1. Introduction

In 1928, John von Neumann found his celebrated minimax theorem [1] and, in 1937, his intersection lemma [2], which was intended to establish easily his minimax theorem and his theorem on optimal balanced growth paths. In 1941, Kakutani [3] 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 [6] and Glicksberg [7] 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 [8] obtained two generalizations of Fan's fixed point theorem [6] and applied them to generalize the von Neumann minimax theorem by following Kakutani's method in [3].

In 1961, Ky Fan [9] obtained his KKM lemma and, in 1964 [10], applied it to another intersection theorem for a finite family of sets having convex sections. This was applied in 1966 [11] 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 [12] extended Fan's intersection theorem [10] 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 [13]. 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 [22], 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 [27].

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.

This paper is a revised and extended version of [22, 27] and a supplement to [24–26], where some other topics on abstract convex spaces can be found.

2. Abstract Convex Spaces and the KKM Spaces

Multimaps are also called simply maps. Let denote the set of all nonempty finite subsets of a set . Recall the following in [24–26].

Definition 2.1. 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 .

Example 2.2. The following are known examples of abstract convex spaces.
(1) A triple is given for the original KKM theorem [34], where is the standard -simplex, is the set of its vertices , and co: is the convex hull operation.
(2) A triple is given, where and are subsets of a t.v.s. such that co and co. Fan's celebrated KKM lemma [9] is for .
(3) 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 [35]. However he obtained several KKM-type theorems w.r.t. .
(4) 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].
(5) Hyperconvex metric spaces due to Aronszajn and Panitchpakdi are particular cases of -spaces; see [37].
(6) Hyperbolic spaces due to Reich and Shafrir [38] 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.
(7) Any topological semilattice with path-connected interval is introduced by Horvath and Llinares [39].
(8) 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 .
For details, see references of [17, 21, 22, 40–42].
(9) 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 [43]. Recently -spaces are called -spaces in [44] and -spaces [43] or simplicial spaces [45] when .
(10) 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 [46]. Then is an abstract convex space.
(11) A topological space with a convexity in the sense of Horvath [47] is another example.
(12) A -space due to Briec and Horvath [30] 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.

Definition 2.3. Let be an abstract convex space. If a multimap satisfies then is called a KKM map.

Definition 2.4. The partial 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.
In our recent works [24–26], we studied the foundations of the KKM theory on abstract convex spaces and noticed that many important results therein are related to the partial KKM principle.

Example 2.5. We give examples of KKM spaces as follows.
(1) Every -convex space is a KKM space [18].
(2) A connected linearly ordered space can be made into a KKM space [26].
(3) The extended long line is a KKM space with the ordinal space ; see [26]. But is not a -convex space.
(4) For a closed convex subset of a complete -tree , and for each , the triple satisfies the partial KKM principle; see [46]. Later we found that is a KKM space [48].
(5) Horvath's convex space with the weak Van de Vel property is a KKM space, where for each ; see [47, 48].
(6) A -space due to Briec and Horvath [30] 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].

Theorem 3.1. 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:(i),(ii) for some ,(iii)for each , there exists a compact -convex subset of relative to some such that and Then

Remark 3.2. 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).

Definition 3.3. 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 [26]. For example, it is equivalent to the Fan-Browder-type fixed point theorem as follows.

Theorem 3.4 (see [26]). 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.

Lemma 3.5. 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.

Theorem 4.1. 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

Proof. 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, .

Corollary 4.2. Under the hypothesis of Theorem 4.1, let Then

Example 4.3. 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 [49].
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 [27].

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 .

Theorem 5.1. Let be the product abstract convex space, and let be four functions, then Suppose that for each , for each and , is -convex; for each and , is -convex, for each , there exists a finite set such that for each , there exists a finite set such that If satisfies the partial KKM principle, then

Proof. Suppose that there exists a real such that For the abstract convex space define two maps by for and , respectively. Then each is nonempty and -convex and is covered by a finite number of open sets 's. Moreover, This implies that for all . Then is a -map. Therefore, by Theorem 3.4, we have such that . Therefore, , a contradiction.

Example 5.2. For convex spaces , and , Theorem 5.1 reduces to that by Cho et al. [50, Theorem 8].

Corollary 5.3. 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 . If satisfies the partial KKM principle, then

Proof. 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.

Example 5.4. (1) Particular or slightly different versions of Corollary 5.3 are obtained by Liu [51], Granas [13, Théorèmes 3.1 et 3.2], and Shih and Tan [52, Theorem 4] for convex subsets of t.v.s.
(2) For , Corollary 5.3 reduces to [27, Theorem 3].

For the case , Corollary 5.3 reduces to the following.

Corollary 5.5 (see [27]). 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 ,(ii)one has

Example 5.6. We list historically well-known particular forms of Corollary 5.5 in chronological order as follows.(1)von Neumann [1], Kakutani [3]. and are compact convex subsets of Euclidean spaces and is continuous.(2)Nikaidô [53]. Euclidean spaces above are replaced by Hausdorff topological vector spaces, and is continuous in each variable.(3)Sion [54]. 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 [17]). 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

Example 5.8. (1) Komiya [55, Theorem 3]. and are compact convex spaces in the sense of Komiya.
(2) Slightly different form of Theorem 5.7 can be seen in [17] with different proof.

6. Collective Fixed Point Theorems

We have the following collective fixed point theorem.

Theorem 6.1. 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 .

Proof. Let be the companion map corresponding to the -map . Define by We show that is a -map with the companion map . In fact, we have where . Since each is open, we have(a)for each , is open.
Note that
and hence, 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.

Example 6.2. (1) If , is a convex space, and , then Theorem 6.1 reduces to the well-known Fan-Browder fixed point theorem; see Park [56].
(2) For the case , Theorem 6.1 for a convex space was obtained by Ben-El-Mechaiekh et al. [69, Theorem 1] and Simons [57, Theorem 4.3]. This was extended by many authors; see Park [56].

We have already the following collective fixed point theorem for arbitrary family of -convex spaces.

Theorem 6.3 (see [40]). 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 .

Example 6.4. 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.

Remark 6.5. 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 [17], 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 .

Theorem 7.1. Let be a finite family of compact abstract convex spaces such that