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Painleve-Kuratowski Convergences for the Solution Sets of Set-Valued Weak Vector Variational Inequalities
Journal of Inequalities and Applications volume 2008, Article number: 435719 (2008)
Abstract
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.
1. Introduction
Since the concept of vector variational inequality (VVI) was introduced by Giannessi [1] in 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 theories (see, e.g., [2–8] and the references cited therein).
The stability analysis of the solution set maps for the parametric (VVI) 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. In [7], 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 perturbed (VVI) problem with a sequence of mappings converging is different from the parametric (VVI) problem 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-Kuratowski (see [11–13]). It is well known that the vector equilibrium problem is a generalization of (VVI) problem. However, if the results obtained for the vector equilibrium problem are to be applied to the (VVI) problem, the required assumptions are on the (VVI) problem as a whole. There is no information about the conditions that are required on the functions defining the (VVI) problem. Clearly, this is unsatisfactory. Our study of the stability properties for the perturbed (VVI) problem with a sequence of converging mappings is under appropriate assumptions on the function defining the (VVI) problem rather than on the (VVI) 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 inequity (SWVVI) with 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 imposed on the sequence of gap functions. Then, we obtain Painleve-Kuratowski lower convergence of the solution sets for
.
The rest of the paper is organized an follows. In Section 2, we introduce problems (SWVVI) and , 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
and then establish Painleve-Kuratowski lower convergence of the solution sets under a key assumption.
2. Preliminaries
Let and
be two Banach spaces and let
be the set of all linear continuous mappings from
to
. The value of a linear mapping
at
is denoted by
. Let
be a closed and convex cone with nonempty interior, that is,
. We define the ordering relations as follows.
For any ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ1_HTML.gif)
Consider the set-valued weak vector variational inequality (SWVVI) problem for finding and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ2_HTML.gif)
where is a nonempty subset and
is a set-valued mapping.
For a sequence of set-valued mappings , we define a sequence of set-valued weak vector variational inequality
problems for finding
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ3_HTML.gif)
where is a sequence of nonempty subsets.
We denote the solution sets of problems (SWVVI) and by
and
, respectively, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ4_HTML.gif)
Throughout this paper, we assume that and
. The stability analysis is to investigate the behaviors of the solution sets
and
.
Now we recall some basic definitions and properties of problems (SWVVI) and . For each
and a subset
, let the open
-neighborhood of
be defined as
. The notation
denotes the open ball with center
and radius
.
In the following, we introduce some concepts of the convergence of set sequences and mapping sequences which will be used in the sequel. Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ5_HTML.gif)
where denotes the set of all positive integer numbers and
is an integer in
.
Definition 2.1 (see [11, 15]).
Let be a normed space. A sequence of sets
is said to converge in the sense of Painleve-Kuratowski (P.K.) to
(i.e.,
) if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ6_HTML.gif)
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ7_HTML.gif)
It is said that the sequence upper converges in the sense of Painleve-Kuratowski to
if
. Similarly, the sequence
is said to lower converge in the sense of Painleve-Kuratowski to
if
.
Definition 2.2 (see [15]).
A set-valued mapping is outer semicontinuous (osc) at
if
with
.
On the other hand, it is inner semicontinuous (isc) at if
with
.
The set-valued mapping is said to be continuous at , written as
as
if it is both outer semicontinuous and inner semicontinuous.
Definition 2.3 (see [15]).
Let be a sequence of set-valued mappings and
be a set-valued mapping. It is said that the sequence
converges continuously to
at
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ8_HTML.gif)
If converges continuously to
at every
, then it is said that
converges continuously to
on
.
Let be a set-valued map, the graph of
is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ9_HTML.gif)
Applying set convergence theory to the graphs of the mappings, we obtain the graphical convergence of the sequence of mappings.
Definition 2.4 (see [15]).
For a sequence of mappings , the graphical outer limit, denoted by
, is the mapping which has as its graph the set
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ10_HTML.gif)
The graphical inner limit, denote by , is the mapping having as its graph the set
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ11_HTML.gif)
If the outer and inner limits of the mappings agree, it is said that their graphical limit,
, exists. In this case, the notation
is used, and the sequence
of mappings is said to converge graphically to
. Clearly,
.
Proposition 2.5 (see [15]).
For any sequence of mappings , it holds that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ12_HTML.gif)
where the unions are taken over all sequences . Thus, the sequence
converges graphically to
if and only if, at each point
, it holds that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ13_HTML.gif)
From Proposition 2.5 and Definition 2.3, the following proposition follows readily.
