Well-Posedness of Generalized Vector Quasivariational Inequality Problems

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Volume 2012, Article ID 582792,17pages doi:10.1155/2012/582792

Research Article

Well-Posedness of Generalized Vector Quasivariational Inequality Problems

Jian-Wen Peng and Fang Liu

School of Mathematics, Chongqing Normal University, Chongqing 400047, China

Correspondence should be addressed to Jian-Wen Peng,[email protected] Received 28 October 2011; Accepted 14 December 2011

Academic Editor: Yeong-Cheng Liou

Copyrightq2012 J.-W. Peng and F. Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We introduce several types of the Levitin-Polyak well-posedness for a generalized vector quasivariational inequality problem with both abstract set constraints and functional constraints.

Criteria and characterizations of these types of the Levitin-Polyak well-posednesses with or without gap functions of generalized vector quasivariational inequality problem are given. The results in this paper unify, generalize, and extend some known results in the literature.

1. Introduction

The vector variational inequality in a finite-dimensional Euclidean space has been introduced in1and applications have been given. Chen and Cheng2studied the vector variational inequality in infinite-dimensional space and applied it to vector optimization problem. Since then, many authors 3–11 have intensively studied the vector variational inequality on diﬀerent assumptions in infinite-dimensional spaces. Lee et al.12,13, Lin et al.14, Konnov and Yao15, Daniilidis and Hadjisavvas16, Yang and Yao17, and Oettli and Schl¨ager18 studied the generalized vector variational inequality and obtained some existence results.

Chen and Li 19 and Lee et al.20introduced and studied the generalized vector quasivariational inequality and established some existence theorems.

On the other hand, it is well known that the well-posedness is very important for both optimization theory and numerical methods of optimization problems, which guarantees that, for approximating solution sequences, there is a subsequence which converges to a solution. The study of well-posedness originates from Tykhonov 21 in dealing with unconstrained optimization problems. Its extension to the constrained case was developed by Levitin and Polyak22. The study of generalized Levitin-Polyak well-posedness for convex scalar optimization problems with functional constraints originates from Konsulova and

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Revalski23. Recently, this research was extended to nonconvex optimization problems with abstract set constraints and functional constraintssee24, nonconvex vector optimization problem with abstract set constraints and functional constraints see 25, variational inequality problems with abstract set constraints and functional constraints see 26, generalized inequality problems with abstract set constraints and functional constraints 27, generalized quasi-inequality problems with abstract set constraints and functional constraints 28, generalized vector inequality problems with abstract set constraints and functional constraints 29, and vector quasivariational inequality problems with abstract set constraints and functional constraints 30. For more details on well-posedness on optimizations and related problems, please also see31–37and the references therein. It is worthy noting that there is no study on the Levitin-Polyak well-posedness for a generalized vector quasi-variational inequality problem.

In this paper, we will introduce four types of Levitin-Polyak well-posedness for a generalized vector quasivariational inequality problem with an abstract set constraint and a functional constraint. InSection 2, by virtue of a nonlinear scalarization function and a gap function for generalized vector quasi-varitional inequality problems, we show equivalent relations between the Levitin-Polyak well-posedness of the optimization problem and the Levitin-Polyak well-posedness of generalized vector quasi-varitional inequality problems. In Section 3, we derive some various criteria and characterizations for thegeneralizedLevitin- Polyak well-posedness of the generalized vector quasi-variational inequality problems. The results in this paper unify, generalize, and extend some known results in26–30.

2. Preliminaries

Throughout this paper, unless otherwise specified, we use the following notations and assumptions.

LetX, · be a normed space equipped with norm topology, and letZ, d1be a metric space. LetX1 ⊂ X,K ⊂ Z be nonempty and closed sets. Let Y be a locally convex space ordered by a nontrivial closed and convex coneCwith nonempty interior intC, that is, y₁ ≤y₂if and only ify₂−y₁ ∈Cfor anyy₁, y₂∈Y. LetLX, Ybe the space of all the linear continuous operators fromXtoY. LetT:X1 → 2^LX,YandS:X1 → 2^X¹be strict set-valued mappingsi.e.,Tx/∅ andSx/∅, for allx ∈ X₁, and letg : X₁ → Zbe a continuous vector-valued mapping. We denote byz,x the valuezx, where z ∈ LX, Y,x ∈ X₁. LetX0 {x ∈ X1 : gx ∈ K}be nonempty. We consider the following generalized vector quasi-variational inequality problem with functional constraints and abstract set constraints.

