0Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 984074,16 pages doi:10.1155/2010/984074
Research Article
Levitin-Polyak Well-Posedness in
Vector Quasivariational Inequality Problems with
Functional Constraints
J. Zhang,1 B. Jiang,2 and X. X. Huang3
1 School of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
2 Department of Systems Engineering and Engineering Management, The Chinese University ofHong Kong, Shatin, Hong Kong
3 School of Economics and Business Administration, Chongqing University, Chongqing 400030, China
Correspondence should be addressed to X. X. Huang, huangxuexiang@cqu.edu.cn Received 17 March 2010; Accepted 6 July 2010 Academic Editor: Lai Jiu Lin
Copyright © 2010 J. Zhang 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.
We introduce several types of Levtin-Polyak well-posedness for a vector quasivariational inequality with functional constraints. Necessary and/or sufficient conditions are derived for them.
1. Introduction
It is well known that, under certain conditions, a Nash equilibrium problem can be formulated and solved as a variational inequality problem. A generalized Nash game is a Nash game in which each player's strategy depends on other players' strategies [1]. The connection between generalized Nash games and quasivariational inequalities was first recognized by Bensoussan [2]. Recently, some researchers [1, 3, 4] found that mathematical models of many real world problems, including some engineering problems, can be formulated as certain kinds of variational inequality problems, including quasivariational inequality problems. However, as noted in [5], compared with variational inequality problems, the study on quasivariational inequality problems is still in its infancy, in particular only a few algorithms have been proposed to solve variational inequalities numerically.
Vector variational inequality problems were introduced by Giannessi [6] and are related to vector network equilibrium problems [7]. Since then, various types of vector
variational inequalities were introduced and studied (see, e.g., [8, 9] and the references therein).
In this paper, we will consider vector quasivariational inequality problems with functional constraints, which are described below.
Let (X, || ■ ||) be a normed space and (Z,d1) a metric space. Let X1 c X, K c Z be nonempty and closed sets. Let Y be a locally convex space and C c Y be a nontrivial closed and convex cone with nonempty interior int C. Define the following order in Y, for any y1,y2 € Y,
yi < y2 ^ y2 - yi € C. (1.1)
Let L(X,Y) be the space of all the linear continuous operators from X to Y. Let F : X1 ^ L(X,Y) and g : X1 ^ Z be two functions. We denote by (F(x), z) the function value F(x)(z), where z € X1. Let S : X1 ^ 2Xi be a strict set-valued map (i.e., S(x) f 0, for all x € X1). Let
Xo = {x € X1 : g(x) € K}. (1.2)
The vector quasivariational inequality problem with functional and abstract set constraints considered in this paper is:
Find x € X0 such that x € S(x) satisfying
_ (VQVI)
(F(x),x - x) € - int C, Vx € S(x).
Denote by X the solution set of (VQVI).
Well-posedness for unconstrained and constrained optimization problems was first studied by Tikhonov [10] and Levitin and Polyak [11]. The issue being considered is that for each approximating solution sequence, there exists a subsequence that converges to a solution of the problem.
In Tikhonov's well posedness, the approximating solution is always feasible. However, it should be noted that many algorithms in optimization and variational inequalities, such as penalty-type methods and augmented Lagrangian methods, terminate when the constraint is approximately satisfied. These methods may generate sequences that may not be necessarily feasible [12].
Up to now, various extensions of these well posednesses have been developed and well studied (see, e.g., [13-18]). Studies on well posedness of optimization problems have been extended to vector optimization problems (see e.g., [19-24]). The study of Levitin-Polyak well posedness for scalar convex optimization problems with functional constraints originates from [25]. Recently, this research was extended to nonconvex optimization problems with abstract and functional constraints [12] and nonconvex vector optimization problems with both abstract and functional constraints [26]. Well-posedness of variational inequality problems, mixed variational inequality problems, and equilibrium problems without functional constraints was investigated in the literature (see, e.g., [27-30]). Well-posedness in variational inequality problems with both abstract and functional constraints was investigated in [31]. Well-posedness of (generalized) quasivariational inequality and
mixed quasivariational-like inequalities has been studied in the literature [32-35]. The study of well posedness for (generalized) vector variational inequality, vector quasiequilibria and vector equilibrium problems can be found in [36-39] and the references therein.
