- Research Article
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Levitin-Polyak Well-Posedness in Vector Quasivariational Inequality Problems with Functional Constraints
Fixed Point Theory and Applications volume 2010, Article number: 984074 (2010)
Abstract
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 be a normed space and
a metric space. Let
be nonempty and closed sets. Let
be a locally convex space and
be a nontrivial closed and convex cone with nonempty interior
. Define the following order in
, for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ1_HTML.gif)
Let be the space of all the linear continuous operators from
to
. Let
and
be two functions. We denote by
the function value
, where
. Let
be a strict set-valued map (i.e.,
).
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ2_HTML.gif)
The vector quasivariational inequality problem with functional and abstract set constraints considered in this paper is:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ3_HTML.gif)
Denote by 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 ,
be two normed spaces. A set-valued map
from
to
is
(i)closed, on , if for any sequence
with
and
with
, one has
;
(ii)lower semicontinuous (l.s.c. in short) at , if
,
, and
imply that there exists a sequence
satisfying
such that
for
sufficiently large. If
is l.s.c. at each point of
, we say that
is l.s.c on
.
Let be a metric space,
, and
. In the sequel, we denote by
the distance function from point
to set
. For a topological vector space
, we denote by
its dual space. For any cone
, we will denote the (positive) polar cone of
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ4_HTML.gif)
Let be fixed. Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ5_HTML.gif)
Throughout this paper, we always assume that the feasible set is nonempty and the function
is continuous on
.
Definition 2.1.
-
(i)
A sequence
is called a type I Levtin-Polyak (LP in short) approximating solution sequence if there exists
with
such that
(2.3)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ7_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ8_HTML.gif)
 (ii) is called a type II LP approximating solution sequence if there exist
with
and
with
such that (2.3)–(2.5) hold and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ9_HTML.gif)
 (iii) is called a generalized type I LP approximating solution sequence if there exists
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ10_HTML.gif)
and (2.4), (2.5) hold.
-
(iv)
is called a generalized type II LP approximating solution sequence if there exist
with
and
with
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 of (VQVI) is nonempty, and for any type I (resp., type II, generalized type I, generalized type II) LP approximating solution sequence
, there exist a subsequence
of
and
such that
.
Remark 2.3.
-
(i)
It is easily seen that if
,
, 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 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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ11_HTML.gif)
where is nonempty and
is proper. The feasible set of (P) is
, where
. The optimal set and optimal value of (P) are denoted by
, respectively. Note that if
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ12_HTML.gif)
then . In this paper, we always assume that
. We note that LP well posedness for the special case, where
is finite valued and l.s.c.,
is closed, has been studied in [12].
Definition 2.4.
-
(i)
A sequence
is called a type I LP minimizing sequence for (P) if
(2.9)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ14_HTML.gif)
 (ii) is called a type II LP minimizing sequence for (P) if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ15_HTML.gif)
and (2.10) hold.
-
(iii)
is called a generalized type I LP minimizing sequence for (P) if
(2.12)
and (2.9) hold.
-
(iv)
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
of (P) is nonempty, and for any type I (resp., type II, generalized type I, generalized type II) LP minimizing sequence
, there exist a subsequence
of
and
such that
.
The Auslender gap function for (VQVI) is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ17_HTML.gif)
From Lemma in [40], we know that
is weak* compact. This fact combined with that
when
implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ18_HTML.gif)
Recall the following nonlinear scalarization function (see, e.g., [9]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ19_HTML.gif)
It is known that is a continuous, (strictly) monotone (i.e., for any
,
,
implies that
and
implies that
), subadditive, and convex function. Moreover, for any
, it holds that
. Furthermore, following the proof of [9, Proposition
], we can prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ20_HTML.gif)
Let be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ21_HTML.gif)
First we have the following lemma.
Lemma 2.6.
Let be defined by (2.14), then
(i), for all
,
(ii) and
if and only if
.
Proof.
-
(i)
Let
, then
. We let
in (2.14) be equal to
, then
.
-
(ii)
Assume that
. Suppose to the contrary that
, then, there exists
such that
(2.18)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ23_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ24_HTML.gif)
Hence, , contradicting the assumption, so
. Conversely, assume that
, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ25_HTML.gif)
As a result, for any , there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ26_HTML.gif)
It follows that . This fact combined with (i) implies that
.
In the rest of this paper, we set in (P) equal to
. Note that if the set-valued map
is closed on
, then
is closed. By Lemma 2.6,
if and only if
minimizes
(defined by (2.26)) over
with
.
The following lemma establishes some relationship between LP approximating solution sequence and LP minimizing sequence.
Lemma 2.7.
