Generalizations of the Nash Equilibrium Theorem in the KKM Theory
© Sehie Park. 2010
Received: 5 December 2009
Accepted: 2 February 2010
Published: 12 April 2010
The partial KKM principle for an abstract convex space is an abstract form of the classical KKM theorem. In this paper, we derive generalized forms of the Ky Fan minimax inequality, the von Neumann-Sion minimax theorem, the von Neumann-Fan intersection theorem, the Fan-type analytic alternative, and the Nash equilibrium theorem for abstract convex spaces satisfying the partial KKM principle. These results are compared with previously known cases for -convex spaces. Consequently, our results unify and generalize most of previously known particular cases of the same nature. Finally, we add some detailed historical remarks on related topics.
In 1928, John von Neumann found his celebrated minimax theorem  and, in 1937, his intersection lemma , which was intended to establish easily his minimax theorem and his theorem on optimal balanced growth paths. In 1941, Kakutani  obtained a fixed point theorem for multimaps, from which von Neumann's minimax theorem and intersection lemma were easily deduced.
In 1950, John Nash [4, 5] established his celebrated equilibrium theorem by applying the Brouwer or the Kakutani fixed point theorem. In 1952, Fan  and Glicksberg  extended Kakutani's theorem to locally convex Hausdorff topological vector spaces, and Fan generalized the von Neumann intersection lemma by applying his own fixed point theorem. In 1972, Himmelberg  obtained two generalizations of Fan's fixed point theorem  and applied them to generalize the von Neumann minimax theorem by following Kakutani's method in .
In 1961, Ky Fan  obtained his KKM lemma and, in 1964 , applied it to another intersection theorem for a finite family of sets having convex sections. This was applied in 1966  to a proof of the Nash equilibrium theorem. This is the origin of the application of the KKM theory to the Nash theorem. In 1969, Ma  extended Fan's intersection theorem  to infinite families and applied it to an analytic formulation of Fan type and to the Nash theorem for arbitrary families.
Note that all of the above results are mainly concerned with convex subsets of topological vector spaces; see Granas . Later, many authors tried to generalize them to various types of abstract convex spaces. The present author also extended them in our previous works [14–28] in various directions. In fact, the author had developed theory of generalized convex spaces (simply, -convex spaces) related to the KKM theory and analytical fixed point theory. In the framework of -convex spaces, we obtained some minimax theorems and the Nash equilibrium theorems in our previous works [17, 18, 21, 22], based on coincidence theorems or intersection theorems for finite families of sets, and in , based on continuous selection theorems for the Fan-Browder maps.
In our recent works [24–26], we studied the foundations of the KKM theory on abstract convex spaces. The partial KKM principle for an abstract convex space is an abstract form of the classical KKM theorem. A KKM space is an abstract convex space satisfying the partial KKM principle and its "open" version. We noticed that many important results in the KKM theory are closely related to KKM spaces or spaces satisfying the partial KKM principle. Moreover, a number of such results are equivalent to each other.
On the other hand, some other authors studied particular types of KKM spaces and deduced some Nash-type equilibrium theorem from the corresponding partial KKM principle, for example, [17, 21, 29–33], explicitly, and many more in the literature, implicitly. Therefore, in order to avoid unnecessary repetitions for each particular type of KKM spaces, it would be necessary to state clearly them for spaces satisfying the partial KKM principle. This was simply done in .
In this paper, we study several stages of such developments from the KKM principle to the Nash theorem and related results within the frame of the KKM theory of abstract convex spaces. In fact, we clearly show that a sequence of statements from the partial KKM principle to the Nash equilibria can be obtained for any space satisfying the partial KKM principle. This unifies previously known several proper examples of such sequences for particular types of KKM spaces. More precisely, our aim in this paper is to obtain generalized forms of the KKM space versions of known results due to von Neumann, Sion, Nash, Fan, Ma, and many followers. These results are mainly obtained by fixed point method, continuous selection method, or the KKM method. In this paper, we follow method and will compare our results to corresponding ones already obtained by method
In Section 2, we state basic facts and examples of abstract convex spaces in our previous works [24–26]. Section 3 deals with a characterization of the partial KKM principle and shows that such principle is equivalent to the generalized Fan-Browder fixed point theorem. In Section 4, we deduce a general Fan-type minimax inequality from the partial KKM principle. Section 5 deals with various von Neumann-Sion-type minimax theorems for abstract convex spaces.
In Section 6, a collective fixed point theorem is deduced as a generalization of the Fan-Browder fixed point theorem. Section 7 deals with the Fan-type intersection theorems for sets with convex sections in product abstract convex spaces satisfying the partial KKM principle. In Section 8, we deduce a Fan-type analytic alternative and its consequences. Section 9 is devoted to various generalizations of the Nash equilibrium theorem and their consequences. Finally, in Section 10, some known results related to the Nash theorem and historical remarks are added.
2. Abstract Convex Spaces and the KKM Spaces
An abstract convex space consists of a topological space , a nonempty set , and a multimap with nonempty values for .
For any , the -convex hull of is denoted and defined by
A subset of is called a -convex subset of relative to if for any we have that , that is, .
When , the space is denoted by . In such case, a subset of is said to be -convex if ; in other words, is -convex relative to . In case , let .
A triple is given for the original KKM theorem , where is the standard -simplex, is the set of its vertices , and co: is the convex hull operation.
A triple is given, where and are subsets of a t.v.s. such that co and co. Fan's celebrated KKM lemma  is for .
