Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2012, Article ID 582792,17 pages 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, jwpeng6@yahoo.com.cn Received 28 October 2011; Accepted 14 December 2011 Academic Editor: Yeong-Cheng Liou

Copyright © 2012 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 in [1] and applications have been given. Chen and Cheng [2] studied 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 Yao [15], Daniilidis and Hadjisavvas [16], Yang and Yao [17], and Oettli and Schlager [18] studied the generalized vector variational inequality and obtained some existence results. Chen and Li [19] and Lee et al. [20] introduced and studied the generalized vector quasi-variational 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 Polyak [22] . The study of generalized Levitin-Polyak well-posedness for convex scalar optimization problems with functional constraints originates from Konsulova and

Revalski [23]. Recently, this research was extended to nonconvex optimization problems with abstract set constraints and functional constraints (see [24]), 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 see [31-37] and 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. In Section 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 the (generalized) Levitin-Polyak well-posedness of the generalized vector quasi-variational inequality problems. The results in this paper unify, generalize, and extend some known results in [26-30] .

2. Preliminaries

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

Let (X, || ■ ||) be a normed space equipped with norm topology, and let (Z,d1) be a metric space. Let X1 c X, K c Z be nonempty and closed sets. Let Y be a locally convex space ordered by a nontrivial closed and convex cone C with nonempty interior int C, that is, y1 < y2 if and only if y2 - y1 e C for any y1, y2 e Y. Let L(X, Y) be the space of all the linear continuous operators from X to Y. Let T : X1 ^ 2L(X,Y) and S : X1 ^ 2Xl be strict set-valued mappings (i.e., T(x) / 0 and S(x) / 0, for all x e X1), and let g : X1 ^ Z be a continuous vector-valued mapping. We denote by (z, x) the value z(x), where z e L(X,Y), x e X1. Let X0 = {x e X1 : g(x) e K} be nonempty. We consider the following generalized vector quasi-variational inequality problem with functional constraints and abstract set constraints.

Find x e X0 such that x e S(x) and there exists z e T(x) satisfying

(z,x - x) / - int C, Vx e S(x). (GVQVI)

Denote by X the solution set of (GVQVI).

Let Z1, Z2 be two normed spaces. A set-valued map F from Z1 to 2Z2 is

(i) closed, on Z3 c Z1, if for any sequence {xn} C Z3 with xn ^ x and yn e F(xn) with yn ^ y, one has y e F(x);

(ii) lower semicontinuous (l.s.c. in short) at x e Z1,if {xn} c Z1, xn ^ x,andy e F(x) imply that there exists a sequence {yn} c Z2 satisfying yn ^ y such that yn e F(xn) for n sufficiently large. If F is l.s.c. at each point of Z1, we say that F is l.s.c. on Z1;

(in) upper semicontinuous (u.s.c. in short) at x e Z1, if for any neighborhood V of F(x), there exists a neighborhood U of x such that F(x') c V, for all X e U. If F is u.s.c. at each point of Z1, we say that F is u.s.c. on Z1.

It is obvious that any u.s.c. nonempty closed-valued map F is closed. Let (P, d) be a metric space, P1 c P, and x e P. We denote by dP1 (x) = inf{d(x, p) : p e P1} the distance from the point x to the set P1. For a topological vector space V, we denote by V* its dual space. For any set O c V, we denote the positive polar cone of O by

O* = {A e V* : A(x) > 0, Vx e O}. (2.1)

Let e e int C be fixed. Denote

C*0 = {A e C : A(e) = 1}. (2.2)

Definition 2.1. (i) A sequence {xn} c X1 is called a type I Levitin-Polyak (LP in short) approximating solution sequence if there exist {en} c R+ = {r > 0|r is a real number} with en ^ 0 and zn e T(xn) such that

dx0 (x (2.3)

xn e S(xn), (2.4)

(zn,x - xn) + ene / - int C, Vx e S(xn). (2.5)

(ii) {xn} c X1 is called a type II LP approximating solution sequence if there exist {en} c R+ with en ^ 0 and zn e T(xn) such that (2.3)-(2.5) hold, and, for any z e T(xn), there exists w(n, z) e S(xn) satisfying

(z,w(n,z) - xn) - ene e-C. (2.6)

(iii) {xn} c X1 is called a generalized type I LP approximating solution sequence if there exist {en} c R+ with en ^ 0 and zn e T(xn) such that

dK(g(xn)) < en (2.7)

and (2.4), (2.5) hold.

