Abstract
In this paper, we introduce symmetric variational relation problems and establish the existence theorem of solutions of symmetric variational relation problems. As the special cases, symmetric (vector) quasi-equilibrium problems and symmetric variational inclusion problems are obtained. Further, we study the notion of essential stability of equilibria of symmetric variational relation problems. We prove that most of symmetric variational relation problems (in the sense of Baire category) are essential and, for any symmetric variational relation problem, there exists at least one essential component of its solution set.
MSC:49J53, 49J40.
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1 Introduction
Let X and Y be real locally convex Hausdorff spaces, and let C and D be nonempty subsets of X and Y, respectively. Let and be set-valued mappings, and be real functions. According to Noor and Oettli [1], the symmetric quasi-equilibrium problem (SQEP) consists in finding such that , , and
The problem is a generalization of equilibrium problem proposed by Blum and Oettli [2]. The equilibrium problem contains as special cases, for instance, optimization problems, problems of Nash equilibria, fixed point problems, variational inequalities, and complementarity problems.
Fu [3] introduced symmetric vector quasi-equilibrium problems (SVQEP). Let Z be a real Hausdorff topological vector space, and let be a closed convex, pointed cone with apex at the origin and with , where intP denotes the interior of P. Let X, Y, C, D, S, T be as above. Let vector mappings be given. The symmetric vector quasi-equilibrium problem (SVQEP) consists in finding such that , , and
Farajzadeh [4] considered symmetric vector quasi-equilibrium problems in the Hausdorff topological vector space by means of a particular technique and answered an open question raised by Fu. The stability of the set of solutions for symmetric vector quasi-equilibrium problems is discussed in [5–7]. Fakhar and Zafarani [8] introduced generalized symmetric vector quasi-equilibrium problems (GSVQEP). Let X, Y, C, D, S, T, f, g be as above. Let P be the set-valued mapping from Z to Z such that for every , is a pointed, closed and convex cone of Z with a nonempty interior and , . The generalized symmetric vector quasi-equilibrium problem (GSVQEP) consists in finding such that , , and
It is well known that the equilibrium problems are unified models of several problems, namely, optimization problems, saddle point problems, variational inequalities, fixed point problems, Nash equilibrium problems etc. Recently, Luc [9] introduced a more general model of equilibrium problems, which is called a variational relation problem (in short, VR). The stability of the solution set of variational relation problems was studied in [10, 11]. Further studies of variational relation problems have been done. Lin and Wang [12] studied simultaneous variational relation problems (SVR) and related applications. Balaj and Luc [13] introduced mixed variational relation problems (MR), and established existence of solutions to a general inclusion problem. Particular cases of variational inclusions and intersections of set-valued mappings were also discussed in [14]. Balaj and Lin [15] brought forward generalized variational relation problems (GVR), and obtained an existence theorem of the solutions for a variational relation problem. An existence theorem for a variational inclusion problem, a KKM theorem and an extension of the well-known Ky Fan inequality have been established as particular cases. Lin and Ansari [16] introduced a system of quasi-variational relations and established the existence of solutions of SQVP by means of the maximal element theorem for a family of multivalued mappings.
Agarwal et al. [17] presented a unified approach in studying the existence of solutions for two types of variational relation problems, which encompass several generalized equilibrium problems, variational inequalities and variational inclusions investigated in the recent literature. Balaj and Lin [18] established existence criteria for the solutions of two very general types of variational relation problems. Moreover, Luc et al. [19] established two main existence conditions for solutions of variational relation problems without convexity, and Pu and Yang [20–22] studied variational relation problems without the KKM property and on Hadamard manifolds.
Motivated and inspired by research works mentioned above, in this paper, we introduce symmetric variational relation problems, and study the existence and essential stability of solutions of symmetric variational relation problems. The results of this paper improve and generalize several known results on variational relation problems and symmetric (vector) quasi-equilibrium problems.
2 Symmetric variational relation
Let X, Y be two nonempty, compact and convex subsets of two normed linear spaces, respectively. Let and be set-valued mappings, and , be two relations linking and . The symmetric variational relation problem consists in finding such that , , and
We recall first some known results concerning set-valued mappings for later use.
Lemma 2.1 (17.35, 5.35 of [23])
(i) Assume that X is a topological space, Y is a locally convex space, and a correspondence is upper semicontinuous at some point . If the set is compact, then the closed convex hull correspondence is upper semicontinuous at . (ii) In a completely metrizable locally convex space, the closed convex hull of a compact set is compact.
Lemma 2.2 (5.29 of [23])
Let be nonempty and compact subsets of the Hausdorff topological linear space. Then is compact.
Lemma 2.3 Let A be a nonempty and compact subset of the compact normed linear space E, and let be open in E. Then .
Proof Since is open in E, coB is also open. Then, for any , there exists an open convex neighborhood of x such that . Since A is nonempty and compact, there is a finite subset of A such that
which implies that
Since is compact for any , by Lemma 2.2, is compact. Hence
□
The following theorem is the main result of this paper.
