Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 234706,23 pages doi:10.1155/2010/234706

Research Article

Generalizations of the Nash Equilibrium Theorem in the KKM Theory

Sehie Park12

1 The National Academy of Sciences, Seoul 137-044, Republic of Korea

2 Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Republic of Korea Correspondence should be addressed to Sehie Park, shpark@math.snu.ac.kr

Received 5 December 2009; Accepted 2 February 2010 Academic Editor: Anthony To Ming Lau

Copyright © 2010 Sehie Park. 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.

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 G-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.

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, G-convex spaces) related to the KKM theory and analytical fixed point theory. In the framework of G-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 (1) fixed point method, (2) continuous selection method, or (3) the KKM method. In this paper, we follow method (3) and will compare our results to corresponding ones already obtained by method (2).

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 (D) denote the set of all nonempty finite subsets of a set D. Recall the following in [24-26].

Definition 2.1. An abstract convex space (E, D; r) consists of a topological space E, a nonempty set D, and a multimap r : (D) ^ E with nonempty values rA := r(A) for A e (D).

For any D' c D, the r-convex hull of D' is denoted and defined by

corD' := U{rA | A e (D')} c E. (2.1)

A subset X of E is called a r-convex subset of (E, D; r) relative to D' if for any N e (D') we have that rN c X, that is, corD' c X.

When D c E, the space is denoted by (E d D; r). In such case, a subset X of E is said to be r-convex if cor(X n D) c X; in other words, X is r-convex relative to D' := X n D. In case E = D, let (E; r) := (E,E; r).

Example 2.2. The following are known examples of abstract convex spaces.

(1) A triple (An d V;co) is given for the original KKM theorem [34], where An is the standard n-simplex, V is the set of its vertices {e^n=0, and co: (V) ^ An is the convex hull operation.

(2) A triple (X d D; r) is given, where X and D are subsets of a t.v.s. E such that co D c X and r := co. Fan's celebrated KKM lemma [9] is for (E d D;co).

(3) A convex space (X d D; r) is a triple where X is a subset of a vector space such that co D c X, and each rA is the convex hull of A e (D) equipped with the Euclidean topology. This concept generalizes the one due to Lassonde for X = D; see [35]. However he obtained several KKM-type theorems w.r.t. (X d D; r).

(4) A triple (X d D; r), is called an H-space if X is a topological space and r = {rA} is a family of contractible (or, more generally, ^-connected) subsets of X indexed by A e (D) such that rA c rB whenever A c B e (D).If D = X, then (X; r) := (X,X; r) is called a c-space by Horvath [36, 37].

(5) Hyperconvex metric spaces due to Aronszajn and Panitchpakdi are particular cases of c-spaces; see [37] .

(6) Hyperbolic spaces due to Reich and Shafrir [38] are also particular cases of c-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 (X, <) with path-connected interval is introduced by Horvath and Llinares [39] .

(8) A generalized convex space or a G-convex space (X, D; r) due to Park is an abstract convex space such that for each A e (D) with the cardinality |A| = n + 1 there exists a continuous function : An ^ r(A) such that J e (A) implies that $a(Aj) c r(J).

Here, Aj is the face of An corresponding to J e (A), that is, if A = {a0, a1,.. .,an} and J = {aio,aii,...,aik} c A,then Aj = co{efo,eh,...,eh}.

For details, see references of [17, 21, 22, 40-42].

(9) A -space (X,D; {$A}Ae(D}) consists of a topological space X, a nonempty set D, and a family of continuous functions : An ^ X (that is, singular n-simplexes) for A e (D) with |A| = n + 1. Every $A-space can be made into a G-convex space; see [43].

Recently -spaces are called GFC-spaces in [44] and FC-spaces [43] or simplicial spaces [45] when X = D.

(10) Suppose that X is a closed convex subset of a complete R-tree H, and for each A e (X), rA := convH(A), where convH(A) is the intersection of all closed convex subsets of H that contain A; see Kirk and Panyanak [46]. Then (H d X; r) is an abstract convex space.

(11) A topological space X with a convexity in the sense of Horvath [47] is another example.

(12) A B-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 G-convex spaces.

Definition 2.3. Let (E, D; r) be an abstract convex space. If a multimap G : D ^ E satisfies

Ta c G(A) := J G(y) VA e (D), (2.2)

then G is called a KKM map.

Definition 2.4. The partial KKM principle for an abstract convex space (E, D; r) is the statement that, for any closed-valued KKM map G : D ^ E, the family {G(y)}yeD 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 G-convex space is a KKM space [18].

(2) A connected linearly ordered space (X, <) can be made into a KKM space [26].

(3) The extended long line L* is a KKM space (L*, D; r) with the ordinal space D := [0, Q]; see [26]. But L* is not a G-convex space.