(iv) {xn} c X1 is called a generalized type II LP approximating solution sequence if there exist {en} c R+ with en ^ 0, zn e T(xn) such that (2.4), (2.5), and (2.7) hold, and, for any z e T(xn), there exists w(n,z) e S(xn) such that (2.6) holds.

Definition 2.2. (GVQVI) is said to be type I (resp., type II, generalized type I, generalized type II) LP well-posed if the solution set X of (GVQVI) is nonempty, and, for any type I (resp., type II, generalized type I, generalized type II) LP approximating solution sequence {xn}, there exists a subsequence {xnj} of {xn} and x e X such that xnj ^ x.

Remark 2.3. (i) It is clear that any (generalized) type II LP approximating solution sequence is a (generalized) type I LP approximating solution sequence. Thus, (generalized) type I LP well-posedness implies (generalized) type II LP well-posedness.

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

(iii) Suppose that g is uniformly continuous functions on a set

Xi(6o) = {x e Xi: dX0 (x) < 60}, (2.8)

for some 60 > 0. Then generalized type I (resp., generalized type II) LP well-posedness of (GVQVI) implies its type I (resp., type II) LP well-posedness.

(iv) If Y = R1, C = R|, then type I (resp., type II, generalized type I, generalized type II) LP well-posedness of (GVQVI) reduces 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 = R|, S(x) = X0 for all x e Xi, then type I (resp., type II, generalized type I, generalized type II) LP well-posedness of (GVQVI) reduces to type I (resp., type II, generalized type I, generalized type II) LP well-posedness of the generalized variational inequality problem defined by Huang, and Yang [27] which contains as special cases for the type I (resp., type II, generalized type I, generalized type II) LP well-posedness of the variational inequality problem in [26] .

(v) If S(x) = X0 for all x e X1, then type I (resp., type II, generalized type I, generalized type II) LP well-posedness of (GVQVI) reduces 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].

(vi) If the set-valued map T is replaced by a single-valued map F, then type I (resp., type II, generalized type I, generalized type II) LP well-posedness of (GVQVI) reduces to type I (resp., type II, generalized type I, generalized type II) LP well-posedness of the vector quasivariational inequality problems defined by Zhang et al. [30].

Consider the following statement:

X = 0 and for any type I ( resp., type II, generalized type I, generalized type II

LP approximating solution sequence {xn}, we have dx(xn) —> 0

Proposition 2.4. If (GVQVI) is type I (resp., type II, generalized type I, generalized type II) LP well-posed, then (2.9) holds. Conversely if (2.9) holds and X is compact, then (1) is type I (resp., type II, generalized type I, generalized type II) LP well-posed.

The proof of Proposition 2.4 is elementary and thus omitted.

To see the various LP well-posednesses of (1) are 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 f (x)

s.t. x e X1 (P)

g (x) e K,

where X1 c X1 is nonempty and f: X1 ^ R1 u{+<x>} is proper. The feasible set of (P) is X0, where X^ = {x e X'1 : g(x) e K}. The optimal set and optimal value of (P) are denoted by X' and v, respectively. Note that if Dom(/) n X'0 = 0, where

Dom(f) = {x e X1 : f (x) < +to}, (2.10)

then v < +<x>. In this paper, we always assume that v > -to.

Definition 2.5. (i) A sequence {xn} c X'1 is called a type I LP minimizing sequence for (P) if

limsupf (xn) < v, (2.11)

dX0 (xn) 0. (2.12)

(ii) {xn} cX'1 is called a type II LP minimizing sequence for (P) if

lim f (xn) = v (2.13)

and (2.12) hold.

(iii) {xn} c X1 is called a generalized type I LP minimizing sequence for (P) if

dK(g(xn)) 0 (2.14)

and (2.11) hold.

(iv) {xn} c X'1 is called a generalized type II LP minimizing sequence for (P) if (2.13) and (2.14) hold.

