a topological characterization of the existence of fixed points and applications

Math. Nachr. 1–9 (2013) / DOI 10.1002/mana.201300003

A topological characterization of the existence of fixed pointsand applications

Phan Quoc Khanh∗1 and Nguyen Hong Quan∗∗2

1 Department of Mathematics, International University of Hochiminh City, Linh Trung, Thu Duc, HochiminhCity, Vietnam

2 Department of Mathematics and Computer Science, University of Science Hochiminh City, 227 Nguyen VanCu Str., Dist.5, Hochiminh City, Vietnam

Received 9 January 2013, revised 16 June 2013, accepted 16 June 2013Published online 6 September 2013

Key words Fixed-points, fixed-component points, coincidence points, KKM-structures, KKM-maps, coerciveconditions, optimization-related problems, Nash equilibriumMSC (2010) 54H25, 91A06, 91A10, 49J53

We prove a topological two-way characterization of the existence of fixed-points, without using linear orconvexity structures and provide applications in optimization-related problems. Such a characterization is alsodemonstrated for a fixed-component point, a slight generalization of a fixed point.

C© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

Fixed-point theorems are the most important and traditional tools for establishing the existence of solutions inmost problems in both pure and applied mathematics (they are even employed directly in other fields of science,technology, and economics). Such theorems inspired and motivated developments of many other important kindsof points like coincidence points, intersection points, sectional points, etc. During a long period in the past,it was believed that the proofs of existence results for all such points required both topological and algebraicmachineries. But, originated from Wu [19] and Horvarth [6], two corresponding directions of dealing with puretopological existence theorems have been intensively developed. The first approach is based on replacing convexityassumptions by connectedness conditions, and the second one on replacing a convex hull by an image of a simplexthrough a continuous map. In this paper, we follow the second direction. There have been a number of existenceresults for various kinds of points and applications, using this idea of replacing convex hulls by such continuousmaps. see, e.g., [3], [4], [7], [8], [10], [14], [15]. Observe that only sufficient conditions for existence wereconsidered in these papers. Of course, in applications, such conditions are usually enough. But, a necessary andsufficient condition (i.e., a full characterization) is clearly the weakest sufficient condition and hence may havemore applications. Inspired by a recent result [9], we develop here a topological (full) characterization of theexistence of fixed-points and applications in optimization-related problems.

The layout of the paper is as follows. A few preliminary facts are included in the rest of this section. Section 2is devoted to fixed points and applications, including also discussions on some other important points. In Section 3we extend the result in Section 2 to a fixed-component point, a slight generalization of a fixed-point, which isshown convenient in applications as well.

Throughout this paper, for a nonempty set X , 〈X〉 stands for the set of all finite subsets of X . ForN = {x0, x1, . . . , xn} ∈ 〈X〉 and M = {xi0 , xi1 , . . . , xik } ⊂ N , �|N | := �n stands for the n-standard simplexof Euclidean space R

n+1 with vertices being unit vectors e0 = (1, 0, . . . , 0), e1 = (0, 1, 0, . . . , 0), . . . , en =(0, 0, . . . , 1), �M denotes the face of �|N | with vertices ei0 , ei1 , . . . , eik . Let H : X ⇒ Y be a set-valued mapbetween nonempty sets X and Y . For x ∈ X and y ∈ Y , an image, a fiber (or inverse image), and a cofiber of His the set H(x), H−1(y) = {x ∈ X | y ∈ H(x)}, and H∗(y) = X \ H−1(y), respectively (shortly, resp).

∗ Corresponding author: e-mail: [email protected], Phone: +84 913917147, Fax: +84 837244271∗∗ e-mail: [email protected], Phone: +84 988942043

C© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

mailto:[email protected]

mailto:[email protected]

2 P. Q. Khanh and N. H. Quan: A topological characterization of the existence of fixed points and applications

Definition 1.1 ([9]) For nonempty sets X and Y , a pair F = (�X ,�Y ) is called a KKM-structure of the pair(X, Y ) if �Y is a topology on Y and �X = {ϕN : �|N | → Y | N ∈ 〈X〉} is a family of maps with all ϕN ∈ �X

being �Y -continuous. In the special case where X = Y , such a F is termed a KKM-structure of X . If �Y iscompact, i.e., Y is �Y -compact, (�X ,�Y ) is called a compact KKM-structure.

