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On the Existence Result for System of Generalized Strong Vector Quasiequilibrium Problems
Fixed Point Theory and Applications volume 2011, Article number: 475121 (2011)
Abstract
We introduce a new type of the system of generalized strong vector quasiequilibrium problems with set-valued mappings in real locally convex Hausdorff topological vector spaces. We establish an existence theorem by using Kakutani-Fan-Glicksberg fixed-point theorem and discuss the closedness of strong solution set for the system of generalized strong vector quasiequilibrium problem. The results presented in the paper improve and extend the main results of Long et al. (2008).
1. Introduction
The equilibrium problem is a generalization of classical variational inequalities. This problem contains many important problems as special cases, for instance, optimization, Nash equilibrium, complementarity, and fixed-point problems (see [1–3] and the references therein). Recently, there has been an increasing interest in the study of vector equilibrium problems. Many results on existence of solutions for vector variational inequalities and vector equilibrium problems have been established (see, e.g., [4–16]).
Let and
be real locally convex Hausdorff space,
a nonempty subset and
be a closed convex pointed cone. Let
be a given set-valued mapping. Ansari et al. [17] introduced the following set-valued vector equilibrium problems (VEPs) to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ1_HTML.gif)
or to find such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ2_HTML.gif)
If is nonempty, and
satisfies (1.1), then we call
a weak efficient solution for (VEP), and if
satisfies (1.2), then we call
a strong solution for VEP. Moreover, they also proved an existence theorem for a strong vector equilibrium problem (1.2) (see [17]).
In 2000, Ansari et al. [5] introduced the system of vector equilibrium problems (SVEPs), that is, a family of equilibrium problems for vector-valued bifunctions defined on a product set, with applications in vector optimization problems and Nash equilibrium problem [11] for vector-valued functions. The (SVEP) contains system of equilibrium problems, systems of vector variational inequalities, system of vector variational-like inequalities, system of optimization problems and the Nash equilibrium problem for vector-valued functions as special cases. But, by using (SVEP), we cannot establish the existence of a solution to the Debreu type equilibrium problem [7] for vector-valued functions which extends the classical concept of Nash equilibrium problem for a noncooperative game. Moreover, Ansari et al. [18] introduced the following concept of system of vector quasiequilibrium problems.
Let be any index set and for each
, let
be a topological vector space. Consider a family of nonempty convex subsets
with
. We denote by
and
. For each
, let
be a topological vector space and let
and
be multivalued mappings and
be a bifunction. The system of vector quasiequilibrium problems (SVQEPs), that is, to find
such that for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ3_HTML.gif)
If for all
, then (SVQEP) reduces to (SVEP) (see [5]) and if the index set
is singleton, then (SVQEP) becomes the vector quasiequilibrium problem. Many authors studied the existence of solutions for systems of (vector) quasiequilibrium problems, see, for example, [19–23] and references therein.
On the other hand, it is well known that a strong solution of vector equilibrium problem is an ideal solution, It is better than other solutions such as efficient solution, weak efficient solution, proper efficient solution and supper efficient solution (see [13]). Thus, it is important to study the existence of strong solution and properties of the strong solution set. In general, the ideal solutions do not exist.
Very recently, the generalized strong vector quasiequilibrium problem (GSVQEPs) is introduced by Long et al. [16]. Let ,
, and
are real locally convex Hausdorff topological vector spaces,
and
are nonempty compact convex subsets, and
is a nonempty closed convex cone. Let
,
and
are three set-valued mappings. They considered the GSVQEP: finding
,
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ4_HTML.gif)
Moreover, they gave an existence theorem for a generalized strong vector quasiequilibrium problem without assuming that the dual of the ordering cone has a weak* compact base.
Motivated and inspired by research works mentioned above, in this paper, we introduce a different kind of systems of generalized strong vector quasiequilibrium problem without assuming that the dual of the ordering cone has a weak* compact base. Let ,
, and
are real locally convex Hausdorff topological vector spaces,
and
are nonempty compact convex subsets, and
is a nonempty closed convex cone. We also suppose that
,
and
are set-valued mappings. We consider the following system of generalized strong vector quasiequilibrium problem (SGSVQEPs): finding
and
,
such that
,
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ5_HTML.gif)
We call this a strong solution for the (SGSVQEP).