Definition 2.6. (P) is said to be type I (resp., type II, generalized type I, generalized type II) LP well-posed if the solution set X of (P) is 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} and x e X such that xnj ^ x.

The Auslender gap function for (GVQVI) is defined as follows:

f (x) = inf sup inf A((z,x - x')), Vx e X1.

zeT (x)xi£S(x)AeC*° (2.15)

Let X2 c X be defined by

X2 = {x e X | x e S(x)}. (2.16)

In the rest of this paper, we set X1 in (P) equal to X1 n X2. Note that if S is closed on X1, then X1 is closed.

Recall the following widely used function (see, e.g., [38])

|: Y R1 : minjt e R1 : y - te e-C}. (2.17)

It is known that £ is a continuous, (strictly) monotone (i.e., for any yi,y2 € Y, yi - y2 € C implies that £(y1) > £(y2) and (y1 - y2 € int C implies that £(y1) > £(y2)), subadditive and convex function. Moreover, it holds that £(te) = t, for all t € R1 and

l(y) = supiec*0l(У), for all y € Y.

Now we given some properties for the function f defined by (2.15).

Lemma 2.7. Let the function f be defined by (2.15), and let the set-valued map T be compact-valued on X1. Then

(i) f (x) > 0, for all x € X1 ;

(ii) for any x € X0, f (x) = 0 if and only if x € X.

Proof. (i) Let x € X'1. Suppose to the contrary that f (x) < 0. Then, there exists a 6 > 0 such that f (x) < -6. By definition, for 6/2 > 0, there exists a z € T(x), such that

sup inf X((z,x - x')) < f (x) + -< --< 0. (2.18)

x'eS(x)AeC*° 2 2

Thus, we have

infl((z,x - x')) < 0, Vx' € S(x), (2.19)

which is impossible when x' = x. This proves (i). (ii) Suppose that x € X'0 such that f (x) = 0.

Then, it follows from the definition of X'0 that x € S(x). And from the definition of f (x) we know that there exist zn € T(x) and 0 < en ^ 0 such that

inf X((zn,x - x')) < f (x) + en = en, Vx'€ S(x), (2.20)

leC*0 v ' '

that is,

¿,((zn,x'- x» >-en, Vx'€ S(x). (2.21) By the compactness of T(x), there exists a sequence [znj} of {zn} and some z € T(x) such that

zn, z. (2.22) This fact, together with the continuity of £ and (2.21), implies that

¿,((z,x'-x» > 0, Vx' € S(x). (2.23)

It follows that x X.

Conversely, assume that x e X. It follows from the definition of X that x e S(x). Suppose to the contrary that f (x) > 0. Then, for any z e T(x),

suP 1inf0A(<z,x - x')) > a (2.24)

x'eS(x)AeC*° v '

Thus, there exist 6 > 0 and x0 e S(x) such that

inf 1((z,x- x0)) > 6.

AeC«°

(2.25)

It follows that

l((z,x0 - x)) <-6 < 0.

(2.26)

As a result, we have

(z,x0 - x) e -int C.

(2.27)

This contradicts the fact that x e X. So, f (x) = 0. This completes the proof.

Lemma 2.8. Let f be defined by (2.15). Assume that the set-valued map T is compact-valued and u.s.c. on X1 and the set-valued map S is l.s.c. on X1. Then f is l.s.c. function from X1 to R1 u {+to}. Further assume that the solution set X of (GVQVI) is nonempty, then Dom(f) = 0.

Proof. First we show that f (x) > -to, for all x e X1. Suppose to the contrary that there exists xo e X1 such that f (x0) = -to. Then, there exist zn e T(x0) and {Mn} c R+ with Mn ^ +to such that

sup inf X((zn,x0 - x')) < -Mn.

x'eS(x0)XeC"°

(2.28)

l({zn,x'- x0» > Mn, VX e S(x0). (2.29)

By the compactness of T(x0), there exist a sequence {znj} c {zn} and some z0 e T(x0) such that znj ^ z0. This fact, together with (2.29) and the continuity of £ on Y, implies that

K(z0,X - X0)) > +to, VX e S(X0) (2.30)

which is impossible, since I is a finite function on Y.

