CUBO A Mathematical Journal

Vol.10, N
o
¯ 04, (137–147). December 2008

A New Version of Fan’s Theorem and its Applications

M. Fakhar

Department of Mathematics, University of Isfahan,

and Institute for Studies in Theoretical Physics and Mathematics (IPM),

Isfahan, 81745-163, Tehran, Iran

email: fakhar@math.ui.ac.ir

and

J. Zafarani

Sheikhbahaee University and University of Isfahan,

Isfahan, 81745-163, Iran

email: jzaf@zafarini.ir

ABSTRACT

In this article, using a generalized version of Ky Fan’s Theorem, we deduce new proofs

for some fixed point theorems and new existence theorems for equilibrium problems.

RESUMEN

Usando una versión generalizada del Teorema de Ky Fan, deducimos nuevas demostra-

ciones para algunos teoremas de punto fijo y nuevos teorema de existencia para prob-

lemas de equilibrio.

Key words and phrases: Fan’s Theorem, fixed point theorem, equilibrium problem, Variational

inequalities.

Math. Subj. Class.: 47H10; 49J53; 54H25.



138 M. Fakhar and J. Zafarani CUBO
10, 4 (2008)

1 Introduction

Many problems in nonlinear analysis can be solved by showing the nonemptyness of the intersec-

tion of certain family of subsets of an underlying set. Each point of the intersection can be a fixed

point, a coincidence point, an equilibrium point, a saddle point or an optimal. The first remark-

able result on nonempty intersection was the celebrated Knaster, Kuratowski and Mazurkiewicz

in 1929 [15], which concerns with certain type of multimaps called the KKM maps later. Fan [11]

proved that the assertion of the KKM theorem for infinite dimensional topological vector space.

Brézis, Nirenberg and Stampacchia [4] improved Fan’s KKM lemma [11] by assuming the closedness

condition only on finite dimensional subspaces, with some topological pseudomonotone condition.

Chowdhury and Tan [5], replacing finite dimensional subspaces by polytopes, restated the Brézis,

Nirenberg and Stampacchia result under weaker assumptions. Ding and Tarafdar [7] obtained the

result of Chowdhury and Tan under weaker compactness condition. The Chowdhury and Tan’s

result was also proved by Kalmoun [14] for transfer closed-valued multi-valued mappings. Our aim

here is to derive a new version of Brézis, Nirenberg and Stampacchia’s result and then apply it to

obtain some fixed point theorems and established the existence solution of equilibrium problems

and generalized variational inequalities.

For the reader’s convenience, we review a few basic definitions and notations from the fixed point

theory. Let X be a Hausdorff topological vector space and K be a nonempty subset of X, then

we denote by < K > the family of all nonempty finite subsets of K. Let K0 be a nonempty

subset of K. A set-valued map Γ : K0 ⇉ K is called a KKM map if for each A ∈< K0 >,

conv(A) ⊆
⋃

x∈A
Γ(x). Let Y be a nonempty set. Then, Γ : Y ⇉ K is said to be transfer closed-

valued if for any (y, x) ∈ Y × K with x 6∈ Γ(y) there exists y′ ∈ Y such that x 6∈ clK Γ(y
′
). If

Y = K, then we will call Γ transfer closed-valued on K. If K0 ⊆ K, then a map Γ : K ⇉ K is

called transfer closed-valued on K0 if the map y 7→ Γ(y)
⋂

K0, y ∈ K0, is transfer closed-valued. A

set-valued map Γ : K ⇉ K is called transfer open-valued on K if the set-valued map Γ̂ : K ⇉ K

defined as follows: Γ̂(x) := K \ Γ(x) is transfer closed-valued on K. Let us recall that a set-valued

map Γ : K ⇉ K has a maximal element, if there exists a point x̄ ∈ K such that Γ(x̄) = ∅.

Suppose that f is a real-valued bifunction on Y × K. Then, we say that f is transfer lower semi-

continuous(l.s.c.) in the second variable if for each (y, x) ∈ Y × K with f (y, x) > 0 there exist

y′ ∈ Y and a neighborhood U (x) of x in K such that f (y′, z) > 0 for all z ∈ U (x). If Y = K and

A ⊆ K, then we call f transfer l.s.c. in the second variable on A, if f |A×A is transfer l.s.c. in the

second variable.