At a quick glance, our required solution seems to be similar to such a thing of Ansari et al. [5, 18], in the case of and
. In fact, however, the main different point comes from the independent choice of coordinate. In this paper, we establish an existence theorem of strong solution set for the system of generalized strong vector quasiequilibrium problem by using Kakutani-Fan-Glicksberg fixed-point theorem and discuss the closedness of the solution set. Moreover, we apply our result to obtain the result of Long et al. [16].
2. Preliminaries
Throughout this paper,we suppose that ,
, and
are real locally convex Hausdorff topological vector spaces,
and
are nonempty compact convex subsets, and
is a nonempty closed convex cone. We also suppose that
,
, and
are set-valued mappings.
Definition 2.1.
Let and
be two topological vector spaces and
a nonempty subset of
and let
be a set-valued mapping.
(i) is called upper
-continuous at
if, for any neighbourhood
of the origin in
, there is a neighbourhood
of
such that, for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ6_HTML.gif)
(ii) is called lower
-continuous at
if, for any neighbourhood
of the origin in
, there is a neighbourhood
of
such that, for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ7_HTML.gif)
Definition 2.2.
Let and
be two topological vector spaces and
a nonempty convex subset of
. A set-valued mapping
is said to be properly
-quasiconvex if, for any
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ8_HTML.gif)
Definition 2.3.
Let and
be two topological vector spaces, and
be a set-valued mapping.
(i) is said to be upper semicontinuous at
if, for any open set
containing
, there exists an open set
containing
such that, for all
,
;
is said to be upper semicontinuous on
if it is upper semicontinuous at all
.
(ii) is said to be lower semicontinuous at
if, for any open set
with
, there exists an open set
containing
such that, for all
,
;
is said to be lower semicontinuous on
if it is lower semicontinuous at all
.
(iii) is said to be continuous on
if, it is at the same time upper semicontinuous and lower semicontinuous on
.
(iv) is said to be closed if the graph,
, of
, that is,
, is a closed set in
.
Lemma 2.4 (see [12]).
Let be a nonempty compact subset of locally convex Hausdorff vector topology space
. If
is upper semicontinuous and for any
,
is nonempty, convex and closed, then there exists an
such that
.
Lemma 2.5 (see [24]).
Let and
be two Hausdorff topological vector spaces and
be a set-valued mapping. Then, the following properties hold:
(i)if is closed and
is compact, then
is upper semicontinuous, where
and
denotes the closure of the set
,
(ii)if is upper semicontinuous and for any
,
is closed, then
is closed,
(iii) is lower semicontinuous at
if and only if for any
and any net
,
, there exists a net
such that
and
.
3. Main Results
In this section, we apply Kakutani-Fan-Glicksberg fixed-point theorem to prove an existence theorem of strong solutions for the system of generalized strong vector quasiequilibrium problem. Moreover, we also prove the closedness of strong solution set for the system of generalized strong vector quasiequilibrium problem.
Theorem 3.1.
For each , let
be continuous set-valued mappings such that for any
,
are nonempty closed convex subsets of
. Let
be upper semi continuous set-valued mappings such that for any
are nonempty closed convex subsets of
and
be set-valued mappings satisfy the following conditions:
(i)for all ,
,
(ii)for all ,
are properly
-quasiconvex,
(iii) are upper
-continuous,
(iv)for all ,
are lower
-continuous.
Then, SGSVQEP has a solution. Moreover, the set of all strong solutions is closed.
Proof.
For any , define set-valued mappings
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ9_HTML.gif)
Step 1.
Show that and
are nonempty.
For any , we note that
and
are nonempty. Thus, for any
, we have
and
are nonempty.
Step 2.
Show that and
are convex subsets of
.
Let and
. Put
. Since
and
is convex set, we have
. By (ii),
is properly
-quasiconvex. Without loss of generality, we can assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ10_HTML.gif)
We claim that . In fact, if
, then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ11_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ12_HTML.gif)
which contradicts to . Therefore
and hence
is a convex subset of
. Similarly, we have
is convex subset of
.
Step 3.
Show that and
are closed subsets of
.
Let be a sequence in
such that
. Thus, we have
. Since
is a closed subset of
, it follows that
. By the lower semicontinuity of
and Lemma 2.5(iii), for any
and any net
, there exists a net
such that
and
. This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ13_HTML.gif)
Since is lower
-continuous, for any neighbourhood
of the origin in
, there is a subnet
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ14_HTML.gif)
From (3.5) and (3.6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ15_HTML.gif)
We claim that . Assume that there exists
and
. Thus, we note that
and
is closed. Hence
is open and
. Since
is a locally convex space, there exists a neighbourhood
of the origin such that
is convex and
. This implies that
, that is,
, which is a contradiction. Therefore
. This mean that
and so
is a closed subset of
. Similarly, we have
is a closed subset of
.