Second, we show that f is l.s.c. on Xi. Let a e R1. Suppose that {xn} c Xi satisfies f (xn) < a, for all n, and xn ^ x0 e X1. It follows that, for each n, there exist zn e T(xn) and 0 <6n 0 such that

-&({zn,y - xn)) < a + 6n, Vy e S(xn).

(2.31)

For any X e S(x0), by the l.s.c. of S, we have a sequence {y„} with {yn} e S(x„) converging to x' such that

-i{(z„,y„ - x„)) < a + 8„. (2.32)

By the u.s.c. of T at x0 and the compactness of T(x0), we obtain a subsequence {z„j} of {z„} and some z0 e T(x0) such that z„j — z0. Taking the limit in (2.32) (with „ replaced by „j), by the continuity of I, we have

-l((z0>x' - x0)) < a, Vx' e S(x0). (2.33)

It follows that f (x0) = infzeT(x0)supx,eS(xo) - l({z,x' - x0)) < a. Hence, f is l.s.c. on X1. Furthermore, if X / 0, by Lemma 2.7, we see that Dom(f) = 0. □

Lemma 2.9. Let the function f be defined by (2.15), and let the set-valued map T be compact-valued on X1. Then,

(i) {x„} c X1 is a sequence such that there exist {e„} c R+ with e„ — 0 and z„ e T(x„) satisfying (2.4) and (2.5) if and only if {x„} c X'i and (2.11) hold with v = 0,

(ii) {x„} c X1 is a sequence such that there exist {e„} c R+ with e„ ^ 0 and z„ e T(x„) satisfying (2.4) and (2.5), and for any z e T (x„), there exists w(„,z) e S(x„) satisfying (2.6) if and only if {x„} c X1 and (2.13) hold with v = 0.

Proof. (i) Let {x„} c X1 be any sequence if there exist {e„} c R+ with e„ ^ 0 and z„ e T(x„) satisfying (2.4) and (2.5), then we can easily verify that

{x„}c X1 > f (x„) < e„. (2.34)

It follows that (2.11) holds with v = 0.

For the converse, let {x„} c X'1 and (2.11) hold with v = 0. We can see that {x„} c X1 and (2.4) hold. Furthermore, by (2.11), we have that there exists {e„ } c R+ with e„ —> 0 such that f (x„) < e„. By the compactness of T(x„), we see that for every „ there exists z„ e T(x„) such that

l{(z„,x'- x„)) > -e„> Vx' e S(x„). (2.35)

It follows that for every „ there exists z„ e T(x„) such that (2.5) holds. (ii) Let {x„} c X1 be any sequence we can verify that

liminff (x„) > 0 (2.36)

holds if and only if there exists {a„} c R+ with a„ — 0 and, for any z e T(x„), there exists w(„> z) e S(x„) such that

(z,w(n, z) - x„) - ane e -C.

(2.37)

From the proof of (i), we know that limsup^^f (xn) < 0 and {xn} c X'1 hold if and only if {xn} c X1 such that there exist {¡n} c R+ with ¡n ^ 0zn e T(xn) satisfying (2.4) and (2.5) (with en replaced by ¡n). Finally, we let en = max{an,fin} and the conclusion follows. □

Proposition 2.10. Assume that X / 0 and T is compact-valued on X1. Then

(i) (GVQVI) is 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 with f (x) defined by (2.15).

(ii) If (GVQVI) is type I (resp., type II) LP well-posed, then (P) is type I (resp., type II) LP well-posed with f (x) defined by (2.15).

Proof. Let f (x) be defined by (2.15). Since X / 0, it follows from Lemma 2.7 that X e X is a solution of (GVQVI) if and only if X is an optimal solution of (5) with v = f (X) = 0.

(i) Similar to the proof of Lemma 2.9, it is also routine to check that a sequence {xn} is a generalized type I (resp., generalized type II) LP approximating solution sequence if and only if it is a generalized type I (resp., generalized type II) LP minimizing sequence of (P). So (GVQVI) is 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 with f (x) defined by (2.15).

(ii) Since X0 c X0, dXo (x) < dX>o (x) for any x. This fact together with Lemma 2.9 implies that a type I (resp., type II) LP minimizing sequence of (P) is a type I (resp., type II) LP approximating solution sequence. So the type I (resp., type II) LP well-posedness of (GVQVI) implies the type I (resp., type II) LP well-posedness of (P) with f (x) defined by (2.15). □

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 of (generalized) LP well-posedness of (GVQVI) defined in Section 2.

