the Solution Sets of Set-Valued Weak Vector Variational Inequalities

Volume 2008, Article ID 435719,14pages doi:10.1155/2008/435719

Research Article

Painleve-Kuratowski Convergences for

the Solution Sets of Set-Valued Weak Vector Variational Inequalities

Z. M. Fang,¹S. J. Li,¹and K. L. Teo²

1College of Mathematics and Science, Chongqing University, Chongqing, 400044, China

2Department of Mathematics and Statistics, Curtin University of Technology, P.O. Box U1987, Perth, WA 6845, Australia

Correspondence should be addressed to S. J. Li,[email protected]

Received 16 July 2008; Revised 11 November 2008; Accepted 10 December 2008 Recommended by Donal O’Regan

Painleve-Kuratowski convergence of the solution sets is investigated for the perturbed set-valued weak vector variational inequalities with a sequence of mappings converging continuously. The closedness and Painleve-Kuratowski upper convergence of the solution sets are obtained. We also obtain Painleve-Kuratowski upper convergence when the sequence of mappings converges graphically. By virtue of a sequence of gap functions and a key assumption, Painleve-Kuratowski lower convergence of the solution sets is established. Some examples are given for the illustration of our results.

Copyrightq2008 Z. M. Fang et al. 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.

1. Introduction

Since the concept of vector variational inequalityVVIwas introduced by Giannessi1in 1980, many important results on various kinds of vector variational inequality problems have been established, such as existence of solutions, relations with vector optimization, stability of solution set maps, gap function, and duality theoriessee, e.g.,2–8and the references cited therein.

The stability analysis of the solution set maps for the parametricVVI problem is of considerable interest amongst researchers in the area. Some results on the semicontinuity of the solution set maps for the parametric VVI problem with the parameter perturbed in the space of parameters are now available in the literature. In 4, Khanh and Luu proved the upper semicontinuity of the solution set map for two classes of parametric vector quasivariational inequalities. In7, Li et al. established the upper semicontinuity property of the solution set map for a perturbed generalized vector quasivariational inequality problem

and also obtained the lower semicontinuity property of the solution set map for a perturbed classical scalar variational inequality. In 9, Cheng and Zhu investigated the upper and lower semicontinuities of the solution set map for a parameterized weak vector variational inequality in a finite dimensional Euclidean space by using a scalarization method. In 6, Li and Chen obtained the closedness and upper semicontinuity of the solution set map for a parametric weak vector variational inequality under weaker conditions than those assumed in 9. Then, under a key assumption, they proved a lower semicontinuity result of the solution set map in a finite dimensional space by using a parametric gap function.

However, there are few investigations on the convergence of the sequence of mappings. In particular, almost no stability results are available for the perturbed VVI problem with the sequence of mappings converging continuously or graphically. It appears that the only relevant paper is 10, where Lignola and Morgan considered generalized variational inequality in a reflexive Banach space with a sequence of operators converging continuously and graphically and obtained the convergence of the solution sets under an assumption of pseudomonotonicity. Since the perturbedVVIproblem with a sequence of mappings converging is different from the parametric VVIproblem with the parameter perturbed in a space of parameters, these results do not apply to the parametric VVI problem with the parameter perturbed in a space of parameters. Thus, it is important to study Painleve-Kuratowski upper and lower convergences of the sequence of solution sets.

In passing, it is worth noting that some stability results are available for the vector optimization and vector equilibrium problems with a sequence of sets converging in the sense of Painleve-Kuratowskisee11–13. It is well known that the vector equilibrium problem is a generalization ofVVIproblem. However, if the results obtained for the vector equilibrium problem are to be applied to theVVIproblem, the required assumptions are on theVVI problem as a whole. There is no information about the conditions that are required on the functions defining theVVIproblem. Clearly, this is unsatisfactory. Our study of the stability properties for the perturbedVVIproblem with a sequence of converging mappings is under appropriate assumptions on the function defining theVVIproblem rather than on theVVI problem as a whole.

