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Some new generalizations of nonempty intersection theorems without convexity assumptions and essential stability of their solution set with applications
Fixed Point Theory and Applications volume 2014, Article number: 3 (2014)
Abstract
As a generalization of the KKM theorem in (Yang and Pu in J. Optim. Theory Appl. 154:17-29 2012), we propose some new nonempty intersection theorems for an infinite family of set-valued mappings without convexity assumptions, and consider generic stability and essential components of solutions of a nonempty intersection theorem for an infinite family of set-valued mappings without convexity assumptions. This paper is an attempt to establish analogue results for the class of equilibria removing convexity assumptions. As applications, we deduce the corresponding results for Ky Fan’s points, Nash equilibrium and variational relations.
MSC:49J53, 49J40.
1 Introduction
The celebrated KKM theorem introduced in 1929 [1] has been extended with various applications to optimization-related problems for many decades. Fan [2] and Browder [3] gave a version of Hausdorff topological vector spaces for this problem. As a generalization of the KKM theorem, Guillerme [4] proved an intersection theorem for an infinite family of set-valued mappings, where index is any set. Moreover, Hou [5] proposed an intersection theorem for an infinite family of set-valued mappings, which was defined on non-compact spaces. Ding [6] introduced product FC-spaces to generalize the KKM theorem, and established the existence of equilibrium for generalized multi-objective games in FC-spaces, where the number of players was finite or infinite, and all payoffs were all set-valued mappings. Recently, Lin [7] brought forward systems of nonempty intersection theorems, and established the existence of solutions of systems of quasi-KKM problems, systems of quasi-variational inclusions and systems of quasi-variational inclusions, as particular cases.
Convexity assumptions or some convexity of mappings played an important role in [1–7]. But, in many works on the theory, some authors replaced the convexity of mappings by more general concepts. For example, two important concepts were marked by the seminal papers of Fan [8, 9] for removing the concavity/quasi-concavity assumptions of functions, and Nishimura and Friedman [10] abandoned concavity completely. Later extensions of the theory were due to Forgo for CF-concavity, Kim and Lee for -concavity, Hou for -quasi-concavity, and others; see [11–15]. Moreover, Pu and Yang [16, 17] studied the KKM theorem without convex hull and variational relation problems without the KKM property.
The method of essential solutions has been widely used in various fields [18–30]. The notation of an essential solution for fixed points was first introduced in [18]. For a fixed point x of a mapping f, if each mapping sufficiently near to f has a fixed point arbitrarily near to x, x is said to be essential. However, it is not true that any continuous mapping has at least one essential fixed point, even though the space has the fixed point property. Instead of considering the essential solution, Kinoshita [19] introduced the notion of essential components of the set of fixed points and proved that, for any continuous mapping of the Hilbert cube into itself, there exists at least one essential component of the set of its fixed points. Kohlberg and Mertens [20] introduced the notions of stable set and essential components of Nash equilibria, and proved that every finite n-person noncooperative game has at least one essential connected component of the set of its Nash equilibrium points. Later, Yu and Xiang [21] brought forward the notion of essential components of the set of Ky Fan’s points, and deduced that every infinite n-person noncooperative game with concave payoff functions has at least one essential component of the set of its equilibrium points.
Motivated and inspired by research works mentioned above, we propose some new nonempty intersection theorems for an infinite family of set-valued mappings without convexity assumptions. Furthermore, we study the notion of essential stability of solutions of a nonempty intersection theorem without convexity assumptions.
2 Nonempty intersection theorem without convexity assumptions
We recall first some definitions and known results concerning set-valued mappings.
Definition 2.1 Let X, Y be two Hausdorff topological spaces. A set-valued mapping is said to be:
-
(1)
upper semicontinuous at if, for any open subset O of Y with , there exists an open neighborhood of x such that for any ;
-
(2)
upper semicontinuous on X if F is upper semicontinuous at each ;
-
(3)
lower semicontinuous at if, for any open subset O of Y with , there exists an open neighborhood of x such that for any ;
-
(4)
lower semicontinuous on X if F is lower semicontinuous at each ;
-
(5)
closed if is a closed subset of .
Definition 2.2 ([31])
Let X be a nonempty subset of a Hausdorff topological space E. X has the fixed point property, if and only if, every continuous mapping has a fixed point.
Throughout this paper, let () stand for the set of nonempty compact (convex) subsets of X, and
A new nonempty intersection theorem for an infinite family of set-valued mappings without convexity assumptions is obtained.