Proposition 2.6.
Let be a sequence of set-valued mappings and
be a set-valued mapping. Then, the sequence
outer converges graphically to
if and only if
outer converges continuously to
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ14_HTML.gif)
Definition 2.7 (see [10]).
Given a sequence of mappings ,
is said to be uniformly bounded if for any sequence
contained in a bounded set, there exists a positive number
such that for any sequence
with
for all
, it holds that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ15_HTML.gif)
Proposition 2.8 (see [16]).
For any fixed ,
,
, and the nonlinear scalarization function
defined by
:
(i) is a continuous and convex function on
;
(ii)
(iii)
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
(i) outer converges continuously to
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ16_HTML.gif)
(ii);
(iii) are uniformly bounded.
Then, , that is to say for any subsequence
of solutions to
, if
, then
is a solution to
.
Proof.
The proof is listed on contradiction arguments. On a contrary, suppose that but
.
From , we have
, where
and
is a subsequence of
. Then, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ17_HTML.gif)
Since , it is clear that for any
, there exists a sequence
with
and
, as
. Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ18_HTML.gif)
Since and
, we have
. Now, we note that
. Thus, for all
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ19_HTML.gif)
From the uniform boundedness of , we may assume, without loss of generality, that
(though a subsequence of
if necessary). By (i), we get
. Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ20_HTML.gif)
It follows from (3.3) and the closedness of that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ21_HTML.gif)
which is a contradiction to (3.4). This completed the proof.
Remark 3.2.
Let and
, where
is a reflexive Banach space and
is its dual. If we take
,
reduce to the generalized variational inequality problems with perturbed operators
considered in [10, Section 3]. The convergence for the solution sets of
was studied under the the pseudomonotonicity assumption in [10]. Furthermore, if
and
are vector-valued mappings, then
reduce to
considered in [10, Section 2]. We also notice that the Painleve-Kuratowski upper convergence of the solution sets of
is obtained under weaker assumptions than these assumed in [10, Proposition 2.1] for obtaining convergence of the solution sets.
From Proposition 2.5 and Theorem 3.1, we obtain readily the following corollary.
Corollary 3.3.
Suppose that
(i) outer converges graphically to
, written as
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ22_HTML.gif)
(ii);
(iii) is uniformly bounded.
Then, .
Remark 3.4.
Let . Then, problems
reduce to the generalized variational inequalities with perturbed operators considered in [10, Proposition 3.1] and the convergence was obtained under the assumption that the operators converge graphically.
Theorem 3.5.
Suppose that
(i) is osc on
, that is, for all
,
;
(ii) and
are compact sets.
Then, is a compact set.
Proof.
First, we prove that is a closed set. Take any sequence
with
. Then, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ23_HTML.gif)
It follows from the compactness of that
. Suppose that
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ24_HTML.gif)
Since is a compact set, without loss of generality, we assume that there exists a
such that
. Thus, we have
. By (i), we get a
. It follows from (3.8) and the closedness of
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ25_HTML.gif)
which contradicts with (3.9). Hence, and
is a closed set. Next, it follows from
and the compactness of
that
is a compact set. The proof is completed.
Similarly, we have the following result.
Theorem 3.6.
For any , suppose that
(i) is osc on
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ26_HTML.gif)
(ii) and
are compact sets.
Then, is 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 that and
are compact sets and that for each
,
and
are compact sets. Let
and
be functions defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ27_HTML.gif)
Since ,
,
, and
are compact sets and
is continuous,
and
are well defined.
Proposition 4.1.
-
(i)
for all
;
-
(ii)
for all
-
(iii)
if and only if
-
(iv)
if and only if
Proof.
Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ28_HTML.gif)
We first prove that . On a contrary, we suppose that this is false. Then, there exist
and
such that
. Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ29_HTML.gif)
which is impossible when . Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ30_HTML.gif)
By the same taken, we can show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ31_HTML.gif)
On the other hand, if , then there exists a
such that
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ32_HTML.gif)
From Proposition 2.8, (4.6) is valid if and only if for any ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ33_HTML.gif)
Clearly, (4.7) holds if and only if for any ,
, that is,
. This proves that (iii) holds.
Similarly, we can show that (iv) holds.
The functions are called the gap functions for
if properties (ii) and (iv) of Proposition 4.1 are satisfied.
