Continuity of the Solution Mappings for Parametric Generalized Strong Vector Equilibrium Problems ()
1. Introduction
The vector equilibrium problem is an interesting and important model in applied mathematics. It is a unified model, includes vector optimization problem, vector variational inequality, the vector complementarity problem and vector saddle point problem. Over the past two decades, this problem has received an increasing interest from many researchers. In the literature, existence and connectedness results for various types of vector equilibrium problems have been investigated intensively, e.g., [1] - [7] and the references therein.
The stability analysis of the solution mappings for vector equilibrium problems is another important topic in optimization theory and has recently received increasing attention from mathematicians with various approaches. The stability of solutions can be understood as lower or upper semicontinuity, continuity in the sense of Berge or Hausdorff, and Lipschitz/Hölder continuity of solution maps. There have been a large number of contributions to the semicontinuity, especially the lower semicontinuity, of solution maps to parametric vector equilibrium problems in the literature, such as [8] - [18] . In addition, many results on the Hölder/Lipschitz continuity of the solution maps are archived, e.g., [19] [20] [21] [22] and the references therein.
We observe that the scalarization technique is one of effective approaches to deal with the lower semicontinuity and the upper semicontinuity of solution mappings to parametric vector equilibrium problems. It is worth noting that the linear scalarization approach to the semicontinuity of solution mappings to parametric generalized vector equilibrium problems always requires (generalized) cone-convexity or strict cone-monotonicity of the objective functions, even the assumptions involve the information about the solution set. To avoid using these assumptions, nonlinear scalarization approaches have been applied for discussing the stability analysis in parametric generalized weak vector equilibrium problems. Recently, Sach [23] [24] has used some nonlinear scalarization functions (generalized versions of Gerstewizt’s function) to investigative the semicontinuity of the solution mappings of parametric generalized weak and non-weak vector equilibrium problems, which are also called parametric generalized Ky Fan inequalities. However, to the best of our knowledge, there are few results in the literature on the continuity of the solution mappings to parametric strong vector equilibrium problems. In particular, so far, there is no work with contribution to the continuity of the solution mappings of parametric strong vector equilibrium problems by using nonlinear scalarization methods.
Hence, motivated by the works reported in [12] [14] [15] [16] [17] [23] [24] , this paper aims to explore an application of the oriented distance function defined in [25] to discuss the lower semicontinuity and the Hausdorff continuity of the solution mapping of a parametric generalized strong vector equilibrium problem. To this end, we firstly establish a density theorem concerned with the solution set to parametric generalized vector equilibrium problem and the solution set of parametric generalized strong vector equilibrium problem by using the nonlinear scalarization method. Then by the density result, the lower semicontinuity and the Hausdorff upper semicontinuity of the solution mapping to the parametric generalized strong vector equilibrium problem are given. Our methods as well as results are different from ones in [23] [24] , since the models discussed in [23] [24] are not parametric generalized strong vector equilibrium problems. Furthermore, some examples are given to illustrate that our results improve the corresponding ones in [12] [14] [15] [16] [17] .
The outline of the paper is as follows. In Section 2, we introduce the parametric generalized strong vector equilibrium problem, and recall some basic concepts and their properties. In Section 3, based on the nonlinear scalarization method, we obtain a new density result. We also discuss the lower semicontinuity of the solution mapping to the parametric generalized strong vector equilibrium problem by means of the density result. In Section 4, we give the upper semicontinuity of the solution mapping to the parametric generalized strong vector equilibrium problem. In Section 5, we give the conclusions of the paper.
2. Preliminaries
Throughout this paper, without special statements, let X, Y and Z be normed vector spaces. We denote by
,
,
,
and
the closure, the complement, the relative interior, the interior and the boundary of a set
, respectively. We also assume that C be a pointed closed convex cone in Y with nonempty interior. Let
be the topological dual space of Y and let
.
Let
be a subset of Z. Let
be a set-valued mapping. Let
be a set-valued mapping. In this paper, we consider the following parametric generalized strong vector equilibrium problem:
(PGSVEP)
For each
, let
denote the solution set of (PGSVEP), i.e.,
For each
, let
denote the solution set of the following parametric generalized weak vector equilibrium problem, i.e.,
For each
, let
denote the solution set of the following parametric generalized vector equilibrium problem, i.e.,
For each
and for each
, let
denote the set of f-efficient solutions to (PGSVEP), i.e.,
It is easy to observe that
. Throughout this paper, we assume that
for all
in
. In this paper, by using the nonlinear scalarization method, we will discuss the lower semicontinuity and the Hausdorff upper semicontinuity of the solution mapping
as a set-valued mapping from the set
to X. Now we recall some basic definitions and their properties.
Definition 2.1. [26] A set-valued mapping
is said to be properly quasiconcave on X If for any
and for any
, one has
Definition 2.2. [27] [28] Let
and
be two topological vector spaces. A set-valued mapping
is said to be
1) Lower semicontinuous (l.s.c.) at
iff, for every open set
with
, there is a neighbourhood
of
, for every
,
.