(4) For a closed convex subset X of a complete R-tree H, and rA := convH(A) for each A e (X), the triple (H d X; r) satisfies the partial KKM principle; see [46]. Later we found that (H d X; r) is a KKM space [48].

(5) Horvath's convex space (X; r) with the weak Van de Vel property is a KKM space, where rA := [[A]] for each A e (X);see [47, 48].

(6) A B-space due to Briec and Horvath [30] is a KKM space. Now we have the following diagram for triples (E, D; r):

simplex convex subset of a t.v.s. Lassonde-type convex space H-space G-convex space ^^ $A-space KKM space

space satisfying the partial KKM principle abstract convex space.

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 (E, D; r) be an abstract convex space.

Recall the following equivalent form of [26, Theorem 8.2].

Theorem 3.1. Suppose that (E, D; r) satisfies the partial KKM principle and a map G : D ^ E satisfies the following.

(1.1) G is closed valued.

(1.2) G is a KKM map (i.e., rA c G(A) for all A e (D)).

(1.3) There exists a nonempty compact subset K of E such that one of the following holds:

(i) K = E,

(ii) k = n(G(z) | z e M} for some M e (D),

(iii) for each N e (D), there exists a compact r-convex subset LN of E relative to some D' c D such that N c D' and

Ln n p| G(z) c K. (3.1)

Then K n n{G(z) | z e D} / 0.

Remark 3.2. Conditions (i)—(iii) in (1.3) are called compactness conditions or coercivity conditions. In this paper, we mainly adopt simply (i), that is, (E, D; r) is compact. However, most of results can be reformulated to the ones adopting (ii) or (iii).

Definition 3.3. For a topological space X and an abstract convex space (E,D; r), a multimap T : X ^ E is called a O-map or a Fan-Browder map provided that there exists a multimap S : X ^ D satisfying the follwing:

(a) for each x e X, corS(x) c T(x) (i.e., N e (S(x)) implies that rN c T(x)),

(b) X = UzeMInt S-(z) for some M e (D).

Here, Int denotes the interior with respect to E and, for each z e D, S-(z) := {x e X | z e S(x)}.

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 (E,D; r) satisfies the partial KKM principle if and only if any O-map T : E ^ E has a fixed point x0 e E, that is, x0 e T (xo).

The following is known.

Lemma 3.5. Let {(Xi,Di; r^}^ be any family of abstract convex spaces. Let X := nieIXi be equipped with the product topology and D = nieIDi. For each i e I, let ni : D ^ Di be the projection. For each A e (D), define r(A) := nieIri(ni(A)). Then (X,D; r) is an abstract convex space.

Let {(Xi, Di; ri) }ieI be a family of G-convex spaces. Then (X, D; r) is a G-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 (E; r) satisfying the partial KKM principle. For example, any compact G-convex space, any compact H-space, or any compact convex space is such a space.

4. The Fan-Type Minimax Inequalities

Recall that an extended real-valued function f : X ^ R, where X is a topological space, is lower [resp., upper] semicontinuous (l.s.c.) (resp., u.s.c.) if {x e X | f (x) > r} (resp., {x e X | f (x) < r}) is open for each r e R.

For an abstract convex space (E d D; r), an extended real-valued function f : E ^ R is said to be quasiconcave (resp., quasiconvex) if {x e E | f (x) > r} (resp., {x e E | f (x) < r}) is r-convex for each r e R.

From the partial KKM principle we can deduce a very general version of the Ky Fan minimax inequality as follows.

Theorem 4.1. Let (X, D; r) be an abstract convex space satisfying the partial KKM principle, f : D x X ^ R,g : X x X ^ R extended real functions, and y e R such that

(3.1) for each z e D, {y e X | f (z,y) < y} is closed,

(3.2) for each y e X, cor{z e D | f (z,y) >j}c{x e X | g(x,y) > j},

(3.3) the compactness condition (1.3) holds for G(z) := {y e X | f (z,y) < y}.

Then either (i) there exists a x e X such that f (z,x) < y for all z e D or (ii) there exists an x0 e X such that g(xo,xo) > j-

Proof. Let G : D ^ X be a map defined by G(z) := {y e X | f (z,y) < j} for z e D. Then each G(z) is closed by (3.1).

Case (i): G is a KKM map.

By Theorem 3.1, we have f)zeD G(z) / 0. Hence, there exists a x e X such that x e G(z) for all z e D, that is, f (z, x) < j for all z e D. Case (ii): G is not a KKM map.

