Discussiones Mathematicae Differential Inclusions 15(2) (1995) 149-162

[BIBTex]

BROWDER-FAN FIXED POINT THEOREM AND RELATED RESULTS

Paul Deguire

Département de Mathématiques, Université de Moncton
Moncton, N.-B., Canada, E1A 3E9

email: deguirp@umoncton.ca

0. Introduction and preliminaries

The starting point of this work is the well known Browder-Fan fixed-point theorem:

Theorem 1.1. Let X be a convex compact subset of a Hausdorff topological vector space and T be a set-valued map from X into itself with non-empty convex values and open fibers. Then T has a fixed point.

This result has been extended in many different directions by various mathematicians in the past 20 years. Most of the improvements consist of weakening the assumptions on the compactness of the space X or on the convexity of the values of the map T. Beginning with Gale and Mas-Colell, see [11], many authors have started to work with families of maps defined in the same domain, each map having its values in a single factor of a product space. Improvements concerning compactness and convexity have also been made in this case.

This work is a short survey of many of the extensions of the Browder-Fan fixed point theorem. It also contains recent results of the author, most of which have been done in collaboration with H. Ben El Mechaiekh and M. Lassonde.

To describe various classes of set-valued maps, we will use the same notations that we used in our works with H. Ben El Mechaiekh and A. Granas (see [3], [4] and [5]).

In this paper we will be concerned with set-valued maps defined in subsets of topological vector spaces, always Hausdorff, real or complex (t.v.s. for short). Then, spaces will be subsets of t.v.s. and convex spaces will be convex subsets of t.v.s.. If T is a set-valued map from X to Y, the value of T at the point y ∈ X is denoted by T(x) and is a subset of Y. The fiber of T at the point y ∈ Y is denoted by T-1(y) and is the following subset of X: T-1(y) = {x ∈ X: y ∈ T(x)}. A fixed point for T is a point x ∈ X such that x ∈ T(x). A polytope is the convex hull of a finite set of points in a t.v.s..

Browder-fan fixed point theorem ...

1. Basic results

The starting point of this paper is the well known Browder-Fan fixed point theorem.

Theorem 1.1. Let X be a compact convex subset of a t.v.s. and let T : X ∪X be a set-valued map with non-empty convex values and open fibers. Then T has a fixed point.

The fixed point formulation of Theorem 1.1, as presented here, has been given by Browder in 1968 [6], and the proof of the theorem was done by showing that the map T has a singled-valued continuous selection taking values in a finite polyhedra. A few years earlier, in 1961, Fan has proved an equivalent result with a different formulation using his extension of the Knaster-Kuratowski-Mazurkiewicz theorem to infinite dimensional spaces [9], [13].

The first theorem is a generalization of Theorem 1.1 in which instead of working with a single map from a space to another space, we have a family of maps from a space into the components of a product space. It is a particular case of various results presented in the past ten years or so. Similar results, more or less general, may be found in: [1, 8, 14, 16, 17, 18, 19].

We will use the following notations: If X = ∏i ∈ I Xi, then for x ∈ X we write x = (xi)i ∈ I where xi ∈ Xi is the projection of x on Xi.

Theorem 1.2. Let I be an arbitrary set of indices and {Ai : X ∪Xi}i ∈ I, be a family of maps that satisfy the following assumptions:

(i) ∀ i ∈ I, Xi is convex and compact
(ii) ∀ x ∈ X, ∀ i ∈ I, Ai(x) is convex
(iii) ∀ x ∈ X, ∃ i ∈ I, with Ai(x) ≠ φ
(iv) ∀ y ∈ X, ∀ i ∈ I, Ai-1(yi) is open.

Then, ∃ x ∈ X, ∃ i ∈ I such that xi ∈ Ai(x).

Remark 1.2.1. The conclusion of Theorem 1.2 is on the existence of a fixed component instead of a fixed point. Theorem 1.1 is a special case of Theorem 1.2 where the index set I has only one element. In that particular case, the notions of fixed point and fixed component are identical. Theorem 1.2 cannot be derived directly from Theorem 1.1 by taking some kind of product map because the values of the maps involved can be empty. For simplicity, in the sequel, we will sometimes refer to fixed points only while generally speaking about either fixed points or fixed components.