If X = Y is a convex subset of a topological vector space E , �X is the topology on X induced by that of E ,and

�X ={

ϕN : �|N | → X | ϕN (e) =∑xi ∈N

λi xi for e =∑

λi ei ∈ �|N |, N ∈ 〈X〉}

,

then the KKM-structure F = (�X ,�X ) is called the natural KKM-structure of X .For i ∈ I (an arbitrary index set), let Xi be a convex subset of a topological vector space Ei , X = ∏

i∈I Xi , �X

the product topology of all topologies �Xi (induced by that of Ei ), and �X the family of maps ϕN : �|N | → X ,

N ∈ 〈X〉 defined by ϕN (e) =(∑n

�=0 λ�x�i

)i∈I

for N = {(x�

i

)i∈I | � = 0, 1, . . . , n

} ∈ 〈X〉 and e = ∑n�=0 λ�e� ∈

�|N |. Then, F = (�X ,�X ) is a KKM- structure and called the natural product KKM-structure of X .

Definition 1.2 ([9]) A set-valued map T : X ⇒ Y between two sets X and Y is called a generalized KKM-map(shortly, GKKM-map) if there exists a KKM-structure F = (�X ,�Y ) of (X, Y ) such that

(i) for all ϕN ∈ �X and M ⊂ N , ϕN (�M) ⊂ ⋃x∈M T (x);

(ii) the values of T are �Y -closed;(iii) there is a finite intersection of values of T which is �Y -compact.

By GKKM(X, Y ) we denote the class of all GKKM-maps of (X, Y ).

Theorem 1.3 ([9]) Let X, Y be nonempty sets and T : X ⇒ Y be a set-valued map. Then,⋂

x∈X T (x) = ∅ ifand only if T is a GKKM-map.

2 Fixed-point theorems

Definition 2.1 Let X be a nonempty set, (�X ,�X ) a KKM-structure of X , and P, Q : X ⇒ X set-valued maps.

(i) Q is said to be �X -convex with respect to (shortly, w.r.t.) P if, for all x ∈ X , ϕN ∈ �X and M ⊂ N ∩ P(x),ϕN (�M) ⊂ Q(x).

(ii) Q is called �X -weak-convex w.r.t. P if, whenever ϕN ∈ �X , M ⊂ N ∩ P(x), and x ∈ ϕN (�M), one hasϕN (�M) ⊂ Q(x).

Clearly, if Q is �X -convex w.r.t. P , then Q is �X -weak-convex w.r.t. P . The reverse is not true.

Example 2.2 Let X = [0, 2] and F = (�X ,�X ) be the natural KKM-structure of X . P(x) = [0, x

3

] ∪ (2x3 , x

)if x ∈ [0, 1) and P(x) = {0} if x ∈ [1, 2], Q(x) = [0, x ] ∪ (1 + x, 2) if x ∈ [0, 1) and Q(x) = {0, 1} if x ∈ [1, 2].We easily see that Q is �X -convex w.r.t. P .

Example 2.3 Let X , FX , and P be as in Example 2.2 and Q(x) = {0, 1} for all x ∈ [0, 2]. For all N ∈ 〈X〉,M ⊂ N , and x ∈ X , we have the following equivalent statements{

x ∈ ϕN (�M),M ⊂ N ∩ P(x) ⇐⇒

{x ∈ coM,

M ⊂ N ∩ P(x)

⇐⇒{

x ∈ [min M, max M ], x ∈ [0, 1),M ⊂ N ∩ ([

0, x3

] ∪ (2x3 , x

)) or

{x ∈ [min M, max M ],x ∈ [1, 2], M ⊂ N ∩ {0}

⇐⇒{

x = 0,

M = {0} ⊂ N .