Findx∈X₀such thatx∈Sxand there existsz∈Txsatisfying

z, x−x ∈ −/ intC, ∀x∈Sx. GVQVI

Denote byXthe solution set ofGVQVI.

LetZ₁, Z₂be two normed spaces. A set-valued mapFfromZ₁to 2^Z²is

iclosed, onZ3⊆Z1, if for any sequence{xn} ⊆Z3withxn → xandyn∈Fxnwith y_n → y, one hasy∈Fx;

iilower semicontinuousl.s.c. in shortatx∈Z₁, if{xn} ⊆Z₁, x_n → x, andy∈Fx imply that there exists a sequence{yn} ⊆ Z2 satisfyingyn → ysuch thatyn ∈ Fxnforn suﬃciently large. IfFis l.s.c. at each point ofZ₁, we say thatFis l.s.c. onZ₁;

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iiiupper semicontinuousu.s.c. in shortatx ∈ Z1, if for any neighborhoodV of Fx, there exists a neighborhoodUofxsuch thatFx⊆ V, for allx ∈U. IfFis u.s.c. at each point ofZ₁, we say thatFis u.s.c. onZ₁.

It is obvious that any u.s.c. nonempty closed-valued mapFis closed.

LetP, dbe a metric space,P1⊂P, andx∈P. We denote bydP1x inf{dx, p:p∈ P₁}the distance from the pointxto the setP₁. For a topological vector spaceV, we denote by V^∗its dual space. For any setΦ⊂V, we denote the positive polar cone ofΦby

Φ^∗{λ∈V^∗:λx≥0, ∀x∈Φ}. 2.1

Lete∈intCbe fixed. Denote

C^∗0{λ∈C^∗:λe 1}. 2.2

Definition 2.1. i A sequence {xn} ⊆ X1 is called a type I Levitin-Polyak LP in short approximating solution sequence if there exist{n} ⊆ R¹ {r ≥0|r is a real number}with _n → 0 andz_n∈Txnsuch that

d_X₀xn≤_n, 2.3

xn∈Sxn, 2.4

zn, x−xn ne /∈ −intC, ∀x∈Sxn. 2.5 ii{xn} ⊆ X1 is called a type II LP approximating solution sequence if there exist {n} ⊆ R¹with _n → 0 andz_n ∈ Txnsuch that2.3–2.5hold, and, for anyz ∈ Txn, there existswn, z∈Sxnsatisfying

z, wn, z−x_n −_ne∈ −C. 2.6 iii{xn} ⊆ X1 is called a generalized type I LP approximating solution sequence if there exist{n} ⊆R¹withn → 0 andzn∈Txnsuch that

dK

gxn

≤n 2.7

and2.4,2.5hold.

iv{xn} ⊆ X₁ is called a generalized type II LP approximating solution sequence if there exist{n} ⊆R¹ withn → 0,zn ∈Txnsuch that2.4,2.5, and2.7hold, and, for anyz∈Txn, there existswn, z∈Sxnsuch that2.6holds.

Definition 2.2. GVQVIis said to be type Iresp., type II, generalized type I, generalized type IILP well-posed if the solution setX ofGVQVIis nonempty, and, for any type Iresp., type II, generalized type I, generalized type II LP approximating solution sequence{xn}, there exists a subsequence{xnj}of{xn}andx∈Xsuch thatx_n_j → x.

