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Essential components of the set of solutions for the system of vector quasi-equilibrium problems
Journal of Inequalities and Applications volume 2012, Article number: 181 (2012)
Abstract
In this paper, we prove that, for every vector quasi-equilibrium problem, there exists at least one essential component of the set of its solutions. As application, we show that, for every system of vector quasi-equilibrium problems, there exists at least one essential component of the set of its solutions in the uniform topological space of objective functions and constraint mappings.
1 Introduction
Essential component has been an important aspect in the study of stability for nonlinear problems. Fort [1] first introduced the notion of essential fixed points of a continuous mapping from a compact metric space into itself and proved that any mapping can be approximately closed by a mapping whose fixed points are all essential. Kinoshita [2] then introduced the notion of essential components of the set of fixed points of a single-valued map. Jiang [3] introduced the notion of essential components of the set of Nash equilibrium points for an n-person non-cooperative game and proved the existence of essential components of the set of Nash equilibrium points. Kohlberg and Mertens [4] studied the stability of Nash equilibrium points and suggested that a satisfactory solution for a non-cooperative game should be set-wise, and they proved that such a solution is just an essential component of Nash equilibrium points. Recently, Yu, Xiang [5], Yu, Luo [6], Isac, Yuan [7], Yang, Yu [8], Lin [9], Chen, Gong [10] introduced the notion of essential components to solution sets of various problems such as Ky Fan point problems, equilibrium problems, coincident point problems, vector optimization problem, and symmetric vector quasi-equilibrium problems. On the other hand, in order to describe the real world and economic behavior better, very recently, much attention has been attracted to multi-criteria equilibrium models. Ansari, Schaible and Yao [11] studied the system of generalized vector equilibrium problems. Ansari, Chan and Yang [12] studied the system of vector quasi-equilibrium problems (briefly, SVQEP). Fang, Huang and Kim [13] studied the system of vector equilibrium problems. Peng, Lee, Yang [14] studied the system of generalized vector quasi-equilibrium problems with set-valued maps (briefly, SGVOEPS). Lin [15] studied the system of generalized vector quasi-equilibrium problems (briefly, SGVQEP) in Banach spaces. Peng, Yang and Zhu [16] studied the system of vector quasi-equilibrium. Lin [9] established essential components of the solution set for SGVQEP under perturbations of the best-reply map. But up to now, no paper has established essential components of the solution set for SVQEP, SGVQEP or SGVQEPS under perturbations of objective functions and constraint mappings. In this paper, we first give a new result of essential components of the solution set for SVQEP under perturbations of objective functions and constraint mappings.
2 Preliminaries and definitions
Let be a finite set which has at least two elements. For each , let and be real Hausdorff topological vector spaces and a nonempty subset of . For each , let be a closed, convex and pointed cone of with , where denotes the interior of . Let . For each , let be a vector-valued mapping and be a set-valued mapping. The SVQEP consists of finding such that for each ,
where denotes the i th component of , and is said to be a solution of the SVQEP. For each , is said to be an objective function of the SVQEP and for each , is said to be a constraint mapping of the SVQEP. The SVQEP includes, as a special case, the following multiobjective generalized game problem:
For each , let be a vector-valued mapping and be a feasible strategy mapping, where . For each , we can write . The multiobjective generalized game problem consists of finding such that for each , and
where is said to be a weakly Pareto-Nash equilibrium point.
For each , setting
the SVQEP coincides with the multiobjective generalized game problem, which has been studied by Yu and Luo [6] but for real functions and Lin [9] but for () for any .
For each , setting , the multiobjective generalized game problem coincides with the multiobjective game problem, which has been studied by Yu and Xiang [5] and Yang and Yu [8].
Definition 2.1 Let X be a real Hausdorff topological space and Y a real Hausdorff topological vector space with a convex cone C. Let be a vector-valued function.
(i) f is said to be C-continuous at if, for any open neighborhood V of the zero element θ in Y, there is an open neighborhood of in X such that
f is said to be C-continuous on X if it is C-continuous at every element of X.
(ii) f is said to be (−C)-continuous at if, for any open neighborhood V of θ in Y, there exists an open neighborhood of in X such that
f is said to be (−C)-continuous on X if it is (−C)-continuous at every point of X.