In this paper, we will introduce and study several types of Levitin-Polyak (LP in short) well posednesses and generalized LP well posednesses for vector quasivariational inequalities with functional constraints. The paper is organized as follows. In Section 2, four types of LP well posednesses and generalized LP well posednesses for vector quasivariational inequality problems will be defined. In Section 3, we will derive various criteria and characterizations for the various (generalized) LP well posednesses of constrained vector quasivariational inequalities.
2. Definitions and Preliminaries
Let Z1, Z2 be two normed spaces. A set-valued map G from Z1 to 2Z2 is
(i) closed, on Z3 c Z1, if for any sequence {xn} c Z3 with xn ^ x e Z3 and yn e G(xn) with yn ^ y, one has y e G(x);
(ii) lower semicontinuous (l.s.c. in short) at x e Z1, if {xn} c Z1, xn ^ x, and y e G(x) imply that there exists a sequence {yn} c Z2 satisfying yn ^ y such that yn e G(xn) for n sufficiently large. If G is l.s.c. at each point of Z1, we say that G is l.s.c on Z1.
Let (P, d2) be a metric space, P1 c P, and p e P .In the sequel, we denote by dP1 (p) = inf{d(p,p') : p' e P1} the distance function from point p to set P1. For a topological vector space V, we denote by V* its dual space. For any cone O c V, we will denote the (positive) polar cone of O by
O* = e V* : $(v) > 0, Vv e O}. (2.1)
Let e e int C be fixed. Denote
C*° = {A e C* : A(e) = 1}. (2.2)
Throughout this paper, we always assume that the feasible set X0 is nonempty and the function g is continuous on X1.
Definition 2.1. (i) A sequence {xn} c X1 is called a type I Levtin-Polyak (LP in short) approximating solution sequence if there exists {en}c R\ with en ^ 0 such that
dxo (x (2.3)
xn e S(xn), (2.4)
{F(xn),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)cR+ with en ^ 0 and {yn} c X1 with yn e S(xn) such that (2.3)-(2.5) hold and
(F{xn),yn - xn )- ene e-C.
(iii) {xn} Ç X1 is called a generalized type I LP approximating solution sequence if there exists [en}ç R with en —> 0 such that
dK (g(xn)) < en, (2.7)
and (2.4), (2.5) hold.
(iv) {xn} c Xi is called a generalized type II LP approximating solution sequence if there exist {en} C R+ with en ^ 0 and {yn} C Xi with yn e S(xn) such that (2.4)-(2.7) hold.
Definition 2.2. (VQVI) is said to be type I (resp., type II, generalized type I, generalized type II) LP well posed if the solution set X of (VQVI) is nonempty, and for any type I (resp., type II, generalized type I, generalized type II) LP approximating solution sequence {xn}, there exist a subsequence {xnj} of {xn} and x e X such that xnj ^ x.
Remark 2.3. (i) It is easily seen that if Y = R1, C = R+, then type I (resp., type II, generalized type I, generalized type II) LP well posedness of (VQVI) reduces to type I (resp., type II, generalized type I, generalized type II) LP well posedness of (QVI) defined in [34].
(ii) 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.
(iii) Each type of LP well posedness of (VQVI) implies that its solution set X is compact.
To see that the various LP well posednesses of (VQVI) 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 Xi (P)
g (x) e K,
where Xi c X1 is nonempty and f : X1 ^ R1 u {+to} is proper. The feasible set of (P) is X0, where X0 = {x e X1 : g(x) e K}. The optimal set and optimal value of (P) are denoted by X and v, respectively. Note that if Dom(f) n X0 = 0, where
Dom(f) = {x e X1 : f (x) < +<»}, (2.8)
then v < +<x>. In this paper, we always assume that v > -to. We note that LP well posedness for the special case, where f is finite valued and l.s.c., X1 is closed, has been studied in [12].
Definition 2.4. (i) A sequence {xn} c X1 is called a type I LP minimizing sequence for (P) if
limsupf (xn) < v, (2.9)
dK (xn) 0. (2.10)
(ii) {xn} CX'1 is called a type II LP minimizing sequence for (P) if
lim f (xn) = v (2.11)
n ^ +<x> x '
and (2.10) hold.