Let the function be defined by (2.14) as follows:
(i) is a sequence such that there exists
with
satisfying (2.4)-(2.5) if and only if
and (2.9) holds with
.
(ii) is a sequence such that there exist
with
and
with
satisfying (2.4)–(2.6) if and only if
and (2.11) holds with
.
Proof.
-
(i)
Let
be any sequence, if there exists
with
satisfying (2.4)-(2.5), then we can easily verify that
(2.23)
It follows that (2.9) holds with .
For the converse, let and (2.9) hold. We can see that
and (2.4) hold. Furthermore, by (2.9), we have that there exists
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ28_HTML.gif)
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ29_HTML.gif)
That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ30_HTML.gif)
Now, we will show that (2.5) holds, otherwise there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ31_HTML.gif)
As a result, for any ,
Since
is a weak* compact set, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ32_HTML.gif)
which contradicts (2.26).
-
(ii)
Let
be any sequence, we can check that
(2.29)
holds if and only if there exists with
and
with
such that (2.6) (with
replaced by
) holds. From the proof of (i), we know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ34_HTML.gif)
and hold if and only if
such that there exists
with
satisfying (2.4)-(2.5) (with
replaced by
). Finally, we set
and the conclusion follows.
The next proposition establishes relationships between the various LP well posednesses of (VQVI) and those of (P) with defined by (2.14).
Proposition 2.8.
Assume that , 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 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 defined by (2.14).
Proof.
By Lemma 2.6, if ,
is a solution of (VQVI) if and only if
is an optimal solution of (P) with
and
defined by (2.14).
-
(i)
Similar to the proof of Lemma 2.7, it is also routine to check that a sequence
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
defined by (2.26).
-
(ii)
Since
,
for any
. 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
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 is closed,
is l.s.c. on
, and
.
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ35_HTML.gif)
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 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 defined for
sufficiently small such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ36_HTML.gif)
With the help of Lemma 2.7, analogously to [35, Theorems , and
], we can prove the following two theorems.
Theorem 3.2.
If (VQVI) is type II LP well posed, the set-valued map is closed valued, then there exists a function c satisfying (3.2) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ37_HTML.gif)
where is defined by (2.14). Conversely, suppose that
is nonempty and compact, and (3.3) holds for some
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 is closed, then there exists a function
satisfying (3.2) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ38_HTML.gif)
where is defined by (2.14). Conversely, suppose that
is nonempty and compact, and (3.4) holds for some
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 be a Banach space. Recall that the Kuratowski measure of noncompactness for a subset
of
is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ39_HTML.gif)
where diam is the diameter of
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ40_HTML.gif)
For any , define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ41_HTML.gif)
Lemma 3.4.
Let be defined by (2.14) and
. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ42_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ43_HTML.gif)
then one has and
.
Proof.
First, we prove the former result. For any satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ44_HTML.gif)
we have and
. We will show that
, for all
. Otherwise, there exists
such that
. By the weak* compactness of
, we have
, which leads to
and gives rise to a contradiction. Furthermore, we observe that
. This fact combined with
implies that
.
Now, we prove the equivalence between and
. Firstly, we can establish the same inclusion for
and
analogously to the proof stated above. Then if
satisfies
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ45_HTML.gif)
It is routine to check that . From (3.11), we know that for each
, there exists
such that
. As a result, we can see that
. Thus, we prove the conclusion.
The next lemma can be proved analogously to ([25, Theorem ]).
Lemma 3.5.
Let be a Banach space. Suppose that
is l.s.c. on
and bounded below on
. 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ46_HTML.gif)
To continue our study, we make some assumptions below.
Assumption.
-
(i)
is a Banach space.
-
(ii)
The set-valued map
is closed, and lower semicontinuous on
.
-
(iii)
The map
is continuous on
.
We have the following lemma concerning the l.s.c. of defined by (2.14).
Lemma 3.6.
Let function be defined by (2.14) and Assumption 1 hold, then
is l.s.c. function from
to
. Further assume that the solution set
of (VQVI) is nonempty, then
.
Proof.
First we show that , for all
. Suppose to the contrary that there exists
such that
, then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ47_HTML.gif)
That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ48_HTML.gif)
Namely,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ49_HTML.gif)
which is impossible since is a finite function on
. Second, we show that
is l.s.c. on
. Note that the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ50_HTML.gif)
is continuous on by the continuity of
on
and the continuity of
. We also note that
. Let
. Suppose that the sequence
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ51_HTML.gif)
and . For any
, by the lower semicontinuity of
and continuity of
, we have a sequence
with
converging to
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ52_HTML.gif)
It follows that . Hence,
is l.s.c. on
. Furthermore, if
, by Lemma 2.6, we see that
.
Theorem 3.7.