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Remark 2.3. iIt is clear that anygeneralizedtype II LP approximating solution sequence is ageneralizedtype I LP approximating solution sequence. Thus,generalizedtype I LP well-posedness impliesgeneralizedtype II LP well-posedness.

ii Each type of LP well-posedness ofGVQVI implies that the solution set X is compact.

iiiSuppose thatgis uniformly continuous functions on a set

X₁δ0 {x∈X₁:d_X₀x≤δ₀}, 2.8 for someδ0 > 0. Then generalized type Iresp., generalized type IILP well-posedness of GVQVIimplies its type Iresp., type IILP well-posedness.

ivIf Y R¹,C R¹, then type I resp., type II, generalized type I, generalized type II LP well-posedness of GVQVIreduces to type I resp., type II, generalized type I, generalized type II LP well-posedness of the generalized quasi-variational inequality problem defined by Jiang et al.28. If Y R¹,C R¹,Sx X₀ for all x ∈ X₁, then type Iresp., type II, generalized type I, generalized type IILP well-posedness ofGVQVI reduces to type Iresp., type II, generalized type I, generalized type IILP well-posedness of the generalized variational inequality problem defined by Huang, and Yang27which contains as special cases for the type Iresp., type II, generalized type I, generalized type II LP well-posedness of the variational inequality problem in26.

vIfSx X₀for allx∈X₁, then type Iresp., type II, generalized type I, generalized type II LP well-posedness of GVQVIreduces to type I resp., type II, generalized type I, generalized type II LP well-posedness of the generalized vector variational inequality problem defined by Xu et al.29.

viIf the set-valued mapT is replaced by a single-valued mapF, then type Iresp., type II, generalized type I, generalized type IILP well-posedness ofGVQVIreduces to type Iresp., type II, generalized type I, generalized type IILP well-posedness of the vector quasivariational inequality problems defined by Zhang et al.30.

Consider the following statement:

X /∅and for any type I

resp., type II,generalized type I, generalized type II

LP approximating solution sequence{xn}, we have d_Xxn−→0

.

2.9

Proposition 2.4. If GVQVIis type I (resp., type II, generalized type I, generalized type II) LP well- posed, then2.9holds. Conversely if 2.9holds andX is compact, then (1) is type I (resp., type II, generalized type I, generalized type II) LP well-posed.

The proof ofProposition 2.4is elementary and thus omitted.

To see the various LP well-posednesses of1are adaptations of the corresponding LP well-posednesses in minimizing problems by using the Auslender gap function, we consider the following general constrained optimization problem:

min fx

s.t. x∈X₁ gx∈K,

P

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whereX₁ ⊆X1is nonempty and f :X1 → R¹∪ {∞}is proper. The feasible set ofP isX₀, whereX₀ {x∈X₁:gx∈K}. The optimal set and optimal value ofPare denoted byXandv, respectively. Note that if Domf∩X₀/∅, where

Dom f

x∈X1 :fx<∞ , 2.10

thenv <∞. In this paper, we always assume thatv >−∞.

Definition 2.5. iA sequence{xn} ⊆X₁ is called a type I LP minimizing sequence forPif lim sup

n→ ∞ fxn≤v, 2.11

d_X₀xn−→0. 2.12

ii{xn} ⊆X₁ is called a type II LP minimizing sequence forPif

nlim→ ∞fxn v 2.13

and2.12hold.

iii{xn} ⊆X₁ is called a generalized type I LP minimizing sequence forPif dK

gxn

−→0 2.14

and2.11hold.

iv{xn} ⊆X₁is called a generalized type II LP minimizing sequence forPif2.13 and2.14hold.

Definition 2.6. P is said to be type I resp., type II, generalized type I, generalized type IILP well-posed if the solution set Xof Pis nonempty, and for any type I resp., type II, generalized type I, generalized type II LP minimizing sequence {xn}, there exists a subsequence{xnj}of{xn}andx∈Xsuch thatx_n_j → x.

The Auslender gap function forGVQVIis defined as follows:

fx inf

z∈Tx sup

x∈Sxinf

λ∈C^∗0λ

z, x−x

, ∀x∈X1. 2.15

LetX₂⊆Xbe defined by

X₂{x∈X|x∈Sx}. 2.16

In the rest of this paper, we setX₁ inPequal toX₁∩X₂. Note that ifSis closed on X₁, thenX₁is closed.

Recall the following widely used functionsee, e.g.,38 ξ:Y −→R¹: min

t∈R¹:y−te∈ −C

. 2.17

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It is known that ξ is a continuous, strictly monotone i.e., for any y1, y2 ∈ Y, y₁ −y₂ ∈ C implies that ξy1 ≥ ξy2 and y1 −y₂ ∈ intC implies that ξy1 > ξy2, subadditive and convex function. Moreover, it holds that ξte t,for allt ∈ R¹ and ξy sup_λ∈C∗0λy,for ally∈Y.