Then there exists N e (D) such that rN/UzeNG(z). Hence there exists an xo e rN such that x0 / G(z) for each z e N, or equivalently f (z,x0) > j for each z e N. Since {z e D | f (z,x0) > j} contains N, by (3.2), we have x0 e rN c {x e X | g(x,x0) > y}, and hence, g(xo,xo) >y. □

Corollary 4.2. Under the hypothesis of Theorem 4.1, let j := supxeXg(x, x). Then

infsupf (z,y) < supg(x,x). (4.1)

yeX zeD xeX

Example 4.3. (1) For a compact convex subset X = D of a t.v.s. and f = g, if f (-,y) is quasiconcave, then (3.2) holds; and if f (x, ■) is l.s.c., then (3.1) holds. Therefore, Corollary 4.2 generalizes the Ky Fan minimax inequality [49].

(2) For a convex space X = D and f = g, Corollary 4.2 reduces to Cho et al. [50, Theorem 9].

(3) There is a very large number of generalizations of the Fan minimax inequality for convex spaces, H-spaces, G-convex spaces, and others. These would be particular forms of Corollary 4.2. For example, see Park [18, Theorem 11], where (X, D; r) is a G-convex space.

(4) Some particular versions of Corollary 4.2 were given in [27].

5. The von Neumann-Sion-Type Minimax Theorems

Let (X; ri) and (Y; r2) be abstract convex spaces. For their product, as in the Lemma 3.5 we can define Txxy(A) := ri(n(A)) x ^(.^(A)) for A e (X x Y).

Theorem 5.1. Let (E; r) := (X x Y; rXxY) be the product abstract convex space, and let f,s,t,g : X x Y ^ R be four functions, then

p := infsupf (x,y), v := supmfg(x,y). (5.1)

yeYxeX xeXyeY

Suppose that

(4.1) f (x,y) < s(x,y) < t(x,y) < g(x,y) for each (x,y) e X x Y,

(4.2) for each r < p and y e Y, {x e X | s(x,y) > r} is r1-convex; for each r > v and x e X, {y e Y | t(x,y) <r} is r2-convex,

(4.3) for each r >v, there exists a finite set {xi}r=1 c X such that

Y = y Int {y e Y | f(xuy)>r}, (5.2)

(4.4) for each r < p, there exists a finite set {yj }n=i c Y such that

X = y Int{x e X | g(x,yj)<r}. (5.3)

If (E; r) satisfies the partial KKM principle, then

p = infsupf x y) < ¡mpmfgtx y) = v. (5.4)

yeYxeX xeXVeY

Proof. Suppose that there exists a real c such that

v = supinfg(x,y) < c < infsupf (x,y) = p. (5.5)

xeXyeY yeY xeX

For the abstract convex space

(E, D; r) := (x x Y, {{xuyj )}..; w), (5.6)

define two maps S : E ^ D,T : E ^ E by

S {xi,y-j) := Int {x e X | g(x,yj) < c} x Int {y e Y | f(xiry) > c), (5.7)

T(x,y) := {x e X | s(x,y) > c) x {y e Y | t(x,y) < c], (5.8)

for (xi,yj) e D and (x,y) e E, respectively. Then each T(x,y) is nonempty and T-convex and E is covered by a finite number of open sets S~(xi,yj)'s. Moreover,

S(x,y) c {{^Vj) | g(x,yj) < c> f(xuy) > c

_ _ _ _ (5.9)

c { (x,y) | s(x,y) > c,t(x,y) < c) c T(x,y).

This implies that corS(x,y) c T(x,y) for all (x,y) e E. Then T is a ®-map. Therefore, by Theorem 3.4, we have (xo,yo) e X x Y such that (xo,yo) e T(x0,y0). Therefore, c < s(x0,y0) < t(x0,y0) < c, a contradiction. □

Example 5.2. For convex spaces X, Y, and f = s = t = g, Theorem 5.1 reduces to that by Cho etal. [50, Theorem 8].

Corollary 5.3. Let (X; r1) and (Y; r2) be compact abstract convex spaces, let (E; r) := (X x Y; rXxY) be the product abstract convex space, and let f,g : X x Y ^ R be functions satisfying the following:

(1) f (x,y) < s(x,y) < t(x,y) < g(x,y) for each (x,y) e X x Y,

(2) for each x e X,f (x, ■) is l.s.c. and t(x, ■) is quasiconvex on Y,

(3) for each y e Y, s(■, y) is quasiconcave and g(■, y) is u.s.c. on X.

If (E; r) satisfies the partial KKM principle, then

minsupf (X y < maxinfg (x y. (5.10)

yeY xeX xeX yeY

Proof. Note that y ^ sup^Xf (x,y) is l.s.c. on Y and x ^ infyeYg(x,y) is u.s.c. on X. 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, Theoremes 3.1 et 3.2], and Shih and Tan [52, Theorem 4] for convex subsets of t.v.s.

(2) For f = s,g = t, Corollary 5.3 reduces to [27, Theorem 3].

For the case f = s = t = g, Corollary 5.3 reduces to the following.