Let Assumption 1 hold and let the solution set of (QVVI) be nonempty and compact, then, (VQVI) is generalized type I LP well posed if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ53_HTML.gif)
Proof.
Note that the function defined by (2.14) is nonnegative on
. By the lower semicontinuity of
and Lemma 3.6,
is l.s.c. on
. Moreover,
is closed, since
is closed on
. 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 defined by (3.8) is closed.
Proof.
Let and
. We show that
. It is obvious that
. Since
and
, by the closedness of
, we have
. Moreover, since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ54_HTML.gif)
hold and is l.s.c., for any
, we can find that
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ55_HTML.gif)
Hence, is closed.
Theorem 3.9.
Let Assumption 1 hold and let be defined by (2.14). Assume that the solution set
of (QVVI) is nonempty and compact, then (VQVI) is type I LP well posed if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ56_HTML.gif)
Proof.
The proof is similar to that of Theorem in [35] and thus omitted.
Example 3.10.
-
(i)
Let
,
,
,
, and
.
maps
into an identical mapping, that is to say
, for any
. The set valued mapping
is defined as follows, given
for some
, then
(3.23)
with , of course
is closed and l.s.c. Now, we can show that, when
,
, which is bounded. Thus,
, by applying Theorem 3.9, we know that (VQVI) is type I LP well posed.
(i)Suppose that is a set-valued mapping from
to
, for fixed
,
implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ58_HTML.gif)
with , obviously
is still closed and l.s.c. If we replace
by
in (i), then
with
, which is unbounded. Therefore,
and the (VQVI) is not LP well posed in sense of type I. Actually, the solution set of this problem is
and thus unbounded.
Definition 3.11.
-
(i)
Let
be a topological space, and let
be nonempty. Suppose that
is an extended real-valued function.
is said to be level compact on
if, for any
, the subset
is compact.
-
(ii)
Let
be a finite dimensional normed space, and let
be nonempty. A function
is said to be level bounded on
if
is bounded or
(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 such that
is compact, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ60_HTML.gif)
(ii)the function defined by (2.14) is level compact on
,
(iii) is finite dimensional and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ61_HTML.gif)
where is defined by (2.14).
(iv)There exists such that
is level-compact on
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 is closed by the closedness of
. Then, we need only to show that for any
, the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ62_HTML.gif)
is bounded since is a finite dimensional space and the function
defined by (2.14) is l.s.c. on
and, thus,
is closed. Suppose to the contrary that there exist
and
such that
and
. From
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ63_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ64_HTML.gif)
which contradicts condition (3.27).
Now, we show that if (iv) holds, then (VQVI) is type I LP well posed. Let be a type I LP approximating solution sequence of (VQVI). Then, there exist
with
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ65_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ66_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ67_HTML.gif)
From (3.32) and (3.33), we can assume without loss of generality that . By Lemma 2.7, we can assume without loss of generality that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ68_HTML.gif)
where is defined by (2.14). By the level compactness of
on
, there exist a subsequence of
of
and
such that
. From this fact and (3.32), we have
. Since
is closed and (3.33) holds, we also have
. That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ69_HTML.gif)
Furthermore, by Lemmas 2.7 and 3.6, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ70_HTML.gif)
We know that by Lemma 2.6, so
. This fact combined with (3.35) and Lemma 2.6 implies that
.
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 such that
is compact, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ71_HTML.gif)
(ii)the function defined by (2.14) is level compact on
,
(iii) is finite dimensional and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ72_HTML.gif)
(iv)There exists such that
is level compact on
defined by (3.37). Then, (VQVI) is generalized type I LP well posed.
Remark 3.14.
If 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 is a normed space,
is a closed and convex cone with nonempty interior
and let
.
Let and denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F984074/MediaObjects/13663_2010_Article_1377_Equ73_HTML.gif)
The next proposition follows immediately from Proposition 2.8(i), Lemma 3.6, and [12, Proposition (iv)].
Proposition 3.15.
Let be a normed space, let
be a closed and convex cone with nonempty interior
and
. Let the set-valued map
be closed and l.s.c on
. Assume that the solution set
of (VQVI) is nonempty. Further assume that there exists
such that the function
defined by (2.14) is level compact on
, then (VQVI) is generalized type I LP well posed.
Remark 3.16.
If is finite dimensional, then the level-compactness condition of
can be replaced by the level boundedness of
.
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This work is supported by the National Science Foundation of China.
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Zhang, J., Jiang, B. & Huang, X. Levitin-Polyak Well-Posedness in Vector Quasivariational Inequality Problems with Functional Constraints. Fixed Point Theory Appl 2010, 984074 (2010). https://doi.org/10.1155/2010/984074
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DOI: https://doi.org/10.1155/2010/984074