In this paper, we should establish Painleve-Kuratowski upper and lower convergences of the solution sets of the perturbed set-valued weak variational inequitySWVVIwith a sequence of converging mappings in a Banach space. We first discuss Painleve-Kuratowski upper convergence and closedness of the solution sets. To obtain Painleve-Kuratowski lower convergence of the solution sets, we introduce a sequence of gap functions based on the nonlinear scalarization function introduced by Chen et al. in 14 and a key assumption Hgimposed on the sequence of gap functions. Then, we obtain Painleve-Kuratowski lower convergence of the solution sets forSWVVI_n.

The rest of the paper is organized an follows. In Section 2, we introduce problems SWVVI and SWVVI_n, and recall some definitions and important properties of these problems. In Section 3, we investigate Painleve-Kuratowski upper convergence and the closedness of the solution sets. In Section 4, we introduce respective gap functions for problems SWVVI and SWVVI_n and then establish Painleve-Kuratowski lower convergence of the solution sets under a key assumption.

2. Preliminaries

LetXandYbe two Banach spaces and letLX, Ybe the set of all linear continuous mappings fromXtoY. The value of a linear mappingt∈LX, Yatx∈Xis denoted byt, x. LetC⊂Y

be a closed and convex cone with nonempty interior, that is, intC /∅. We define the ordering relations as follows.

For anyy₁, y₂∈Y,

y1≤intCy2⇐⇒y2−y1∈intC, y1/≤intCy2 ⇐⇒y2−y1/∈intC.

2.1

Consider the set-valued weak vector variational inequality SWVVI problem for findingx∈Kandt∈Txsuch that

t, y−x ∈Y\ −intC ∀y∈K, 2.2

whereK⊂Xis a nonempty subset andT :K → 2^LX,Yis a set-valued mapping.

For a sequence of set-valued mappingsTn : Kn → 2^LX,Y, we define a sequence of set-valued weak vector variational inequalitySWVVI_n problems for findingx_n ∈K_nand t_n∈T_nxnsuch that

tn, y−xn

∈Y\ −intC ∀y∈Kn, 2.3

whereKn⊂Xis a sequence of nonempty subsets.

We denote the solution sets of problemsSWVVIandSWVVI_nbyITandITn, respectively, that is,

IT

x∈K| ∃t∈Tx,s.t.t, y−x ∈Y\ −intC∀y∈K , ITn

xn∈Kn| ∃tn∈Tnxn,s.t.

tn, y−xn

∈Y\ −intC∀y∈Kn

.

2.4

Throughout this paper, we assume thatIT/∅andITn/∅. The stability analysis is to investigate the behaviors of the solution setsITandITn.

Now we recall some basic definitions and properties of problems SWVVI and SWVVI_n. For eachε >0 and a subsetA⊂ X, let the openε-neighborhood ofAbe defined asUA, ε {x ∈X | ∃a ∈ A,s.t.a−x < ε}. The notationBλ, δdenotes the open ball with centerλand radiusδ >0.

In the following, we introduce some concepts of the convergence of set sequences and mapping sequences which will be used in the sequel. Define

N_∞:{N⊂ N | N \N finite}

{subsequences ofNcontaining alln∈ Nbeyond somen}, N_∞:{N⊂ N |N infinite}

{all subsequences ofN},

2.5

whereNdenotes the set of all positive integer numbers andnis an integer inN.

Definition 2.1see11,15. LetXbe a normed space. A sequence of sets{Dn⊂X :n∈N}is said to converge in the sense of Painleve-KuratowskiP.K.toDi.e.,D_n −→^P.K. Dif

lim sup

n→ ∞ D_n⊂D⊂lim inf

n→ ∞ D_n 2.6

with

lim inf

n→ ∞ D_n:

x| ∃N∈ N_∞,∃xn∈D_nn∈Nwith x_n−→x , lim sup

n→ ∞ Dn:

x| ∃N∈ N∞,∃xn∈Dnn∈Nwith xn−→x

. 2.7

It is said that the sequence{Dn}upper converges in the sense of Painleve-Kuratowski toD if lim sup_{n→ ∞}Dn⊂D. Similarly, the sequence{Dn}is said to lower converge in the sense of Painleve-Kuratowski toDifD⊂lim infn→ ∞Dn.