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Math. Nachr. (2013) / www.mn-journal.com 3

Clearly, ϕN (�{0}) = co{0} ⊂ Q(0). Thus, Q is �X -weak-convex w.r.t. P . Note that Q is not �X -convexw.r.t. P .

Definition 2.4 Let X be a nonempty set and (�X ,�X ) a KKM-structure of X . We say that a subset B of Xis �X -convex if, for all ϕN ∈ �X and M ⊂ N ∩ B, ϕN (�M) ⊂ B. In case X is a topological vector space and(�X ,�X ) is the natural KKM-structure of X , the set B is �X -convex if and only if it is convex. For C ⊂ X , thesmallest �X -convex set containing C , denoted by �X -coC , is called �X -convex hull of C . It is not hard to checkthat �X -coC = ⋃

N∈〈C〉 �X -coN .

Coercive conditions. Let X be a nonempty set, P : X ⇒ X , and F = (�X ,�X ) be a KKM-structure of X .The following conditions are kinds of coercive conditions for P .

(F1) There exist N ∈ 〈X〉 and a nonempty �X -compact subset K of X such that X \ ⋃x∈N int�X P−1(x)

⊂ K .(F2) There exists a nonempty �X -compact subset K of X such that, for each N ∈ 〈X〉, there exists a �X -compact

and a �X -convex subset L N ⊂ X containing N such that L N ∩ (X \ ⋃

x∈L Nint�X P−1(x)

) ⊂ K .

If the fibers of P are �X -open, then conditions (F1) and (F2) are rewritten as follows.

(F1’) There exist N ∈ 〈X〉 and a nonempty �X -compact subset K of X such that, for each x ∈ X\K , there is any ∈ N with y ∈ P(x).

(F2’) There exists a nonempty �X -compact subset K of X such that, for each N ∈ 〈X〉, there exists a �X -compactand �X -convex subset L N ⊂ X containing N such that L N \K ⊂ ⋃

x∈L NP−1(x).

Theorem 2.5 Let X be a nonempty set and Q : X ⇒ X. Then, Q has a fixed point if and only if there exista KKM-structure F = (�X ,�X ) of X and P : X ⇒ X such that X = ⋃

x∈X int�X P−1(x), Q is �X -weak-convexw.r.t. P, and either of the conditions (F1) and (F2) holds for P.

P r o o f. Necessity. Assume that x is a fixed point of Q. Let P : X ⇒ X be defined by P(x) = {x} for allx ∈ X , and a KKM-structure F = (�X ,�X ) of X be defined as follows: �X = {U ⊂ X | x /∈ U } ∪ {X} and�X = {

ϕN : �|N | → X | ϕN (e) = x for all e ∈ �|N |, N ∈ 〈X〉}. Then, it is not hard to see that the conditionsstated in Theorem 2.5 are satisfied.

Sufficiency. Case 1: Condition (F1) holds. We define T : X ⇒ X by T (x) = X \ int�X P−1(x). Then, Thas �X -closed values as (ii) of Definition 1.2 requires. Condition (F1) implies that

⋂x∈N T (x) ⊂ K . Hence,

clearly⋂

x∈N T (x) is compact, i.e., condition (iii) of Definition 1.2 for T is fulfilled. Because⋂

x∈X T (x) =X \ ⋃

x∈X int�X P−1(x) = ∅, Theorem 1.3 yields that assumption (i) of this definition must be false, i.e., thereexist ϕN ∈ �X and M ⊂ N such that ϕN (�M) ⊂ ⋃

x∈M T (x) = X \ ⋂x∈M int�X P−1(x). Therefore, there is

a ∈ ϕN (�M) with a ∈ ⋂x∈M int�X P−1(x) ⊂ ⋂

x∈M P−1(x). Then, a ∈ ϕN (�M) and M ⊂ P(a). Since Q is�X -weak-convex w.r.t. P , a ∈ ϕN (�M) ⊂ Q(a). Thus, a is a fixed point of Q.

