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Sensitivity analysis for generalized quasi-variational relation problems in locally G-convex spaces
Fixed Point Theory and Applications volume 2012, Article number: 158 (2012)
Abstract
In this paper, we study generalized quasi-variational relation problems in locally G-convex spaces. Using the Kakutani-Fan-Glicksberg fixed-point theorem for upper semicontinuous set-valued mapping with nonempty closed acyclic values, we establish an existence theorem of a solution set for these problems. Moreover, the stability and closedness of the solution set for these problems are also obtained. The results presented in the paper improve and extend the main results in the literature.
MSC:47J20, 49J40.
1 Introduction and preliminaries
The generalized quasi-variational relation problems include, as special cases, the generalized variational inclusion problems, the generalized vector equilibrium problems, the generalized vector variational inequality problems etc. In recent years, a lot of results for the existence and stability of solutions for variational relation problems, vector equilibrium problems and vector variational inequality problems have been established by many authors in different ways. For example, variational relation problems [1–4], vector equilibrium problems [5–18], vector variational inequality problems [19, 20] and the references therein.
For a set X, we shall denote by and the families of all subsets of X and the family of all nonempty finite subsets of X, respectively. For each , denotes the cardinality of A. Let denote the standard n-dimensional simplex in with vertices , that is,
where is the i th unit vector in .
For any nonempty subset J of , we denote by the convex hull of the vertices .
A convex set A in a vector space is called a convex space if it is equipped with a topology which includes the Euclidean topology on convex hulls of any nonempty finite subsets of A.
The notion of a G-convex space was introduced by Park and Kim in [21]. Let X be a topological space, be a nonempty subset and a function be such that the following conditions hold:
-
(a)
for each , if ,
-
(b)
for each with , there exists a continuous mapping such that, for each , , where denotes the face of corresponding to .
Then is called a generalized convex space (or a G-convex space). If , we omit A simply write .
For a G-convex space , a subset B of X is said to be G-convex if, for each , implies . A space X is said to have a G-convex structure if and only if X is a G-convex space. A G-convex X is said to be a locally G-convex space if X is a uniform topological space with uniformity U, which has an open base of symmetric entourages such that for each , the set is a G-convex set for each .
Now, we pass to our problem setting. Let X, Y, Z be real locally G-convex Hausdorff topological vector spaces, , and be nonempty compact convex subsets. Let , , be multifunctions and be a relation linking , and . We adopt the following notations (see [2]). Letters w, m and s are used for weak, middle and strong kinds of considered problems respectively. For subsets U and V under consideration, we adopt the following notations:
Let and . We consider the following for a generalized quasi-variational relation problem (in short, (QVR α )):
(QVR α ): Find such that and satisfying
Let be the solution set of (QVR α ).
Special cases of the problem (QVR α ) are as follows:
-
(I)
If we let A, D, B, X, Y, Z, , , T be as in (QVR α ) and be a multifunction, the relation R is defined as follows:
Then (QVR α ) becomes the generalized quasi-variational inclusion problem:
Find such that and satisfying
-
(II)
If we let A, D, B, X, Y, Z, , , T, R be as in (QVR α ) and and be multifunctions, the relation R is defined as follows:
Then (QVR α ) becomes the generalized quasi-variational inclusion problem:
Find such that and satisfying
-
(III)
If we let A, D, B, X, Y, Z, , , T be as in (QVR α ) and , be multifunctions such that is a closed convex cone with , the relation R is defined as follows:
Then (QVR α ) becomes the generalized vector quasi-equilibrium problem:
Find such that and satisfying
-
(IV)
If we let A, D, B, X, Y, Z, , be as in (QVR α ), be a vector function, and be a multifunction such that is a closed convex cone with , the relation R is defined as follows:
Then (QVR α ) becomes the vector quasi-equilibrium problem:
Find such that and satisfying
-
(V)
If we let D, , , Z, , , T be as in (QVR α ), be the space of all linear continuous operators from X to Z and , , be continuous single-valued mappings, be a multifunction such that is a closed convex cone with , the relation R is defined as follows:
Then (QVR α ) becomes the generalized mixed vector quasi-variational inequality problem:
Find such that and satisfying
Let X, Y be two topological vector spaces, A be a nonempty subset of X and be a multifunction.
-
(i)
F is said to be lower semicontinuous (lsc) at if for some open set implies the existence of a neighborhood N of such that , . F is said to be lower semicontinuous in A if it is lower semicontinuous at all .
-
(ii)
F is said to be upper semicontinuous (usc) at if for each open set , there is a neighborhood N of such that , . F is said to be upper semicontinuous in A if it is upper semicontinuous at all .
-
(iii)
F is said to be continuous in A if it is both lsc and usc in A.
-
(iv)
F is said to be closed if is a closed subset in .
Definition 2 ([22])
Let X, Y be two topological vector spaces, A be a nonempty subset of X, be a multifunction and be a nonempty closed convex cone.