Definition 2.2see15. A set-valued mappingS:X → 2^Y is outer semicontinuousoscat xif lim sup_x→xSx⊂Sxwith lim sup_x→xSx:

xn→xlim sup_{n→ ∞}Sxn.

On the other hand, it is inner semicontinuousiscatxif lim infx→xSx⊃Sxwith lim inf_x→xSx:

xn→xlim inf_{n→ ∞}Sxn.

The set-valued mapping is said to be continuous at x, written as Sx → Sx as x → xif it is both outer semicontinuous and inner semicontinuous.

Definition 2.3see15. LetS_n:X → 2^Y be a sequence of set-valued mappings andS:X → 2^Y be a set-valued mapping. It is said that the sequence{Sn}converges continuously toSat xif

lim sup

n→ ∞ S_nxn⊂Sx⊂lim inf

n→ ∞ S_nxn ∀sequencex_n−→x. 2.8 If {Sn} converges continuously to S at everyx ∈ X, then it is said that {Sn} converges continuously toSonX.

LetS:X → 2^Y be a set-valued map, the graph ofSis defined as gphS

x, u|u∈Sx

. 2.9

Applying set convergence theory to the graphs of the mappings, we obtain the graphical convergence of the sequence of mappings.

Definition 2.4see15. For a sequence of mappingsS_n :X → 2^Y, the graphical outer limit, denoted byg−lim sup_nS_n, is the mapping which has as its graph the set lim sup_ngphS_n:

gph

g−lim sup

n

S_n lim sup

n

gphS_n ,

g−lim sup

n

Sn x

u| ∃N∈ N_∞, xn−→N x, un−→N u, un∈Sn

xn

.

2.10

The graphical inner limit, denote byg−lim infnSn, is the mapping having as its graph the set lim inf_ngphS_n:

gph

g−lim inf

n Sn lim inf

n

gphSn

,

g−lim inf

n Sn x

u| ∃N∈ N∞, xn N

−→x, un N

−→u, un∈Sn

xn

.

2.11

If the outer and inner limits of the mappingsSnagree, it is said that their graphical limit,g− lim_nS_n, exists. In this case, the notationS_n →^g Sis used, and the sequence{Sn}of mappings is said to converge graphically toS. Clearly,S_n →^g S⇔gphS_n −→^P.K. gphS.

Proposition 2.5see15. For any sequence of mappingsSn:X → 2^Y, it holds that

g−lim inf

n Sn x

{xn→x}

lim inf

n→ ∞ Sn

xn

,

g−lim sup

n

Sn x

{xn→x}

lim sup

n→ ∞ Sn

xn

,

2.12

where the unions are taken over all sequencesxn → x. Thus, the sequence{Sn}converges graphically toSif and only if, at each pointx∈X, it holds that

{xn→x}

lim sup

n→ ∞ S_n x_n

⊂Sx⊂

{xn→x}

lim inf

n→ ∞ S_n x_n

. 2.13

FromProposition 2.5andDefinition 2.3, the following proposition follows readily.

Proposition 2.6. Let Sn : X → 2^Y be a sequence of set-valued mappings andS : X → 2^Y be a set-valued mapping. Then, the sequence{Sn}outer converges graphically toSif and only if{Sn} outer converges continuously toS, that is,

g−lim sup

n

Sn⊂S⇐⇒lim sup

n

Sn

xn

⊂Sx for anyx∈X, ∀sequencesxn−→x. 2.14

Definition 2.7 see 10. Given a sequence of mappingsSn, {Sn} is said to be uniformly bounded if for any sequencexncontained in a bounded set, there exists a positive numberk such that for any sequenceu_nwithu_n∈S_nxnfor alln∈N, it holds that

un≤k ∀n∈N. 2.15 Proposition 2.8see16. For any fixede∈intC, y∈Y,r ∈R, and the nonlinear scalarization functionξ_e:Y → Rdefined byξ_ey min{t∈R:y∈te−C}:

iξ_eis a continuous and convex function onY; iiξey< r⇔y∈re−intC;

iiiξ_ey≥r⇔y /∈re−intC.