Theorem 2.1 Let I be any index set. For each , let be a nonempty and compact subset of a Hausdorff topological space , let have the fixed point property, and let be a set-valued mapping. Assume that:
-
(i)
for any , is closed in X;
-
(ii)
for any finite subset of X, there exists a continuous mapping such that, for any , there exists for which for each .
Then
Proof Define the set-valued mapping by
Let be arbitrarily fixed. By (ii), for the set , there exists a continuous mapping such that , hence F has nonempty closed values.
Using again (ii), we infer that, for every finite subset of X, there exists a continuous mapping such that, for any , there exists for which . Thus F satisfies all the conditions of Theorem 2.1 in [16]. Hence there exists such that
This completes the proof. □
If I is a singleton, then Theorem 2.1 collapses Theorem 2.1, the main result of [16].
Theorem 2.2 Let X be a nonempty and compact subset of a Hausdorff topological space E, let X have the fixed point property, and let be a set-valued mapping. Assume that:
-
(i)
for any , is nonempty and closed in X;
-
(ii)
for any finite subset of X, there exists a continuous mapping such that, for any , there exists for which .
Then
Next, we obtain a generalized nonempty intersection theorem for an infinite family of set-valued mappings without convexity assumptions.
Theorem 2.3 Let I be any index set. For each , let be a nonempty and compact subset of a locally convex topological linear space , let have the fixed point property, and let , be two set-valued mappings. Assume that:
-
(i)
is closed in ;
-
(ii)
is continuous with nonempty compact values;
-
(iii)
for any finite subset of X, there exists a continuous mapping such that, for any , there exists for which and for each .
Then there exists such that, for each , and
Proof Let be the locally convex topological vector space containing , and let be a basis of open neighborhoods of . For every , consider the set-valued mapping
Since is continuous with nonempty compact values, then is closed in X, and is open in X for any .
For any and each , define the mapping by
Clearly, (i) for any , is closed in X; (ii) for any finite subset of X, there exists a continuous mapping such that, for any , there exists for which for each . By Theorem 2.1, there exists such that
i.e., for each , and for any . Since X is compact, we may assume without loss of generality that . Then for each .
Suppose that there are and such that , i.e., . Since G is continuous, there is such that . Since is closed in , , which implies that and . It is a contradiction. Hence, for each , and
□
3 Essential stability
In this section, we study the essential stability of solutions of a nonempty intersection theorem without convexity assumptions. Let I be a finite set. For each , let be a nonempty, convex and compact subset of a normed linear space , and let be a set of continuous mappings. Denote by ℳ the set of such that the following conditions hold: (i) for any , is closed in ; (ii) is continuous with nonempty convex compact values; (iii) for any finite subset of X, there exists a continuous mapping such that, for any , there exists for which and for each .
By Theorem 2.3, for each , there exists such that, for each , and for any , which is called a solution of . The solution set of , denoted by , is nonempty. The solution correspondence is well defined. Moreover, to analyze the stability of solutions, some topological structure in the collection ℳ is also needed. For each , we define
where is the Hausdorff distance defined on , and is the Hausdorff distance defined on . Then ℳ becomes a metric space.
Definition 3.1 Let . An is said to be an essential point of if, for any open neighborhood of x in X, there is a positive δ such that for any with . w is said to be essential if all 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 It is easy to see that the problem is essential, if and only if, the mapping is lower semicontinuous at w.
First of all, let us introduce some mathematical tools for the following proof.
Lemma 3.1 ([32])
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 semicontinuous.
Lemma 3.2 ([33])
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 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 distance defined on Y.
Lemma 3.4 ([26])
Let X, Y, Z be three metric spaces, and be two set-valued mappings. Suppose that there exists at least one essential component of for each , and there exists a continuous single-valued mapping such that for each . Then there exists at least one essential component of for each .
Lemma 3.5 ([27])
Let C, D be two nonempty, convex and compact subsets of a linear normed space Y. Then
where , , and h is the Hausdorff distance defined on Y.
Theorem 3.1 is a complete metric space.
Proof Let be any Cauchy sequence in ℳ, i.e., for any , there exists such that for any . Then, for each and , and are two Cauchy sequences in and , converging to and . Denote . We will show that .
-
(i)
Clearly, under the metric ρ.