Now we given some properties for the functionfdefined by2.15.

Lemma 2.7. Let the functionfbe defined by2.15, and let the set-valued mapT be compact-valued onX1. Then

ifx≥0, for allx∈X₁;

iifor anyx∈X₀,fx 0 if and only ifx∈X.

Proof. iLetx ∈X₁. Suppose to the contrary thatfx < 0. Then, there exists aδ > 0 such thatfx<−δ. By definition, forδ/2>0, there exists az∈Tx, such that

sup

x∈Sxinf

λ∈C^∗0λ

z, x−x

≤fx δ 2 <−δ

2 <0. 2.18

Thus, we have

λ∈Cinf^∗0λ

z, x−x

<0, ∀x∈Sx, 2.19

which is impossible whenxx. This provesi.

iiSuppose thatx∈X₀such thatfx 0.

Then, it follows from the definition ofX₀ thatx ∈ Sx. And from the definition of fxwe know that there existzn∈Txand 0< n → 0 such that

λ∈Cinf^∗0λ

z_n, x−x

≤fx _n_n, ∀x∈Sx, 2.20

that is,

ξ

zn, x−x

≥ −n, ∀x∈Sx. 2.21

By the compactness ofTx, there exists a sequence{znj}of{zn}and somez∈Txsuch that

z_n_j −→z. 2.22

This fact, together with the continuity ofξand2.21, implies that ξ

z, x−x

≥0, ∀x∈Sx. 2.23

It follows thatx∈X.

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Conversely, assume that x ∈ X. It follows from the definition of X thatx ∈ Sx.

Suppose to the contrary thatfx>0. Then, for anyz∈Tx, sup

x∈Sx λ∈Cinf^∗0λ

z, x−x

>0. 2.24

Thus, there existδ >0 andx0∈Sxsuch that

λ∈Cinf^∗0λz, x−x0 ≥δ. 2.25 It follows that

ξz, x0−x ≤ −δ <0. 2.26

As a result, we have

z, x0−x ∈ −intC. 2.27

This contradicts the fact thatx∈X. So,fx 0. This completes the proof.

Lemma 2.8. Letf be defined by2.15. Assume that the set-valued mapT is compact-valued and u.s.c. onX₁and the set-valued mapSis l.s.c. onX₁. Thenfis l.s.c. function fromX₁toR¹∪ {∞}.

Further assume that the solution setXofGVQVIis nonempty, then Domf/∅.

Proof. First we show thatfx>−∞, for allx∈X₁. Suppose to the contrary that there exists x0∈X1 such thatfx0 −∞. Then, there existzn ∈Tx0and{Mn} ⊂R¹withMn → ∞ such that

sup

x∈Sx0 λ∈Cinf^∗0λ

z_n, x₀−x

≤ −Mn. 2.28

Thus,

ξ

zn, x−x0

≥Mn, ∀x∈Sx0. 2.29

By the compactness ofTx0, there exist a sequence{znj} ⊂ {zn}and somez0 ∈Tx0such thatznj → z0. This fact, together with2.29and the continuity ofξonY, implies that

ξ

z₀, x−x₀

≥∞, ∀x∈Sx0 2.30

which is impossible, sinceξis a finite function onY.

Second, we show thatf is l.s.c. onX₁. Leta ∈ R¹. Suppose that{xn} ⊂ X₁ satisfies fxn≤a,for alln, andxn → x0 ∈X1. It follows that, for eachn, there existzn ∈Txnand 0< δn → 0 such that

−ξ

z_n, y−x_n

≤aδ_n, ∀y∈Sxn. 2.31

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For any x ∈ Sx0, by the l.s.c. of S, we have a sequence{yn} with {yn} ∈ Sxn converging toxsuch that

−ξ

zn, yn−xn

≤aδn. 2.32

By the u.s.c. ofTatx₀and the compactness ofTx0, we obtain a subsequence{znj}of {zn}and somez0 ∈Tx0such thatznj → z0. Taking the limit in2.32 withnreplaced by n_j, by the continuity ofξ, we have

−ξ

z₀, x−x₀

≤a, ∀x∈Sx0. 2.33

It follows thatfx0 inf_z∈Tx₀sup_x∈Sx0−ξz, x−x0 ≤a. Hence,f is l.s.c. onX1. Furthermore, ifX /∅, byLemma 2.7, we see that Domf/∅.