Definition 1.1 Let f : K × K → R. We recall that:

(i) f is pseudomonotone if, for all (x, y) ∈ K × K, f (x, y) ≥ 0 implies f (y, x) ≤ 0;

(ii) f is called 0-segmentary closed if ∀x, y ∈ K, when (yα) be a net on K converging to y, then

the following implication holds, if f (u, yα) ≤ 0 for all u ∈ [x, y], then f (x, y) ≤ 0.



CUBO
10, 4 (2008)

A New Version of Fan’s Theorem and its Applications 139

(iii) f (., y) is upper sign continuous if the following implication holds for every x ∈ K,

f (u, y) ≥ 0, ∀u ∈ ]x, y[⇒ f (x, y) ≥ 0,

We note that if f is hemicontinuous function, then f and −f both are upper sign continuous.

2 Brézis, Nirenberg and Stampacchia type theorem

In [8, 9], the authors refined the Ding and Tarafdar’s result [7] and the Kalmoun’s result [14].

Based on the Remark 2 in [4], recently the authors obtain a short and direct proof of the following

Brézis, Nirenberg and Stampacchia version of Fan’s KKM Theorem [10]

Lemma 2.1 Let K be a nonempty and convex subset of a Hausdorff t.v.s. X. Suppose that

Γ : K ⇉ K is a set-valued mapping such that the following conditions are satisfied:

(i) Γ is a KKM map;

(ii) for all A ∈ < K >, Γ is transfer closed-valued on conv(A);

(iii) for all x, y ∈ K, clK (
⋂

u∈[x,y]
Γ(u)) ∩ [x, y] = (

⋂

u∈[x,y]
Γ(u)) ∩ [x, y];

(iv) there is a nonempty compact convex set B ⊆ K, such that clK (
⋂

x∈B
Γ(x)) is compact.

Then,
⋂

x∈K
Γ(x) 6= ∅.

Based on the above Lemma, here we obtain another new version of the above result.

Theorem 2.1. Let K be a nonempty convex subset of a Hausdorff t.v.s. X. Suppose that Γ :

K ⇉ K is a set-valued mapping such that the following conditions are satisfied:

(H1) Γ is a KKM map;

(H2) ∀A ∈ 〈K〉, Γ is transfer closed-valued on conv(A);

(H3) ∀x, y ∈ K, clK (
⋂

u∈[x,y]
Γ(u)) ∩ [x, y] = (

⋂

u∈[x,y]
Γ(u)) ∩ [x, y];

(H4) there exist a nonempty compact convex subset B of K and a nonempty compact subset D of

K such that, for each y ∈ K \ D there exists x ∈ conv(B ∪ {y}) such that y 6∈ Γ(x).

Then,
⋂

x∈K
Γ(x) 6= ∅.

Proof. Suppose that A ∈< K > and LA = conv(A
⋃

B), then LA is compact. Let ΓA : LA ⇉ LA

be defined as ΓA(x) = Γ(x) ∩ LA. Then, from Lemma 2.1, we have

⋂

x∈LA

ΓA(x) 6= ∅.



140 M. Fakhar and J. Zafarani CUBO
10, 4 (2008)

Now, we show that
⋂

x∈LA

ΓA(x) ⊆ D.

Suppose that this claim is not true, then there exists y ∈
⋂

x∈LA
ΓA(x) such that y ∈ K \ D. But

by assumption (H4) there exists x ∈ conv(B ∪ {y}) such that y 6∈ Γ(x). Therefore, x 6∈ LA. But

since y ∈ LA, then conv(B ∪ {y}) ⊆ LA which contradicts (H4).

Assume that

MA =
⋂

x∈LA

Γ(x) for any A ∈< K >, (1)

then

MA ⊆ D for all A ∈< K > . (2)

If M = {MA : A ∈< K >}, then by (1) one can see that the class M has the finite intersection

property. Therefore, from (2), we have
⋂

A∈<K>
clK MA 6= ∅.

If x̄ ∈
⋂

A∈<K>
clK MA, x ∈ X and A0 = {x̄, x}, then conv(A0) = [x̄, x] and

x̄ ∈ clK MA0 = clK





⋂

u∈LA0

Γ(u)



 ⊆ clK





⋂

u∈[x̄,x]

Γ(u)





Hence, by condition (H3)

x̄ ∈ clK





⋂

u∈[x̄,x]

Γ(u)



 ∩ [x̄, x] =





⋂

u∈[x̄,x]

Γ(u)



 ∩ [x̄, x].