2) Upper semicontinuous (u.s.c.) at
iff, for every open set
with
, there is a neighbourhood
of
, for every
,
.
3) Hausdorff upper semicontinuous (H-u.s.c.) at
iff, for each neighbourhood U of
, there is a neighbourhood
of
such that for any
,
.
4) G is called closed at
iff for each sequence
,
, it follows that
.
We say that G is l.s.c. (resp. u.s.c.) on
, if it is l.s.c. (resp. u.s.c.) at each
. G is said to be continuous on
if it is both l.s.c. and u.s.c. on
.
In the following Proposition 2.1, let
and
be two normed vector spaces.
Proposition 2.1. [28] [29]
1) G is l.s.c. at
if and only if for any sequence
with
and for any
, there exists
such that
.
2) If G has compact values at
, then G is u.s.c. at
if and only if for any sequence
with
and any
, there exist
and a subsequence
of
such that
.
Definition 2.3. [25] For a set
, the oriented distance function
is defined as
where
,
, and
denotes the norm of y in Y.
Proposition 2.2. (See Proposition 3.2 in [30] ) If the set A is nonempty and
. Then
1)
is real valued;
2)
is 1-Lipschitzian;
3)
;
4)
;
5)
;
6) If A is a closed and convex cone, then
is nonincreasing with respect to the ordering relation induced on Y, i.e., the following is true: if
, then
if A has a nonempty interior, then
3. Lower Semicontinuity
In this section, we discuss the lower semicontinuity of the solution mapping
to (PGSVEP).
Firstly, we define the function
as follows.
(1)
Proposition 3.1. Assume that the following conditions hold.
1)
is continuous with nonempty compact values on
;
2)
is continuous with nonempty compact values on
.
Then,
is continuous on
.
Proof. We define
by
Then
(2)
It follows from assumption (2), the continuity of
, Proposition 19 and Proposition 21 in [28] that
is continuous on
. Furthermore, combining this with the assumption (1), (2), Proposition 19 and Proposition 21 in [28] , we see that
is continuous on
.
Remark 3.1. It is worth of being observed that if
is l.s.c. on
and
is continuous with nonempty compact values on
, then we can get that
is lower semicontinuous on
by Proposition 19 in [28] .
Proposition 3.2. Assume that
is properly C-quasiconcave on X for each
and for each
. Then
is quasiconcave on X for each
.
Proof. For any given
and
, since
is properly C-quasiconcave on
, for any
, we have either
or
Thus, in terms of Proposition 2.2 (vi), for any
, one has
This together with the arbitrariness of
shows that
or
For any given
, the above inequalities give us that either
or
Therefore,
is quasiconcave on X for each
.
Lemma 3.1. Assume that the following conditions are satisfied.
1)
is l.s.c. on
;
2)
is continuous with nonempty compact values on
.
Then
is l.s.c. on
.
Proof. By Proposition 2.2 (v), it is not hard to see that
To prove the result by contradiction, suppose that there exists
such that
is not l.s.c. at
. Then by Proposition 2.1 (1), there exist a sequence
with
and
such that for any
, we have
.
From
, we have
. As
is l.s.c. at
, there exists
such that
. By the above contradiction assumption, there must exist subsequence
such that,
with
, i.e.,
(3)
It follows from Remark 3.1 that
is lower semicontinuous at
. This together with (3) implies that
which contradicts
. Thus
is l.s.c. on
.
Lemma 3.2. (See Theorem 1.1.2 in [31] ) Let
be a convex set. If
, then
.
Lemma 3.3. For each
, suppose the following conditions are satisfied.
1)
is continuous with nonempty compact values on
;
2)
is continuous with nonempty compact values on
;
3)
is properly C-quasiconcave on X for each
.
Then, we have
Proof. Taking into consideration that
, we only need to show that
(4)
In fact, it follows from Proposition 2.2 (3) that
for each
. In addition, by means of the assumption (2) and Proposition 3.1, we know that
is continuous on
. Hence, it is easy to see that
is closed for each
.
Next, we need to prove that the set
is convex for each
. Indeed, it follows from the assumption (3) and Proposition 3.2 that
is properly quasi-concave on X for each
. Hence, for any
, and for any
, we have either
or
This shows that for each
,
for any
and so
is convex for each
. Taking into account
for each
and with the help of Lemma 3.2, we see that (4) is valid.
Remark 3.2. It is worth mentioning that Lemma 3.2 shows that the solution set
is dense in the solution set
for each
. The density result is obtained with the help of nonlinear scalarization method, which is better than ones derived in [12] [14] [15] [16] [17] by using linear scalarization methods.
Theorem 3.1. Suppose the following conditions are satisfied:
1)
is continuous with nonempty compact values on
;
2)
is continuous with nonempty compact values on
;
3)
is properly C-quasiconcave on X for each
.
Then
is l.s.c. at
.
Proof. By virtue of Lemma 3.3, we have
For any given
, we claim that
is l.s.c. at
. Indeed, for any
and any neighborhood
of x, where
is a neighborhood of
in X. Since
we have
By Lemma 3.1, we have
is l.s.c at
. For the above
, there exists a neighborhood
of
such that
Taking into account
for each
, we arrive at
This states that
is l.s.c. at
.
