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Upper Semicontinuity of Solution Maps for a Parametric Weak Vector Variational Inequality
Journal of Inequalities and Applications volume 2010, Article number: 482726 (2010)
Abstract
This paper investigates the upper semicontinuity of the solution map for a parametric weak vector variational inequality associated to a -hemicontinuous and weakly
-pseudomonotone operator.
1. Introduction and Preliminaries
A vector variational inequality (VVI, in short) was first introduced by Giannessi [1] in the setting of finite-dimensional Euclidean space. Later on, it was studied and generalized to infinite-dimensional spaces. Existences of the solutions for VVI have been studied extensively in various versions; see [2–6] and references therein.
Stability of the solution map for VVI or vector equilibrium problems is an important topic in optimization theory. A great deal of papers have been denoted to study the semicontinuity and continuity of the solution maps; see [7–18] and references therein. All results of the stability of the solution map for VVI in the literature are obtained based on continuity of the operator. It is well known that -hemicontinuity is weaker than continuity. In this paper, our aim is to investigate the upper semicontinuity of the solution map for a parametric weak vector variational inequality associated to a
-hemicontinuous and weakly
-pseudomonotone operator.
Let ,
, and
(the spaces of parameters) be Banach spaces and let
be a pointed closed and convex cone with nonempty interior
. Let
be the space of all linear continuous operators from
to
. The value of a linear operator
at
is denoted by
. Consider the following weak vector variational inequality problem:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482726/MediaObjects/13660_2010_Article_2165_Equ1_HTML.gif)
where is a nonempty subset and
is a vector-valued mapping.
When is perturbed by a parameter
, which varies over a nonempty set
, for a given
, we can define the parametric weak vector variational inequality problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482726/MediaObjects/13660_2010_Article_2165_Equ2_HTML.gif)
where is a nonempty subset and
is a vector-valued mapping.
For each , we denote the solution map of (PWVVI) by
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482726/MediaObjects/13660_2010_Article_2165_Equ3_HTML.gif)
Throughout the paper, we always assume that is nonempty for all
in a neighborhood of
. Now, we recall some basic definitions and their properties.
Let be a nonempty convex subset of
and let
be an operator.
is said to be
-hemicontinuous if and only if, for every
,
and
, the mapping
is continuous at
.
Definition 1.2 (see [6]).
Let be a nonempty subset of
and
be an operator.
is weakly
-pseudomonotone on
if, for every pair of points
,
, one has that
implies that
.
Proposition 1.3 (see [6, Generalized Linearization Lemma]).
Let be a nonempty convex subset of
and let
be an operator. Consider the following problems:
(I) such that
for all
,
(II) such that
for all
.
Then the following are obtained.
(i)Problem (I) implies Problem (II) if is weakly
-pseudomonotone.
(ii)Problem (II) implies Problem (I) if is
-hemicontinuous.
Let be a set-valued mapping, given that
.
Definition 1.4 (see [19, 20]).
(i) is called lower semicontinuous (l.s.c) at
if, for any open set
satisfying
, there exists
such that for every
,
.
(ii) is called upper semicontinuous (u.s.c) at
if, for any open set
satisfying
, there exists
such that, for every
,
.
We say is l.s.c (resp., u.s.c) on
, if it is l.s.c (resp., u.s.c) at each
.
is said to be continuous on
if it is both l.s.c and u.s.c on
.
Proposition 1.5 (see [19, 21]).
(i) is l.s.c at
if and only if, for any sequence
with
and any
, there exists
such that
.
-
(ii)
If
has compact values (i.e.,
is a compact set for each
), then
is u.s.c at
if and only if, for any sequence
with
and for any
, there exist
and a subsequence
of
such that
.
2. Main Results
In this section, we mainly discuss the upper semicontinuity of the solution map for (PWVVI).
Lemma 2.1.
Let be a nonempty compact convex subset of
. Suppose that, for any
,
is
-hemicontinuous and weakly
-pseudomonotone on
. Then,
has compact values on
, that is,
is a compact set for each
.
Proof.