In this paper we will mostly present results of a geometrical nature. Those results will be either variants or generalization of Theorem 1.1. We will present one analytical result in the last section of the paper. The text is organized as follow:

In Sections 2, 3, 4 and 5, we present many extensions of Theorems 1.1 and 1.2 in which assumptions on compactness or convexity are relaxed. Most results appear in two versions:

1) A : X ∪Y, is a single map from a space X to a space Y, we will call this case the classical case.
2) {Ai : X ∪Yi}i ∈ I is a family of maps from a space X into the components of a product space Y. We will call this case the product case. The index set I will always be arbitrary.

We will see that the topological assumptions in the product case are similar to the assumptions used in the classical case in most of the results. It is not known whether we can always pass from the classical case to the product case without strenghtening the assumptions. In Section 6, we will present some abstract theorems, extending to the product case some results of Ben El Mechaiekh, Deguire and Granas in the classical case (see [5]). Finally, in Section 7, we will present the analytical formulation of Theorem 1.2. This result is a minimax inequality for a family of numerical applications.

The following definitions will be useful throughout the next five sections.

Definition 1.1. A map A from a space X to a convex space Y is said to be of type F* if the values A(x) are non-empty, convex and the fibers A-1(y) are open, for all x in X and y in Y. For short, we will speak about F*-maps.

Definition 1.2. A family of maps {Ai : X∪Yi}i ∈ I, is said to be of type F* if the following assumptions are satisfied:

(i) for all i ∈ I, the space Yi is convex,
(ii) ∀ x ∈ X, ∀ i ∈ I, Ai(x) is convex,
(iii) ∀ x ∈ X, ∃ i ∈ I with Ai(x) ≠ φ,
(iv) ∀ y ∈ X, ∀ i ∈ I, Ai-1(yi) is open.

For short, we will speak about F*-families. Observe that the members of a F*-family of maps need not be F*-maps. But if a F*-family consists of just one map, it must be a F*-map.

Remark. All the results that we are going to present in this paper hold true if we replace F*-maps and F*-families by the more general φ*-maps and φ*-families. φ*-maps are maps having convex values and admitting a set-valued selection with non-empty values and open fibers. φ*-families are defined in a similar way. We will not use those slightly more general classes of maps in this paper. For more details, see [1] and [4].

We finish this section with two basic results concerning selection properties of F*-maps and F*-families of maps. Those properties play a crucial role in finding fixed points for those maps or fixed components for those families of maps. The new notion of selecting families has been presented in our joint paper with Lassonde (see [8]). It is the basic tool used to obtain most results of the product case in the following sections.

Theorem 1.3. (on selection properties)

A) If A : X ∪Y is a F*-map and X is paracompact, then there exists a continuous single-valued map g : X ∪Y such that, for any x in X, g(x) ∈ A(x).
B) If {Ai : X ∪Yi}i ∈ I is a F*-family of maps and X is paracompact, then there exists a family of continuous valued maps, {gi : X →Yi}i ∈ I, satisfying the following property:

For any x in X, there exists i in I, such that gi(x) ∈ Ai(x).

For Theorem 1.3 B), see [8, Theorem 3.3]. Theorem 1.3 B) reduces to Theorem 1.3 A) when I is a singleton.

Note. Part A of Theorem 1.3 is implicitely contained in the proof of the Michael Selection Theorem (see Lemma 4.1 in [15]).

Remark 1.3.1. The family of single-valued map {gi : X∪Yi}i ∈ I satisfying the conclusion of the Theorem 1.3 B), will be called a selecting family. This notion of selecting family reduces to the well known notion of continuous selection when the index set I has only one element.