A convex space is a triple where is a subset of a vector space such that co , and each is the convex hull of equipped with the Euclidean topology. This concept generalizes the one due to Lassonde for ; see . However he obtained several KKM-type theorems w.r.t. .
A triple is called an -space if is a topological space and is a family of contractible (or, more generally, -connected) subsets of indexed by such that whenever . If , then is called a -space by Horvath [36, 37].
Hyperconvex metric spaces due to Aronszajn and Panitchpakdi are particular cases of -spaces; see .
Hyperbolic spaces due to Reich and Shafrir  are also particular cases of -spaces. This class of metric spaces contains all normed vector spaces, all Hadamard manifolds, the Hilbert ball with the hyperbolic metric, and others. Note that an arbitrary product of hyperbolic spaces is also hyperbolic.
Any topological semilattice with path-connected interval is introduced by Horvath and Llinares .
A generalized convex space or a -convex space due to Park is an abstract convex space such that for each with the cardinality there exists a continuous function such that implies that .
Here, is the face of corresponding to , that is, if and , then .
A -space consists of a topological space , a nonempty set , and a family of continuous functions (that is, singular -simplexes) for with . Every -space can be made into a -convex space; see . Recently -spaces are called -spaces in  and -spaces  or simplicial spaces  when .
Suppose that is a closed convex subset of a complete -tree , and for each , , where is the intersection of all closed convex subsets of that contain ; see Kirk and Panyanak . Then is an abstract convex space.
A topological space with a convexity in the sense of Horvath  is another example.
A -space due to Briec and Horvath  is an abstract convex space.
Note that each of (2)–(12) has a large number of concrete examples and that all examples (1)–(9) are -convex spaces.
then is called a KKM map.
Thepartial KKM principle for an abstract convex space is the statement that, for any closed-valued KKM map , the family has the finite intersection property. The KKM principle is the statement that the same property also holds for any open-valued KKM map.
An abstract convex space is called a KKM space if it satisfies the KKM principle.
Every -convex space is a KKM space .
A connected linearly ordered space can be made into a KKM space .
The extended long line is a KKM space with the ordinal space ; see . But is not a -convex space.
A -space due to Briec and Horvath  is a KKM space.
Now we have the following diagram for triples :
It is not known yet whether there is a space satisfying the partial KKM principle that is not a KKM space.
3. The KKM Principle and the Fan-Browder Map
Let be an abstract convex space.
Recall the following equivalent form of [26, Theorem 8.2].
Suppose that satisfies the partial KKM principle and a map satisfies the following.
is closed valued.
is a KKM map (i.e., for all ).
There exists a nonempty compact subset of such that one of the following holds:
(ii) for some ,
Conditions (i)–(iii) in are called compactness conditions or coercivity conditions. In this paper, we mainly adopt simply (i), that is, is compact. However, most of results can be reformulated to the ones adopting (ii) or (iii).
For a topological space and an abstract convex space , a multimap is called a -map or a Fan-Browder map provided that there exists a multimap satisfying the follwing:
(a)for each (i.e., implies that ),
(b) Int for some .
Here, Int denotes the interior with respect to and, for each , .
There are several equivalent formulations of the partial KKM principle; see . For example, it is equivalent to the Fan-Browder-type fixed point theorem as follows.
Theorem 3.4 (see ).
An abstract convex space satisfies the partial KKM principle if and only if any -map has a fixed point , that is, .
The following is known.
Let be any family of abstract convex spaces. Let be equipped with the product topology and . For each , let be the projection. For each , define . Then is an abstract convex space.
Let be a family of -convex spaces. Then is a -convex space.
It is not known yet whether this holds for KKM spaces.
From now on, for simplicity, we are mainly concerned with compact abstract convex spaces satisfying the partial KKM principle. For example, any compact -convex space, any compact -space, or any compact convex space is such a space.
4. The Fan-Type Minimax Inequalities
Recall that an extended real-valued function , where is a topological space, is lower [resp., upper] semicontinuous (l.s.c.) (resp., u.s.c.) if (resp., ) is open for each .
For an abstract convex space , an extended real-valued function is said to be quasiconcave (resp., quasiconvex) if (resp., ) is -convex for each .
From the partial KKM principle we can deduce a very general version of the Ky Fan minimax inequality as follows.
Let be an abstract convex space satisfying the partial KKM principle, extended real functions, and such that
(3.1)for each , is closed,
(3.2)for each , ,
(3.3)the compactness condition (1.3) holds for .
Then either (i) there exists a such that for all or (ii) there exists an such that
Let be a map defined by for . Then each is closed by (3.1).
Ca (i): is a KKM map.
By Theorem 3.1, we have . Hence, there exists a such that for all , that is, for all .
Ca (ii): is not a KKM map.
Then there exists such that . Hence there exists an such that for each , or equivalently for each . Since contains , by we have and hence, .
For a compact convex subset of a t.v.s. and , if is quasiconcave, then (3.2) holds; and if is l.s.c., then (3.1) holds. Therefore, Corollary 4.2 generalizes the Ky Fan minimax inequality .
For a convex space and , Corollary 4.2 reduces to Cho et al. [50, Theorem 9].
There is a very large number of generalizations of the Fan minimax inequality for convex spaces, -spaces, -convex spaces, and others. These would be particular forms of Corollary 4.2. For example, see Park [18, Theorem 11], where is a -convex space.
Some particular versions of Corollary 4.2 were given in .
5. The von Neumann-Sion-Type Minimax Theorems
Let and be abstract convex spaces. For their product, as in the Lemma 3.5 we can define for .
for each ,
for each and , is -convex; for each and , is -convex,
for each , there exists a finite set such that
This implies that for all . Then is a -map. Therefore, by Theorem 3.4, we have such that . Therefore, , a contradiction.