Case 2: Condition (F2) holds. Let K be given in the coercive condition (F2). Wehave K ⊂ X = ⋃

x∈X int�X P−1(x). Since K is �X -compact, one finds N ∈ 〈X〉 such that K ⊂⋃x∈N int�X P−1(x) ⊂ ⋃

x∈L Nint�X P−1(x). Then, by (F2), L N = (L N \ K ) ∪ K ⊂ ⋃

x∈L Nint�X P−1(x). Hence,

L N = ⋃x∈L N

(int�X P−1(x) ∩ L N

). Let �L N be the topology induced by �X on L N and �L N := {ϕM ∈

�X | M ∈ 〈L N 〉}. Then, (�L N ,�L N ) is a KKM-structure of L N . Let Q′, P ′ : L N ⇒ L N be defined, resp, byQ′(a) = Q(a) ∩ L N and P ′(a) = P(a) ∩ L N for all a ∈ L N . We check the �L N -weak-convexity of Q′ w.r.t.P ′. Let ϕN ′ ∈ �L N , M ′ ⊂ N ′ ∩ P ′(a), and a ∈ ϕN ′(�M ′). Since M ′ ⊂ N ′ ⊂ L N , ϕN ′ ∈ �L N ⊂ �X , and L N is�X -convex, we have ϕN ′(�M ′) ⊂ L N . Moreover, M ′ ⊂ N ′ ∩ P ′(a) = N ′ ∩ P(a) ∩ L N ⊂ N ′ ∩ P(a). Hence,from the �X -weak-convexity of Q w.r.t. P , we have ϕN ′(�M ′) ⊂ Q(a). Therefore,

ϕN ′(�M ′) ⊂ Q(a) ∩ L N = Q′(a).

Thus, Q′ is �L N -weak-convex w.r.t. P ′. Now, by the same argument as in Case 1 with L N , (�L N ,�L N ), Q′, andP ′ replacing X , (�X ,�X ), Q and P , resp, we have a ∈ L N such that a ∈ Q(a) ∩ L N ⊂ Q(a). �

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Remark 2.6 (i) Let us consider the following conditions.

(i1) Q is �X -weak-convex w.r.t. P .(i2) Q is �X -convex w.r.t. P .(i3) �X -coP(x) ⊂ Q(x) for each x ∈ X .(i4) P(x) ⊂ Q(x) and Q(x) is �X -convex for each x ∈ X .

Obviously, (i2) ⇒ (i1). One also has (i3) ⇒ (i1). Indeed, for all x ∈ X , ϕN ∈ �X , and M ⊂ N ∩ P(x), wehave ϕN (�M) ⊂ �X -coP(x) ⊂ Q(x), i.e., Q is �X -weak-convex w.r.t. P . Furthermore, (i4) ⇒ (i1). Indeed,for all x ∈ X , ϕN ∈ �X and M ⊂ N ∩ P(x), we have M ⊂ P(x) ⊂ Q(x). Since Q(x) is �X -convex, one hasϕN (�M) ⊂ Q(x). Thus, Q is �X -weak-convex w.r.t. P .

Observe that, in Theorem 2.5 and Remark 2.6 (i2)–(i4), the condition X = ⋃x∈X int�X P−1(x) is satisfied if

P has nonempty values and �X -open fibers. Furthermore, if in Remark 2.6 (i1)–(i4), P and F = (�X ,�X ) aregiven, then (i4) ⇒ (i3) ⇒ (i2)⇒ (i1).

(ii) If we replace the conditions “X = ⋃x∈X int�X P−1(x), Q is �X -weak-convex w.r.t. P , and either of the

conditions (F1) and (F2) holds for P” of Theorem 2.5 by “X = ⋃x∈X int�X Q−1(x), and either of the conditions

(F1) and (F2) holds for Q”, then �X -coQ(·) has a fixed point. Indeed, applying the statement in (i4) with P andQ replaced by Q and �X -coQ(·), resp, we are done.

