=Y=R,A=B=== [, ],C=R+,ε∈intC,ξ> ,γ= ,Hbe
an identity map, and letK,K:A×→A,T:A×→L(X,Y), andη:A×A×→A
be defined by
K(x,γ) =K(x,γ) = [γ,γ + ],
η(y,x,γ) =γ+–ε,
T(y,γ) =
γ+ .
We show that the assumptions (i) and (ii) of Theorem . are easily seen to be fulfilled and
α(γ,λ,ε) = ⎧ ⎨ ⎩
[γ,γ+ ], ifγ ∈(, ],
[, ], ifγ = ,
and γλ
α (ξ,ε) ⊆[, ]. Hence, ζ(αγλ(ξ,ε))→ as (ξ,ε)→(, ), and the family
{(MQVIPγ λ) : (γ,λ)∈×}is well-posed in the generalized sense at (, ).
Competing interests
Acknowledgements
The author is grateful to Prof. Phan Quoc Khanh and Prof. Lam Quoc Anh for their help in the research process. The author also thanks the three anonymous referees for their valuable remarks and suggestions, which helped them to improve considerably the article.
Received: 13 August 2013 Accepted: 25 April 2014 Published:12 May 2014
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