For convex spaces , and , Theorem 5.1 reduces to that by Cho et al. [50, Theorem 8].
Let and be compact abstract convex spaces, let be the product abstract convex space, and let be functions satisfying the following:
(1) for each ,
(2)for each is l.s.c. and is quasiconvex on
(3)for each is quasiconcave and is u.s.c. on .
Note that is l.s.c. on and is u.s.c. on . Therefore, both sides of the inequality exist. Then all the requirements of Theorem 5.1 are satisfied.
For the case , Corollary 5.3 reduces to the following.
Corollary 5.5 (see ).
Let and be compact abstract convex spaces and let be an extended real function such that
(1)for each is l.s.c. and quasiconvex on ,
(2)for each is u.s.c. and quasiconcave on .
If satisfies the partial KKM principle, then
(i) has a saddle point ,
We list historically well-known particular forms of Corollary 5.5 in chronological order as follows.
(2)Nikaidô . Euclidean spaces above are replaced by Hausdorff topological vector spaces, and is continuous in each variable.
(3)Sion . and are compact convex subsets of topological vector spaces in Corollary 5.5.
(4)Komiya [55, Theorem 3]. and are compact convex spaces in the sense of Komiya.
(5)Horvath [36, Proposition 5.2]. and are -spaces with being compact and without assuming the compactness of .
In these two examples, Hausdorffness of is assumed since they used the partition of unity argument.
(6)Bielawski [29, Theorem (4.13)]. and are compact spaces having certain simplicial convexities.
(7)Park [17, Theorem 5]. and are -convex spaces.
In 1999, we deduced the following von Neumann–Sion type minimax theorem for -convex spaces based on a continuous selection theorem:
Theorem 5.7 (see ).
Let and be -convex spaces, Hausdorff compact, an extended real function, and . Suppose that
(5.1) is l.s.c. on and is -convex for each and ,
(5.2) is u.s.c. on and is -convex for each and . Then
6. Collective Fixed Point Theorems
We have the following collective fixed point theorem.
Let be a finite family of compact abstract convex spaces such that satisfies the partial KKM principle, and for each is a -map. Then there exists a point such that that is, for each .
where . Since each is open, we have
(a)for each , is open.
Therefore, we have
(b)for each implies that .
Moreover, let . Since is the companion map corresponding to the -map , for each , there exists such that
where Since is compact, we have
(c) for some .
Since satisfies the partial KKM principle, by Theorem 3.4, the -map has a fixed point.
If , is a convex space, and , then Theorem 6.1 reduces to the well-known Fan-Browder fixed point theorem; see Park .
We have already the following collective fixed point theorem for arbitrary family of -convex spaces.
Theorem 6.3 (see ).
Let be a family of compact Hausdorff -convex spaces, , and for each let be a -map. Then there exists a point such that that is, for each .
In case when are all -spaces, Theorem 6.3 reduces to Tarafdar [58, Theorem 2.3]. This is applied to sets with -convex sections [58, Theorem 3.1] and to existence of equilibrium point of an abstract economy [58, Theorem 4.1 and Corollary 4.1]. These results also can be extended to -convex spaces and we will not repeat then here.
Each of Theorems 6.1, 7.1, 8.1, 9.1, and 9.4, respectively, in this paper is based on the KKM method and concerns with finite families of abstract convex spaces such that their product satisfies the partial KKM principle. Each of them has a corresponding Theorems 6.3, 7.3, 8.3, 9.2 and 9.6, respectively, based on continuous selection method for infinite families of Hausdorff -convex spaces. Note that for finite families the Hausdorffness is redundant in these corresponding theorems.
7. Intersection Theorems for Sets with Convex Sections
In our previous work , from a -convex space version of the Fan-Browder fixed point theorem, we deduced a Fan-type intersection theorem for subsets of a cartesian product of compact -convex spaces. This was applied to obtain a von Neumann-sion-type minimax theorem and a Nash-type equilibrium theorem for -convex spaces.
In the present section, we generalize the abovementioned intersection theorem to product abstract convex spaces satisfying the partial KKM principle.
The collective fixed point theorem in Section 6 can be reformulated to a generalization of various Fan-type intersection theorems for sets with convex sections as follows.
Let be a family of sets, and let be fixed. Let
If and , then let denote the th coordinate of . If and , then let be defined as follows: its th coordinate is and for the th coordinate is . Therefore, any can be expressed as for any , where denotes the projection of in .
Let be a finite family of compact abstract convex spaces such that satisfies the partial KKM principle and, for each , let and be subsets of satisfying the following.
For each .
For each is open in .
We apply Theorem 6.1 with multimaps given by and for each . Then for each we have the following.
(a)For each , we have .
(b)For each , we have
Note that is open in and that is a -map. Therefore, by Theorem 6.1, there exists such that for all . Hence .
For convex spaces , particular forms of Theorem 7.1 have appeared as follows:
Fan [10, Théeorème 1]. for all .
Fan [11, Theorem ]. and for .
From these results, Fan  deduced an analytic formulation, fixed point theorems, extension theorems of monotone sets, and extension theorems for invariant vector subspaces.
For particular types of -convex spaces, Theorem 7.1 was known as follows.
Bielawski [29, Proposition and Theorem ]. have the finitely local convexity.
Kirk et al. [32, Theorem 5.2]. are hyperconvex metric spaces.