Theorem 2.1 Assume that
-
(i)
X, Y are two nonempty, compact and convex subsets of two normed linear spaces;
-
(ii)
S, T are continuous with nonempty convex compact values;
-
(iii)
and are closed;
-
(iv)
for any , any finite subset of X and any , there is such that holds;
-
(v)
for any , any finite subset of Y and any , there is such that holds.
Then the symmetric variational relation problem has at least one solution.
Proof Firstly, denote
Then and are closed in and , respectively. Next, define two mappings and by
where and are the distance on and , respectively. Then (i) f, g are continuous; (ii) if and only if holds, and if and only if holds; (iii) for any , and for any .
Now, define the mappings and by
Since S, T are continuous with nonempty convex compact values, and f, g are continuous, it follows from the well-known Berge maximum theorem that H and F are upper semicontinuous with nonempty compact values. By Lemma 2.1, and are upper semicontinuous with nonempty convex compact values. Thus, define the mapping by
By the well-known Fan-Glicksberg fixed point theorem, there exists such that , i.e.,
Suppose that the result is false, without loss of generality, there is such that does not hold. Then . By the definition of H, we have
for any , i.e.,
Since is closed, is open in X. By Lemma 2.3,
Since , , i.e., there is a finite subset such that . By condition (iv), there is such that holds, which contradicts the fact that does not hold for any . This completes the proof. □
Remark 2.1 By Theorem 2.1, we obtain the following typical examples.
-
(1)
Let and be two real-valued functions. Define the variational relations R, Q as follows:
The symmetric quasi-equilibrium problem (SQEP) consists in finding such that , , and
By virtue of Theorem 2.1, problem (SQEP) has at least one solution.
-
(2)
Let X, Y, S, T be as above, Z be a real Hausdorff topological vector space, and be a closed convex, pointed cone with apex at the origin and with , where intP denotes the interior of P. Let vector mappings be given. Define the variational relations R, Q as follows:
The symmetric vector quasi-equilibrium problem (SVQEP) consists in finding such that , , and
By virtue of Theorem 2.1, problem (SVQEP) has at least one solution.
-
(3)
Let X, Y, S, T be as above, Z be a real Hausdorff topological vector space, and be two multivalued mappings. Define the variational relations R, Q as follows:
The symmetric variational inclusion of type (I) consists in finding such that , , and
By virtue of Theorem 2.1, the symmetric variational inclusion of type (I) has at least one solution.
-
(4)
Define the variational relations R, Q as follows:
The symmetric variational inclusion of type (II) problem consists in finding such that , , and
By virtue of Theorem 2.1, the symmetric variational inclusion of type (II) has at least one solution.
-
(5)
Define the variational relations R, Q as follows:
The symmetric variational inclusion of type (III) problem consists in finding such that , , and
By virtue of Theorem 2.1, the symmetric variational inclusion of type (III) has at least one solution.
3 Essential stability
Essential components play an important role in the study of stability. Now, let us start studying the essential stability of solution set of symmetric variational relation problems. First, denote by ℳ the collection of symmetric variational relation problems such that all conditions of Theorem 2.1 hold. For each , denote by the solution set of q. Thus, a set-valued correspondence is well defined. To analyze the stability of in ℳ, some topological structure in the collection ℳ is also needed. For each , define the distance on ℳ by
where is the Hausdorff distance defined on X, is the Hausdorff distance defined on Y, h is the Hausdorff distance defined on , and is the Hausdorff distance defined on . Clearly, is a metric space.
Definition 3.1 Let . An is said to be an essential point of if, for any open neighborhood of in , there is a positive δ such that for any with . q is said to be essential if each is essential.
Definition 3.2 Let . A nonempty closed subset of is said to be an essential set of if, for any open set U, , there is a positive δ such that for any with .
Definition 3.3 Let . An essential subset is said to be a minimal essential set of if it is a minimal element of the family of essential sets ordered by set inclusion. A component is said to be an essential component of if is essential.
Remark 3.1 (1) It is easy to see that the problem is essential if and only if the mapping is lower semicontinuous at q. (2) For two closed , if is essential, then is also essential.
First of all, let us introduce some mathematical tools for the following proof, which can be found in [23–26].
Lemma 3.1 ([23])
Let X and Y be two topological spaces with Y compact. If F is a closed set-valued mapping from X to Y, then F is upper semi-continuous.
Lemma 3.2 ([24])
If X, Y are two metric spaces, X is complete and is upper semicontinuous with nonempty compact values, then the set of points, where F is lower semicontinuous, is a dense residual set in X.
Lemma 3.3 ([25])
Let C, D be two nonempty, convex and compact subsets of linear normed space E. Then
where h is the Hausdorff distance defined on E, and , .