Therefore, x̄ ∈ Γ(x) for all x ∈ X and the proof is complete.

Remark 2.2. (a) By a similar proof as that of the above theorem, we can obtain some other

versions of Fan’s KKM Theorem.

Let K0 be a nonempty subset of K and Γ : K0 ⇉ K satisfying the following conditions:

(i) Γ is a KKM map,

(ii) for each A ∈ F(K0), Γ : A ⇉ conv(A) is transfer closed valued,

(iii) for each x, y ∈ K0 ,

clK





⋂

u∈[x,y]∩K0

Γ(u)



 ∩ [x, y] =





⋂

u∈[x,y]∩K0

Γ(u)



 ∩ [x, y],

(iv) there exists a nonempty convex compact subset B of K such that for each y ∈ K \ B there

exists x ∈ conv(B ∪ {y}) ∩ K0 such that y 6∈ Γ(x).



CUBO
10, 4 (2008)

A New Version of Fan’s Theorem and its Applications 141

Then,
⋂

x∈K0
Γ(x) 6= ∅.

(b) Instead of assumptions (ii) and (iii) in part (a) we can assume that Γ is transfer closed

valued. Furthermore, in this case, condition (iv) of part (a) can replaced by the following condition:

(iv)′ there exist a nonempty compact convex subset B of K and a nonempty compact subset

D of K such that, for each y ∈ K \ D there exists x ∈ conv(B ∪ {y}) ∩ K0 such that y 6∈ Γ(x).

3 Fixed point Theorems

In this section, we deduce slight generalizations of known fixed point theorems from Theorem 2.1.

Theorem 3.1. Let K be a nonempty convex subset of a (t.v.s.) X and S : K ⇉ K a set-valued

map such that:

(i) for each A ∈< K >, S− is transfer open valued on conv(A);

(ii) for each x, y ∈ K;

int





⋃

z∈[x,y]

S−(z)



 ∩ [x, y] =





⋃

z∈[x,y]

S−(z)



 ∩ [x, y];

(iii) there exist a nonempty convex compact subset B of K and a nonempty compact subset D of

K such that, for each y ∈ K \ D there exists x ∈ conv(B ∪ {y}) such that x ∈ S(y).

Then, either S has a maximal element or convS has a fixed point.

Proof. Suppose that S has no maximal element, then
⋃

y∈X
S−(y) = K. If Γ(x) = K \ S−(x) for

all x ∈ X, then
⋂

x∈K

Γ(x) = ∅.

Therefore, one of the assumptions of Theorem 2.1 does not hold for Γ. By condition (i), Γ is

transfer closed-valued on convA for any A ∈< K >. Now suppose that x, y ∈ K, z ∈ [x, y]

and z 6∈ ∩u∈[x,y]Γ(u). Then z ∈ (
⋃

u∈[x,y]
S−(u)) ∩ [x, y] and so z ∈ int(

⋃

u∈[x,y]
S−(u)) ∩ [x, y].

Therefore, there exists an open neighborhood U of z in K such that U ⊆
⋃

u∈[x,y]
S−(u). Hence,

U
⋂

(∩u∈[x,y]Γ(u)) = ∅.

That is z 6∈ clK
(

⋂

u∈[x,y]
Γ(u)

)

∩[x, y] and so condition (H3) is satisfied. Also condition (iii) implies

condition (H4). Therefore, Γ is not KKM map. Thus, there exists a finite subset A = {x1, ..., xn}

of K such that

conv(A) *

n
⋃

i=1

Γ(xi).



142 M. Fakhar and J. Zafarani CUBO
10, 4 (2008)

This implies that there is a point x ∈ conv(A) such that x ∈ S−(xi) for all i = 1, ..., n. Therefore,

for all i = 1, ..., n, xi ∈ S(x) and x ∈ convS(x).

Remark 3.2. When K is compact, then trivially condition (iii) holds. Furthermore, if S− is

transfer open valued on K, then we can replace S−(z) in condition (ii) by intS−(z). Hence, in this

case conditions (ii) and (iii) are fulfilled.

Now we deduce the following version of Theorem 1.2 in Ansari and Yao [1] as a corollary of

our Theorem 3.1.