For any , take any sequence
with
; we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482726/MediaObjects/13660_2010_Article_2165_Equ4_HTML.gif)
By Proposition 1.3 and the weakly -pseudomonotonicity of
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482726/MediaObjects/13660_2010_Article_2165_Equ5_HTML.gif)
From , we have
as
. It follows from the closedness of
and (2.2) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482726/MediaObjects/13660_2010_Article_2165_Equ6_HTML.gif)
Moreover, by Proposition 1.3 and the -hemicontinuity of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482726/MediaObjects/13660_2010_Article_2165_Equ7_HTML.gif)
That is . Thus,
is a closed set. Furthermore, it follows from
and the compactness of
that
is a compact set. The proof is complete.
Theorem 2.2.
Let be a nonempty compact convex subset of
. Suppose that the following conditions are satisfied.
(i)For any ,
is
-hemicontinuous on
,
(ii)For any ,
is weakly
-pseudomonotone on
,
(iii)For any ,
is continuous on
.
Then, is u.s.c on
.
Proof.
For any , any sequences
with
, and
, we have
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482726/MediaObjects/13660_2010_Article_2165_Equ8_HTML.gif)
Since is a compact set, there are an
and a subsequence
such that
. Particularly, from (2.5), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482726/MediaObjects/13660_2010_Article_2165_Equ9_HTML.gif)
By Proposition 1.3 and (iii), we can obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482726/MediaObjects/13660_2010_Article_2165_Equ10_HTML.gif)
Since is continuous and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482726/MediaObjects/13660_2010_Article_2165_Equ11_HTML.gif)
we get , as
. It follows from the closedness of
and (2.7) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482726/MediaObjects/13660_2010_Article_2165_Equ12_HTML.gif)
Moreover, by Proposition 1.3 and (ii), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482726/MediaObjects/13660_2010_Article_2165_Equ13_HTML.gif)
that is .
Thus, for any sequence with
and for any
, there exist
and a subsequence
of
such that
. By Proposition 1.5 and Lemma 2.1, we have
is u.s.c at
. From the arbitrariness of
, we can get
is u.s.c on
. The proof is complete.
Remark 2.3.
In [7–10], the upper semicontinuity of the solution map for has been discussed based on the continuity of the operator. Note that
-hemicontinuity is weaker than continuity. Moreover, together with the assumption of weakly
-pseudomonotonicity,
-hemicontinuity may not derive the continuity of the operator. Thus, it is necessary to investigate the upper semicontinuity of the solution map for
associated to a
-hemicontinuous and weakly
-pseudomonotone operator. Now we give an example to illustrate our result.
Example 2.4.
Let ,
,
,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482726/MediaObjects/13660_2010_Article_2165_Equ14_HTML.gif)
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482726/MediaObjects/13660_2010_Article_2165_Equ15_HTML.gif)
It is clear that conditions (ii) and (iii) of Theorem 2.2 are satisfied. For any ray ,
is continuous. Thus,
is
-hemicontinuous on
and condition (i) of Theorem 2.2 is satisfied. By Theorem 2.2, we conclude that
is u.s.c on
. In fact,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482726/MediaObjects/13660_2010_Article_2165_Equ16_HTML.gif)
Then, by the definition of upper semicontinuity, it follows readily that the solutions map is u.s.c on
.
However, for with
, we have
, but
. Thus, for any
,
is not continuous at
. Therefore, the theorems concerning the upper semicontinuity in the literatures are not applicable.
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Acknowledgments
The authors would like to thank two anonymous referees for their valuable comments and suggestions, which helped to improve the paper. This research was partially supported by the National Natural Science Foundation of China (Grant no. 10871216) and Chongqing University Postgraduates Science and Innovation Fund (Project no. 201005B1A0010338).
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Fang, Z., Li, S. Upper Semicontinuity of Solution Maps for a Parametric Weak Vector Variational Inequality. J Inequal Appl 2010, 482726 (2010). https://doi.org/10.1155/2010/482726
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DOI: https://doi.org/10.1155/2010/482726