Park [17, Theorem 4], [18, Theorem 19]. In , from a -convex space version of the Fan-Browder fixed point theorem, we deduced a Fan-type intersection theorem for subsets of a cartesian product of compact -convex spaces. This was applied to obtain a von Neumann-Sion-type minimax theorem and a Nash-type equilibrium theorem for -convex spaces.
Park [27, Theorem 4]. We gave a different proof.
In , a collective fixed point theorem was reformulated to a generalization of various Fan-type intersection theorems for arbitrary number of sets with convex sections as follows.
Theorem 7.3 (see ).
Let be a family of Hausdorff compact -convex spaces and, for each , let and be subsets of satisfying the following.
For each is open in .
For convex subsets of topological vector spaces, particular forms of Theorem 7.3 have appeared as follows.
Ma [12, Theorem 2]. The case for all with a different proof is given.
Park [19, Theorem 4.2]. are convex spaces.
Note that if is finite in Theorem 7.3, the Hausdorffness is redundant by Theorem 7.1.
8. The Fan-Type Analytic Alternatives
From the intersection Theorem 7.1, we can deduce the following equivalent form of a generalized Fan-type minimax inequality or analytic alternative. Our method is based on that of Fan [9, 10] and Ma .
Let be a finite family of compact abstract convex spaces such that satisfies the partial KKM principle and, for each , let be real functions satisfying
for each is quasiconcave on
for each is l.s.c. on .
Let be a family of real numbers. Then either
(a)there exist an and an such that
or (b)there exists an such that
for each . Then
(2)for each , is -convex,
(3)for each , is open in .
Therefore, by Theorem 7.1, there exists an . This is equivalent to (b).
Fan [9, Théorème 2], [10, Theorem 3]. are convex subsets of t.v.s., and for all . From this, fan [9, 10] deduced Sion's minimax theorem , the Tychonoff fixed point theorem, solutions to systems of convex inequalities, extremum problems for matrices, and a theorem of Hardy-Littlewood-Pólya.
From the intersection Theorem 7.3, we can deduce the following equivalent form of a generalized Fan-type minimax inequality or analytic alternative.
Theorem 8.3 (see ).
Let be a family of compact Hausdorff -convex spaces and, for each , let be real functions as in Theorem 8.1. Then the conclusion of Theorem 8.1 holds.
We obtained Theorem 8.1 from Theorem 7.1. As was pointed out by Fan  for his case, we can deduce Theorem 7.1 from Theorem 8.1 by considering the characteristic functions of the sets and .
The conclusion of Theorems 8.1 and 8.3 can be stated as follows
9. The Nash-Type Equilibrium Theorems
From Theorem 8.1, we obtain the following form of the Nash-Fan-type equilibrium theorems in  with different proofs.
Let be a finite family of compact abstract convex spaces such that satisfies the partial KKM principle and, for each , let be real functions such that
(9.0) for each
(9.1) for each , is quasiconcave on
(9.2) for each , is u.s.c. on
(9.3) for each , is l.s.c. on .
Since each is compact, by for any , exists for all and all . Hence Theorem 8.1(a) does not hold. Then by Theorem 8.1(b), there exists an such that for all . Since is arbitrary, the conclusion follows.
This is not comparable to the following generalized Nash-Ma type theorem:
Theorem 9.2 (see ).
Park [19, Theorem 8.2]. are convex spaces.
From Theorem 9.1 for , we obtain the following form of the Nash-Fan-type equilibrium theorem for abstract convex spaces.
Let be a finite family of compact abstract convex spaces such that satisfies the partial KKM principle and, for each , let be a function such that
(10.1) for each , is quasiconcave on
(10.2) for each , is u.s.c. on
(10.3) for each , is l.s.c. on .
For continuous functions , a number of particular forms of Theorem 9.4 have appeared for convex subsets of Hausdorff topological vector spaces as follows:
(1)Nash [5, Theorem 1] where are subsets of Euclidean spaces,
(2)Nikaido and Isoda [60, Theorem 3.2],
(3)Fan [10, Theorem 4],
(4)Tan et al. [61, Theorem 2.1].
For particular types of -convex spaces and continuous functions , particular forms of Theorem 9.4 have appeared as follows.
(5)Bielawski [29, Theorem 4.16]. have the finitely local convexity.
(6)Kirk et al. [32, Theorem ]. are hyperconvex metric spaces.
(8)Park [21, Theorem 4.7]. A variant of Theorem 9.4 is under the hypothesis that is a compact -convex space and are continuous functions.
(9)González et al. . Each is a compact, sequentially compact -space and each is continuous as in 8.
(10)Briec and Horvath [30, Theorem ]. Each is a compact -convex set and each is continuous as in 8.
From Theorem 9.2, we obtain the following generalization of the Nash-Ma-type equilibrium theorem for -convex spaces.
Theorem 9.6 (see ).
For continuous functions and for convex subsets of Hausdorff topological vector spaces, Theorem 9.6 was due to Ma [12, Theorem 4].
The point in the conclusion of Theorems 9.4 or 9.6 is called a Nash equilibrium. This concept is a natural extension of the local maxima and the saddle point as follows.
In case is a singleton, we obtain the following.
Let be a closed bounded convex subset of a reflexive Banach space and a quasiconcave u.s.c. function. Then attains its maximum on , that is, there exists an such that for all .
Let be equipped with the weak topology. Then, by the Hahn-Banach theorem, is still u.s.c. because is quasiconcave, and is still closed. Being bounded, is contained in some closed ball which is weakly compact. Since any closed subset of a compact set is compact, is (weakly) compact. Now, by Theorem 9.4 for a single family, we have the conclusion.