Corollary 3.3. Let K be a nonempty convex subset of a Hausdorff topological vector space X.

Suppose that S, T : K ⇉ K are two set-valued maps with nonempty values such that

(a) for each x ∈ K, A ∈ 〈S(x)〉, conv(A) ⊂ T (x);

(b) for each A ∈< K >, S transfer open valued on conv(A);

(c) for each x, y ∈ K,

int





⋃

z∈[x,y]

S
−
(z)



 ∩ [x, y] =





⋃

z∈[x,y]

S
−

(z)



 ∩ [x, y];

(e) there exist a nonempty convex compact subset B of K and a nonempty compact subset D

of K such that, for each y ∈ K \ D there exists x ∈ conv(B ∪ {y}) such that x ∈ S(y).

Then T has a fixed point.

From Theorem 3.1, we also deduce Theorem 1.1 of Djafari-Rouhani, Tarafdar and Watson [6].

Corollary 3.4. Let K be a nonempty convex subset of a Hausdorff topological vector space X.

Suppose that S, T : K ⇉ K are two set-valued maps with nonempty values that

(a) for each x ∈ K, A ∈ 〈S(x)〉, conv(A) ⊂ T (x).

(b) for each y ∈ K, S−(y) contains an open set Oy, which may be empty such that K = ∪{Oy :

y ∈ K}

(c) there exist a nonempty convex compact subset B of K and a points {x̂1, x̂2, ..., x̂n} in K such

that
⋂

x∈B

Ocx ⊆
n
⋃

i=1

Ox̂i ,

where Ocx is the complement of Ox in K. Then T has a fixed point.

Proof. From (b-c), we obtain that

⋂

x∈B

{K \ S−(x)} ⊆
⋂

x∈B

Ocx ⊆
n
⋃

i=1

Ox̂i ⊆
n
⋃

i=1

S−(x̂i).



CUBO
10, 4 (2008)

A New Version of Fan’s Theorem and its Applications 143

Let C = B ∪ {x̂1, x̂2, ..., x̂n}. Then

K =
⋃

x∈C

S−(x).

Let H = conv(C), then H is compact and convex and moreover,

H =
⋃

x∈C

S−(x) ∩ H ⊆
⋃

x∈H

S−(x).

Now we define the set-valued mapping Γ : H ⇉ H as

Γ(x) = H \ S−(x).

Then from Remark 3.2, we conclude the proof.

As another consequence of Theorem 3.1, we obtain the following fixed point theorem of Ansari

and Lin [2].

Corollary 3.5. Let K be a nonempty convex subset of a Hausdorff topological vector space X.

Suppose that S, T : K ⇉ K are two mutivalued maps such that

(a) for each x ∈ K, A ∈ 〈S(x)〉, conv(A) ⊂ T (x);

(b) there exist a nonempty convex compact subset B of K and a points {x̂1, x̂2, ..., x̂n} in K

such that
⋂

x∈B

(K \ intK S
−

(x)) ⊆
n
⋃

i=1

intK S
−

(x̂i).

Then T has a fixed point.

Proof. It is enough in the above corollary to set Ox = intK S
−

(x) for all x ∈ K.

Remark 3.6. By the same argument as in the above corollary,one can also obtain a proof for

Theorem 2.1 in [2].

4 Equilibrium Problems

We now give some new applications of Theorem 2.1 in obtaining existence results of equilibrium

problem.



144 M. Fakhar and J. Zafarani CUBO
10, 4 (2008)

Theorem 4.1. Let K be a nonempty convex subset of a Hausdorff (t.v.s.) X. Suppose f is a

pseudomomtone real-valued on K × K such that:

(A1) f (x, x) = 0 for any x ∈ X;

(A2) for each x, y, z ∈ X if f (x, y) < 0 and f (x, z) ≤ 0, then f (x, u) < 0 for all u ∈]y, z[;

(A3) for each A ∈< K >, f is transfer l.s.c. in the second variable on conv(A);

(A4) f is 0-segmentary closed;

(A5) there exist a nonempty compact subset D ⊆ K and a nonempty convex compact subset

B of K such that for each x ∈ K \ D, there exists y ∈ conv(B ∪ {x}) such that f (x, y) > 0.

Then, there exists x̄ ∈ X such that f (y, x̄) ≤ 0 for all y ∈ X.