-
(ii)
Assume that , then there are a finite set of X and such that, for any , there is for which , i.e., , or . Since under the metric ρ, , i.e., , or for enough large m, which contradicts the fact that . This completes the proof. □
Theorem 3.2 The corresponding 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 in converging to such that for any n. Then, for each , and for any and any . Since and , then for each .
Suppose that there are and such that , then there exists a sequence of such that and . Since , and , for enough large n, which implies and . It is a contradiction. Hence, for each , and for any . This completes the proof. □
Theorem 3.3 There exists a dense residual subset of ℳ such that for each , w is essential. In other words, there are most of the problems, whose solutions are all essential.
Proof Since is complete, and is upper semicontinuous with nonempty compact values, by Lemma 3.2, there is a dense residual subset of ℳ, where w is lower semicontinuous. Hence w is essential for each . □
Theorem 3.4 For each , there exists at least one minimal essential subset of .
Proof Since is upper semicontinuous with nonempty compact values, then, 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 was not connected, then there exist two non-empty compact subsets , with , and there exist two disjoint open subsets , in X 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
Here, we choose two open sets , such that
and, for each , denote , , which are open in , and .
Since and it is essential, there exists such that for any with . Since is the minimal essential set, neither nor is essential. Then, for , there exist two such that
Thus .
We define by
where
Now we will show that and .
-
(i)
Clearly, is closed in for any .
-
(ii)
is continuous with nonempty convex compact values.
-
(iii)
Assume that , then there are a finite subset of X and such that, for any , there is for which , or . Since and ,
Hence is false.
Since , or . Without loss of generality, we may assume that . Since
then . Therefore , which contradicts the fact that . Hence .
-
(iv)
By Lemma 3.3 and Lemma 3.5,
Hence .
Since , we assume without loss of generality, i.e., there exists such that . It follows from the definition of that , which contradicts the fact that . This completes the proof. □
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. Hence C is an essential component of . □
Denote by the set of F, when I is a singleton and . The following results are obtained.
Theorem 3.7 There exists a dense residual subset Φ of such that, for each , F is essential.
Theorem 3.8 For each , there exists at least one minimal essential subset of .
Theorem 3.9 For each , every minimal essential subset of is connected.
Theorem 3.10 For each , there exists at least one essential component of .
Remark 3.2 Theorems 3.7-3.10 are generalizations of the results of [25], where convexity assumptions of KKM mappings are necessary.
Remark 3.3 Khanh and Quan [34] obtained generic stability and essential components of generalized KKM points. Thus it is worth comparing the results in Section 3 of this paper with the results of [34].
-
(i)
This paper is a multiplied version of the KKM theorem.
-
(ii)
If (1) I is a singleton, and for any in this paper; (2) in [34], is a nonempty and compact subset of a metric space, and holds the fixed property; (3) in [34], T is the identity mapping, Theorems 3.1-3.6 coincide with Section 3 and Section 4 of [34].
4 Application (I): Ky Fan’s points
To discuss the essential components of Ky Fan’s points without convexity assumptions, we need the following definitions.
Definition 4.1 ([15])
Let X be a Hausdorff topological space, let be a set of continuous mappings. A function is said to be -quasi-concave on X if, for any finite subset of X, one has for any .
Definition 4.2 Let X be a nonempty and compact subset of a metric space having the fixed point property, and let be a set of continuous mappings. Denote by the set of all functions such that the following conditions hold: (i) for each fixed , is lower semicontinuous; (ii) for each fixed , is -quasi-concave on X; (iii) for all .
For each , we denote , which is nonempty and compact (see [16]). Furthermore, points in are called Ky Fan’s points of φ (see [21]). The solution mapping is well defined. For each , we define the corresponding by
Clearly, for each . It is easy to see that the single-valued mapping by is isometric. Furthermore, . For any , define the distance on by .
Theorem 4.1 For each , there exists at least one essential component of .
Proof Since is an isometric mapping, it is continuous. Since there exists at least one essential component of for each , by Lemma 3.4, there exists at least one essential component of for each . □
Remark 4.1 In [21], X is a nonempty, convex and compact subset of a normed linear space. Denote by the set of all functions such that the following conditions hold: (1) for each fixed , is lower semicontinuous; (2) for each fixed , is concave; (3) for all ; (4) . Clearly, . For each , for each fixed , is -quasi-concave, not only concave, and is unnecessary. In [21], the notion of essential components is based on the metric , which is defined by
Next we will explain that the metric is neither stronger nor weaker than even in the same space .
Example 4.1 Let , for all . Then and for all .