In the case where X is compact, Theorem 1.3 is of course true and the following properties are also satisfied:

Theorem 1.4. (on the meaning of the compactness)

A) If A : X∪Y is a F*-map and if X is compact, then there exists a polytope C in Y such that for any x in X, A(x)∩C ≠ φ.
B) If {Ai : X ∪Yi}i ∈ I is a F*-family of maps and if X is compact, then there is a subset C = ∏i ∈ ICi of Y such that for any x in X, there exists i in I with Ai(x) ∩Ci ≠ φ. ∀ i ∈ I, Ci is a polytope. All of those polytopes, but a finite number, consist of a single point.

Remark 1.4.1. The selection g in the conclusion of Theorem 1.3 A) takes its values in C. In the particular case where X = Y, we can restrict the range and domain of the map A to the set C and work with this finite dimensional space.

The selecting family of the conclusion of Theorem 1.3 B) consists of maps taking their values in the factors Ci of C. In the particular case where X = Y, we can restrict the ranges and the domain of the maps Ai, i in I, to the sets C and Ci, i in I, and work in a space which behaves like a finite dimensional one. In both cases, classical and product, this enables us to use the Brouwer fixed point theorem and to work in general topological vector spaces. For Theorem 1.4 B), see Proposition 2.5 in [8]. Theorem 1.4 B) reduces to Theorem 1.4 A) when I is a singleton.

2. Fixed point for composition with compact maps

This section is concerned with composition of compact upper semi-continuous maps with F*-maps or F*-families of maps. We recall that a map A from X into Y is upper-continuous (u.s.c.) if for any open set U in Y, the subset of all x in X for which A(x) is contained in U is open. Moreover, the map A is said to be compact, if the closure of A(X) is a compact subset of Y.

We first define the class of compact u.s.c. maps that we will need for the next result.

Definition 2.1. A map A from a set X into a topological space Y belongs to the class V*(X,Y) if the following assumptions are satisfied:

(i) A is u.s.c. and has non-empty compact values.
(ii) If C ⊆ X is a polytope and s is a single-valued continuous function from Y into C, then the restriction of the composition s°A from C to C has a fixed point.

V*(X,Y) is A very large class of map. Any u.s.c. map A whose values are non-empty and acyclic (with respect to the ech cohomology with rational coefficients), belongs to this class. Another examples are the continuous single-valued maps (Brouwer fixed point theorem) and the u.s.c. non-empty convex valued maps (Kakutani maps).

Theorem 2.1. (composition with compact maps)

A) Let X be a space and Y be a convex space. If A is a F*-map from X to Y and B ∈ V*(Y,X) is compact, then there exists y ∈ Y such that y ∈ A°B(y).
B) Let { Ai : X ∪Yi}i ∈ I, be a F*-family of maps from a space X into the (convex) factors of the space Y. If B ∈ V*(Y,X) is compact, then there exists y in Y and i in I, such that yi ∈ Ai°B(y).

Proof. The idea of the proof of Theorem 2.1 is straightforward when we consider the definition of the class V*(Y,X). In the classical case, one uses Theorem 1.3 A) to provide a continuous selection of the map A, this selection can be chosen to satisfy the Theorem 1.4 A). The composition of this selection with the map B has a fixed point which verifies the conclusion of the theorem. In the more delicate product case, one follows the same idea, using Theorems 1.3 B) and 1.4 B) (see Theorem 4.1 in [8]).

3. Fixed point with weak compactness assumptions

In this section we present a fixed point theorem in which the compactness of the domain is replaced by the following weaker coercitivity assumption:

Definition 3.1.

A) A map A from X to Y is said to verify the K condition, if there exist a compact subset K of X and a compact convex subset C of Y such that for any x ∈ X \K, A(x)∩C ≠ φ.
B) A family of maps {Ai : X∪Yi}i ∈ I is said to verify the K condition, if there exists a compact subset K of X, and for all i ∈ I a compact convex subset Ci of Yi, such that for any x ∈ X \K, there exists i in I with Ai(x)∩Ci ≠ φ

The K condition, and some other weak coercitivity conditions, have been used by many authors in the classical case (e.g. Fan; Lassonde; Tarafdar; Ben El Mechaiekh, Deguire and Granas). In this paper, and in our recent paper with Lassonde, see [8], we use this condition in the product case. This is possible because of the simplicity of our method based on generalized selections (selecting families). In some recent papers where other methods are used, much stronger coercitivity condition is assumed. The sets K and Ci, i ∈ I, are the same, but instead of asking that Ai(x) intersects Ci for at least one index i ∈ I it is asked that Ai(x) intersects Ci for all indices i ∈ I. Moreover, the space X has to be paracompact.