Lemma 2.9. Let the functionfbe defined by2.15, and let the set-valued mapT be compact-valued onX₁. Then,

i{xn} ⊆ X₁is a sequence such that there exist{n} ⊆ R¹ with_n → 0 andz_n ∈Txn satisfying2.4and2.5if and only if{xn} ⊆X₁ and2.11hold withv0,

ii{xn} ⊆ X₁ is a sequence such that there exist{n} ⊆ R¹with _n → 0 andz_n ∈Txn satisfying2.4and2.5, and for anyz∈Txn, there existswn, z∈Sxnsatisfying 2.6if and only if{xn} ⊆X₁ and2.13hold withv0.

Proof. iLet{xn} ⊆X₁be any sequence if there exist{n} ⊆R¹with_n → 0 andz_n∈Txn satisfying2.4and2.5, then we can easily verify that

{xn} ⊆X₁, fxn≤_n. 2.34 It follows that2.11holds withv0.

For the converse, let{xn} ⊆X₁ and2.11hold withv0. We can see that{xn} ⊆X₁ and2.4hold. Furthermore, by2.11, we have that there exists{n} ⊆R¹ with _n → 0 such thatfxn≤n. By the compactness ofTxn, we see that for everynthere existszn ∈Txn such that

ξ

zn, x−xn

≥ −n, ∀x∈Sxn. 2.35

It follows that for everynthere existsz_n∈Txnsuch that2.5holds.

iiLet{xn} ⊆X1be any sequence we can verify that lim inf

n→ ∞ fxn≥0 2.36

holds if and only if there exists{αn} ⊆R¹ withα_n → 0 and, for anyz∈ Txn, there exists wn, z∈Sxnsuch that

z, wn, z−x_n −α_ne∈ −C. 2.37

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From the proof ofi, we know that lim sup_n_{→ ∞}fxn≤0 and{xn} ⊆X₁hold if and only if{xn} ⊆X₁such that there exist{βn} ⊆R¹withβ_n → 0z_n∈Txnsatisfying2.4and 2.5 with_nreplaced byβ_n. Finally, we let_nmax{αn, β_n}and the conclusion follows.

Proposition 2.10. Assume thatX /∅andT is compact-valued onX₁. Then

i GVQVIis generalized type I (resp., generalized type II) LP well-posed if and only if P is generalized type I (resp., generalized type II) LP well-posed withfxdefined by2.15.

iiIf GVQVIis type I (resp., type II) LP well-posed, thenPis type I (resp., type II) LP well-posed withfxdefined by2.15.

Proof. Letfxbe defined by2.15. SinceX /Ø, it follows fromLemma 2.7thatx∈Xis a solution ofGVQVIif and only ifxis an optimal solution of5withvfx 0.

iSimilar to the proof ofLemma 2.9, it is also routine to check that a sequence{xn}is a generalized type Iresp., generalized type IILP approximating solution sequence if and only if it is a generalized type Iresp., generalized type IILP minimizing sequence ofP. SoGVQVIis generalized type I resp., generalized type IILP well-posed if and only if Pis generalized type Iresp., generalized type II LP well-posed withfxdefined by2.15.

iiSince X₀ ⊆ X0, dX0x ≤ d_X₀x for any x. This fact together with Lemma 2.9 implies that a type Iresp., type IILP minimizing sequence ofPis a type Iresp., type IILP approximating solution sequence. So the type Iresp., type IILP well- posedness ofGVQVIimplies the type Iresp., type IILP well-posedness ofP withfxdefined by2.15.

3. Criteria and Characterizations for Generalized LP Well-Posedness of (GVQVI)

In this section, we shall present some necessary and/or suﬃcient conditions for the various types ofgeneralizedLP well-posedness ofGVQVIdefined inSection 2.