(iii) {xn} CX1 is called a generalized type I LP minimizing sequence for (P) if
dK(g(xn)) 0. (2.12)
and (2.9) hold.
(iv) {xn} C X1 is called a generalized type II LP minimizing sequence for (P) if (2.11) and (2.12) hold.
Definition 2.5. (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 exist a subsequence {xnj} of {xn } and x e X such that xnj ^ x.
The Auslender gap function for (VQVI) is
r, ^ ■ r 1(F(x),x - x')
f (x) = rn , Vx e X1. (2.13)
From Lemma 1.1 in [40], we know that C*° is weak* compact. This fact combined with that 1(e) = 1 when 1 e C*0 implies that
f (x) = sup min1(F(x),x - x'), Vx e Xi. (214)
x' eS(x)1eC'° ( . )
Recall the following nonlinear scalarization function (see, e.g., [9]):
|: Y R1 : £(y) = min{t e R1 : y - te e -C}. (2.15)
It is known that I is a continuous, (strictly) monotone (i.e., for any y1, y2 e Y, y1 - y2 e C implies that Ky\) > Kyi) and y1 - y2 e int C implies that Ky\) > Kyi)), subadditive, and convex function. Moreover, for any t e R1, it holds that Kte) = t. Furthermore, following the proof of [9, Proposition 1.44], we can prove that
1(V) = sup 1$- = max01(y), Vy e Y. (2.16)
1eC*0 1(e) 1 eC
Let X2 C X be defined by
X2 = {x e X1 | x e S(x)}.
(2.17)
First we have the following lemma. Lemma 2.6. Let f be defined by (2.14), then
(i) f (x) > 0, for all x e X2 n X0,
(ii) f (x) = 0 and x e X2 n X0 if and only if x e X.
Proof. (i) Let x e X2 n Xo, then x e S(x). We let x' in (2.14) be equal to x, then f (x) > 0.
(ii) Assume that f (x) = 0. Suppose to the contrary that x / X, then, there exists x0 e S(x) such that
(F(x),x0 - x) e- int C. (2.18)
!(F(x),x - xo) > 0, V! e C
(2.19)
It follows that
min X(F(x),x- x0) > 0. (2 20)
XeC*10 v ' '
Hence, f (x) > 0, contradicting the assumption, so x e X. Conversely, assume that x e X, then we have
x e X2 n X0, (F(x),x'-x) / - int C, Vx'e S(x). (2.21)
As a result, for any x' e S(x), there exists X e C*° such that
X(F(x),x - x')<0. (2.22)
It follows that f (x) < 0. This fact combined with (i) implies that f (x) = 0. □
In the rest of this paper, we set X1 in (P) equal to X2. Note that if the set-valued map S is closed on X2, then X1 is closed. By Lemma 2.6, x e X if and only if x minimizes f (x) (defined by (2.26)) over X0 n X2 with f (x) = 0.
The following lemma establishes some relationship between LP approximating solution sequence and LP minimizing sequence.
Lemma 2.7. Let the function f be defined by (2.14) as follows:
(i) {xn} c Xi is a sequence such that there exists [en] c R\ with en ^ 0 satisfying (2.4)-(2.5) if and only if {xn} c X1 and (2.9) holds with v = 0.
(ii) {xn} c X1 is a sequence such that there exist {en} c R| with en ^ 0 and {yn} c X1 with yn e S(xn) satisfying (2.4)-(2.6) if and only if {xn} c X1 and (2.11) holds with v = 0.
Proof. (i) Let {x„} c X1 be any sequence, if there exists {e„} c R+ with e„ —> 0 satisfying (2.4)-(2.5), then we can easily verify that
{x„}C X1, f (x„) < e„. (2.23)
It follows that (2.9) holds with v = 0.