We will present only a fixed point theorem using this coercitivity condition. It is possible, in the product case as well as in the classical case, to obtain some coincidence and intersection theorems with the same compactness assumptions, see [7] and [8].

Theorem 3.1. (on a fixed point for non-compact spaces)

A) Let X be a convex space. If A is a F*-map from X to itself satisfying the coercitivity condition K then there exists x ∈ X such that x ∈ A(x).
B) Let {Ai : X ∪Xi}i ∈ I, be a F*-family of maps from a space X into its convex factors Xi, (i ∈ I), satisfying the coercitivity condition K. Then there exists x in X and i in I, such that xi ∈ Ai(x)

Proof of (3.2 B). K being compact, for all i in I, there exists a polytope Di in Xi such that for all x in K, we have that Ai(x) intersects Di for some index i (Theorem 1.4 B). For all x in X \K, Ai(x) intersects Ci for some index i (by assumption). If Hi is the convex hull of Ci∪Di, then Hi is convex compact and Ai(x) intersects Hi for all x in X. By Theorem 1.2 with the sets X and Xi replaced by the sets H = ∏i ∈ IHi and Hi respectively, for all i in I, we conclude that 3.2 B) is true.

4. Fixed points for compact maps

In this section, we look at the situation where the compactness of the spaces is replaced by the compactness of the maps. Section 4, like the preceding Section 3, uses a coercitivity condition that is automatically satisfied when the spaces are compact. But, there is no relation between the coercitivity condition K and the assumption that the maps are compact. The techniques are different and so are some of the results. In this section, we will only present a result in the classical case and we will make some comments on open problems in both the classical and the product cases.

Theorem 4.1. (on a fixed point for compact maps)

Let X be convex in a locally convex vector space and let A be a compact F*-map from X into itself. Then, the map A has a fixed point.

Proof. Let K compact such that A(X) ⊆ K ⊆ X and let Y be the convex hull of the compact K. Then Y is paracompact. The map A restricted to Y has a continuous selection s (by Theorem 1.3) that has a fixed point by the Schauder - Tychonoff fixed point theorem. This point satisfies the conclusion of the theorem.

Remark 4.1.1. In our previous papers, as well as in other papers, the space X is assumed to be either closed or paracompact. But the fact that the convex hull of a compact space is paracompact (σ-compact to be more precise, but this is stronger than paracompact) makes those assumptions unnecessary.

Remark 4.1.2. The proof of Theorem 4.1 does not work in the product case, because the selecting family provided by Theorem 1.3 B) is not composed of compact maps. We can obtain some results in the product case if we add some assumptions that are not necessary in the classical case (see Section 5 in [8]).

Open problems.

1) It is not known if Theorem 4.1 remains true when the topological vector spaces involved are not locally convex. This problem, because of Theorem 1.3 (existence of continuous single-valued selection), may be related to the Schauder conjecture that says that compact convex subsets of any topological vector spaces have the fixed point property for continuous single-valued maps.

2) It is not known if we can obtain, in the product case, some results using the same assumptions that we use in the classical case. For instance, at this point we are unable to obtain a result in the product case that would exactly reduce to Theorem 4.1 if the index set I had only one element.

5. Fixed points without convexity

In this section we present a result of Horvath that generalizes the Browder-Fan fixed point theorem by replacing the convexity of the values by some weaker contractibility assumptions. We also present the extension of this result to the product case. The sets involved in this section do not have to be subsets of linear spaces.