The following example illustrates Theorem 2.5 and Remark 2.6 (i2).

Example 2.7 Let X , F , P , and Q be as in Example 2.2. For x ∈ X , we have

P−1(x) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

[0, 2] if x = 0,

(x, 1) if 0 < x < 13 ,(

x, 3x2

)if 1

3 ≤ x ≤ 23 ,

∅ if 23 < x ≤ 2,

is open in X . Thus, by Remark (i2), Q has a fixed point. From the formula of Q, we see directly that x = 0 is itsfixed point. Furthermore, by Theorem 2.5, Q in Example 2.3 also has a fixed point, and x = 0 is seen directly aswell to be a fixed point of this Q.

The next example shows that the implications stated by the end of Remark 2.6 (i) are not reverse.

Example 2.8 Let X = [0, 2], F = (�X ,�X ) with �X being the usual topology on [0, 2], �X = {ϕN : �|N | →

X | ϕN (e) = 0 for all e ∈ �|N |, N ∈ 〈X〉}, P(x) = [0, x) if x ∈ (0, 2] and P(x) = {0} if x = 0, Q(x) = {0, x}for all x ∈ [0, 2]. We easily see that the condition (i2) in Remark 2.6 holds, but (i3) does not because �X -coP(x) = [0, x) ⊂ Q(x) for all x ∈ (0, 2].

Now let X and �X be as above, �X = {ϕN : �|N | → X | ϕN (e) = min N+max N

2 for all e ∈ �|N |, N ∈ 〈X〉},P(x) = [

0, x3

)if x ∈ (0, 2] and P(x) = {0} if x = 0, and Q(x) = [

0, x3

] ∪ [2x3 , 2

]for all x ∈ [0, 2]. Since the

values of Q are not �X -convex, the condition (i4) in Remark 2.6 does not hold, while the one in (i3) holds.

Corollary 2.9 Let X, Y be nonempty sets and Q : X ⇒ X. Assume that there exist a KKM-structure F =(�X ,�X ) of X and T, H : X ⇒ Y such that X = ⋃

y∈H(X) int�X T −1(y), {x ′ ∈ X | T (x) ∩ H(x ′) = ∅} ⊂ Q(x),Q(x) is �X -convex for each x ∈ X, and the following condition holds:

(F3) there exists a nonempty �X -compact subset K of X such that, for each N ∈ 〈X〉, there exists a �X -compactand �X -convex subset L N ⊂ X containing N such that L N \ K ⊂ ⋃

x ′∈L Nint�X {x ∈ X | T (x) ∩ H(x ′) =

∅}.

Then, Q has a fixed point.



P r o o f. Apply Theorem 2.5 when (i1) is replaced with (i4) and P(x) = {x ′ ∈ X | T (x) ∩ H(x ′) = ∅} andnote that

X =⋃x ′∈X

⋃y∈H(x ′)

int�X T −1(y) ⊂⋃x ′∈X

int�X

( ⋃y∈H(x ′)

T −1(y)

)=

⋃x ′∈X

int�X P−1(x ′).

�

Remark 2.10 The sufficiency part of Theorem 2.5 together with Remark 2.6 and Corollary 2.9 imply manyknown Browder type fixed-point theorems. For instance, Remark 2.6 (i2) includes Theorem 4 of [18]; (i4) containsTheorem 2.1 of [16], Theorem 1 of [14], Theorem 4.2 of [5], Browder’s fixed-point theorem [2] and Tarafdar’sfixed-point theorem [17]; Remark 2.6 (ii) contains Theorem 1 of [11]; Corollary 2.9 implies Theorem 1 of [1].