For the converse, let {x„} c X1 and (2.9) hold. We can see that {x„} c X1 and (2.4) hold. Furthermore, by (2.9), we have that there exists
{e„} c R+ with e„ -— 0 (2.24)
such that
f (x„) < e„. (2.25)
That is,
sup inf X(F(x„),x„ - x') < e„.
x'eS(Xn)^C•^ ' (2.26)
Now, we will show that (2.5) holds, otherwise there exists x0 S(x„) such that
(F(x„),x0 - x„) + e„e e -int C. (2.27)
As a result, for any X e C*0, X(F(x„),x„ - x0) > e„. Since C*0 is a weak* compact set, we have
inf X(F(x„),x„ - x0) > e„, (2.28)
XeC*0 v '
which contradicts (2.26).
(ii) Let {x„} c X1 be any sequence, we can check that
liminf f (x„) > 0, (2.29)
holds if and only if there exists {a„} c R+ with a„ — 0 and {y„} c X1 with y„ e S(x„) such that (2.6) (with e„ replaced by a„) holds. From the proof of (i), we know that
limsupf (x„) < 0 (2.30)
„ — +<x>
and {x„} c X1 hold if and only if {x„} c X1 such that there exists {p„} c R+ with ¡3„ — 0 satisfying (2.4)-(2.5) (with e„ replaced by ¡„). Finally, we set e„ = max{an,¡n} and the conclusion follows. □
The next proposition establishes relationships between the various LP well posed-nesses of (VQVI) and those of (P) with f (x) defined by (2.14).
Proposition 2.8. Assume that X / 0, then
(i) (VQVI) 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 (2.14).
(ii) If (VQVI) is type I (resp., type II) LP well posed, (P) is type I (resp., type II) LP well posed with f (x) defined by (2.14).
Proof. By Lemma 2.6, if X ^0, x is a solution of (VQVI) if and only if x is an optimal solution of (P) with v = f (x) = 0 and f (x) defined by (2.14).
(i) Similar to the proof of Lemma 2.7, 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 (VQVI) 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 (2.26).
(ii) Since X0 c X0, dX0 (x) < dX>0 (x) for any x. This fact together with Lemma 2.7 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 type I (resp., type II) LP well posedness of (VQVI) implies type I (resp., type II) LP well posedness of (P) with f (x) defined by (2.26). □
To end this section, we note that all the results in [12] for the well posedness hold for (P) so long as X1 is closed, f is l.s.c. on X[, and Dom(f) n X0 = 0.
3. Criteria and Characterizations for Various LP Well-Posedness of (VQVI)
In this section, we give necessary and/or sufficient conditions for the various types of (generalized) LP well posednesses defined in Section 2. 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].
The next proposition can be straightforwardly proved.
Proposition 3.1. If (VQVI) is type I (resp., type II, generalized type I, generalized type II) LP well posed, then (3.1) holds. Conversely, if (3.1) holds and X is compact, then (VQVI) is type I (resp., type II, generalized type I, generalized type II) LP well posed.
Now, we 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.
With the help of Lemma 2.7, analogously to [35, Theorems 3.1, and 3.2], we can prove the following two theorems.
Theorem 3.2. If (VQVI) is type II LP well posed, the set-valued map S is closed valued, then there exists a function c satisfying (3.2) such that
\f (x)\ > c(dX(x),dxo(x),ds(X)(x)) Vx e Xi, (3.3)
where f (x) is defined by (2.14). Conversely, suppose that X is nonempty and compact, and (3.3) holds for some c satisfying (3.2), then (VQVI) is type II LP well posed.
Theorem 3.3. If (VQVI) is type II LP well posed in the generalized sense, the set-valued mapping S is closed, then there exists a function c satisfying (3.2) such that
\f (x)\> c(dX(x),dK(g(x)),ds(x)(x)) Vx e Xi, (3.4)
where f (x) is defined by (2.14). Conversely, suppose that X is nonempty and compact, and (3.4) holds for some c satisfying (3.2), then (VQVI) is generalized type II LP well posed.
Next we give Furi-Vignoli type characterizations [41] for the (generalized) type I LP well posednesses of (VQVI).
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 je > 0 : H c (JHi, diam(Hi) < e, i = 1,...,n J, (3.5)
where diam(Hi) is the diameter of Hi defined by
diam(Hi) = sup{|x1 -x2|| : x1,x2 e Hi}. (3.6)
For any e > 0, define
¥1(e) = \x e X1 : f (x) < v + e,dX (x) < e),
1 0 ' (3.7)
¥2(e) = {x e X1 : f (x) < v + e,dK(g(x)) < e}.