Now consider a real-valued function c ct, s, rdefined for t, s, r ≥ 0 suﬃciently small, such that

ct, s, r≥0, ∀t, s, r, c0,0,0 0,

s_n−→0, t_n≥0, r_n0, ctn, s_n, r_n−→0 imply thatt_n−→0,

3.1

Theorem 3.1. Let the set-valued mapT be compact-valued onX₁. If GVQVIis type II LP well- posed, the set-valued mapSis closed-valued, then there exist a functioncsatisfying3.1such that

fx≥c

d_Xx, dX0x, dSxx

, ∀x∈X₁, 3.2

wherefxis defined by2.15. Conversely, suppose thatXis nonempty and compact and3.2holds for somecsatisfying3.1. ThenGVQVIis type II LP well-posed.

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Proof. Define

ct, s, r inffx:x∈X1, d_Xx t, dX0x s, d_Sxx r . 3.3 Since X /Ø, it is obvious that c0,0,0 0. Moreover, if s_n → 0, t_n ≥ 0, r_n 0, and ctn, sn, rn → 0, then there exists a sequence{xn} ⊆X1withd_Xxn tn,d_Sx_nxn rn0,

d_X₀xn s_n−→0, 3.4

such that

fxn−→0. 3.5 SinceS is closed-valued, xn ∈ Sxnfor any n. This fact, combined with3.4and 3.5andLemma 2.9iiimplies that{xn}is a type II LP approximating solution sequence of GVQVI. ByProposition 2.4, we have thatt_n → 0.

Conversely, let {xn} be a type II LP approximating solution sequence ofGVQVI.

Then, by3.2, we have

fxn≥c

d_Xxn, dX0xn, d_Sx_nxn

. 3.6

Let

tnd_Xxn, sndX0xn, rn d_Sx_nxn. 3.7 Then sn → 0 and rn 0, for all n ∈ N. Moreover, by Lemma 2.9, we have that

|fx| → 0. Then,ctn, sn, rn → 0. These facts together with the properties of the functionc imply thatt_n → 0. ByProposition 2.4, we see thatGVQVIis type II LP well-posed.

Theorem 3.2. Let the set-valued mapT be compact-valued onX1. If GVQVIis generalized type II LP well-posed, the set-valued mapSis closed, then there exist a functioncsatisfying3.1such that

fx≥c

d_Xx, dK

gx

, d_Sxx

, ∀x∈X₁, 3.8

wherefxis defined by2.15. Conversely, suppose thatXis nonempty and compact and3.8holds for somecsatisfying3.4and3.5. Then,GVQVIis generalized type II LP well-posed.

Proof. The proof is almost the same as that ofTheorem 3.1. The only diﬀerence lies in the proof of the first part ofTheorem 3.1. Here we define

ct, s, r inffx:x∈X1, d_Xx t, dK

gx

s, d_Sxx r . 3.9 Next we give the Furi-Vignoli-type characterizations39for thegeneralizedtype I LP well-posedness ofGVQVI.

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LetX,·be a Banach space. Recall that the Kuratowski measure of noncompactness for a subsetHofXis defined as

μH inf

>0 :H⊆

H_i,diamHi< , i1, . . . , n

, 3.10

where diamHiis the diameter ofHidefined by

diamHi sup{x1−x₂:x₁, x₂∈H_i}. 3.11

Given two nonempty subsets A and B of a Banach space X, · , the Hausdorﬀ distance between A and B is defined by

hA, B max

sup{dBa:a∈A},sup{dAb:b∈B} . 3.12 For any ≥ 0, two types of approximating solution sets for GVQVI are defined, respectively, by

Ω1

x∈X₁:x∈Sx, dX0x≤,∃z∈Tx,s.t.

z, x−x

e /∈ −intC, ∀x∈Sx , Ω2

x∈X₁:x∈Sx, dK

gx

≤,∃z∈Tx,s.t.

z, x−x

e /∈ −intC, ∀x∈Sx . 3.13

Theorem 3.3. Assume thatT is u.s.c. and compact-valued onX1andS is l.s.c. and closed on X1. Then

a GVQVIis type I LP well-posed if and only if

lim→0μΩ1 0, 3.14

b GVQVIis generalized type I LP well-posed if and only if

lim→0μΩ2 0. 3.15

Proof. aFirst we show that, for every > 0, Ω1 is closed. In fact, letxn ∈ Ω1and xn → x0. Then2.4and the following formula hold:

d_X₀xn≤,

∃zn∈Txn, s.t.