2.1 Coincidence points and alternative principles

Let X and Y be nonempty sets, F : X ⇒ Y and G : Y ⇒ X . A coincidence point of F and G is a point(x, y) ∈ X × Y such that x ∈ G(y) and y ∈ F(x).

Definition 2.11 Let X, Y be nonempty sets, (�X×Y ,�X×Y ) a KKM-structure of X × Y , F : X ⇒ Y , G : Y ⇒X , and H : X × Y ⇒ X × Y .

(a) (F, G) is said to be �X×Y -convex w.r.t. H if, for all (x, y) ∈ X × Y , ϕN ∈ �X×Y and M ⊂ N ∩ H(x, y),ϕN (�M) ⊂ G(y) × F(x).

(b) (F, G) is called �X×Y -weak-convex w.r.t. H if, whenever ϕN ∈ �X , M ⊂ N and (x, y) ∈ X × Y satisfying(x, y) ∈ ϕN (�M) and M ⊂ H(x, y), one has ϕN (�M) ⊂ G(y) × F(x).

Let X, Y be nonempty sets, (�X×Y ,�X×Y ) a KKM-structure of X × Y , and H : X × Y ⇒ X × Y . We willuse the following coercive conditions.

(C1) There exist N ∈ 〈X × Y 〉 and a nonempty �X×Y -compact subset K of X × Y such that X × Y \ K ⊂⋃(x,y)∈N int�X×Y H−1(x, y).

(C2) There exists a nonempty �X×Y -compact subset K of X × Y such that, for each N ∈ 〈X × Y 〉, thereexists a �X×Y -compact and �X×Y -convex subset L N ⊂ X × Y containing N such that L N \ K ⊂⋃

(x,y)∈L Nint�X×Y H−1(x, y).

Theorem 2.12 Let X and Y be nonempty sets, F : X ⇒ Y , and G : Y ⇒ X. Then, F and G have a coincidencepoint if and only if there exist a KKM-structure (�X×Y ,�X×Y ) of X × Y and H : X × Y ⇒ X × Y such thatX × Y = ⋃

(x,y)∈X×Y int�X×Y H−1(x, y), (F, G) is �X×Y -weak-convex w.r.t. H, and either of the conditions (C1)and (C2) holds.

P r o o f. Let D : X × Y ⇒ X × Y be defined by D(x, y) := G(y) × F(x) for all (x, y) ∈ X × Y . Then,(x, y) is a coincidence point of F and G if and only if (x, y) ∈ G(y) × F(x) = D(x, y). Replacing X , Q and Pin Theorem 2.5 by X × Y , D and H , resp, we are done. �

Remark 2.13 (i) Each of the following conditions is sufficient for the condition “(F, G) is �X×Y -weak-convexw.r.t. H” in Theorem 2.12.

(i1) �X×Y -coH(x, y) ⊂ G(y) × F(x) for all (x, y) ∈ X × Y (this is clear).(i2) H(x, y) ⊂ G(y) × F(x) and G(y) × F(x) is �X×Y -convex for all (x, y) ∈ X × Y (this implies (i1)).

(ii) In Theorem 2.12 and Remark 2.13 (i), if X and Y are convex subsets of topological vector spaces E1 andE2, resp, then using the natural product KKM-structures we can obtain some interesting results. For instance, wecan deduce from (i2) that, for compact convex sets X and Y , if F : X ⇒ Y and G : Y ⇒ X have open fibers andnonempty, convex values, then they have a coincidence point.



Corollary 2.14 Let X and Y be nonempty sets, F : X ⇒ Y , and G : Y ⇒ X. Assume that there exists a KKM-structure F = (�X ,�X ) of X such that X = ⋃

y∈G−1(X) int�X F−1(y), the set{

x ′ ∈ X | F(x) ∩ G−1(x ′) = ∅}is

�Y -convex for all x ∈ X, and the following condition holds:

(C3) there exists a nonempty �X -compact subset K of X such that, for each N ∈ 〈X〉, there exists a �X -compact and �X -convex subset L N ⊂ X containing N such that L N \ K ⊂ ⋃

x ′∈L Nint�X

{x ∈ X | F(x) ∩

G−1(x ′) = ∅}.