3. Painleve-Kuratowski upper convergence of the solution sets

In this section, our focus is on the Painleve-Kuratowski upper convergence and the closedness of the solution sets.

Theorem 3.1. Suppose that

iT_nouter converges continuously toT, that is, lim sup

n→ ∞ T_n x_n

⊂Tx for any sequence x_n

with x_n−→x; 3.1

iiK_n −→^P.K. K;

iiiTnare uniformly bounded.

Then, lim sup_{n→ ∞}ITn⊂IT, that is to say for any subsequence{xnk}of solutions toSWVVI_n, ifxnk → x, thenxis a solution toSWVVI.

Proof. The proof is listed on contradiction arguments. On a contrary, suppose that ∃x ∈ lim sup_{n→ ∞}ITnbutx /∈IT.

Fromx∈lim sup_{n→ ∞}ITn, we havexlimk→ ∞xnk, wherexnk ∈ITnkand{nk}is a subsequence ofN. Then, there existstnk ∈Tnkxnksuch that

tnk, z−xnk

∈Y \ −intC ∀z∈Knk. 3.2

SinceK ⊂ lim inf_{n→ ∞}K_n, it is clear that for anyz ∈ K, there exists a sequence{znk}with {znk} ⊂Knk andznk → z, ask → ∞. Thus,

tnk, znk−xnk

∈Y\ −intC. 3.3

Since lim sup_{n→ ∞}K_n ⊂ Kandx_nk ∈K_nk, we havex ∈K. Now, we note thatx /∈IT. Thus, for allt∈Tx, there existsz_t∈Ksuch that

t, zt−x ∈ −intC. 3.4

From the uniform boundedness ofTn, we may assume, without loss of generality, that t_nk → t₀though a subsequence of{tnk}if necessary. Byi, we gett₀∈Tx. Thus,

t_nk, z_nk−x_nk

−→

t₀, z−x

, ask−→∞. 3.5

It follows from3.3and the closedness ofY\ −intCthat t₀, z−x

∈Y \ −intC ∀z∈K, 3.6

which is a contradiction to3.4. This completed the proof.

Remark 3.2. LetX EandY E^∗, whereEis a reflexive Banach space andE^∗ is its dual.

If we take C R, SWVVI_n reduce to the generalized variational inequality problems with perturbed operators GVI_n considered in 10, Section 3. The convergence for the solution sets ofGVI_nwas studied under the the pseudomonotonicity assumption in10.

Furthermore, if T and Tn are vector-valued mappings, then SWVVI_n reduce to VI_n considered in10, Section 2. We also notice that the Painleve-Kuratowski upper convergence of the solution sets ofSWVVI_nis obtained under weaker assumptions than these assumed in10, Proposition 2.1for obtaining convergence of the solution sets.

FromProposition 2.5andTheorem 3.1, we obtain readily the following corollary.

Corollary 3.3. Suppose that

iT_nouter converges graphically toT, written asg−lim sup_nT_n⊂T,that is, lim sup

n→ ∞

gphT_n

⊂gphT; 3.7

iiK_n −→^P.K. K;

iii{Tn}is uniformly bounded.

Then, lim sup_{n→ ∞}ITn⊂IT.

Remark 3.4. LetXY R^m. Then, problemsSWVVI_nreduce to the generalized variational inequalities with perturbed operators considered in10, Proposition 3.1and the convergence was obtained under the assumption that the operators converge graphically.

Theorem 3.5. Suppose that

iTis osc onK, that is, for allx∈K, lim sup_{n→ ∞}Txn⊂Txfor any sequencex_n → x;

iiKandTKare compact sets.

Then,ITis a compact set.

Proof. First, we prove thatITis a closed set. Take any sequencex_n ∈ ITwithx_n → x.