For , Theorem 9.4 reduces to Corollary 5.5 as follows.
Proof of Corollary 5.5 from Theorem 9.4.
Therefore, we have the conclusion.
10. Historical Remarks on Related Results
As we have seen in Sections 1–3, we have three methods in our subject as follows:
fixed point method—applications of the Kakutani theorem and its various generalizations (e.g., acyclic-valued multimaps, admissible maps, or better admissible maps in the sense of Park); see [3–8, 10, 12, 14–16, 19, 20, 23, 28, 42, 53, 64–68] and others,
continuous selection method—applications of the fact that Fan-Browder-type maps have continuous selections under certain assumptions like Hausdorffness and compactness of relevant spaces; see [17, 22, 36, 39, 40, 58, 69, 70] and others,
the KKM method—as for the Sion theorem, direct applications of the KKM theorem, or its equivalents like the Fan-Browder fixed point theorem for which we do not need the Hausdorffness; see [9, 11, 17, 21, 24–27, 30, 31, 33, 35–37, 39, 43, 45, 47–50, 54, 55, 57, 59, 71, 72] and others.
An upper semicontinuous (u.s.c.) multimap with nonempty compact convex values is called a Kakutani map. The Fan-Glicksberg theorem was extended by Himmelberg  in 1972 for compact Kakutani maps instead of assuming compactness of domains. In 1990, Lassonde  extended the Himmelberg theorem to multimaps factorizable by Kakutani maps through convex sets in Hausdorff topological vector spaces. Moreover, Lassonde applied his theorem to game theory and obtained a von Neumann-type intersection theorem for finite number of sets and a Nash-type equilibrium theorem comparable to Debreu's social equilibrium existence theorem .
Since 1996 , many authors have published some results of the present paper for hyperconvex metric spaces. For example, Kirk et al. in 2000  established the KKM theorem, its equivalent formulations, fixed point theorems, and the Nash theorem for hyperconvex metric spaces. However, already in 1993, Horvath  found that hyperconvex metric spaces are a particular type of -spaces.
In 1998 , an acyclic version of the social equilibrium existence theorem of Debreu is obtained. This is applied to deduce acyclic versions of theorems on saddle points, minimax theorems, and the following Nash equilibrium theorem.
Corollary 10.1 (see ).
Let be a family of acyclic polyhedra, , and for each , a continuous function such that
(0)for each and each , the set
is empty or acyclic.
Then there exists a point such that
In the present paper, for abstract convex spaces, we notice that the partial KKM principle the Fan-Browder fixed point theorem the Nash equilibrium theorem, with or without additional intermediate steps. This procedure can be called "from the KKM principle to the Nash equilibria" (simply, "K to N"); see .
In 2000  and 2002 , we applied our fixed point theorem for compact compositions of acyclic maps on admissible (in the sense of Klee) convex subsets of a t.v.s. to obtain a cyclic coincidence theorem for acyclic maps, generalized von Neumann-type intersection theorems, the Nash type equilibrium theorems, and the von Neumann minimax theorem.
The following examples are generalized forms of quasi equilibrium theorem or social equilibrium existence theorems which directly imply generalizations of the Nash-Ma-type equilibrium existence theorem.
Theorem 10.2 (see ).
Let be a family of convex sets, each in a t.v.s. , a nonempty compact subset of , a closed map, and u.s.c. functions for each .
Suppose that, for each ,
(i) for each ,
(ii)the function defined on by
is l.s.c., and
(iii)for each , the set
If is admissible in , then there exists an such that, for each ,
Theorem 10.3 (see ).
Let be a topological space and let be a family of convex sets, each in a t.v.s. . For each , let be a closed map with compact values, and u.s.c. real-valued functions.
Suppose that, for each ,
(i) for each ,
(ii)the function defined by
is l.s.c., and
(iii)for each , the set
If is admissible in and if all the maps are compact except possibly and is u.s.c., then there exists an equilibrium point , that is,
In 2001 , we obtained generalized forms of the von Neumann-Sion-type minimax theorem, the Fan-Ma intersection theorem, the Fan-Ma type analytic alternative, and the Nash-Ma-equilibrium theorem for -convex spaces. In , all -convex spaces were assumed to be Hausdorff because the results are based on a selection theorem in , where Hausdorffness was indispensable.
In 2001, for any topological semilattice with path-connected interval introduced by Horvath and Llinares , the KKM theorem, the Fan-Browder theorem, and the Nash theorem are shown by Luo . Note that such semilattice is known to be a -convex space.
Cain and González  considered relationship among some subclasses of the class of -convex spaces and introduced a subclass of the so-called -spaces. In 2007, González et al.  repeated to show that -convex spaces and -spaces satisfy the partial KKM principle. They added that -spaces satisfy the properties of the Fan type minimax inequality, Fan-Browder-type fixed point, and the Nash-type equilibrium. All of such results are already known for more general -convex spaces.
In 2008, Kulpa and Szymanski  introduced a series of theorems called Infimum Principles in simplicial spaces. As for applications, they derive fixed point theorems due to Schauder, Tychnoff, Kakutani, and Fan-Browder: minimax theorems, the Nash equilibrium theorem, the Gale-Nikaido-Debreu theorem, and the Ky Fan minimax inequality. Their study is based on and utilizes the techniques of simplicial structure and the Fan-Browder map. Recall that for any abstract convex spaces satisfying abstract KKM principle we can deduce such classical theorems without using any Infimum Principles. Moreover, we note that the newly defined -spaces in  are particular types of abstract convex spaces satisfying the abstract KKM principle.