Now consider a real-valued function c = c(t,s,r) defined for t,s,r > 0 sufficiently small, such that

c(t, s, r) > 0, Vt, s, r, c(0,0,0) = 0,

sn —> 0, tn > 0, rn = 0, c(tn,sn,rn) —> 0 imply that tn —> 0,

Theorem 3.1. Let the set-valued map T be compact-valued on X1. If (GVQVI) is type II LP well-posed, the set-valued map S is closed-valued, then there exist a function c satisfying (3.1) such that

|f (x)| > c(dX(x),dX0(x),dS(x)(x)), Vx e X1, (3.2)

where f (x) is defined by (2.15). Conversely, suppose that X is nonempty and compact and (3.2) holds for some c satisfying (3.1). Then (GVQVI) is type II LP well-posed.

Proof. Define

c(t,s,r) = inf{|f (x)| : x e X1,dx(x) = t,dX0(x) = s,dS(x)(x) = (3.3)

Since X / 0, it is obvious that c(0> 0,0) = 0. Moreover, if s„ — 0>t„ > 0>r„ = 0, and c(t„> s„> r„) — 0, then there exists a sequence {x„} cX1 with (x„) = t„, dS(x„) (x„) = r„ = 0,

dx0 (x„) = s„ —> 0> (3.4)

such that

|f (x„) | -— 0. (3.5)

Since S is closed-valued, x„ e S(x„) for any „. This fact, combined with (3.4) and (3.5) and Lemma 2.9 (ii) implies that {x„} is a type II LP approximating solution sequence of (GVQVI). By Proposition 2.4, we have that t„ — 0.

Conversely, let {x„} be a type II LP approximating solution sequence of (GVQVI). Then, by (3.2), we have

|f(x„)| > c(dX(x„)>dx0(x„)>dS(x„)(x„)). (3.6)

t„ = dx(x„)> s„ = dX0 (x„)> r„ = dS(x„) (x„). (3.7)

Then s„ — 0 and r„ = 0> for all „ e N. Moreover, by Lemma 2.9, we have that \f (x)| — 0. Then, c(t„>s„>r„) — 0. These facts together with the properties of the function c imply that t„ — 0. By Proposition 2.4, we see that (GVQVI) is type II LP well-posed. □

Theorem 3.2. Let the set-valued map T be compact-valued on X1. If (GVQVI) is generalized type II LP well-posed, the set-valued map S is closed, then there exist a function c satisfying (3.1) such that

|f (x)| > c(dX(x)>dK(g(x))>dS(x)(x))> Vx e X1, (3.8)

where f (x) is defined by (2.15). Conversely, suppose that X is nonempty and compact and (3.8) holds for some c satisfying (3.4) and (3.5). Then, (GVQVI) is generalized type II LP well-posed.

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

c(t>s>r) = inf{|f (x)| : x e X1>dX(x) = t>dK(g(x)) = s>dS(x)(x) = r}. (3.9)

Next we give the Furi-Vignoli-type characterizations [39] for the (generalized) type I LP well-posedness of (GVQVI).

Let (X, || ■ ||) be a Banach space. Recall that the Kuratowski measure of noncompactness for a subset H of X is defined as

¡i(H) = inf | e > 0 : H c [J Hi, diam(Hi) < e,i = 1,...,n}, (3.10)

where diam(Hi) is the diameter of Hi defined by

diam(Hi) = sup{||x1 -x2|| : x1,x2 e Hi}. (3.11)

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

h(A,B) = max{sup{dB(a) : a e A},sup{dA(b) : b e B}}. (3.12)

For any e > 0, two types of approximating solution sets for (GVQVI) are defined, respectively, by

Q1 (e) = {x e X1 : x e S(x), dX0 (x) < e, 3z e T(x), s.t.(z, x' - x) + ee / - int C, Vx' e S(x)}, Q2(e) = {x e X1 : x e S(x),dK(g(x)) < e, 3z e T(x),s.t.(z,x' - x) + ee / - int C, Vx' e S(x)}.