Lemma 3.4. Let f (x) be defined by (2.14) and v = 0. Let
Q1 (e) = {x e X1 : x e S(x),dX0 (x) < e, {F(x),x' - x) + ee / - int C, Vx' e S(x)}, (3.8) □2(e) = {x e X1 : x e S(x), dK(g(x)) < e, {F(x),x' - x) + ee / - int C, Vx' e S(x)}, (3.9)
then one has ¥1(e) c □1(e) and ¥2(e) = (e).
Proof. First, we prove the former result. For any x € X[ satisfying
f (x) < e, dX'o (x) < e, (3.10)
we have x € Xi and x € S(x). We will show that (F(x),x' - x) + ee / - int C, for all x' € S(x). Otherwise, there exists x' € S(x) such that (F(x),x' - x) + ee € - int C. By the weak* compactness of C*0, we have infXeC.0X(F(x),x-x') > e, which leads to f (x) > e and gives rise to a contradiction. Furthermore, we observe that X0 c X0. This fact combined with dX>o (x) < e implies that dXo (x) < e.
Now, we prove the equivalence between ¥2(e) and Q2(e). Firstly, we can establish the same inclusion for ¥2(e) and Q2(e) analogously to the proof stated above. Then if x € X1 satisfies x € S(x), dK (x) < e and
(F(x),x' - x) + ee / - int C, Vx'€ S(x). (3.11)
It is routine to check that x € X'1. From (3.11), we know that for each x' € S(x), there exists X € C*0 such that X(F(x),x- x') < e. As a result, we can see that f (x) < e. Thus, we prove the conclusion. □
The next lemma can be proved analogously to ([25, Theorem 5.5]).
Lemma 3.5. Let (X, || ■ ||) be a Banach space. Suppose that f is l.s.c. on X'1 and bounded below on X0. Assume that the optimal solution set of (P) is nonempty and compact, then, (P) is (generalized) type I LP well posed if and only if
(lW¥2(e)) = 0.) limu(¥1 (e)) = 0. (3.12)
\e ^ 0 /e ^ 0
To continue our study, we make some assumptions below.
Assumption 1. (i) X is a Banach space.
(ii) The set-valued map S is closed, and lower semicontinuous on X1.
(iii) The map F is continuous on X1.
We have the following lemma concerning the l.s.c. of f defined by (2.14).
Lemma 3.6. Let function f be defined by (2.14) and Assumption 1 hold, then f is l.s.c. function from X1 to R1 u Further assume that the solution set X of (VQVI) is nonempty, then Dom(f) f 0.
Proof. First we show that f (x) > -to, for all x € X1. Suppose to the contrary that there exists x0 € X1 such that f (x0) = -to, then,
inf X((F(x0),x0 - x)) = -to, Vx € S(x0). (3.13)
XeC*°
That is,
supX((F(x0),x0 - x)) = +to, Vx € S(x0).
XeC*°
(3.14)
Fixed Point Theory and Applications Namely,
£((F(x0),x0 - x)) = +œ, Vx e S(x0), (3.15)
which is impossible since £ is a finite function on Y. Second, we show that f is l.s.c. on X1. Note that the function
h{x,y) = mm\({F(x),x - y)) = -l({F{x),y - x)) (3.16)
is continuous on X1 x X2 by the continuity of F on X1 and the continuity of £. We also note that f (x) = supyeS(x)h(x,y). Let t e R1. Suppose that the sequence {xn} Ç X1 satisfies
f (xn) < t (3.17)
and xn ^ x* e Xi. For any y e S(x*), by the lower semicontinuity of S and continuity of h, we have a sequence {yn} with yn e S(xn) converging to y such that
h(x*,y) = lim h(xn,y^\ < liminf f (xn) < t. (3.18)
^ ' Vt —i -l-rv-i ^ ' Vt —i -l-rv-> ^ '
It follows that f (x*) = supyeS(x)h(x*,y) < t. Hence, f is l.s.c. on X1. Furthermore, if X = 0, by Lemma 2.6, we see that Dom(f ) = 0. □
Theorem 3.7. Let Assumption 1 hold and let the solution set X of(QVVI) be nonempty and compact, then, (VQVI) is generalized type I LP well posed if and only if
lim p(Q2(e))= 0. (3.19)
e ^ 0 x '
Proof. Note that the function f (x) defined by (2.14) is nonnegative on X'0. By the lower semicontinuity of S and Lemma 3.6, f is l.s.c. on Xi = X1 n X2. Moreover, Xi is closed, since S is closed on X1 n X2. By Proposition 2.8, Lemmas 3.4 and 3.5, the conclusion follows. □
Although the type I (type II) LP well posedness of (VQVI) is not equivalent to the type I (type II) LP well posedness of (P), we can still establish the same characterization for type I (type II) LP well posedness of (VQVI) as Theorem 3.7. We need the next lemma.