Then, F and G have a coincidence point.

P r o o f. Applying Corollary 2.9 with Q(x) = {x ′ ∈ X | F(x) ∩ G−1(x ′) = ∅}

, T (x) = F(x), and H(x) =G−1(x), we have a point x ∈ Q(x), i.e., F(x) ∩ G−1(x) = ∅. Then, one has y ∈ Y such that y ∈ F(x) ∩ G−1(x).Hence (x, y) is a coincidence point of F and G. �

Corollary 2.15 Let X, Y be nonempty sets and F, G : X ⇒ Y . Assume that there exists a KKM-structureF = (�X ,�X ) of X such that the fibers of F are �X -open, F(x) ⊂ G(x) and the set {x ′ ∈ X | F(x) ⊂ G(x ′)} is�X -convex for each x ∈ X, and the following condition holds:

(C4) there exists a nonempty �X -compact subset K of X such that, for each N ∈ 〈X〉, there exists a �X -compactand �X -convex subset L N ⊂ X containing N such that L N \ K ⊂ ⋃

x ′∈L Nint�X {x ∈ X | F(x) ⊂ G(x ′)}.

Then, at least one of the following assertions holds:

(a) there exists x ∈ X such that F(x) = ∅;(b)

⋂x∈X G(x) = ∅.

P r o o f. We shall apply Theorem 2.5 when P = Q, with Q : X ⇒ X defined by Q(x) = {x ′ ∈ X |F(x) ⊂ G(x ′)}. Suppose to the contrary that, for all x ∈ X , F(x) = ∅ and Y = ⋃

x ′∈X (Y \ G(x ′)). It follows that

X =⋃y∈Y

F−1(y) =⋃x ′∈X

⋃y∈Y\G(x ′)

F−1(y) =⋃x ′∈X

⋃y∈Y\G(x ′)

int�X F−1(y)

⊂⋃x ′∈X

int�X

( ⋃y∈Y\G(x ′)

F−1(y)

)=

⋃x ′∈X

int�A Q−1(x ′).

Thus, from Theorem 2.5, via Remark 2.6 (i4), we have a point x ∈ X such that x ∈ Q(x), i.e., F(x) ⊂ G(x)which contradicts the assumption. �

Corollary 2.16 Let X, Y be nonempty sets and F, G : X ⇒ Y . Assume that there exists a KKM-structure F =(�X ,�X ) of X such that the fibers of F are �X -open, F(x) ∩ G(x) = ∅ and the set {x ′ ∈ X | F(x) ∩ G(x ′) = ∅}is �X -convex for each x ∈ X, and the following condition holds:

(C5) there exists a nonempty �X -compact subset K of X such that, for each N ∈ 〈X〉, there exists a �X -compactand �X -convex subset L N ⊂ X containing N such that L N \ K ⊂ ⋃

x ′∈L Nint�X {x ∈ X | F(x) ∩ G(x ′) =

∅}.

Then, at least one of the following assertions holds:

(a) there exists x ∈ X such that F(x) = ∅;(b) there exists y ∈ Y such that G−1(y) = ∅.

P r o o f. Apply Corollary 2.15 when the set-valued mapping G is replaced by its complementary, Gc(x) =Y \ G(x) for all x ∈ X . �



2.2 Applications to optimization

For a nonempty set X and f : X → R, we consider a general constrained minimization problem:(P) Find x ∈ X such that f (x) = minx∈X f (x).