Proof. Assume that Γ̂ : K ⇉ K, Γ : K ⇉ K are defined by:

Γ̂(y) = {x ∈ X : f (x, y) ≥ 0},

Γ(y) = {x ∈ X : f (y, x) ≤ 0}.

Then, as f is pseudomonotone, Γ̂(y) ⊆ Γ(y) for all y ∈ K. By (A1) and (A2) Γ̂ is a KKM map,

so Γ is a KKM map. Condition (A5) implies condition (H4). Therefore, the conditions (H1) and

(H4) are fulfilled by the set-valued map Γ. The condition (A3) implies that the condition (H2)

holds for Γ. For condition (H3), suppose

z ∈ clK





⋂

u∈[x,y]

Γ(u)



 ∩ [x, y].

Then, there exists a net (zα) converging to z such that f (u, zα) ≤ 0 for all u ∈ [x, y]. Since

z ∈ [x, y], for each v ∈ [u, z], we have also f (v, zα) ≤ 0, hence by (A4), we obtain f (u, z) ≤ 0 for

each u ∈ [x, y]. Thus, we have

z ∈





⋂

u∈[x,y]

Γ(u)



 ∩ [x, y].

Hence Γ satisfies also the condition (H3). Therefore, from Theorem 2.1, we have

⋂

y∈X

Γ(y) 6= ∅.

Thus, any point x̂ in this intersection is a solution for our problem.

Corollary 4.2. In Theorem 4.1, if f (., y) is upper sign continuous for every y ∈ K, then there

exists x̄ ∈ K such that f (x, x̄) ≥ 0 for all x ∈ K.

Proof. By Theorem 4.1, suppose that x̄ ∈ K such that f (y, x̄) ≤ 0 for all y ∈ X. Assume that

there exists ȳ ∈ K such that f (x̄, ȳ) < 0. By our assumption on x̄ we have also f (ȳ, x̄) ≤ 0. We

will show that f (u, ȳ) ≥ 0 for all u ∈ ]x̄, ȳ[. Indeed, if f (u, ȳ) < 0 for some u ∈ ]x̄, ȳ[, then as



CUBO
10, 4 (2008)

A New Version of Fan’s Theorem and its Applications 145

f (u, x̄) ≤ 0, we obtain from (A2) that f (u, u) < 0, which contradicts (A1). Now by upper sign

continuity of f, we have f (x̄, ȳ) ≥ 0, which is a contradiction.

The following example shows that Corollary 4.2 improves the corresponding results in [3] and

[13].

Example 4.3. Assume that K = R and f : K × K → R is defined as follows:

f (x, y) :=



































−y if x = 0,

1 − y if x = 1,

1 if x = 3
2
, 5 > |y| ≥ 3,

−2 + y if x = 2,

0 otherwise.

Let D = [−1, 2] and B = [1, 2]. For y < −1, we have f (1, y) > 0. If y > 2, then f (2, y) > 0.

Therefore, f satisfies all of the condition of Corollary 4.2. But for x = 3/2, the set

{y ∈ R : f (3/2, y) ≤ 0} =] − 3, 3[∪] − ∞, −5[∪]5, ∞[,

which is not convex and K is not compact.

As a consequence of our results, we conclude a new version of Theorem 15 in [12] and its

corollary for existence of variational inequalities problem.

Corollary 4.3. Let K be a nonempty convex subset of a Hausdorff (t.v.s.) X and T : K ⇉ X∗.

Suppose that

(i) T is upper semicontinuous from conv(A) of any A ∈< K > to X∗ endowed with w*-topology

and for each x ∈ K, T (x) is convex w*-compact;

(ii) f (x, y) := infy∗∈T (y)〈y
∗, y − x〉 is 0-segmentary closed;

(iii) there exist a nonempty compact subset D and a nonempty convex compact subset B of K

such that, for each y ∈ K\D, there exists x ∈ conv(B ∪ {y}) such that

inf
y∗∈T (y)

〈y∗, y − x〉 > 0.

Then, there exist ȳ ∈ K and y∗
0
∈ T (ȳ) such that 〈y∗

0
, x − ȳ〉 ≥ 0, ∀x ∈ K.

Proof. Let

f (x, y) = inf
y∗∈T (y)

〈y∗, y − x〉.