Step 4.
Show that and
are upper semicontinuous.
Let be given such that
, and let
such that
. Since
and
is upper semicontinuous, it follows by Lemma 2.5(ii) that
. We now claim that
. Assume that
. Then, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ16_HTML.gif)
which implies that there is a neighbourhood of the origin in
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ17_HTML.gif)
Since is upper
-continuous, for any neighbourhood
of the origin in
, there exists a neighbourhood
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ18_HTML.gif)
Without loss of generality, we can assume that . This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ19_HTML.gif)
Thus there is such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ20_HTML.gif)
which contradicts to . Hence
and, therefore,
is a closed mapping. Since
is a compact set and
is a closed subset of
, we note that
is compact. Then,
is also compact. Hence, by Lemma 2.5(i),
is an upper semicontinuous mapping. Similarly, we note that
is an upper semicontinuous mapping.
Step 5.
Show that SGSVQEP has a solution.
Define the set-valued mapping and
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ21_HTML.gif)
Then, and
are upper semicontinuous and, for all
,
, and
are nonempty closed convex subsets of
.
Define the set-valued mapping by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ22_HTML.gif)
Then, is also upper semicontinuous and, for all
,
is a nonempty closed convex subset of
. By Lemma 2.4, there exists a point
such that
, that is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ23_HTML.gif)
This implies that ,
,
, and
. Then, there exists
and
,
such that
,
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ24_HTML.gif)
Hence SGSVQEP has a solution.
Step 6.
Show that the set of solutions of SGSVQEP is closed.
Let be a net in the set of solutions of SGSVQEP such that
. By definition of the set of solutions of SGSVQEP, we note that there exist
,
,
, and
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ25_HTML.gif)
Since and
are continuous closed valued mappings, we obtain
and
. Let
and
. Since
and
are upper semicontinuous closed valued mappings, it follows by Lemma 2.5(ii) that
and
are closed. Thus, we note that
and
. Since
and
are lower
-continuous, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ26_HTML.gif)
This means that belongs to the set of solutions of SGSVQEP. Hence the set of solutions of SGSVQEP is closed set. This completes the proof.
If we take ,
, and
. Then, from Theorem 3.1, we derive the following result.
Corollary 3.2.
Let be a continuous set-valued mapping such that for any
,
is nonempty closed convex subset of
. Let
be an upper semicontinuous set-valued mapping such that for any
,
is a nonempty closed convex subset of
and
be set-valued mapping satisfy the following conditions:
(i)for all ,
,
(ii)for all ,
is properly
-quasiconvex,
(iii) is an upper
-continuous,
(iv)for all ,
is a lower
-continuous,
(v)if and
then
.
Then, GSVQEP has a solution. Moreover, the set of all solution of GSVQEP is closed.
Now we give an example to explain that Theorem 3.1 is applicable.
Example 3.3.
Let ,
, and
. For each
, let
,
and
,
. We consider the set-valued mappings
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ27_HTML.gif)
Then, it is easy to check that all of condition (i)–(iv) in Theorem 3.1 are satisfied. Hence, by Theorem 3.1, SGSVQEP has a solution. Let be the set of all strong solutions for SGSVQEP. Then, we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F475121/MediaObjects/13663_2010_Article_1406_Equ28_HTML.gif)
References
Bianchi M, Schaible S: Generalized monotone bifunctions and equilibrium problems. Journal of Optimization Theory and Applications 1996,90(1):31–43. 10.1007/BF02192244
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994,63(1–4):123–145.