z_n, x−x_n

e /∈ −intC, ∀x∈Sxn. 3.16

Sincexn → x0, by the closedness ofSand2.4, we havex0∈Sx0. From3.16, we get

d_X₀x0≤, 3.17

∃zn∈Txn, s.t. ξ

z_n, x−x_n

≥ −, ∀x∈Sxn. 3.18

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For anyv∈Sx0, by the lower semi-continuity ofSand3.18, we can findvn∈Sxnwith v_n → vsuch that

ξzn, v_n−x_n ≥ −. 3.19

By the u.s.c. ofTatx₀and the compactness ofTx0, there exist a subsequence{znj} ⊂ {zn}and somez₀∈Tx0such that

znj −→z0. 3.20

This fact, together with the continuity ofξand3.19, implies that

ξz0, v−x0 ≥ − ∀v∈Sx0. 3.21

It follows that

z0, v−x0 e /∈ −intC ∀v∈Sx0. 3.22 Hence,x0∈Ω1.

Second, we show thatX

>0Ω1. It is obvious thatX⊆

>0Ω1. Now suppose that_n>0 with_n → 0 andx^∗∈

>0Ω1n. Then

dX0x^∗≤n, ∀n, 3.23

x^∗∈Sx^∗, 3.24

∃z∈Tx^∗, s.t.z, x−x^∗ _ne /∈ −intC, ∀x∈Sx^∗. 3.25

From3.23, we have

x^∗∈X₀. 3.26

From3.25, we have

z, x−x^∗

∈ −/ intC, ∀x∈Sx^∗, 3.27 that isx^∗∈X. Hence,X

>0Ω1.

Now we assume thatGVQVIis type I LP well-posed. ByRemark 2.3, we know that the solutionXis nonempty and compact. For every positive real number, sinceX∈Ω1, one gets

Ω1/∅, h

Ω1, X max

sup

u∈Ω1d_Xu,sup

v∈X

d_Ω₁v

sup

u∈Ω1d_Xu. 3.28

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For everyn∈N, the following relations hold:

μΩ1≤2h

Ω1, X μ

X 2h

Ω1, X

, 3.29

whereμX 0 sinceX is compact. Hence, in order to prove that lim_→₀μΩ1 0, we only need to prove that

lim→0h

Ω1, X lim

→0 sup

u∈Ω1

d_Xu 0. 3.30

Suppose that this is not true, then there exist β > 0, n → 0, and sequence {un}, un∈Ω1n, such that

d_Xun> β, 3.31

fornsuﬃciently large.

Since{un}is type I LP approximating sequence forGVQVI, it contains a subsequence {unk}conversing to a point ofX, which contradicts3.31.

For the converse, we know that, for every >0, the setΩ1is closed,X

>0Ω1, and lim

→0μΩ1 0. The theorem on Page. 412 in40,41can be applied, and one concludes that the setXis nonempty, compact, and

lim→0h

Ω1, X

0. 3.32

If{xn}is type I LP approximating sequence forGVQVI, then there exists a sequence {n}of positive real numbers decreasing to 0 such thatxn∈Ω1n, for everyn∈N. SinceX is compact and

nlim→∞d_Xxn≤ lim

n→∞h

Ω1n, X

0, 3.33

byProposition 2.4,GVQVIis type I LP well-posed.

bThe proof is Similar to that ofa, and it is omitted here. This completes the proof.

Definition 3.4. i LetZbe a topological space, and let Z₁ ⊆ Zbe nonempty. Suppose that h:Z → R¹∪ {∞}is an extended real-valued function.his said to be level-compact onZ1

if, for anys∈R¹, the subset{z∈Z1:hz≤s}is compact.

ii Let X be a finite-dimensional normed space, and let Z₁ ⊂ Z be nonempty. A functionh:Z → R¹∪ {∞}is said to be level-bounded onZ1ifZ1is bounded or

z∈Z1,z →∞lim hz ∞. 3.34

Now we establish some suﬃcient conditions for type Iresp., generalized I typeLP well-posedness ofGVQVI.