-
(1)
For each n, we define for all . Then , for all , and , . Then under the metric , while under the metric .
-
(2)
For each n, we define
Then and
Hence
Then under the metric , while under the metric .
5 Application (II): Nash equilibrium
An n-person non-cooperative game Γ is a tuple , where , the i th player has a strategy set , and is his payoff function. Denote , , , . A point is said to be a Nash equilibrium point if, for each , . Denote by the set of Nash equilibrium points of Γ.
Definition 5.1 Denote by the set of all games such that the following conditions hold: (i) for each , is a nonempty and compact subset of a metric space , has the fixed point property, and is a set of continuous mappings; (ii) for each , is upper semicontinuous on X, and is lower semicontinuous on for any (iii) for any finite subset of X and any , there exists such that , .
Theorem 5.1 For any , .
Proof For any and any , define the corresponding by
Clearly, satisfies all the conditions of Theorem 2.1. Hence . □
Clearly, for each . It is easy to see that the single-valued mapping by is isometric. Furthermore, . For any , define the distance on by .
Theorem 5.2 For each , there exists at least one essential component of .
Proof Since is an isometric mapping, it is continuous. Since, there exists at least one essential component of for each , by Lemma 3.4, there exists at least one essential component of for each . □
Remark 5.1 In [21], denote by the set of games such that the following conditions hold: (1) for any , is a nonempty, compact and convex subset of a normed linear space; (2) is upper semicontinuous on X; (3) for any and any , is lower semicontinuous on ; (4) for any , is concave on X; (5) . Clearly, . Games in have concave and uniform bounded payoffs, which are invalid for games in . In [21], the notion of essential components is based on the metric , which is defined by
It is neither stronger nor weaker than even in the same space .
Example 5.1 Let , , , , . Then
(1) For each n, we define , . Then ,
Then under the metric , while under the metric .
(2) For each n, we define
Then and
Hence
Then under the metric , while under the metric .
6 Application (III): variational relations
Luc [35] introduced a more general model of equilibrium problems, which is called a variational relation problem (in short, VRP). Further studies of variational relation problems were done in [36–43]. Let A, B and C be nonempty sets, , be set-valued mappings with nonempty values, and be a relation linking elements , and .
(VRP) Find such that:
-
(i)
;
-
(ii)
holds for any and any .
Definition 6.1 ([35])
Let A and B be nonempty subsets of topological spaces and , respectively, and be a relation linking and . For each fixed , we say that is closed in the first variable if, for every net converges to some a, and holds for any α, then the relation holds.
Let X be a nonempty and compact subset of a metric space having the fixed point property, and let be a set of continuous mappings. Denote by the set of variational relations such that the following conditions hold: (i) is closed; (ii) for any , and is open in X for any ; (iii) for any fixed , is lower semicontinuous; (iv) for any fixed , is closed; (v) for any finite subset of X and any , there exists such that holds for any ; if for any , then .
For any , denote by the solution set of q, which is nonempty and compact. The solution mapping is well defined. Moreover, define the mapping by
Clearly, for each . It is easy to see that the single-valued mapping by is isometric. Furthermore, . For any , define the distance on by .
Theorem 6.1 For each , there exists at least one essential component of .
Proof Since is an isometric mapping, it is continuous. Since there exists at least one essential component of for each , by Lemma 3.4, there exists at least one essential component of for each . □
Remark 6.1 As convexity assumptions are not necessary to variational relation problem in , Theorem 6.1 includes properly Theorem 3.4 of [43].
7 Conclusion
As a generalization of the KKM theorem in [16], we propose a new nonempty intersection theorem for an infinite family of set-valued mappings without convexity assumptions, and study the notion of essential stability of a solution set of the nonempty intersection theorem without convexity assumptions. We show that most of problems (in the sense of Baire category) are essential and, for any problem, there exists at least one essential component of its solution set. This paper is the attempt to establish analogue results for the class of equilibria removing convexity assumptions. As applications, we deduce the corresponding results for Ky Fan’s points, Nash equilibrium and variational relations.
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Supported by an open project of Key Laboratory of Mathematical Economics (SUFE), Ministry of Education (Project Number: 201309KF02).
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Yang, Z. Some new generalizations of nonempty intersection theorems without convexity assumptions and essential stability of their solution set with applications. Fixed Point Theory Appl 2014, 3 (2014). https://doi.org/10.1186/1687-1812-2014-3
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DOI: https://doi.org/10.1186/1687-1812-2014-3