Then, there existst_n∈Txnsuch that t_n, z−x_n

∈Y \ −intC ∀z∈K. 3.8

It follows from the compactness ofKthatx∈K. Suppose thatx /∈IT, we have

∀t∈Tx, ∃z0 ∈K,s.t.

t, z₀−x

∈ −intC. 3.9

SinceTKis a compact set, without loss of generality, we assume that there exists a t₀ such thatt_n → t₀. Thus, we havetn, z−x_n → t0, z−x. Byi, we get at₀ ∈Tx. It follows from3.8and the closedness ofY\ −intCthat

t0, z−x

∈Y \ −intC ∀z∈K, 3.10

which contradicts with3.9. Hence,x0∈ITandITis a closed set. Next, it follows from IT⊂Kand the compactness ofKthatITis a compact set. The proof is completed.

Similarly, we have the following result.

Theorem 3.6. For anyn, suppose that i T_nis osc onK_n, that is,∀x∈K_n

lim sup

m→ ∞ Tn

xm

⊂Tnx for any sequencexm−→x; 3.11

ii K_nandT_nKnare compact sets.

Then,ITnis a compact set.

4. Painleve-Kuratowski lower convergence of the solution sets

In this section, we focus on the lower convergence of the solution sets. Assume thatKand Knare compact sets and that for eachx∈X,TxandTnxare compact sets. Letg:K → R andgn:Kn → Rbe functions defined by

gx max

t∈Tx min

y∈K ξe

t, y−x

, x∈K, gnxn max

tn∈Tnxn min

y∈Kn

ξe

tn, y−xn

, xn∈Kn. 4.1

SinceK, Kn, Tx, andTnxare compact sets andξe·is continuous,gxandgnxnare well defined.

Proposition 4.1. igx≤0 for allx∈K;

iignxn≤0 for allxn ∈Kn; iiigx0 0 if and only ifx₀ ∈IT;

ivgnxn 0 if and only ifxn∈ITn. Proof. Define

gx, t min

y∈Kξe

t, y−x

, x∈K, t∈Tx. 4.2

We first prove thatgx, t≤0. On a contrary, we suppose that this is false. Then, there existx∈Kandt∈Txsuch thatgx, t>0. Thus,

0< gx, t≤ξe

t, y−x

∀y∈K, 4.3

which is impossible whenyx. Therefore, gx max

t∈Txgx, t≤0 ∀x∈K. 4.4

By the same taken, we can show that g_nxn max

tn∈Tnxnmin

y∈Kn

ξ_e

t_n, y−x_n

≤0 ∀xn∈K_n. 4.5

On the other hand, ifgx0 0, then there exists at0 ∈ Tx0such thatgx0, t0 0, that is,

miny∈Kξe

t0, y−x0

0, x0∈K. 4.6

FromProposition 2.8,4.6is valid if and only if for anyy∈K, ξe

t0, y−x0

≥0. 4.7

Clearly,4.7holds if and only if for anyy ∈ K,t0, y−x₀ ∈Y \ −intC, that is,x₀ ∈IT. This proves thatiiiholds.

Similarly, we can show thativholds.

The functionsg_nare called the gap functions forSWVVI_n if propertiesiiandiv ofProposition 4.1are satisfied.

In view of hypothesisHgof6,17,18, we introduce the following key assumption:

Hg: given the sequence{Tn}for any > 0, there exist anα > 0 and an n such that g_nxn≤ −αfor alln > nand for allx_n∈K_n\UITn, .

Geometrically, the hypothesis Hg means that given a sequence of mappings{Tn}, we can find for any small positive number >0, a small positive numberα >0 and a large- enough positive numbern >0 such that for alln > n, if a feasible pointx_nis away from the solution setsITnby distance of at least, then the values of all gap functions forSWVVI_n is less than or equal to at least some “−α.”

To illustrate assumptionHg, we give the following example.

Example 4.2. Let

XR, Y R²,

Tnx

⎛

⎜⎜

⎝

1

1,1 1 nx²

⎞

⎟⎟

⎠,

Tx

1 1,1x²

,

KKn 0,1, CR².