Definition 2.2 Let K be a nonempty convex subset of a vector space X, let Y be a vector space with a convex pointed cone C. Let be a mapping. f is said to be C-convex if, for any and ,
Definition 2.3 Let X and Y be two Hausdorff topological spaces, let be a set-valued mapping. F is said to be upper semicontinuous (in short, u.s.c.) at if, for any neighborhood of , there exists a neighborhood of such that
F is said to be upper semicontinuous on X if F is u.s.c. at every point .
F is said to be lower semicontinuous (in short, l.s.c.) at if, for any and any neighborhood of , there exists a neighborhood of such that
F is said to be lower semicontinuous on X if it is lower semicontinuous at every .
F is said to be continuous on X if it is both u.s.c. and l.s.c. on X.
F is said to be a closed mapping if is a closed set in .
F is an usco mapping if F is u.s.c. on X and is compact for every .
Let be a linear metric space. Denote by all nonempty convex compact subsets of X. Define the Hausdorff metric h on as follows.
For any , let
where
and
Theorem 2.1[17]
Let Y be a real Hausdorff topological vector space, andbe a closed convex pointed cone with. Let K be a nonempty compact convex subset of a real locally convex Hausdorff topological vector space X. Let the set-valued mappingbe continuous with nonempty compact convex values. Ifsatisfies the following conditions:
(i) is (−C)-continuous;
(ii) for any fixed, is C-convex;
(iii) for any, .
Then, there exists an elementsuch thatand
3 Essential components of the solution set for the system of vector quasi-equilibrium problems
Throughout this section, let be a finite set which has at least two elements. For each , let be a real normed linear space and a Banach space with ; let be a nonempty compact convex subset of , and let be a closed convex pointed cone of with and . Let and .
Let Φ be the collection of all vector-valued functions such that such that: (i) is -continuous on ; (ii) for each fixed , is -convex; (iii) for any , , where θ is the zero element of ; (iv) .
Let M be the collection of all set-valued mappings such that: (i) for each , is convex and closed; (ii) S is continuous on K.
Let . For any , , define
where is the norm on and h is the Hausdorff metric defined on . Clearly, is a metric space.
For any , by Theorem 2.1, there exists a solution to the vector quasi-equilibrium problem: and
For each , define
Thus for any and indeed defines a set-valued mapping from H to K.
The following lemma can be found in [18].
Lemma 3.1 Let X be a metric space andbe the family of all nonempty compact subsets of X. Let () satisfy the condition that for each open set O containing A, there exists an integer N such that whenever, we have. Then for any sequencewith (), there exists a subsequence which converges to a point in A.
Lemma 3.2is an usco mapping.
Proof Since K is compact, by [19], it suffices to prove that is closed. Let with , where and . Since , we have
For any open neighborhood O of in K, since is compact, by [[19], p.108], there is such that
where . Since , , and S is u.s.c. at , there is N such that for any , we have
and
So whenever , we have
Since belongs to , and and are compact, by Lemma 3.1, there exists a subsequence of such that . Since , we have
Since S is l.s.c. at , for any , by [19], there exists such that . Since , there exists a subsequence of such that
Thus, there exists a subsequence of such that
which implies that there exists such that
where is a subsequence of . Since and (), we have that (). As , we have
Now we need to show that
If the conclusion is false, then , which implies that there is such that
where B denotes the open unit ball in . Since ψ is -continuous on , and , for above , there is a positive integer such that
On the other hand, since , there is a positive integer with , such that
By (7), (6) and (5), we have
This contradicts (3). Hence (4) holds. Then by the arbitrariness of , we obtain that
By (2) and (8), we have that . Hence, is closed. is also closed, for all . By the compactness of K, we know that F is a set-valued mapping with compact values. Hence, F is an usco mapping. The proof is completed. □
For each , the component of a point is the union of all the connected subsets of containing x. Note that the components are connected closed subsets of , and thus are connected and compact, see [20]. It is easy to see that the components of two distinct points of either coincide or are disjoint, so that all components constitute a decomposition of into connected pairwise disjoint compact subsets, i.e.,
where Λ is an index set for each , is a nonempty connected compact subset of and, for any (), .
Definition 3.1 Let and m be a nonempty closed subset of . m is said to be an essential set of if, for each open set , there exists such that for any with , . If a component of is an essential set, then is said to be an essential component of . An essential set m of is said to be a minimal essential set of if m is a minimal element of the family of essential sets in ordered by set inclusion.