Lemma 3.8. Let Assumption 1 hold, then Q1(e) defined by (3.8) is closed.
Proof. Let xn e Q1(e) and xn ^ x0. We show that x0 e Q1(e). It is obvious that dXo(x0) < e. Since xn e S(xn) and xn ^ x0, by the closedness of S, we have x0 e S(x0). Moreover, since
(zn,x'- xn) + ee e- int C, Vx' e S(xn) (3.20)
hold and S is l.s.c., for any y e S(x0), we can find that yn e S(xn) with {yn} ^ y such that (F(x0),y - x0> = lim (F(xn),yn - x^ + ee/ - intC. (3.21)
Hence, Q1 is closed. □
Theorem 3.9. Let Assumption 1 hold and let Q1(e) be defined by (2.14). Assume that the solution set X of(QVVI) is nonempty and compact, then (VQVI) is type I LP well posed if and only if
lim ^(Qi(e))= 0. (3.22)
e ^ 0 x '
Proof. The proof is similar to that of Theorem 3.4 in [35] and thus omitted. □
Example 3.10. (i) Let X = Y = R2, C = R+, e = (1,1)T, Xi = R2, and Xo = R+. F maps R+ into an identical mapping, that is to say (F(x),y) = (y1 ,y2)T, for any x,y e R2. The set valued mapping S is defined as follows, given y e S(x) for some x,y e R2, then
I [xi, 1], if Xi < 1,
yi = ^ (3.23)
[[2xj - 1,3xj - 2], if Xj> 1,
with i e {1,2}, of course S is closed and l.s.c. Now, we can show that, when 0 < e < 1, Q1(e) c {x | -e < x1 < 1, -e < x2 < 1}, which is bounded. Thus, ^(Q(e)) = 0, by applying Theorem 3.9, we know that (VQVI) is type I LP well posed.
(i) Suppose that G is a set-valued mapping from R2 to 2R2, for fixed x e R2, y e G(x) implies that
( [xi, 1], if xi < 1, yi = i i (3.24)
[[1,2xj - 1], if Xj> 1,
with i e {1,2}, obviously G is still closed and l.s.c. If we replace S by G in (i), then Q1(e) 2 {x | -e < x1 < 1+e, x2 > 0} with0 < e < 1, which is unbounded. Therefore, lime^0^(Q1(e)) / 0and the (VQVI) is not LP well posed in sense of type I. Actually, the solution set of this problem is {x | 0 < x1 < 1,x2 > 0}u{x | 0 < x2 < 1,x1 > 0} and thus unbounded.
Definition 3.11. (i) Let Z be a topological space, and let Z1 c Z be nonempty. Suppose that h : Z ^ R1 u {+<x>} 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 Z be a finite dimensional normed space, and let Z1 c Z be nonempty. A function h : Z ^ R1 u {+<x>} is said to be level bounded on Z1 if Z1 is bounded or
lim h(z) = +». (3.25)
The following proposition presents some sufficient conditions for type I LP well posedness of (VQVI)
Proposition 3.12. Let Assumption 1 hold. Further assume that one of the following conditions holds.