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Proposition 3.5. Suppose that the solution setXof GVQVIis nonempty and set-valued mapSis l.s.c. and closed onX₁, the set-valued map T is u.s.c. and compact-valued onX₁. Suppose that one of the following conditions holds:

(i) there exists 0< δ₁≤δ₀such thatX₁δ1is compact, where

X₁δ1 {x∈X₁∩X₂ :d_X₀x≤δ₁}; 3.35 (ii) the functionfdefined by2.15is level-compact onX1∩X2;

(iii)Xis finite-dimensional and

x∈X1∩Xlim2,x →∞max

fx, dX0x ∞, 3.36

wherefis defined by2.15;

(iv) there exists 0< δ1 ≤ δ0such thatf is level-compact onX1δ1defined by3.35. Then GVQVIis type I LP well-posed.

Proof. First, we show that each ofi,ii, andiiiimpliesiv. Clearly, either ofiandii impliesiv. Now we show thatiiiimpliesiv. Indeed, we need only to show that, for any t∈R¹, the set

A

x∈X₁δ1:fx≤t 3.37

is bounded sinceXis finite-dimensional space and the functionfdefined by2.15is l.s.c. on X1 and thusAis closed. Suppose to the contrary that there existst ∈R¹and{x_n} ⊆X1δ1 such thatx_n → ∞andfx_n≤t. From{x_n} ⊆X₁δ1, we haved_X₀x_n≤δ₁.

Thus,

max f

x_n , dX0

x_n ≤max{t, δ1}, 3.38

which contradicts3.36.

Therefore, we only need to we show that ifivholds, thenGVQVIis type I LP well- posed. Let{xn}be a type I LP approximating solution sequence forGVQVI. Then, there exist{n} ⊆R¹with_n → 0 andz_n ∈Txnsuch that2.3,2.4, and2.5hold. From2.3 and2.4, we can assume without loss of generality that{xn} ⊆ X₁δ1. ByLemma 2.9, we can assume without loss of generality that{xn} ⊆ {x ∈ X1δ1 : fx ≤ 1}. By the level- compactness off onX1δ1, we can find a subsequence {xnj}of{xn}andx ∈ X1δ1such thatx_n_j → x. Taking the limit in2.3 withx_nreplaced byx_n_j, we havex∈X₀. SinceSis closed and2.4holds, we also havex∈Sx.

Furthermore, from the u.s.c. ofT at xand the compactness ofTx, we deduce that there exist a subsequence{znj}of{zn}and somez∈Txsuch thatznj → z. From this fact, together with2.5, we have

z, x−x

∈ −/ intC, ∀x∈Sx. 3.39 Thus,x∈X.

The next proposition can be proved similarly.

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Proposition 3.6. Suppose that the solution setXof GVQVIis nonempty and set-valued mapSis l.s.c. and closed onX₁, the set-valued map T is u.s.c. and compact-valued onX₁. Suppose that one of the following conditions holds:

(i) there exists 0< δ1≤δ0such thatX2δ1is compact, where X2δ1

x∈X1∩X2 :dK

gx

≤δ1 ; 3.40

(ii) the functionfdefined by2.15is level-compact onX₁∩X₂; (iii)Xis finite-dimension and

x∈X1∩Xlim2,x →∞ max

fx, dKgx ∞, 3.41

wherefis defined by2.15,

(iv) there exists 0< δ₁ ≤ δ₀such thatf is level-compact onX₂δ1defined by3.40. Then GVQVIis generalized type II LP well-posed.

Remark 3.7. IfXis finite-dimensional, then the “level-compactness” condition in Propositions 3.1 and3.6can be replaced by “level boundedness” condition.

Remark 3.8. It is easy to see that the results in this paper unify, generalize and extend the main results in26–30and the references therein.

Acknowledgments

This research was supported by the National Natural Science Foundation of ChinaGrant no. 11171363 and Grant no. 10831009, the Natural Science Foundation of Chongqing Grant No. CSTC, 2009BB8240and the special fund of Chongqing Key LaboratoryCSTC 2011KLORSE01.

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