4.8

Consider problemsSWVVI_n. From direct computation, we obtainITn {0}. To check assumptionHg, we takee 1,1^T ∈intR². Then,

gn

xn

max

tn∈Tnxn min

y∈Kn

ξe

tn, y−xn

max

tn∈Tnxn min

y∈Kn

max1≤i≤2

tn, y−xn

i

max

zn∈1,11/nxn²min

y∈Kn

max

y−xn, zn

y−xn

−xn.

4.9

For any given 0 < , we takeα > 0 and N 1. Then, for all n > Nand for allxn ∈ Kn\ ∪ITn, , we havegnxn −xn≤ −α. Hence, assumptionHgis valid.

Lemma 4.3. Suppose that

iT_ninner converges continuously toT, that is, Tx⊂lim inf

n→ ∞ Tnxn for any sequence{xn}with xn−→x; 4.10

iiK_n −→^P.K. K;

iii_∞

n1Knis a compact set.

Then, for any δ ≥ 0, x₀ ∈ K and sequence {xn} with x_n ∈ K_n and x_n → x₀, there exists a subsequence{xnl}of{xn}andN >0 such thatg_nlxnl≥gx0−δfor alll≥N.

Proof. Letg:K×LX, Y → Rbe a function defined by

gx, t min

y∈Kξ_e

t, y−x

, x∈K, t∈Tx. 4.11

From the continuity ofξ_et, y−xwith respect tox, t, y, the compactness ofKand19, Chapter 3, Section 1, Proposition 23, we have thatgx, t is continuous with respect tox, t.

Thus, from the compactness ofTx0, there exists at₀∈Tx0such that g

x0

max

t∈Tx0min

y∈Kξe

t, y−x0

max

t∈Tx0gx 0, t min

y∈Kξe

t0, y−x0

. 4.12

From assumptioni, there exists a sequence{tn}satisfyingtn∈Tnxnsuch that

t_n−→t₀. 4.13

It follows from the compactness ofKnthat there exists{yn}withyn∈Knsuch that miny∈Kn

ξ_e

t_n, y−x_n ξ_e

t_n, y_n−x_n

. 4.14

Since_∞

n1Knis compact, we assume, without loss of generality, thatyn → y0. Thus, it follows fromiithaty₀∈K.Consequently,

nlim→ ∞ξ_e

t_n, y_n−x_n ξ_e

t₀, y₀−x₀

≥min

y∈Kξ_e

t₀, y−x₀ g

x₀

. 4.15

So, for anyδ >0, there exists anN >0 such thatξ_etn, y_n−x_n≥gx0−δfor alln≥N. By 4.14, we have

gn

xn

max

tn∈Tnxnmin

y∈Kn

ξe

tn, y−xn

≥min

y∈Kn

ξ_e

t_n, y−x_n ξ_e

t_n, y_n−x_n

≥g x₀

−δ ∀n≥N.

4.16

Hence, the result holds.

SetT₀TandK₀K. We have the following lemma.

Lemma 4.4. Suppose that forn0,1,2, . . . , Tnis osc onKn, that is, forn0,1,2, . . . , lim sup

m→ ∞ T_n x_m

⊂T_nx for any sequence x_m

withx_m−→x. 4.17

Then,IT⊂lim inf_{n→ ∞}ITnif and only if for all >0,∃N >0 such thatIT⊂UITn, for alln > N.

Proof. We assumeIT ⊂ lim inf_{n→ ∞}ITn, but there exists an > 0 such that for allN >0, there exists an N_n ≥ N satisfying IT/⊂UITNn, . Then, there exists a sequence {xn} withxn ∈IT, butxn/∈UITNn, . FromTheorem 3.5, we note thatITis a compact set.

Without loss of generality, we assumex_n → xandx ∈ IT. Thus, for any sequence {yn} satisfyingy_n → ywith y_n ∈ ITn, we have yNn −x_n ≥ > 0. Lettingn → ∞, we get y−x ≥ >0. Therefore, there does not exist any sequenceyn ∈ITnsatisfyingyn → x.