In view of hypothesis of [6, 17, 18], we introduce the following key assumption:
(H
g
):given the sequence for any
, there exist an
and an
such that
for all
and for all
.
Geometrically, the hypothesis means that given a sequence of mappings
, we can find for any small positive number
, a small positive number
and a large-enough positive number
such that for all
, if a feasible point
is away from the solution sets
by distance of at least
, then the values of all gap functions for
is less than or equal to at least some "
."
To illustrate assumption , we give the following example.
Example 4.2.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ34_HTML.gif)
Consider problems . From direct computation, we obtain
. To check assumption
, we take
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ35_HTML.gif)
For any given , we take
and
. Then, for all
and for all
, we have
. Hence, assumption
is valid.
Lemma 4.3.
Suppose that
(i) inner converges continuously to
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ36_HTML.gif)
(ii);
(iii) is a compact set.
Then, for any ,
and sequence
with
and
, there exists a subsequence
of
and
such that
for all
.
Proof.
Let be a function defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ37_HTML.gif)
From the continuity of with respect to
, the compactness of
and [19, Chapter 3, Section 1, Proposition 23], we have that
is continuous with respect to
. Thus, from the compactness of
, there exists a
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ38_HTML.gif)
From assumption (i), there exists a sequence satisfying
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ39_HTML.gif)
It follows from the compactness of that there exists
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ40_HTML.gif)
Since is compact, we assume, without loss of generality, that
. Thus, it follows from (ii) that
Consequently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ41_HTML.gif)
So, for any , there exists an
such that
for all
. By (4.14), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ42_HTML.gif)
Hence, the result holds.
Set and
. We have the following lemma.
Lemma 4.4.
Suppose that for is osc on
, that is, for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ43_HTML.gif)
Then, if and only if for all
,
such that
for all
.
Proof.
We assume , but there exists an
such that for all
there exists an
satisfying
. Then, there exists a sequence
with
, but
. From Theorem 3.5, we note that
is a compact set. Without loss of generality, we assume
and
. Thus, for any sequence
satisfying
with
, we have
. Letting
, we get
. Therefore, there does not exist any sequence
satisfying
. This is a contradiction to
.
Conversely, suppose that for any such that
for all
. From Theorem 3.6, we note that
is compact for all
. Thus, for any
, there exists
such that
for all
. So, we have
and
. 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 assumption holds and that the following conditions are satisfied:
(i) is osc on
for
that is, for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ44_HTML.gif)
(ii) inner converge continuously to
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ45_HTML.gif)
(iii) is a compact set;
(iv).
Then, .
Proof.
We prove the result via contradiction. On a contrary, we assume, by Lemma 4.4, that there exists an such that for any
, we have
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ46_HTML.gif)
that is, there exists a sequence satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ47_HTML.gif)
From the compactness of , we can assume, without loss of generality, that
. Then, there exists an
such that
. It is clear that
for any positive integer
. Since
, there exist a sequence
satisfying
. Then, there exists an
such that
for all
.
Now, we note that . Otherwise, there would exist a sequence
with
such that
. Thus, for
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ48_HTML.gif)
This implies that , which contradicts with (4.21). Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ49_HTML.gif)
By hypothesis , there exist, for any
, an
and an
such that for all
and for all
,
. In particular, it follows from (4.23) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ50_HTML.gif)
By virtue of Lemma 4.3, there exists, for any , a subsequence
of
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ51_HTML.gif)
We can take such that
. Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ52_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ53_HTML.gif)
So, for any ,
. Thus, there exists a
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F435719/MediaObjects/13660_2008_Article_1816_Equ54_HTML.gif)
Consequently, by Proposition 2.8, we have , which shows that
. This contradicts with
. Therefore, our result follows readily.
Now, we explain the applicability of Theorem 4.5 through an example.
Example 4.6.
Consider Example 4.2. It follows from a direct computation that . It is easy to testify that assumption
holds and so are conditions (i)–(v) of Theorem 4.5. Obviously, the solution sets of problem
lower converge in the sense of Painleve-Kuratowski.
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Acknowledgment
This research was partially supported by the National Natural Science Foundation of China (Grants no. 10871216 and no. 60574073).
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Fang, Z.M., Li, S.J. & Teo, K.L. Painleve-Kuratowski Convergences for the Solution Sets of Set-Valued Weak Vector Variational Inequalities. J Inequal Appl 2008, 435719 (2008). https://doi.org/10.1155/2008/435719
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DOI: https://doi.org/10.1155/2008/435719