Theorem 5.1. (on a fixed point for maps without convex values)
A) (Horvath 1983) Let X be a contractible compact space and let A be a map from X to X with open fibers and non-empty values such that, for any open subset U of X, the following set ∩x ∈ UA(x) is contractible or empty.
Then the map A has a fixed point.
B) Let {Xi}i ∈ I be a family of compact contractible spaces and let {Ai : X∪X}i ∈ I with X = ∏i ∈ I Xi, be a family of maps satisfying the following:
(i) For all i, the map Ai has open fibers.
(ii) For all x ∈ X, there exist i ∈ I with Ai(x) non-empty.
(iii) For all i, for any open subset U of X, ∩x ∈ UAi(x) is contractible or empty.

Then there exist x ∈ X, i ∈ I, such that xi ∈ Ai(x).

For Theorem 5.1 B), see [1]. Of course, Theorem 5.1 B) reduces to Theorem 5.1 A) when I is A singleton.

6. Abstract results

In our third paper on fixed point and coincidence theorem for set-valued maps, with Ben-El-Mechaiekh and Granas, we defined the abstract class of all the maps whose restrictions on compact sets admit a continuous selection with values in a polytope. This abstract class of maps has many known subclasses, including the class of F*-maps. In this section, we will extend our definition to the product case and derive some fixed point theorems having as special case Theorem 1.2 and some other results not covered by the previous sections of this paper.

Definition 6.1. (See [5]). We say that the map A from X into Y is a M*-map, if the following conditions are satisfied:

Y is convex and for any compact subset K of X, there exist a polytope C ⊆ Y and a continuous single-valued map s from K to C such that s(x) ∈ A(x), for all x ∈ K.

This class of maps was presented, together with many theorems illustrating its properties, in [5]. We now extend this definition to the product case.

Definition 6.2. Let X be a space and {Yi}i ∈ I be convex spaces with Y = ∏i ∈ IYi. A family of maps {Ai : X∪Yi}i ∈ I, will be called a M*-family if the following conditions are satisfied:

For any compact subset K of X and for any i ∈ I, there exist a polytope Hi in Yi, (Hi being a single point for all but a finite number of indices i ∈ I) and a single-valued continuous map si from K to Hi, such that for any x in K, there exists i ∈ I with si(x) ∈ Ai(x). The family {si}I is a selecting family for the restrictions {Ai restricted to K:K∪ Yi}I.

Observe, as we already did for F*-families, any member of a M*-family of maps is not necessarily a M*-map but if a M*-family of maps contains just one map, then this map is a M*-map.

Proposition 6.1.

A) Any F*-map is a M*-map.

B) Any F*-family of maps is a M*-family of maps.

This proposition is evident when we consider Theorem 1.3 and Theorem 1.4. The following fixed point theorem follows directly from the definition of M*-families.

Theorem 6.1. (a fixed component theorem for M*-family of maps)

Let X = ∏i ∈ IXi be a product of compact convex spaces. If {Ai : X∪Xi}i ∈ I is a M*-family of maps, then there exist x in X and i in I such that xi ∈ Ai(x).

Remark 6.1.1. Theorem 1.2 is a particular case of Theorem 6.1 where M*-families are replaced by F*-families.

In the classical case, one of the nicest property of the class of M*-maps is that this class is closed under finite compositions. This helped us to obtain, for instance, fixed point theorems for compositions of F*-maps.

Theorem 6.2. Let Y = ∏i ∈ IYi, X = ∏i ∈ I Xi and for i in I, Xi = ∏j ∈ JXi,j be products of compact and convex spaces. If { Ai : X∪Yi}i,j and, for all i ∈ I {Ai,j : Yi∪Xi,j}Ji are M*-families of maps, then there exist x in X, i in I and j in Ji such that : xi,j ∈ Ai,j(Ai(x)).

Proof. We have to show that the family { Ai,j°: X∪Xi,j}I,Ji is a M*-family of maps. Then, we apply to this family Theorem 6.1.