-
(i)
F is called upper C-continuous at if for any neighborhood U of the origin in Y, there is a neighborhood V of such that
-
(ii)
F is called lower C-continuous at if for any neighborhood U of the origin in Y, there is a neighborhood V of such that
Definition 3 ([23])
Let X and Y be two topological vector spaces and A be a nonempty convex subset of X. A set-valued mapping is said to be properly C-quasiconvex if for any and , we have
Lemma 4 ([23])
Let X, Y be two topological vector spaces, A be a nonempty convex subset of X and be a multifunction.
-
(i)
If F is upper semicontinuous at with closed values, then F is closed at .
-
(ii)
If F is closed at and Y is compact, then F is upper semicontinuous at .
-
(iii)
If F has compact values, then F is usc at if and only if, for each net which converges to and for each net with , there are and a subnet of such that .
Definition 5 ([24])
Let X be a topological space. A subset A of X is called contractible at , if there is a continuous such that for all and for all .
A topological space X is said to be acyclic if all of its reduced Čech homology groups over the rationals vanish. In particular, each contractible space is acyclic, and thus any nonempty convex or star-shaped set is acyclic. Moreover, by the definition of a contractible set, we see that each convex space is contractible.
We now have the following fixed-point theorem in locally G-convex spaces given by Yuan [25] which is a generalization of the Fan-Glickberg-type fixed-point theorem for an upper semicontinuous set-valued mapping with nonempty closed acyclic values.
Theorem 6 ([25], Theorem 2.1)
Let X be a compact locally G-convex space and be an upper semicontinuous set-valued mapping with nonempty closed acyclic values. Then F has a fixed-point; that is, there exists an such that .
2 Existence of solutions
In this section, we apply the Kakutani-Fan-Glicksberg fixed-point theorem for upper semicontinuous set-valued mapping with nonempty closed acyclic values to establish sufficient conditions for the existence of a solution set of generalized quasi-variational relation problems. Moreover, the closedness of the solution set for these problems is obtained.
Definition 7 Let X be a topological vector space, A be a nonempty convex subset of X and be a relation linking . We say that R is quasiconvex at if , such that holds and holds, we have
R is said to be quasiconvex in A if it is quasiconvex at all .
Remark 8 In the Definition 7, if we let , and let mapping , then the relation R defined by holds iff . We have , , if , , then . This means that R is modified 0-level quasiconvex, since the classical quasiconvexity says that , ,
Theorem 9 Assume for the problem (QVR α ) that
-
(i)
is upper semicontinuous in A with nonempty closed contractible values, and is lower semicontinuous A with nonempty closed values;
-
(ii)
T is upper semicontinuous in A with nonempty closed acyclic values if (or ) and lower semicontinuous in A with nonempty acyclic values if ;
-
(iii)
for all , holds;
-
(iv)
for all , is quasiconvex in A;
-
(v)
the set is closed.
Then, the (QVR α ) has a solution, i.e., there exist such that and satisfying
Moreover, the solution set of the (QVR α ) is closed.
Proof Since , we have in fact three cases. However, the proof techniques are similar. We present only the proof for the case where .
Indeed, for all , define a set-valued mapping: by
Since for any , , are nonempty. Thus, by assumption (iii), we have .
-
(I)
We show that is acyclic.
Since every contractible set is acyclic, it is enough to show that is contractible. Let , thus and , . Since is contractible, there exists a continuous mapping such that for all and for all . Now, we set for all . Then F is a continuous mapping, and we see that for all and for all . Let , we need to prove that . Since , and is contractible, thus, for , it follows that
and
By (iv), is quasiconvex in A, we have
i.e., . Therefore, is contractible.
-
(II)
We will prove is upper semicontinuous in with nonempty closed values.
Since A is a compact set and . Hence is compact. We need to show that is a closed mapping. Indeed, let a net such that , and let such that . Now, we need only prove that . Since and is upper semicontinuous at with nonempty closed values, by Lemma 4(i), we have is closed at , thus . Suppose, to the contrary, . Then such that
By the lower semicontinuity of , there is a net with such that . Since , we have
By the condition (v) and (2), we have
There is a contradiction between (3) and (1). Thus, . Hence, is upper semicontinuous in with nonempty closed values.
-
(III)
Now, we shall show that the solution set .
Define the set-valued mapping by
Then is upper semicontinuous in and , is a nonempty closed convex subset of . By Theorem 6, there exists a point such that , that is,
which implies that there exists and such that and
i.e., .
-
(IV)
Next, we prove that is closed.
Let a net . We need to prove that . Indeed, by the lower semicontinuity of , for any , there exists such that . As , there exists such that
Since is upper semicontinuous with nonempty closed values, by Lemma 4(i), we have is closed. Thus, . Since T is upper semicontinuous in A and is compact, there exists such that . By the condition (v), we have
This means that . Thus is a closed set. □
Remark 10 If we let , D, , Z, , T, R, as in (QVR α ), be a multifunction and be a nonempty closed convex cone, the relation R is defined as follows:
and
Then, (QVR α ) becomes the generalized strong vector quasi-equilibrium problem of type (I) and (II) (in short, (GSVQEP I) and (GSVQEP II)) studied in [17].