(3.13) □

Theorem 3.3. Assume that T is u.s.c. and compact-valued on X1and S is l.s.c. and closed on X1. Then

(a) (GVQVI) is type I LP well-posed if and only if

1^/(^1 (e)) = C1, (3.14)

(b) (GVQVI) is generalized type I LP well-posed if and only if

lim^2(e)) = 0. (3.15)

Proof. (a) First we show that, for every e > 0, Q1(e) is closed. In fact, let xn e Q1(e) and Xn ^ X0. Then (2.4) and the following formula hold:

, dXo {x:} -e' (3.16)

3Zn e T(X:), ) + ee / - int C, Vx e S(xn).

Since xn ^ xo, by the closedness of S and (2.4), we have xo e S(xo). From (3.16), we get

dx0 (xo) - e, (3.17)

3zn e T(xn), s.t. &({zn,x' - xn)) > -e, Vx' e S(xn). (3.18)

For any v e S(x0), by the lower semi-continuity of S and (3.18), we can find v„ e S(x„) with v„ — v such that

l({z„,v„ - x„)) >-e. (3.19)

By the u.s.c. of T at x0 and the compactness of T(x0), there exist a subsequence {znj} c {z„} and some z0 e T(x0) such that

z„j -— z0. (3.20) This fact, together with the continuity of ¿, and (3.19), implies that

K(z0>v - x0)) >-e Vv e S(x0). (3.21)

It follows that

(z0>v - x0) + ee / -int C Vv e S(x0). (3.22)

Hence, x0 e Q1(e).

Second, we show that X = f)e>0 Q1(e). It is obvious that X c f)e>0 Q1(e). Now suppose that e„> 0 with e„ — 0 and x* e f)e>0 ^1(e„). Then

dX0 (x*) < e„> Vn> (3.23)

x* e S(x*)> (3.24)

3z e T(x*)> s.t. (z>x'- x*) + e„e / - int C> Vx' e S(x*). (3.25)

From (3.23), we have

x* e X0. (3.26)

From (3.25), we have

(z> x'- x*) / - int C> Vx' e S(x*)> (3.27) that is x* e X. Hence, X = f|e>0 Q1(e).

Now we assume that (GVQVI) is type I LP well-posed. By Remark 2.3, we know that the solution X is nonempty and compact. For every positive real number e, since X e ^1(e), one gets

□1(e) = 0> h(Q1(e)>X^ = maxj sup d^(u)>sup do1(e)(vw = sup d^(u). (3.28)

^MeQ1(e) veX J ueQ1(e)

For every „ e N, the following relations hold:

y(^1(e)) < 2h(^1(e)>X + = 2h(fl1(e)>X)> (3.29)

where |(X) = 0 since X is compact. Hence, in order to prove that lime—0|(Q1(e)) = 0, we only need to prove that

□1(e)>X) = lim suP dX(u) = 0. (3.30)

Suppose that this is not true, then there exist ¡5 > 0, e„ — 0, and sequence {u„}, u„ e □1(e„), such that

dX(u„) > ¡> (3.31)

for n sufficiently large.

Since {u„} is type I LP approximating sequence for (GVQVI), it contains a subsequence {u„k} conversing to a point of X, which contradicts (3.31).

For the converse, we know that, for every e > 0, the set □1(e) is closed, X = □1(e),

and limu(Q1(e)) = 0. The theorem on Page. 412 in [40, 41] can be applied, and one concludes

e — 0

that the set X is nonempty, compact, and

lim h( Q1(e)>X) = 0. (3.32)

If {x„} is type I LP approximating sequence for (GVQVI), then there exists a sequence {e„} of positive real numbers decreasing to 0 such that x„ e □1(e„), for every n e N. Since X is compact and

lim dX(x„) < lim h(□1(e„)>X = 0> (3.33)

„ — +<x> X „ — +<x> \ J

by Proposition 2.4, (GVQVI) is type I LP well-posed.

(b) The proof is Similar to that of (a), and it is omitted here. This completes the proof.

Definition 3.4. (i) Let Z be a topological space, and let Z1 c Z be nonempty. Suppose that h : Z — R1 u {+to} is an extended real-valued function. h is said to be level-compact on Z1 if, for any s e R1, the subset {z e Z1 : h(z) < s} is compact.