Corollary 5.5 (see [27]). Let (X;ri) and (Y;r2) be compact abstract convex spaces and let f : X x Y ^ R bean extended real function such that

(1) for each x e X,f (x, ■) is l.s.c. and quasiconvex on Y,

(2) for each y e Y,f (■,y) is u.s.c. and quasiconcave on X.

If (X x Y; rXxY) satisfies the partial KKM principle, then (i) f has a saddle point (x0, y0) e X x Y, ( ii) one has

max min f fx, y) = min max f fx, y). (5.11)

xeX yeYJ y J' yeY xeXJ y J'

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]. X and Y are compact convex subsets of Euclidean spaces and f is continuous.

(2) Nikaido [53]. Euclidean spaces above are replaced by Hausdorff topological vector spaces, and f is continuous in each variable.

(3) Sion [54]. X and Y are compact convex subsets of topological vector spaces in Corollary 5.5.

(4) Komiya [55, Theorem 3]. X and Y are compact convex spaces in the sense of Komiya.

(5) Horvath [36, Proposition 5.2]. X and Y are c-spaces with Y being compact and without assuming the compactness of X.

In these two examples, Hausdorffness of Y is assumed since they used the partition of unity argument.

(6) Bielawski [29, Theorem (4.13)]. X and Y are compact spaces having certain simplicial convexities.

(7) Park [17, Theorem 5]. X and Y are G-convex spaces.

In 1999, we deduced the following von Neumann-Sion type minimax theorem for G-convex spaces based on a continuous selection theorem:

Theorem 5.7 (see [17]). Let (X, ri) and (Y,r2) be G-convex spaces, Y Hausdorff compact, f : X x Y ^ R an extended real function, and p := supxeX infyeYf (x,y). Suppose that

(5.1) f (x, ■) is l.s.c. on Y and {y e Y | f (x,y) < r} is r2-convexfor each x e X and r > p,

(5.2)f (-,y) is u.s.c. on X and {x e X | f (x,y) > r} is r 1-convex for each y e Y and r > p.

sup minf (x,y) = min supf (x,y). (5.12)

xeX yeY yeY xeX

Example 5.8. (1) Komiya [55, Theorem 3]. X and Y 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 {(Xi; ri) }"=1 be a finite family of compact abstract convex spaces such that (X; r) = (I! n=1Xi; r) satisfies the partial KKM principle, and for each i, T : X ^ X{ is a O-map. Then there

exists a point x e X such that x e T(x) := Yin=1Ti(x), that is, Xi = ni(x) e Ti(x) for each i = 1,2,... ,n.

Proof. Let Si : X ^ Xi be the companion map corresponding to the ®-map Ti. Define S : X ^ X by

S(x) := Si(x) for each x e X. (6.1)

We show that T is a ®-map with the companion map S. In fact, we have

x e S~(y) ^^ y e S(x) ^^ yi e Si(x) for each i ^^ x e S-(y) for each i, (6.2)

where y = {y1,...,yn}. Since each S'-(yi) is open, we have

(a) for each y e X, S~(y) = f)i=1 S-(yi) is open. Note that

M e (S(x)) n(M) e (Si(x)) Ti(xi(M)) c Ti(x), (6.3)

and hence,

Tm ^ri(ni(M)) c]JTi(x) = T(x). (6.4)

i=1 i=1

Therefore, we have

(b) for each x e X,M e (S(x)) implies that Tm c T(x).

Moreover, let x e X. Since Si : X ^ Xi is the companion map corresponding to the

®-map Ti, for each i, there exists j = j(i) such that

x e S-( yj ^ yirj e Si(x) ^ y e Si(x) = S(x) ^ x e S~( y), (6.5)

where y := (yi ,j(i),... ,yn,j(n)). Since X is compact, we have

(c) X = UzeM S-(z) for some M e (X).

Since (X; r) satisfies the partial KKM principle, by Theorem 3.4, the ®-map T has a fixed point. □

Example 6.2. (1) If n = 1, X is a convex space, and S = T, then Theorem 6.1 reduces to the well-known Fan-Browder fixed point theorem; see Park [56].

(2) For the case n = 1, Theorem 6.1 for a convex space X 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 G-convex spaces.

Theorem 6.3 (see [40]). Let {(X^ rf) }ieJ be a family of compact Hausdorff G-convex spaces, X = YlieIXi, and for each i e I, let Ti : X ^ Xi be a O-map. Then there exists a point x e X such that x e T(x) := nieITi(x), that is, xi = ni(x) e Ti(x) for each i e I.