This is a contradiction toIT⊂lim infn→ ∞ITn.

Conversely, suppose that for any > 0,∃N > 0 such thatIT ⊂ UITn, for all n ≥ N. FromTheorem 3.6, we note thatITnis compact for alln. Thus, for anyx ∈ IT, there existsxn∈ITnsuch thatxn−xdx, ITn≤for alln≥N. So, we havexn → x andIT⊂lim inf_{n→ ∞}ITn. Therefore, the result of the lemma follows readily.

Now, we are in a position to state and prove our main result in the following theorem.

Theorem 4.5. Suppose that assumptionHgholds and that the following conditions are satisfied:

iT_nis osc onK_nforn0,1,2, . . . ,that is, forn0,1,2, . . . , lim sup

m→ ∞ Tn

xm

⊂Tnx for any sequence xm

with xm−→x; 4.18

iiTninner converge continuously toT, that is, Tx⊂lim inf

n→ ∞ Tn

xn

for any sequence xn

with xn−→x; 4.19

iii_∞

n1K_nis a compact set;

ivK_n −→^P.K. K.

Then,IT⊂lim infn→ ∞ITn.

Proof. We prove the result via contradiction. On a contrary, we assume, byLemma 4.4, that there exists an >0 such that for anyN >0, we haveNn ≥Nsatisfying

IT/⊂U I

TNn

,

, 4.20

that is, there exists a sequence{xNn}satisfying x_Nn ∈IT\U

I T_Nn

,

. 4.21

From the compactness ofIT, we can assume, without loss of generality, thatx_Nn → x ∈ IT. Then, there exists anN1 > 0 such thatxNn −x ≤ /4 foralln > N1. It is clear that Bx, /Nn∩K /∅for any positive integern. SinceK⊂lim inf_{n→ ∞}K_n, there exist a sequence {yNn} ⊂KNn satisfyingyNn → x. Then, there exists anN2>0 such thatyNn ∈Bx, /Nn∩ K_Nn for alln > N₂.

Now, we note that yNn/∈UITNn, /4. Otherwise, there would exist a sequence {zNn} withz_Nn ∈ ITNn such thatyNn −z_Nn < /4. Thus, forN₀ max{N1, N₂}, we have

xNn−zNn≤xNn−xx−yNnyNn −zNn≤

4

N_n

4 < ∀n > N0. 4.22 This implies thatxNn ∈UITNn, , which contradicts with4.21. Thus,

y_Nn ∈K_Nn\U

I T_Nn

,

4 . 4.23

By hypothesisHg, there exist, for any > 0, anα > 0 and anN such that for all n > Nand for allx∈K_n\UITn, ,g_nx≤ −α. In particular, it follows from4.23that

gNn

yNn

≤ −α fornlarge enough. 4.24

By virtue ofLemma 4.3, there exists, for anyδ >0, a subsequence{yN_nk}of{yNn}andN > 0 such that

gN_nk

yN_nk

≥gx−δ ∀k >N. 4.25

We can takeδsuch that−αδ <0. Thus, gx≤g_Nnk

y_Nnk

δ≤ −αδ <0, 4.26

that is,

t∈Txmax min

y∈Kξe

t, y−x

<0. 4.27

So, for anyt∈Tx, miny∈Kξet, y−x<0. Thus, there exists ay∈Ksuch that ξe

t, y−x

<0. 4.28

Consequently, byProposition 2.8, we havet, y−x ∈ −intC, which shows thatx /∈IT. This contradicts withx∈IT. Therefore, our result follows readily.

Now, we explain the applicability ofTheorem 4.5through an example.

Example 4.6. ConsiderExample 4.2. It follows from a direct computation thatITn IT {0}. It is easy to testify that assumption Hg holds and so are conditions i–v of Theorem 4.5. Obviously, the solution sets of problemSWVVI_nlower converge in the sense of Painleve-Kuratowski.

Acknowledgment

This research was partially supported by the National Natural Science Foundation of China Grants no. 10871216 and no. 60574073.

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