Theorem 6.2 looks very easy, but it is nevertheless a useful result because it contains a fixed point theorem for composition of F*-families and those compositions are not in general F*-families themselves. In general, the maps involved in such compositions do not have convex values. Of course, Theorem 6.2 can be extended to composition of any finite number of M*-families.

The next theorem shows that, in the product case, we can get the fixed point results for composition of maps in two different ways.

Theorem 6.3. Let { Ai : X∪Yi}i ∈ I and {Bj : Y ∪Xj}j ∈ J be two M*-families, with Y = ∏i ∈ IYi and X = ∏j ∈ JXj being compact and convex. Then there exist x ∈ X, y ∈ Y, i ∈ I and j ∈ J such that: yi ∈ Ai(x) and xj ∈ Bj(y).

Proof. If {si}I and {tj}J are selecting families for {Ai}I and {Bj}J respectively, then let s from X to Y and t from Y to X be defined by s(x) = (si(x))I, for x ∈ X and t(y) = (tj(y))J, for y ∈ Y. There exist a subset C of Y and a subset D of X, such that:

(i) C and D are homeomorphic to a finite product of polytopes.
(ii) s takes its value in C and t takes its value in D.

The composition t°s from D to D has a fixed point by the Brouwer fixed point theorem. Let x = t(s(x)) and y = s(x).
{si}I and {tj}J being selecting families, there exist i ∈ I and j ∈ J with yi = si(x) ∈ Ai(x) and xj = tj(y) ∈ Bj(y).

Remark 6.3.1. This theorem can either be seen as a fixed point theorem for the composition of two M*-families of maps, or as a coincidence theorem between those two families. The fixed point formulation and the coincidence formulation are the same when there are only two families of maps involved. When more than two families are involved, the fixed point results for the composition and the intersection results are distinct.

7. A minimax inequality for a family of numerical maps

Fixed point and coincidence theorem for F*-maps and F*-families of maps can be formulated as non-linear alternative and minimax theorems. Those analytical results have numerous applications in different areas of applied mathematics (e.g.see [2]).

In this short section, we will present the analytical formulation of Theorem 1.2. We recall that a numerical map f:X ∪IR is said to be lower semi-continuous, i.e. l.s.c. (respectively quasi-concave) if, for all t ∈ IR, the following set { x ∈ X : f(x) >t } is open (respectively convex).

Theorem 7.1. Let X = ∏i ∈ IXi be a product of compact convex spaces. If for all i ∈ I the map fi : X×XiIR satisfies the following conditions:

(i) yi ∪fi(x,yi) is quasiconcave, ∀ x ∈ X
(ii) x∪fi(x, yi) is l.s.c. ∀ yi ∈ Xi

A) Then, ∀ λ ∈ IR, at least one of the following statements is satisfied:

1) ∃ x ∈ X : supi ∈ Isupyi ∈ Xi fi(x, yi) ≤ λ, or
2) ∃ x ∈ X, ∃ i ∈ I, : fi(x, xi) >λ.

B) The following minimax inequality holds:
infx ∈ Xsupi ∈ Isupyi ∈ Xi fi(x, yi) ≤ supi ∈ Isupx ∈ X fi(x, xi)

Proof. This non-linear alternative is equivalent to Theorem 1.2. An element x for wich the assumption 1.2 (iii) would not be satisfied is called by the mathematical economists a maximal element. Under all assumptions of Theorem 1.2 except assumption 1.2 (iii), one can say that either there is a maximal element or a fixed component. This alternative translates into part A of Theorem 7.1. The correspondence follows from this definition:

For each i in I and for λ ∈ IR, we define the application Fi : X∪Xi at the point x in X by Fi(x) = {yi ∈ Xi : fi(x, yi) > λ}.

Part B follows easily from part A if one takes λ = supi ∈ Isupx ∈ X fi(x, xi). In this case, the statement 2) of part A is impossible and the statement 1) provides the minimax inequality of part B.

For more examples see [1] and [8].

Remark 7.1.1. In the special case where the index set I has only one element, this result reduces to a non-linear alternative that is equivalent to the well known Fan minimax inequality (e.g. see [10, 2]).

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