(GSVQEP I): Find and such that and
and
(GSVQEP II): Find and such that and
The following example shows that all the assumptions of Theorem 9 are satisfied, but Theorem 3.1 in [17] does not work. The reason is that F is not lower (−C)-continuous.
Example 11 Let , , , and
and
We let the relation R be defined by holds iff . We can show that all the assumptions of Theorem 9 are satisfied. However, F is not lower (−C)-continuous at . Also, Theorem 3.1 in [17] does not work.
The following example shows that all the assumptions of Theorem 9 are satisfied, but Theorem 3.1 in [17] is not fulfilled. The reason is that F is not upper C-continuous.
Example 12 Let A, B, D, X, Y, Z, K, C be as in Example 11 and and
We let the relation R be defined by holds iff . It is easy to check that all the assumptions of Theorem 9 are satisfied. So, (QVR α ) has a solution. However, F is not upper C-continuous at . Also, Theorem 3.1 in [17] does not work.
The following example shows that all assumptions of Theorem 9 are satisfied, but Theorem 3.1 in [17] is not fulfilled. The reason is that F is not C-quasiconvex.
Example 13 Let A, B, D, X, Y, Z, K, C, T be as in Example 12 and
We let the relation R be defined by holds iff . It is easy to check that all the assumptions of Theorem 9 are satisfied. However, F is not C-quasiconvex at . Thus, it gives also cases where Theorem 9 can be applied but Theorem 3.1 in [17] does not work.
If we let X, Y, Z be real locally convex Hausdorff topological vector spaces, then we have the following corollary.
Corollary 14 Assume for problem (QVR α ) that
-
(i)
is upper semicontinuous in A with nonempty closed convex values, and is lower semicontinuous A with nonempty closed values;
-
(ii)
T is upper semicontinuous in A with nonempty closed convex values if (or ) and lower semicontinuous in A with nonempty convex values if ;
-
(iii)
for all , holds;
-
(iv)
for all , is quasiconvex in A;
-
(v)
the set is closed.
Then the (QVR α ) has a solution, i.e., there exist such that and satisfying
Moreover, the solution set of the (QVR α ) is closed.
Remark 15
-
(i)
If we let X, Y, Z be real locally convex Hausdorff topological vector spaces, then (GSVQEP I) becomes the problem (GSVQEP) studied in [15].
-
(ii)
If , , Z, , R as in (QVR α ) and , is a multifunction, is a nonempty closed convex cone, the relation R is defined as follows:
Then (QVR α ) becomes strong vector quasi-equilibrium problem (in short, (SVQEP)) studied in [18].
Find such that and
Remark 16
3 Stability
In this section, we discuss the stability of the solutions for (QVR α ). Throughout this section, let X, Y, Z be Banach spaces, N be a real locally G-convex Hausdorff topological vector space. Let , and be nonempty compact convex subsets, , be multifunctions, and be a relation linking , and . Now, we let
Let , be compact sets in a normed space. Recall that the Hausdorff metric is defined by
where and .
For , define
where , are the appropriate Hausdorff metrics. Obviously, is a metric space.
Assume that R satisfies the conditions of Theorem 9. Then for each , (QVR α ) has a solution , i.e., there exists such that and satisfying
For , let
Then , and so defines a set-valued mapping from Ξ into A.
Lemma 17 ([26])
Let Z be a metric space and let M, () be compact sets in Z. Suppose that for any open set , there exists such that , . Then any sequence satisfying has a convergent subsequence with limit in M.
Theorem 18 is upper semicontinuous with compact values.
Proof Similar arguments can be applied to three cases. We present only the proof for the cases where . Indeed, since A is compact, we need only show that is a closed mapping. Let a sequence be given such that . We now show that .
For any n, since , we have that and , such that
For any open set , since is a compact set, there exists such that
where .
Since , and T is upper semicontinuous at , such that
From (5), (6) and (7), we have
Since and , we can apply Lemma 17. There exists a subsequence of such that convergent to , it follows that . By using the same argument as above, we can show that .
Next, we need only show that . Since and K is upper semicontinuous at , is closed, there exists such that (taking a subsequence if necessary).
Since , we can chose a subsequence of such that
Thus,
This implies that there exist , such that
As
and so we have . Since , and , applying (5), we have
Assumption (v) yields that
Since and and (10) yields that and so is closed. Therefore, is closed. Since A is a compact set and . Hence has a compact valued mapping. □
Remark 19 Theorem 18 improves and extends Theorems 3.1 and 3.3 in [17], Theorem 3.1 in [15].
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Hung, N.V. Sensitivity analysis for generalized quasi-variational relation problems in locally G-convex spaces. Fixed Point Theory Appl 2012, 158 (2012). https://doi.org/10.1186/1687-1812-2012-158
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DOI: https://doi.org/10.1186/1687-1812-2012-158