146 M. Fakhar and J. Zafarani CUBO
10, 4 (2008)

We will show that all of the conditions of Theorem 4.1 and its corollary are fulfilled by f . Lemma

2.2 in [7] implies that for each fixed x ∈ K, the function y → infy∗∈T (y)〈y
∗, y−x〉 is l.s.c. on conv(A)

of any A ∈< K >, hence we trivially have condition(A3) of Theorem 4.1. Since f (x, x) = 0, for

every x ∈ Kand f is affine in the first argument, hence f satisfies conditions (A1) and (A2).

Trivially f (., y) is upper sign continuous, thus from Corollary 4.2, there exists ȳ ∈ K such that

inf
y∗∈T (ȳ)

〈y∗, ȳ − x〉 ≤ 0, ∀x ∈ K.

Now, let g : K × T (ȳ) → R be defined as follows:

g(x, y∗) = 〈y∗, ȳ − x〉,

then since T (ȳ) is convex, by Kneser’s minimax theorem, we have

inf
y∗∈T (ȳ)

sup
x∈K

〈y∗, ȳ − x〉 = sup
x∈K

inf
y∗∈T (ȳ)

〈y∗, ȳ − x〉 ≤ 0.

Therefore, there exists a point y∗0 ∈ T (ȳ) such that

sup
x∈K

〈y∗0 , ȳ − x〉 ≤ 0.

Acknowledgment. The authors were partially supported by the Center of Excellence for Mathe-

matics (University of Isfahan). The first author was partially supported by a grant from IPM (No.

86470016).

Received: May 2008. Revised: May 2008.

References

[1] Ansari, Q.H. and Yao, J.C., A fixed point theorem and its applications to a system of

variational inequalities, Bull. Austral. Math. Soc., 59 (1999), no. 3, 433–442.

[2] Ansari, Q.H. and Lin, Y.C., Fixed point and maximal element theorems with application

to abstract economy, Nonlinear Stud., 13 (2006), no. 1, 43–52.

[3] Bianchi, M. and Pini,R., Coercivity conditions for equilibrium problems, J. Optim. Theory

Appl., 124 (2005), 79–92.

[4] Brézis, H., Nirenberg, L. and Stampacchia, G., A Remark on Ky Fan’s minimax

principle, Bolletino della Unione Matematica Italiana, 6 (1972), 293–300.



CUBO
10, 4 (2008)

A New Version of Fan’s Theorem and its Applications 147

[5] Chowdhury, M.S.R. and Tan, K.K., Generalization of Ky Fan’s minimax inequality with

applications to generalized variational inequalities for pseudomonotone operators and fixed-

point theorem, J. Math. Anal. Appl., 204 (1996), 910–929.

[6] Djafari Rouhani, B., Tarafdar, E. and Watson, P.J., Existence of solutions to some

equilibrium problems, J. Optim. Theory Appl., 126 (2005), 97–107.

[7] Ding, X.P. and Tarafdar, E., Generalized variational-like inequalities with pseudomono-

tone set-valued mappings, Archiv. Math., 74 (2000), 302–313.

[8] Fakhar, M. and Zafarani, J., Generalized equilibrium problems for quasimonotone and

pseudomonotone bifunctions, J. Optim. Theory Appl., 123 (2004), 349–364.

[9] Fakhar, M. and Zafarani, J., Generalized vector equilibrium problems for pseudomono-

tone multi-valued bifunctions, J. Optim. Theory Appl., 126 (2005), 109–124.

[10] Fakhar, M. and Zafarani, J., On generalized variational inequalities, to appear in J.

Global Optim.

[11] Fan, K., A Generalization of Tychonoff’s fixed-point theorem, Math. Ann., 142 (1961), 305–

310.

[12] Inoan, D. and Kolumban, J., On pseudomonotone set-valued mappings, Nonlinear Anal.,

68 (2008), 47–53.

[13] Iusem, A.N. and Sosa, W., New existence results for equilibrium problems, Nonlinear.

Anal., 52 (2003), 621–635.

[14] Kalmoun, E.M., On Ky Fan’s minimax inequalities, mixed equilibrium problems and hemi-

variational inequalities, JIPAM. J. Inequal. Pure Appl. Math., 2 (2001), 1–13.

[15] Knaster, B., Kuratowski, C. and Mazurkiewicz, S., Ein Bewies des Fixpunktsatzes

für n-dimensional simplexe, Fund. Math., 14 (1929), 132–137.


	N11-Fans-FaZa