Example 6.4. In case when (Xi; ri) are all H-spaces, Theorem 6.3 reduces to Tarafdar [58, Theorem2.3]. This is applied to sets with H-convex sections [58, Theorem3.1] 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 G-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 G-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 G-convex space version of the Fan-Browder fixed point theorem, we deduced a Fan-type intersection theorem for n subsets of a cartesian product of n compact G-convex spaces. This was applied to obtain a von Neumann-sion-type minimax theorem and a Nash-type equilibrium theorem for G-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 {Xi}ieI be a family of sets, and let i e I be fixed. Let

X = ^Xj, Xi = n Xj. (7.1)

jeI jeI\{i}

If xi e Xi and j e I \ {i}, then let xj denote the jth coordinate of xi. If xi e Xi and xi e Xi, then let [xi,xi] e X be defined as follows: its ith coordinate is xi and for j / i the j th coordinate is xj. Therefore, any x e X can be expressed as x = [xi,xi] for any i e I, where xi denotes the projection of x in Xi.

Theorem 7.1. Let {(Xi; ri)}f=1 be a finite family of compact abstract convex spaces such that (X; r) = (11 n=iX r) satisfies the partial KKM principle and, for each i, let Ai and Bi be subsets of X satisfying the following.

(7.1) For each xi e Xi, 0 = cori Bi(xi) c Ai(xi) := {yi e Xi | [xi,yi] e Ai}.

(7.2) For each yi e Xi,Bi(yi) := {xi e Xi | [x\yi] e Bi} is open in Xi. Then nn=1 Ai = 0.

Proof. We apply Theorem 6.1 with multimaps Si,Ti : X ^ Xi given by Si(x) := Bi(xi) and Ti(x) := Ai(xi) for each x e X. Then for each i we have the following.

(a) For each x e X, we have 0 / cori Si(x) c Ti(x).

(b) For each yi e Xi, we have

x e S-(y^ yi e Si(x) = B^x^ [xi,y^ e B{ c Xi x Xi = X. (7.2)

Hence,

S-(yi) = {x = [x^x] e X | ^ e B(yl),xi e XJ = Bi(yi) x Xi. (7.3)

Note that S-(yi) is open in X = Xi x Xi and that Ti is a ®-map. Therefore, by Theorem 6.1, there exists x e X such that xi e Ti(x) = Ai(xi) for all i. Hence x = [xi,xi] e f)"=1 Ai / 0. □

Example 7.2. For convex spaces Xi, particular forms of Theorem 7.1 have appeared as follows:

(1) Fan [10, Theeoreme 1]. Ai = Bi for all i.

(2) Fan [11, Theorem 1']. n = 2 and A{ = Bi for i = 1,2.

From these results, Fan [11] deduced an analytic formulation, fixed point theorems, extension theorems of monotone sets, and extension theorems for invariant vector subspaces. For particular types of G-convex spaces, Theorem 7.1 was known as follows.

(3) Bielawski [29, Proposition (4.12) and Theorem (4.15)]. Xi have the finitely local convexity.

(4) Kirk et al. [32, Theorem 5.2]. Xi are hyperconvex metric spaces.

(5) Park [17, Theorem4], [18, Theorem 19]. In [17], from a G-convex space version of the Fan-Browder fixed point theorem, we deduced a Fan-type intersection theorem for n subsets of a cartesian product of n compact G-convex spaces. This was applied to obtain a von Neumann-Sion-type minimax theorem and a Nash-type equilibrium theorem for G-convex spaces.

(6) Park [27, Theorem 4]. We gave a different proof.

In [22], 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 [22]). Let {(Xi; ri) }ieI be a family ofHausdorff compact G-convex spaces and, for each i e I, let Ai and Bi be subsets of X = n ieIXi satisfying the following.

(7.1)' For each xi e Xi, 0 = cori Bi(xi) c Ai(xi) := {yi e Xi | [xi,yi] e Ai}.(7.2)' For each yi e Xi, Bi(yi) := {xi e Xi | [xi,yi] e Bi} is open in Xi. Then nteiAi = 0.

Example 7.4. For convex subsets Xi of topological vector spaces, particular forms of Theorem 7.3 have appeared as follows.

(1) Ma [12, Theorem 2]. The case Ai = Bi for all i e I with a different proof is given.

(2) Chang [59, Theorem 4.2] obtained Theorem 7.3 with a different proof. The author also obtained a noncompact version of Theorem 7.3 as in [59, Theorem 4.3].

(3) Park [19, Theorem 4.2]. Xi are convex spaces.

Note that if I 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 [12].

Theorem 8.1. Let {(Xi; ri)}JL1 be a finite family of compact abstract convex spaces such that (X; r) = (11 n=1Xi; r) satisfies the partial KKM principle and, for each i, let fi,gi : X = X1 x Xi ^ R be real functions satisfying

(8.1) fi(x) < gi(x) for each x e X,

(8.2) for each xj e X1, xi ^ gi [x\ xi] is quasiconcave on Xi,

(8.3) for each xi e Xi,xi ^ fi[xi,xi] is l.s.c. on Xj.