In 2008, for -spaces, Briec and Horvath  showed that some theorems mentioned in this paper hold, that is, Fan-Browder fixed point theorem, Himmelberg-type (in fact, Browder-type and Kakutani-type) fixed point theorems, Fan type minimax inequality, existence of Nash equilibria, and others. Note that -spaces are KKM spaces [30, Corollary 2.2], and their authors depend the Pelegs type multiple KKM theorem.
Finally, recall that there are several hundred published works on the KKM theory and we can cover only a part of them. For more historical background for the related fixed point theory and for more involved or related results in this paper, see the references [24–27, 41, 43, 48] and the literature therein.
- von Neumann J: Zur theorie der gesellschaftsspiele. Mathematische Annalen 1928,100(1):295–320. 10.1007/BF01448847MathSciNetView ArticleMATHGoogle Scholar
- von Neumann J: Über ein ökonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes. Ergebnisse eines Mathematischen Kolloquiums 1937, 8: 73–83.MATHGoogle Scholar
- Kakutani S: A generalization of Brouwer's fixed point theorem. Duke Mathematical Journal 1941, 8: 457–459. 10.1215/S0012-7094-41-00838-4MathSciNetView ArticleMATHGoogle Scholar
- Nash JF Jr.: Equilibrium points in -person games. Proceedings of the National Academy of Sciences of the United States of America 1950, 36: 48–49. 10.1073/pnas.36.1.48MathSciNetView ArticleMATHGoogle Scholar
- Nash J: Non-cooperative games. Annals of Mathematics 1951, 54: 286–295. 10.2307/1969529MathSciNetView ArticleMATHGoogle Scholar
- Fan K: Fixed-point and minimax theorems in locally convex topological linear spaces. Proceedings of the National Academy of Sciences of the United States of America 1952, 38: 121–126. 10.1073/pnas.38.2.121MathSciNetView ArticleMATHGoogle Scholar
- Glicksberg IL: A further generalization of the Kakutani fixed theorem, with application to Nash equilibrium points. Proceedings of the American Mathematical Society 1952, 3: 170–174.MathSciNetMATHGoogle Scholar
- Himmelberg CJ: Fixed points of compact multifunctions. Journal of Mathematical Analysis and Applications 1972, 38: 205–207. 10.1016/0022-247X(72)90128-XMathSciNetView ArticleMATHGoogle Scholar
- Fan K: A generalization of Tychonoff's fixed point theorem. Mathematische Annalen 1961, 142: 305–310. 10.1007/BF01353421MathSciNetView ArticleMATHGoogle Scholar
- Fan K: Sur un théorème minimax. Comptes Rendus des Séances de l'Académie des Sciences. Série I 1964, 259: 3925–3928.MATHGoogle Scholar
- Fan K: Applications of a theorem concerning sets with convex sections. Mathematische Annalen 1966, 163: 189–203. 10.1007/BF02052284MathSciNetView ArticleMATHGoogle Scholar
- Ma T: On sets with convex sections. Journal of Mathematical Analysis and Applications 1969, 27: 413–416. 10.1016/0022-247X(69)90058-4MathSciNetView ArticleMATHGoogle Scholar
- Granas A: Sur quelques méthodes topologiques én analyse convexe. In Méthodes Topologiques en Analyse Convexe. Volume 110. Presses Université de Montréal, Montreal, Canada; 1990:11–77.Google Scholar
- Idzik A, Park S: Leray-Schauder type theorems and equilibrium existence theorems. In Differential Inclusions and Optimal Control, Lecture Notes in Nonlinear Analysis. Volume 2. University in Toruń, Toruń, Poland; 1998:191–197.Google Scholar
- Park S: Applications of the Idzik fixed point theorem. Nonlinear Functional Analysis and Its Applications 1996, 1: 21–56.Google Scholar
- Park S: Remarks on a social equilibrium existence theorem of G. Debreu. Applied Mathematics Letters 1998,11(5):51–54. 10.1016/S0893-9659(98)00079-2MathSciNetView ArticleMATHGoogle Scholar
- Park S: Minimax theorems and the Nash equilibria on generalized convex spaces. In NLA98: Convex Analysis and Chaos, Josai Mathematical Monographs. Volume 1. Josai University, Sakado, Japan; 1999:33–45.Google Scholar
- Park S: Elements of the KKM theory for generalized convex spaces. The Korean Journal of Computational & Applied Mathematics 2000,7(1):1–28.MathSciNetMATHGoogle Scholar
- Park S: Fixed points, intersection theorems, variational inequalities, and equilibrium theorems. International Journal of Mathematics and Mathematical Sciences 2000,24(2):73–93. 10.1155/S0161171200002593MathSciNetView ArticleMATHGoogle Scholar
- Park S: Acyclic versions of the von Neumann and Nash equilibrium theorems. Journal of Computational and Applied Mathematics 2000,113(1–2):83–91. 10.1016/S0377-0427(99)00246-0MathSciNetView ArticleMATHGoogle Scholar
- Park S: New topological versions of the Fan-Browder fixed point theorem. Nonlinear Analysis: Theory, Methods & Applications 2001,47(1):595–606. 10.1016/S0362-546X(01)00204-8MathSciNetView ArticleMATHGoogle Scholar
- Park S: Generalizations of the Nash equilibrium theorem on generalized convex spaces. Journal of the Korean Mathematical Society 2001,38(4):697–709.MathSciNetMATHGoogle Scholar