(i) There exists 0 < 61 <60 such that X1(61) is compact, where
X1(S1) = {x e X1 n X2 : dx0 (x) < 61}, (3.26)
(ii) the function f defined by (2.14) is level compact on X1 n X2,
(iii) X is finite dimensional and
v maxf(x),dX0(x)} = +<xv (3.27)
xeX1nX2,|xy^ +<x> v '
where f is defined by (2.14).
(iv) There exists 0 < 61 < 60 such that f is level-compact on X1(61) defined by (3.26). Then, (VQVI) is type I LP well posed.
Proof. First, we show that each one of (i), (ii), and (iii) implies (iv). Clearly, either of (i) and (ii) implies (iv). Now, we show that (iii) implies (iv). We notice that the set X1 n X2 is closed by the closedness of S. Then, we need only to show that for any t e R1, the set
A = {x e X1(61) : f (x) < t} (3.28)
is bounded since X is a finite dimensional space and the function f defined by (2.14) is l.s.c. on X1 and, thus, A is closed. Suppose to the contrary that there exist t e R1 and {x'n} C X1(61) such that \\x'n\\ ^ +<x> and f (x'n) < t. From {x'n} C X1(61), we have
dX0 (x'n) < 61. (3.29)
max{f (x'n),dX0(x'n)} < max{t,61}, (3.30)
which contradicts condition (3.27).
Now, we show that if (iv) holds, then (VQVI) is type I LP well posed. Let {xn} be
a type I LP approximating solution sequence of (VQVI). Then, there exist {en} c R+ with en ^ 0 and zn e T(xn) such that
(F(xn),x - xn) + en / int C, Vx' e S(xn), (3.31)
dX0 (x n < en, (3.32)
xn e S(xn). (3.33)
From (3.32) and (3.33), we can assume without loss of generality that {xn} c X1(61). By Lemma 2.7, we can assume without loss of generality that
{xn}c {x e Xi(6i) : f (x) < 1}, (3.34)
where f is defined by (2.14). By the level compactness of f on X1(61), there exist a subsequence of {xH]} of {xn} and x e X1(61) such that xnj ^ x. From this fact and (3.32), we have x e X0. Since S is closed and (3.33) holds, we also have x e S(x). That is,
x e Xo n X2 = X0. (3.35)
Furthermore, by Lemmas 2.7 and 3.6, we have
f (x) < liminf fixn) < limsup f(xni) < 0. (3.36)
n ^ +» \ v n ^ +<x, v '
We know that f (x) > 0 by Lemma 2.6, so f (x) = 0. This fact combined with (3.35) and Lemma 2.6 implies that x e X. □
Similarly, we can prove the next proposition.
Proposition 3.13. Let Assumption 1 hold. Further assume that one of the following conditions holds.
(i) There exists 0 < 61 < 60 such that X2(61) is compact, where
X2(6i) = {x e Xi n X2 : dK(g(x)) < 61}, (3.37)
(ii) the function f defined by (2.14) is level compact on X1 n X2,
(iii) X is finite dimensional and
yJim„ max{f (x),dK (g(x))} = +OT. (3.38)
(iv) There exists 0 < 61 < 60 such that f is level compact on X2(61) defined by (3.37). Then, (VQVI) is generalized type I LP well posed.
Remark 3.14. If X is finite dimensional, then the "level-compactness" condition in Propositions 3.12 and 3.13 can be replaced by the "level-boundedness" condition.
Now, we consider the case when Y is a normed space, K is a closed and convex cone with nonempty interior int K and let e e int K. Let t > 0 and denote
Xs(i) = {x e X1 n X2 : g(x) e K - te}. (3.39)
The next proposition follows immediately from Proposition 2.8(i), Lemma 3.6, and [12, Proposition 2.3(iv)].
Proposition 3.15. Let Y be a normed space, let K be a closed and convex cone with nonempty interior int K and e e int K. Let the set-valued map S be closed and l.s.c on X1. Assume that the solution set X of (VQVI) is nonempty. Further assume that there exists t1 > 0 such that the function f (x) defined by (2.14) is level compact on X3(t1), then (VQVI) is generalized type I LP well posed.
Remark 3.16. If X is finite dimensional, then the level-compactness condition of f can be replaced by the level boundedness of f.
Acknowledgment
This work is supported by the National Science Foundation of China.
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