Lemma 3.3[7]
Let A, B and C be nonempty convex compact subsets of a normed linear space X. Then, where h is the Hausdorff metric defined on, , , and.
The following theorem is the exist theorem of an essential component of the set of solutions for the vector quasi-equilibrium problem.
Theorem 3.1
(a) For any, there exists at least one minimal essential set of, and every minimal essential set ofis connected;
(b) For any, there exists at least one essential connected component of.
Proof Since F is upper semicontinuous, following the idea of Lemma 2.2 in [5], we can easily obtain that there exists one minimal essential set of for each . Now, for each minimal essential set of , as Yang and Yu did in [8], we prove that each minimal essential set of is connected. Let be one minimal essential set of . If is not connected, then there exist two nonempty closed sets and of and two open sets and in K such that
Since is a minimal essential set of , neither nor is essential. Thus, there exist two open sets and such that, for any , there exist with
Set , and . Then both and are open sets and , and . Since and are a closed subset of the compact set , and are a compact set, there exist two open sets and such that
Since is essential and , there exists such that for any with , we have
Since and , for above , by (9), there exist such that
Since , we have and . Now we define and by
and
respectively, where
It is obvious that λ and μ are continuous functions on K with , , and for any .
We can see that: (i) is -continuous on ; (ii) for each fixed , is -convex in y; (iii) for all ; (iv) ; (v) for each , is convex and compact; (vi) is continuous on K. Hence . By Lemma 3.3, we have
and
Thus, by (11), we have
Using (10), we have
If , then , , and for all . If , then and for all . Since and for all , we have . This contradicts (11). Thus, we have . Similarly, we can show that . This contradicts (12). Hence, is connected, so the conclusion holds.
For any , by , there exists at least one essential connected set m of . There exists a component of such that . It is obvious that is essential. □
Now, for each , let be a vector-valued mapping and a set-valued mapping. Let
Let
Let . For any
define
where h is the Hausdorff metric defined on . Clearly, is a metric space.
Let . It is clear that K is a nonempty compact convex subset of . For any , and , define the mapping by
and the mapping by
Since and , we can see that is continuous with nonempty convex and compact valued, (i) is -continuous on ; (ii) for each fixed , is -convex; and (iii) for all . It is clear that . Since for each , by Theorem 2.1, there exists an element such that and
That is
For each , by the arbitrariness of , , , take , and by assumption , , and , we obtain that and
Since and , it follows that
Thus, there exists such that for each , and
For each , denote by all solutions to the SVQEP. By (13), there exists , thus . Similar to Definition 3.1, we can define the minimal essential set and essential component of .
Lemma 3.4 For each, define the mappingby
where
and
Then T is continuous.
Proof It is easy to check that for each , .
For any , , if , then by the definition of , we have
This completes the proof of the lemma. □
The following lemma can be found in [8].
Lemma 3.5 Let U, Y and Z be three metric spaces, be an usco mapping andbe a set-valued mapping. Suppose that there exists a continuous mappingsuch thatfor each. Furthermore, suppose that there exists at least one essential component offor each. Then there exists at least one essential component offor each.
As application of Theorem 3.1, now we will show that, for every system of vector quasi-equilibrium problems, there exists at least one essential component of the set of its solutions in the uniform topological space of objective functions and constraint mappings.
Theorem 3.2 For each, there exists at least one essential component of.
Proof For any , define by , where
and
By Lemma 3.4, T is continuous. Now we need to prove that for each , , where F is defined by (1). If , then , and
That is
For each , by the arbitrariness of , , , take , and by assumption , , and , we obtain that and
Since and , it follows that
Hence and hence . Thus, by Lemma 3.2, Theorem 3.1 and Lemma 3.5, there exists at least one essential component of . □
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Acknowledgements
This research was partially supported by the National Natural Science Foundation of China (Grant No. 11061023) and the Natural Science Foundation of Jiangxi Province (2010GZS0176), China.
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This work was carried out in collaboration between all authors. X-HG gave the ideas of the problems in this research and interpreted the results. J-CC proved the theorems, interpreted the results and wrote the article. All authors defined the research theme, read and approved the manuscript.
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Gong, XH., Chen, JC. Essential components of the set of solutions for the system of vector quasi-equilibrium problems. J Inequal Appl 2012, 181 (2012). https://doi.org/10.1186/1029-242X-2012-181
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DOI: https://doi.org/10.1186/1029-242X-2012-181