- Park S: Remarks on acyclic versions of generalized von Neumann and Nash equilibrium theorems. Applied Mathematics Letters 2002,15(5):641–647. 10.1016/S0893-9659(02)80018-0MathSciNetView ArticleMATHGoogle Scholar
- Park S: Elements of the KKM theory on abstract convex spaces. Journal of the Korean Mathematical Society 2008,45(1):1–27.MathSciNetView ArticleMATHGoogle Scholar
- Park S: Equilibrium existence theorems in KKM spaces. Nonlinear Analysis: Theory, Methods & Applications 2008,69(12):4352–5364. 10.1016/j.na.2007.10.058MathSciNetView ArticleMATHGoogle Scholar
- Park S: New foundations of the KKM theory. Journal of Nonlinear and Convex Analysis 2008,9(3):331–350.MathSciNetMATHGoogle Scholar
- Park S: From the KKM principle to the Nash equilibria. International Journal of Mathematics and Statistics 2010,6(S10):77–88.MathSciNetGoogle Scholar
- Park S, Park JA: The Idzik type quasivariational inequalities and noncompact optimization problems. Colloquium Mathematicum 1996,71(2):287–295.MathSciNetMATHGoogle Scholar
- Bielawski R: Simplicial convexity and its applications. Journal of Mathematical Analysis and Applications 1987,127(1):155–171. 10.1016/0022-247X(87)90148-XMathSciNetView ArticleMATHGoogle Scholar
- Briec W, Horvath C: Nash points, Ky Fan inequality and equilibria of abstract economies in Max-Plus and -convexity. Journal of Mathematical Analysis and Applications 2008,341(1):188–199. 10.1016/j.jmaa.2007.09.056MathSciNetView ArticleMATHGoogle Scholar
- González L, Kilmer S, Rebaza J: From a KKM theorem to Nash equilibria in -spaces. Topology and Its Applications 2007,155(3):165–170. 10.1016/j.topol.2007.10.003MathSciNetView ArticleMATHGoogle Scholar
- Kirk WA, Sims B, Yuan GX-Z: The Knaster-Kuratowski and Mazurkiewicz theory in hyperconvex metric spaces and some of its applications. Nonlinear Analysis: Theory, Methods & Applications 2000,39(5):611–627. 10.1016/S0362-546X(98)00225-9MathSciNetView ArticleMATHGoogle Scholar
- Luo Q: KKM and Nash equilibria type theorems in topological ordered spaces. Journal of Mathematical Analysis and Applications 2001,264(2):262–269. 10.1006/jmaa.2001.7624MathSciNetView ArticleMATHGoogle Scholar
- Knaster B, Kuratowski K, Mazurkiewicz S: Ein beweis des fixpunktsatzes für n-dimensionale simplexe. Fundamenta Mathematicae 1929, 14: 132–137.MATHGoogle Scholar
- Lassonde M: On the use of KKM multifunctions in fixed point theory and related topics. Journal of Mathematical Analysis and Applications 1983,97(1):151–201. 10.1016/0022-247X(83)90244-5MathSciNetView ArticleMATHGoogle Scholar
- Horvath CD: Contractibility and generalized convexity. Journal of Mathematical Analysis and Applications 1991,156(2):341–357. 10.1016/0022-247X(91)90402-LMathSciNetView ArticleMATHGoogle Scholar
- Horvath CD: Extension and selection theorems in topological spaces with a generalized convexity structure. Toulouse Annales Faculté des Sciences 1993,2(2):253–269. 10.5802/afst.766MathSciNetView ArticleMATHGoogle Scholar
- Reich S, Shafrir I: Nonexpansive iterations in hyperbolic spaces. Nonlinear Analysis: Theory, Methods & Applications 1990,15(6):537–558. 10.1016/0362-546X(90)90058-OMathSciNetView ArticleMATHGoogle Scholar
- Horvath CD, Llinares Ciscar JV: Maximal elements and fixed points for binary relations on topological ordered spaces. Journal of Mathematical Economics 1996,25(3):291–306. 10.1016/0304-4068(95)00732-6MathSciNetView ArticleMATHGoogle Scholar
- Park S: Continuous selection theorems in generalized convex spaces. Numerical Functional Analysis and Optimization 1999,20(5–6):567–583. 10.1080/01630569908816911MathSciNetView ArticleMATHGoogle Scholar
- Park S: Ninety years of the Brouwer fixed point theorem. Vietnam Journal of Mathematics 1999,27(3):187–222.MathSciNetMATHGoogle Scholar
- Park S: Fixed points of better admissible maps on generalized convex spaces. Journal of the Korean Mathematical Society 2000,37(6):885–899.MathSciNetMATHGoogle Scholar
- Park S: Generalized convex spaces, -spaces, and -spaces. Journal of Global Optimization 2009,45(2):203–210. 10.1007/s10898-008-9363-1MathSciNetView ArticleMATHGoogle Scholar
- Khanh PQ, Quan NH, Yao J-C: Generalized KKM-type theorems in GFC-spaces and applications. Nonlinear Analysis: Theory, Methods & Applications 2009,71(3–4):1227–1234. 10.1016/j.na.2008.11.055MathSciNetView ArticleMATHGoogle Scholar
- Kulpa W, Szymanski A: Applications of general infimum principles to fixed-point theory and game theory. Set-Valued Analysis 2008,16(4):375–398. 10.1007/s11228-007-0055-7MathSciNetView ArticleMATHGoogle Scholar
- Kirk WA, Panyanak B: Best approximation in -trees. Numerical Functional Analysis and Optimization 2007,28(5–6):681–690. 10.1080/01630560701348517MathSciNetView ArticleMATHGoogle Scholar