(ii) Let X be a finite-dimensional normed space, and let Z1 c Z be nonempty. A function h : Z — R1 u {+to} is said to be level-bounded on Z1 if Z1 is bounded or

lim h(z) = +». (3.34)

Now we establish some sufficient conditions for type I (resp., generalized I type) LP well-posedness of (GVQVI).

Proposition 3.5. Suppose that the solution set X of (GVQVI) is nonempty and set-valued map S is l.s.c. and closed on X1, the set-valued map T is u.s.c. and compact-valued on X1. Suppose that one of the following conditions holds:

(i) there exists 0 < 61 < 60 such that X1(61) is compact, where

X1(61) = {x e X1 n X2 : dX0 (x) < 61}; (3.35)

(ii) the function f defined by (2.15) is level-compact on X1 n X2;

(iii) X is finite-dimensional and

lim max{f (X),dX0 (x)} = +a>, (3.36)

where f is defined by (2.15);

(iv) there exists 0 < 61 < 60 such that f is level-compact on X1(61) defined by (3.35). Then (GVQVI) is type I LP well-posed.

Proof. First, we show that each of (i), (ii), and (iii) implies (iv). Clearly, either of (i) and (ii) implies (iv). Now we show that (iii) implies (iv). Indeed, we need only to show that, for any t e R1, the set

A = {x e X1(61) : f (x) < t} (3.37)

is bounded since X is finite-dimensional space and the function f defined by (2.15) is l.s.c. on X1 and thus A is closed. Suppose to the contrary that there exists t e R1 and {Xn} c X1(61) such that Hx'nH ^ +<x> and f (x'n) < t. From {x'n} c X1(61), we have dX0(x'n) < 61.

ma^{f{Xn),dX0(x'n)} < max{t,61}, (3.38)

which contradicts (3.36).

Therefore, we only need to we show that if (iv) holds, then (GVQVI) is type I LP well-posed. Let {xn} be a type I LP approximating solution sequence for (GVQVI). Then, there exist {en} c R+ with en ^ 0 and zn e T(xn) such that (2.3), (2.4), and (2.5) hold. From (2.3) and (2.4), we can assume without loss of generality that {xn} c X1(61). By Lemma 2.9, we can assume without loss of generality that {xn} c {x e X1(61) : f (x) < 1}. By the level-compactness of f on X1(61), we can find a subsequence {xH]} of {xn} and x e X1(61) such that xnj ^ x. Taking the limit in (2.3) (with xn replaced by xnj), we have x e X0. Since S is closed and (2.4) holds, we also have x e S(x).

Furthermore, from the u.s.c. of T at x and the compactness of T(X), we deduce that there exist a subsequence {znj} of {zn} and some z e T(X) such that znj ^ z. From this fact, together with (2.5), we have

(z, X - X) / - int C, VX e S(X). (3.39)

Thus, X e X.

The next proposition can be proved similarly. □

Proposition 3.6. Suppose that the solution set X of (GVQVI) is nonempty and set-valued map S is l.s.c. and closed on X1, the set-valued map T is u.s.c. and compact-valued on X1. Suppose that one of the following conditions holds:

(i) there exists 0 < 61 < 60 such that X2(61) is compact, where

X2(61) = {x e X1 n X2 : dK(g(x)) < 61}; (3.40)

(ii) the function f defined by (2.15) is level-compact on X1 n X2;

(iii) X is finite-dimension and

lim maxi f (x)>dKg(x)} = +<x>> (3 41)

xeX1nX2r\\x\\ — +<x> ^ ' '

where f is defined by (2.15),

(iv) there exists 0 < 61 < 60 such that f is level-compact on X2(61) defined by (3.40). Then (GVQVI) is generalized type II LP well-posed.

Remark 3.7. If X is finite-dimensional, then the "level-compactness" condition in Propositions 3.1 and 3.6 can 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 in [26-30] and the references therein.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant no. 11171363 and Grant no. 10831009), the Natural Science Foundation of Chongqing (Grant No. CSTC, 2009BB8240) and the special fund of Chongqing Key Laboratory (CSTC 2011KL0RSE01).

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