Oettli W, Schläger D: Existence of equilibria for monotone multivalued mappings. Mathematical Methods of Operations Research 1998,48(2):219–228. 10.1007/s001860050024
Ansari QH, Oettli W, Schläger D: A generalization of vectorial equilibria. Mathematical Methods of Operations Research 1997,46(2):147–152. 10.1007/BF01217687
Ansari QH, Schaible S, Yao JC: System of vector equilibrium problems and its applications. Journal of Optimization Theory and Applications 2000,107(3):547–557. 10.1023/A:1026495115191
Bianchi M, Hadjisavvas N, Schaible S: Vector equilibrium problems with generalized monotone bifunctions. Journal of Optimization Theory and Applications 1997,92(3):527–542. 10.1023/A:1022603406244
Debreu G: A social equilibrium existence theorem. Proceedings of the National Academy of Sciences of the United States of America 1952, 38: 886–893. 10.1073/pnas.38.10.886
Fu J-Y: Generalized vector quasiequilibrium problems. Mathematical Methods of Operations Research 2000,52(1):57–64. 10.1007/s001860000058
Giannessi F: Vector Variational Inequilities and Vector Equilibria, Mathematical Theories. Volume 38. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2000:xiv+523.
Gong X: Strong vector equilibrium problems. Journal of Global Optimization 2006,36(3):339–349. 10.1007/s10898-006-9012-5
Gong XH: Efficiency and Henig efficiency for vector equilibrium problems. Journal of Optimization Theory and Applications 2001,108(1):139–154. 10.1023/A:1026418122905
Holmes RB: Geometric Functional Analysis and Its Application, Graduate Texts in Mathematics, no. 2. Springer, New York, NY, USA; 1975:x+246.
Hou SH, Gong XH, Yang XM: Existence and stability of solutions for generalized Ky Fan inequality problems with trifunctions. Journal of Optimization Theory and Applications 2010,146(2):387–398. 10.1007/s10957-010-9656-7
Huang NJ, Li J, Thompson HB: Stability for parametric implicit vector equilibrium problems. Mathematical and Computer Modelling 2006,43(11–12):1267–1274. 10.1016/j.mcm.2005.06.010
Li SJ, Teo KL, Yang XQ: Generalized vector quasiequilibrium problems. Mathematical Methods of Operations Research 2005,61(3):385–397. 10.1007/s001860400412
Long X, Huang N, Teo K: Existence and stability of solutions for generalized strong vector quasiequilibrium problem. Mathematical and Computer Modelling 2008,47(3–4):445–451. 10.1016/j.mcm.2007.04.013
Ansari QH, Oettli W, Schiager D: A generalization of vectorial equilibria. In Proceedings of the 2nd International Symposium on Operations Research and Its Applications (ISORA '96), December1996, Guilin, China. Volume 1114. World Publishing; 181–185.
Ansari QH, Chan WK, Yang XQ: The system of vector quasiequilibrium problems with applications. Journal of Global Optimization 2004,29(1):45–57.
Ansari QH: Existence of solutions of systems of generalized implicit vector quasiequilibrium problems. Journal of Mathematical Analysis and Applications 2008,341(2):1271–1283. 10.1016/j.jmaa.2007.11.033
Ansari QH, Khan Z: System of generalized vector quasiequilibrium problems with applications. In Mathematical Analysis and Applications. Volume 7. Edited by: Nanda S, Rajasekhar GP. Narosa, New Delhi, India; 2004:1–13.
Ansari QH, Lin LJ, Su LB: Systems of simultaneous generalized vector quasiequilibrium problems and their applications. Journal of Optimization Theory and Applications 2005,127(1):27–44. 10.1007/s10957-005-6391-6
Ansari QH, Schaible S, Yao J-C: The system of generalized vector equilibrium problems with applications. Journal of Global Optimization 2002,22(1–4):3–16.
Lin L-J: System of generalized vector quasiequilibrium problems with applications to fixed point theorems for a family of nonexpansive multivalued mappings. Journal of Global Optimization 2006,34(1):15–32. 10.1007/s10898-005-4702-y
Aubin J-P, Ekeland I: Applied Nonlinear Analysis, Pure and Applied Mathematics. John Wiley & Sons, New York, NY, USA; 1984:xi+518.
Acknowledgments
The authors would like to thank the referees for the insightful comments and suggestions. S. Plubtieng the Thailand Research Fund for financial support under Grants no. BRG5280016. Moreover, K. Sitthithakerngkiet would like to thanks the Office of the Higher Education Commission, Thailand for supporting by grant fund under Grant no. CHE-Ph.D-SW-RG/41/2550, Thailand.
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Plubtieng, S., Sitthithakerngkiet, K. On the Existence Result for System of Generalized Strong Vector Quasiequilibrium Problems. Fixed Point Theory Appl 2011, 475121 (2011). https://doi.org/10.1155/2011/475121
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DOI: https://doi.org/10.1155/2011/475121