Let {ii}"=1 be a family of real numbers. Then either (a) there exist an i and an xi e Xi such that

fijx^yi] < ti Vyi e Xi,

or (b) there exists an x e X such that

gi(x) >ti Vi = 1,2,..

Proof. Suppose that (a) does not hold, that is, for any i and any xj e X1, there exists an xi e Xi such that fi[xi,xi] > ti. Let

Ai := {x e X | gi(x) > ti}, Bi = {x e X | fi(x) > ti}

for each i. Then

(1) for each xj e Xj, 0 = Bj(xj) c Aj(xj),

(2) for each xj e X1, Ai(xi) is ^-convex,

(3) for each yi e Xi, Bi(yi) is open in Xj.

Therefore, by Theorem 7.1, there exists an x e f)"=i Ai. This is equivalent to (b). □

Example 8.2. Fan [9, Theoreme 2], [10, Theorem3]. Xi are convex subsets of t.v.s., and fi = gi for all i. From this, fan [9, 10] deduced Sion's minimax theorem [54], the Tychonoff fixed point theorem, solutions to systems of convex inequalities, extremum problems for matrices, and a theorem of Hardy-Littlewood-Polya.

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 [22]). Let {(x,; }ieI be a family of compact Hausdorff G-convex spaces and, for each i e I, let fi,gi : X = X* x X* ^ R be real functions as in Theorem 8.1. Then the conclusion of Theorem 8.1 holds.

Example 8.4. (1) Ma [12, Theorem 3]. Xi are convex subsets of t.v.s. and fi = gi for all i e I. (2) Park [19, Theorem 8.1]. Xi are convex spaces.

Remark 8.5. (1) We obtained Theorem 8.1 from Theorem 7.1. As was pointed out by Fan [9] for his case, we can deduce Theorem 7.1 from Theorem 8.1 by considering the characteristic functions of the sets Ai and Bi.

( 2) The conclusion of Theorems 8.1 and 8.3 can be stated as follows

minsup fi\xi,xi\>ti Vi, (8.4)

xieXiXi, eXi, L J

then (b) holds; see Fan [9,10].

(3) For I = (1,2}, Theorems 8.1 and 8.3 imply the Fan minimax inequality.

9. The Nash-Type Equilibrium Theorems

From Theorem 8.1, we obtain the following form of the Nash-Fan-type equilibrium theorems in [27] with different proofs.

Theorem 9.1. Let {(Xi; r¿) }"=1 be a finite family of compact abstract convex spaces such that (X; r) = (IIn=1Xi; r) satisfies the partial KKM principle and, for each i, let fi,gi : X = X1 x Xi ^ R be real functions such that

(9.0)fi(x) < gi(x) for each x e X,

(9.1) for each x1 e X1, xi ^ gi[xi,xi] is quasiconcave on Xi,

(9.2) for each x1 e X1, xi ^ fi[xi,xi] is u.s.c. on Xi,

(9.3) for each xi e Xi, x1 ^ fi[xi,xi] is l.s.c. on X1. Then there exists a point x e X such that

gi(x) > maxfi fx1,yÁ Vi = 1,2,...,n. (9.1)

yieXi L J

Proof. Since each Xi is compact, by (9.2), for any s > 0, ti := maxyieXifi[xi,yi] - e exists for all x1 e X1 and all i. Hence Theorem 8.1(a) does not hold. Then by Theorem 8.1(b), there exists an x e E such that gi(x) > ti = maxyieXifi[xi,yi] - e for all i e I. Since e > 0 is arbitrary, the conclusion follows. □

This is not comparable to the following generalized Nash-Ma type theorem:

Theorem 9.2 (see [22]). Let {(Xi; ri) }ieI be a family of compact Hausdorff G-convex spaces and, for each i e I, let fi,gi : X = X1 x Xi ^ R be real functions satisfying (9.0)-(9.3). Then there exists a point x e X such that

gi(x) > maxfi fx1, yÁ Vi e I. (9.2)

yieXi L J

Example 9.3. Park [19, Theorem 8.2]. Xi are convex spaces.

From Theorem 9.1 for fi = gi, we obtain the following form of the Nash-Fan-type equilibrium theorem for abstract convex spaces.

Theorem 9.4. Let {(Xi; ri) }"=1 be a finite family of compact abstract convex spaces such that (X; r) = d! i=1Xi; r) satisfies the partial KKM principle and, for each i e I, let fi: X ^ R be a function such that

(10.1) for each xi e Xi, xi ^ fi[xi,xi] is quasiconcave on Xi,

(10.2) for each xi e Xi, xi ^ fi[xi,xi] is u.s.c. on Xi,

(10.3) for each xi e Xi, xi ^ fi[xi,xi] is l.s.c. on Xi. Then there exists a point x e X such that

fi(x) = maxfi \xi, Vil Vi = 1,2,..., n. (9.3)

yieXi L J

Example 9.5. For continuous functions fi, a number of particular forms of Theorem 9.4 have appeared for convex subsets Xi of Hausdorff topological vector spaces as follows:

(1) Nash [5, Theorem 1] where Xi are subsets of Euclidean spaces,

(2) Nikaido and Isoda [60, Theorem 3.2],

(3) Fan [10, Theorem 4],

(4) Tanetal. [61, Theorem 2.1].