Lemma 3.4 ([26])
Let be a metric space, and be two nonempty compact subsets of Y, and be two nonempty disjoint open subsets of Y. If , then
where h is the Hausdorff metric defined on Y.
Theorem 3.1 is a complete metric space.
Proof Let be any Cauchy sequence in ℳ, then, for any , there is such that for any , i.e.,
for any .
-
(1)
Clearly, there exist set-valued mappings , , and a closed subset A of , a closed subset B of such that , for any , and S, T are continuous with nonempty convex compact values, and , .
-
(2)
Further, define the following symmetric variational relation by
Clearly, under the distance ρ. We will show that .
-
(i)
Clearly, and are closed.
-
(ii)
Suppose that there exist , a finite subset of X and such that does not hold for each , then for each . Since , then for each and for enough large m, which implies that does not hold for each . It is a contradiction. Thus, for any , any finite subset of X and any , there is such that holds. Similarly, for any , any finite subset of Y and any , there is such that holds. Hence and is complete. □
Theorem 3.2 The solution mapping is upper semicontinuous with nonempty compact values.
Proof The desired conclusion follows from Lemma 3.1 as soon as we show that is closed. Let be a sequence converging to such that for any n. Then and , and hold for any and any . Since , it follows that and .
Suppose that there exists such that does not hold, then there exists such that , and . For enough large n, we have , i.e., does not hold. It is a contradiction. Therefore holds for any . Similarly, holds for any . Hence . □
Theorem 3.3 There exists a dense residual subset of ℳ such that q is essential for each .
Proof Since the metric space is complete (by Theorem 3.1), and the mapping is upper semicontinuous with compact values (by Theorem 3.2), by Lemma 3.2, Λ is lower semicontinuous on a dense residual subset of ℳ. Thus, by Remark 3.1(1), q is essential for each . □
Theorem 3.4 For each , there exists at least one minimal essential subset of .
Proof By Theorem 3.2, is upper semicontinuous with compact values, that is, for each open set , there exists such that for any with . Hence is an essential set of itself. Let Θ denote the family of all essential sets of ordered by set inclusion. Then Θ is nonempty and every decreasing chain of elements in Θ has a lower bound (because by the compactness the intersection is in Θ); therefore, by Zorn’s lemma, Θ has a minimal element and it is a minimal essential set of . □
Theorem 3.5 For each , every minimal essential subset of is connected.
Proof For each , let be a minimal essential subset of . Suppose that is not connected, then there exist two non-empty compact subsets , with , and there exist two disjoint open subsets , of such that , . Since is a minimal essential set of , neither nor is essential. There exist two open sets , such that, for any , there exist with
Denote , , we know that , are open, , , and we may assume that , . Denote and , .
Since is essential and , there exists such that for any with . Since is a minimal essential set of , neither nor is essential. Thus, for , there exist two such that
Thus
Next, define the symmetric variational relation problem by
where
Easily, we can check that
-
(i)
, are continuous with nonempty compact convex values.
-
(ii)
Since and are closed in , A is closed in , which implies that is closed. Similarly, is closed.
-
(iii)
Suppose that there exist , a finite subset and such that does not hold for any , then
As , without loss of generality, we may assume that . Therefore,
i.e.,
which implies that for any . Thus does not hold for any , which is a contradiction. Hence, for any , any finite subset of X and any , there is such that holds. Similarly, for any , any finite subset of Y and any , there is such that holds. Hence .
-
(iv)
Further, by Lemma 3.3 and Lemma 3.4, we have
Hence
Thus and .
Since
we assume without loss of generality. Then there exists such that , , , and and hold for any and any . Since ,
which implies that . Hence , which contradicts . Thus is connected. □
Theorem 3.6 For each , there exists at least one essential component of .
Proof By Theorem 3.5, there exists at least one connected minimal essential subset of . Thus, there is a component C of such that . It is obvious that C is essential by Remark 3.1(2). Thus C is an essential component. □
Remark 3.2 Our paper has improved the results of [5]. (i) The symmetric (vector) quasi-equilibrium problem is a special case of symmetric variational relation problem; (ii) In [5], the existence of essential connected components is based on disturbance of S, T for fixed , (see Section 4 in [5]), but the existence of essential connected components in our paper is based on disturbance of S, T, R, Q, which are more general.
4 Conclusion
In this paper, symmetric variational relation problems are introduced, and we establish the existence theorem of solutions of symmetric variational relation problems. Further, we study the notion of essential stability of equilibria of symmetric variational relation problems. Our paper improves the results of [5].
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Supported by the open project of Key Laboratory of Mathematical Economics (SUFE), Ministry of Education (Project Number: 201309KF02).
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Yang, Z. On existence and essential stability of solutions of symmetric variational relation problems. J Inequal Appl 2014, 5 (2014). https://doi.org/10.1186/1029-242X-2014-5
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DOI: https://doi.org/10.1186/1029-242X-2014-5