The following example illustrates that Theorem 3.1 can be applied, while the corresponding lower semicontinuity results in [12] [14] [15] [16] [17] are not applicable.
Example 3.1. Let
,
,
and
. Let
for each
and
It follows form a direct computation that
and so
for all
. It is not hard to check that the assumptions (1)-(3) in Theorem 3.1 are satisfied. However,
(
) is not strictly C-monotone on
, that is, the assumption (2) in Theorem 2.1 of [12] and the assumption (3) in Theorem 3.2 of [14] are violated at
. Indeed, there exist
and
in
such that
The assumption (5) in Theorem 3.1 of [15] is not valid at
for
. Indeed, for each
,
, we have
The assumption (4) (i.e., f-property) in Theorem 3.1 of [16] is also violated at
for
. Indeed, for any
and
with
, we have
, but
,
.
The assumption (5) in Theorem 3.1 of [17] does not hold at
for
. Indeed,
.
As a consequence, Theorem 2.1 of [12] , Theorem 3.2 of [14] , Theorem 3.1 of [15] [16] [17] are not applicable.
4. Hausdorff Upper Semicontinuity
In the section, we give the Hausdorff upper semicontinuity of the solution mapping
to (PGSVEP).
Lemma 4.1. Assume that
1)
is continuous with nonempty compact values on
;
2)
is continuous with nonempty compact values on
.
Then
is u.s.c. with compact values on
.
Proof. By Proposition 2.2 (3), it is not hard to see that
(5)
where
is given in (1). Suppose to the contrary that there exists
such that
is not u.s.c. at
. Then there exist an open set
with
and a sequence
with
such that
. This implies that there is
such that
(6)
As
is u.s.c. with compact values at
and
, by Proposition 2.1 (2), there exist
and a subsequence
of
such that
. Without loss of generality, we assume that
.
Now, we claim that
. Indeed, suppose that
. Then it follows from (5) that
(7)
With the help of Proposition 3.1, we see that
is continuous on
. As a result, it follows from (7) that
which contradicts
because of (5). Thus,
. This together with
and
shows that
for n large enough, which contradicts (6). Therefore,
is u.s.c. on
.
Next, we show that
is compact for each
. Indeed, by (5) and Proposition 3.1, we can see that
is closed. This together with the compactness of
and
for each
gives us desired result.
Theorem 4.1. Assume that the following conditions hold.
1)
is continuous with nonempty compact values on
;
2)
is continuous with nonempty compact values on
;
3)
is properly C-quasiconcave on X for each
.
Then
is H-u.s.c. on
.
Proof. To prove the result by contradiction, suppose that there exists
such that
is not H-u.s.c. at
. Then there exists an open set
of
such that for any neighborhood
of
and there exists
with
. Hence, there exists a sequence
with
such that
. This implies that there is
such that
(8)
Taking into consideration that
and
, we have
,
. Due to Lemma 4.1, we know that
is u.s.c. with compact values at
. Then by Proposition 2.1 (2), there exist a subsequence
of
and
such that
. It follows from the closedness of
and Lemma 3.3 that
Consequently,
This together with
indicates that
for k large enough, which contradicts (8). Therefore,
is H-u.s.c. at
.
The example is given to show that Theorem 4.1 is applicable, but Theorem 4.1 of [15] and Theorem 4.1 of [17] are not applicable.
Example 4.1. Let
,
,
and
. Let
and
It follows form a direct computation that
and so
for all
. Hence
is H-u.s.c. on
. It is not hard to see that the assumptions (1)-(3) in Theorem 4.1 are satisfied. However, the assumption (5) in Theorem 4.1 of [15] is violated at
for
. Indeed, for each
,
, we have
The assumption (5) in Theorem 4.1 of [17] is also violated at
for
. Indeed,
,
. Therefore, Theorem 4.1 of [15] and Theorem 4.1 of [17] are not applicable.
5. Conclusions
In this paper, we established a density result in regard to the solution set to parametric generalized vector equilibrium problem and the solution set of parametric generalized strong vector equilibrium problem by using the nonlinear scalarization method. Then by using the density result, we obtained the lower semicontinuity and the Hausdorff upper semicontinuity of the solution mapping to the parametric generalized strong vector equilibrium problem. Additionally, some examples were given to illustrate that our results improve ones in [12] [14] [15] [16] [17] .
The multi-criteria traffic network equilibrium model is used to evaluate the traffic flow pattern and the travel costs, and it has played an important role in the traffic network programming and the traffic control. The topic has received increasing interest from many researchers, e.g., [32] [33] [34] [35] . It is worth noting that the multi-criteria traffic network equilibrium model can be shifted to a vector equilibrium problem. Therefore, it would be interesting to discuss the stability of the multi-criteria traffic network equilibrium model.
Acknowledgements
This research was supported by the Science and Technology Research Project of Education Department of Jiangxi Province (Grant number: GJJ191330).