For particular types of G-convex spaces Xi and continuous functions fi, particular forms of Theorem 9.4 have appeared as follows.

(5) Bielawski [29, Theorem 4.16]. Xi have the finitely local convexity.

(6) Kirk et al. [32, Theorem 5.3]. Xi are hyperconvex metric spaces.

(7) Park [17, Theorem 6.1], [18, Theorem 20]. Xi are G-convex spaces.

(8) Park [21, Theorem 4.7]. A variant of Theorem 9.4 is under the hypothesis that (X; r) is a compact G-convex space and f\,...,fn : X ^ R are continuous functions.

(9) Gonzalez et al. [31]. Each Xi is a compact, sequentially compact L-space and each fi is continuous as in 8.

(10) Briec and Horvath [30, Theorem 3.2]. Each Xi is a compact B-convex set and each fi is continuous as in 8.

From Theorem 9.2, we obtain the following generalization of the Nash-Ma-type equilibrium theorem for G-convex spaces.

Theorem 9.6 (see [22]). Let {(Xi; ri) }ieI be a family of compact Hausdorff G-convex spaces and, for each i e I, let fi : X ^ R be a function satisfying conditions (10.1M10.3). Then there exists a point x e X such that

fi(x) = maxfi \xi, Vil Vi e I. (9.4)

yieXi L J

Example 9.7. For continuous functions fi and for convex subsets Xi of Hausdorff topological vector spaces, Theorem 9.6 was due to Ma [12, Theorem 4].

The point x 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 I is a singleton, we obtain the following.

Corollary 9.8. Let X be a closed bounded convex subset of a reflexive Banach space E and f : X ^ R a quasiconcave u.s.c. function. Then f attains its maximum on X, that is, there exists an x e X such that f (x) > f (x) for all x e X.

Proof. Let E be equipped with the weak topology. Then, by the Hahn-Banach theorem, f is still u.s.c. because f is quasiconcave, and X is still closed. Being bounded, X is contained in some closed ball which is weakly compact. Since any closed subset of a compact set is compact, X is (weakly) compact. Now, by Theorem 9.4 for a single family, we have the conclusion. □

Corollary 9.8 is due to Mazur and Schauder in 1936. Some generalized forms of Corollary 9.8 were known by Park et al. [62, 63].

For I = (1,2}, Theorem 9.4 reduces to Corollary 5.5 as follows.

Proof of Corollary 5.5 from Theorem 9.4. Let fi(x,y) := -f (x,y) and f2(x,y) := f (x,y). Then all of the requirements of Theorem 9.4 are satisfied. Therefore, by Theorem 9.4, there exists a point (x0,y0) e X x Y such that

fi( x0,yci) = maxfi(xc,y), fi(xo,yc) = maxf2(x,y0). (9.5)

yeY xeX

Therefore, we have

-f (xo,yo) = fi(xc,yc) > fi(xo,y) = -f(xc,y) Vy e Y, (9.6)

f(xc,yc) = fi{xc,yc) > fi{x,yc) = f{x,yc) Vx e X. (9.7)

f{x,yc) < f{xc,yc) < f(xc,y) V(x,y) e X x Y. (9.8)

Therefore

max/(x,yo) < f(xo,yo) < minf (xo,y). (9.9)

xeX yeX

This implies that

minmaxf (x,y) < f(x0,y0) < maxminf x,y). (9.10)

yeX xeX xeX yeX

On the other hand, we have trivially

minf(x, v) <maxf(x, v), (9.11)

and hence

maxminf (x,v) < minmaxf (x,v)- (9.12)

xeX yx ^ J/ v^X x(XJ y J'

Therefore, we have the conclusion. □

10. Historical Remarks on Related Results

(I) As we have seen in Sections 1-3, we have three methods in our subject as follows:

(1) 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,

(2) 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,

(3) 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, 4750,54,55,57,59, 71, 72] and others.

For Case (1), we will study it elsewhere and, in this paper, we are mainly concerned with Cases (2) and (3).

(II) 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 [8] in 1972 for compact Kakutani maps instead of assuming compactness of domains. In 1990, Lassonde [67] 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 [66].

Fixed point theorems extending the Kakutani theorem can be applied to particular forms of results in this paper. Since such extended theorems usually assume Hausdorffness and certain (abstract) local convexity of the related space, their applicability is restrictive.