- Horvath CD: Topological convexities, selections and fixed points. Topology and Its Applications 2008,155(8):830–850. 10.1016/j.topol.2007.01.024MathSciNetView ArticleMATHGoogle Scholar
- Park S: Remarks on the partial KKM principle. Nonlinear Analysis Forum 2009, 14: 51–62.MathSciNetMATHGoogle Scholar
- Fan K: A minimax inequality and applications. In Inequalities, III. Edited by: Shisha O. Academic Press, New York, NY, USA; 1972:103–113.Google Scholar
- Cho YJ, Kim JK, Lee BY: Remarks on KKM maps and applications. The Journal of Advanced Research in Applied Mathematics 2009,1(1):1–8.MathSciNetGoogle Scholar
- Liu FC: A note on the von Neumann-Sion minimax principle. Bulletin of the Institute of Mathematics. Academia Sinica 1978,6(2, part 2):517–524.MathSciNetMATHGoogle Scholar
- Shih MH, Tan K-K: Noncompact sets with convex sections. II. Journal of Mathematical Analysis and Applications 1986,120(1):264–270. 10.1016/0022-247X(86)90214-3MathSciNetView ArticleMATHGoogle Scholar
- Nikaidô H: On von Neumann's minimax theorem. Pacific Journal of Mathematics 1954, 4: 65–72.MathSciNetView ArticleMATHGoogle Scholar
- Sion M: On general minimax theorems. Pacific Journal of Mathematics 1958, 8: 171–176.MathSciNetView ArticleMATHGoogle Scholar
- Komiya H: Convexity on a topological space. Fundamenta Mathematicae 1981,111(2):107–113.MathSciNetMATHGoogle Scholar
- Park S: Foundations of the KKM theory via coincidences of composites of upper semicontinuous maps. Journal of the Korean Mathematical Society 1994,31(3):493–519.MathSciNetMATHGoogle Scholar
- Simons S: Two-function minimax theorems and variational inequalities for functions on compact and noncompact sets, with some comments on fixed-point theorems. In Nonlinear Functional Analysis and Its Applications, Part 2, Proceedings of Symposia in Pure Mathematics. Volume 45. American Mathematical Society, Providence, RI, USA; 1986:377–392.View ArticleGoogle Scholar
- Tarafdar E: Fixed point theorems in -spaces and equilibrium points of abstract economies. Journal of Australian Mathematical Society 1992,53(2):252–260. 10.1017/S1446788700035825MathSciNetView ArticleMATHGoogle Scholar
- Chang SY: A generalization of KKM principle and its applications. Soochow Journal of Mathematics 1989,15(1):7–17.MathSciNetMATHGoogle Scholar
- Nikaidô H, Isoda K: Note on non-cooperative convex games. Pacific Journal of Mathematics 1955, 5: 807–815.MathSciNetView ArticleMATHGoogle Scholar
- Tan K-K, Yu J, Yuan X-Z: Existence theorems of Nash equilibria for non-cooperative -person games. International Journal of Game Theory 1995,24(3):217–222. 10.1007/BF01243152MathSciNetView ArticleMATHGoogle Scholar
- Park S: Variational inequalities and extremal principles. Journal of the Korean Mathematical Society 1991,28(1):45–56.MathSciNetMATHGoogle Scholar
- Park S, Kim SK: On generalized extremal principles. Bulletin of the Korean Mathematical Society 1990,27(1):49–52.MathSciNetMATHGoogle Scholar
- Bohnenblust HF, Karlin S: On a theorem of Ville. In Contributions to the Theory of Games, Annals of Mathematics Studies, no. 24. Princeton University Press, Princeton, NJ, USA; 1950:155–160.Google Scholar
- Dantzig GB: Constructive proof of the min-max theorem. Pacific Journal of Mathematics 1956, 6: 25–33.MathSciNetView ArticleMATHGoogle Scholar
- 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.38.10.886MathSciNetView ArticleMATHGoogle Scholar
- Lassonde M: Fixed points for Kakutani factorizable multifunctions. Journal of Mathematical Analysis and Applications 1990,152(1):46–60. 10.1016/0022-247X(90)90092-TMathSciNetView ArticleMATHGoogle Scholar
- Torres-Martínez JP: Fixed points as Nash equilibria. Fixed Point Theory and Applications 2006, 2006:-4.Google Scholar
- Ben-El-Mechaiekh H, Deguire P, Granas A: Une alternative non linéaire en analyse convexe et applications. Comptes Rendus des Séances de l'Académie des Sciences. Série I 1982,295(3):257–259.MathSciNetMATHGoogle Scholar
- Browder FE: The fixed point theory of multi-valued mappings in topological vector spaces. Mathematische Annalen 1968, 177: 283–301. 10.1007/BF01350721MathSciNetView ArticleMATHGoogle Scholar
- Cain GL Jr., González L: The Knaster-Kuratowski-Mazurkiewicz theorem and abstract convexities. Journal of Mathematical Analysis and Applications 2008,338(1):563–571. 10.1016/j.jmaa.2007.05.050MathSciNetView ArticleMATHGoogle Scholar
- Khamsi MA: KKM and Ky Fan theorems in hyperconvex metric spaces. Journal of Mathematical Analysis and Applications 1996,204(1):298–306. 10.1006/jmaa.1996.0438MathSciNetView ArticleMATHGoogle Scholar
- Eilenberg S, Montgomery D: Fixed point theorems for multi-valued transformations. American Journal of Mathematics 1946, 68: 214–222. 10.2307/2371832MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.