(III) In 1946, the Kakutani theorem was extended for acyclic maps by Eilenberg and Montgomery [73]. This result was applied by Debreu [66] to the social equilibrium existence theorem and related results.

(IV) Since 1996 [72], many authors have published some results of the present paper for hyperconvex metric spaces. For example, Kirk et al. in 2000 [32] established the KKM theorem, its equivalent formulations, fixed point theorems, and the Nash theorem for hyperconvex metric spaces. However, already in 1993, Horvath [37] found that hyperconvex metric spaces are a particular type of c-spaces.

(V) In 1998 [16], 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 [16]). Let {Xj}"=1 be a family of acyclic polyhedra, X = YXi=\Xi, and for each i, fi : X ^ R a continuous function such that

(0) for each xi e Xi and each a e R, the set

(VI) 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 [27].

In 1999 [17], we obtained a "K to N" for G-convex spaces. These results extended and unified a number of known results for particular types of G-convex spaces; see also [1821, 42]. Therefore, the procedure also holds for Lassonde type-convex spaces, Horvath's c-space, hyperconvex metric spaces, and others.

(VII) In 2000 [20] and 2002 [23], 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 [20]). Let {Xi }i=1 be a family of convex sets, each in a t.v.s. Ei, Ki a nonempty compact subset of Xi, Si : X ^ Ki a closed map, and fi,gi : X = Xi x Xi ^ R u.s.c. functions for each i.

Suppose that, for each i,

(i) gi(x) < fi(x) for each x e X,

(ii) the function Mi defined on X by

(10.1)

is empty or acyclic.

Then there exists a point a e X such that

(10.2)

Mi(x) = max gi(xl,y for x e X

(10.3)

is l.s.c., and

(in) for each x e X, the set

[y e Si(x) | fi(xl,y) > Mt(x)} (10.4)

is acyclic.

If X is admissible in E = TVj=iEj, then there exists an x e K such that, for each i,

xi e Si(x), f^xi,x^ > gi(xi/y^ Vy e Si(x). (10.5)

Theorem 10.3 (see [23]). Let X0 be a topological space and let (Xi}"=1 be a family of convex sets, each in a t.v.s. Ei. For each i = 0,1,...,n, let Si : Xi ^ Xi be a closed map with compact values, and fi/gi : X = n n=0Xi ^ R u.s.c. real-valued functions. Suppose that, for each i,

(i) gi(x) < fi(x) for each x e X,

(ii) the function Mi : Xi ^ R defined by

Mdx0 = max gdxi,yA for xi e Xi (10.6)

V ' yieSi(xi) V '

is l.s.c., and

(iii) for each xi e Xi, the set

[xi e Sifx1) I fift/x^ > Mi(xi)} (10.7)

is acyclic.

If X0 is admissible in E0 = Tl^Ej and if all the maps Si are compact except possibly Sn and Sn is u.s.c., then there exists an equilibrium point x e X, that is,

xi e Si[xl), fi(x) > max gi{xl,y) Vi e Zn+1. (10.8)

V ' yie Si(xi) v '

The following are particular results of ours for G-convex spaces or KKM spaces. (VIII) In 2001 [22] , 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 G-convex spaces. In [22], all G-convex spaces were assumed to be Hausdorff because the results are based on a selection theorem in [40], where Hausdorffness was indispensable.

As we have seen already in Theorems 7.3, 8.3, 9.2, and 9.6, results following the selection theorem in [40] hold for arbitrary family of Hausdorff G-convex spaces. But corresponding results following the partial KKM principle hold for finite families of abstract convex spaces. At present, we have no way to unify these three cases, namely, (1) fixed point method, (2) selection method, and (3) KKM method.

(IX) In 2001, for any topological semilattice (X, <) with path-connected interval introduced by Horvath and Llinares [39], the KKM theorem, the Fan-Browder theorem, and the Nash theorem are shown by Luo [33]. Note that such semilattice is known to be a G-convex space.

(X) Cain and Gonzalez [71] considered relationship among some subclasses of the class of G-convex spaces and introduced a subclass of the so-called L-spaces. In 2007, Gonzalez et al. [31] repeated to show that G-convex spaces and L-spaces satisfy the partial KKM principle. They added that L-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 G-convex spaces.

(XI) In 2008, Kulpa and Szymanski [45] 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 L*-spaces in [45] are particular types of abstract convex spaces satisfying the abstract KKM principle.

(XII) In 2008, for B-spaces, Briec and Horvath [30] showed that some theorems mentioned in this paper hold, that is, Fan-Browder fixed point theorem, Himmelbergtype (in fact, Browder-type and Kakutani-type) fixed point theorems, Fan type minimax inequality, existence of Nash equilibria, and others. Note that B-spaces are KKM spaces [30, Corollary 2.2], and their authors depend the Pelegs type multiple KKM theorem.

(XIII) 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.

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