Theorem 2.17 The constrained minimization problem (P) has a solution if and only if there exist a KKM-structure F = (�X ,�X ) of X such that the following conditions hold:

(i) for any N ∈ 〈X〉, M ⊂ N, and x ∈ X, one has: [∃y ∈ ϕN (�M), f (y) ≥ f (x)] implies [x /∈ ϕN (�M) or∃z ∈ M: f (z) ≥ f (x)];

(ii) the set {x ∈ X : f (y) < f (x)} is �X -open for each y ∈ X;(iii) there exists a nonempty �X -compact subset K of X such that, for each N ∈ 〈X〉, there exists a �X -compact

and �X -convex subset L N ⊂ X containing N such that L N ∩ ⋂y∈L N

{x ∈ X : f (y) ≥ f (x)} ⊂ K .

P r o o f. Necessity. If x ∈ X is a solution, then we easily check that the KKM-structure F = (�X ,�X ) definedin the proof of Theorem 2.5 satisfies all (i)–(iii).

Sufficiency. Reasoning by contraposition, suppose for each x ∈ X , there exists y ∈ X such that f (y) < f (x).We define Q : X ⇒ X by Q(x) = {y ∈ X : f (y) < f (x)} for all x ∈ X . Then, Q has nonempty values, andQ−1(y) = {x ∈ X : f (y) < f (x)} is �X -open for each y ∈ X . Hence X = ⋃

y∈X int�X Q−1(y). By (i), wheneverϕN ∈ �X , M ⊂ N ∩ Q(x), and x ∈ ϕN (�M), one has ϕN (�M) ⊂ Q(x), i.e, Q is �X -weak-convex w.r.t. Q.For K , N , and L N given in (iii), we have L N ∩ (

X \ ⋃y∈L N

int�X Q−1(y)) = L N ∩ (

X \ ⋃y∈L N

{x ∈ X : f (y) <

f (x)}) = L N ∩ ⋂y∈L N

{x ∈ X : f (y) ≥ f (x)} ⊂ K . Thus, applying Theorem 2.5 under (F2) with P = Q wehave a point x ∈ X such that x ∈ Q(x) = {y ∈ X : f (y) < f (x)}, i.e., f (x) < f (x), a contradiction. �

Corollary 2.18 Let X be nonempty convex subset of a topological vector space and f : X → R be given.Assume that

(i) for each N ∈ 〈X〉 and x ∈ X, if f (y) ≥ f (x) for some y ∈ coN then x /∈ coN or ∃z ∈ N, f (z) ≥ f (x);(ii) for each y ∈ X, the set {x ∈ X : f (y) < f (x)} is open;

(iii) there exists a nonempty compact subset K of X satisfying that, for each N ∈ 〈X〉, there exists a compactand convex subset L N ⊂ X containing N such that L N ∩ ⋂

y∈L N{x ∈ X : f (y) ≥ f (x)} ⊂ K .

Then, problem (P) has a solution.

P r o o f. Employ Theorem 2.17 with the KKM-structure F = (�X ,�X ) being the natural KKM-structureof X . �

Observe that if X is compact and f is continuous, the conditions (i)–(iii) are clearly satisfied, and hence wereobtained the classical result.

3 Fixed-component theorems

We define a product KKM-structure as follows. Let I be a finite index set, Xi be nonempty sets andFi = (�Xi ,�Xi )be a KKM-structure on Xi . Let X = ∏

i∈I Xi , �X be the Tikhonov product topology of topologies �Xi onX . Let N ∈ 〈X〉 with N = {(

x1i

)i∈I ,

(x2

i

)i∈I , . . . ,

(xn

i

)i∈I

}. We define the “components” Ni of N by Ni ={

x1i , x2

i , . . . , xni

} ∈ 〈Xi 〉 and denote N = ⊗i∈I Ni . Each element x = (xi )i∈I ∈ X is also denoted by x = ⊗

i∈I xi .Let

�X ={

ϕN : �|N | → X | ϕN (e) =⊗i∈I

ϕNi (e) for e ∈ �|N |, N =⊗i∈I

Ni ∈ 〈X〉}

.

Then, F = (�X ,�X ) is a KKM-structure on X , called the product KKM-structure of the KKM-structures Fi =(�Xi ,�Xi ), and denoted by F = ∏

i∈I Fi = ∏i∈I (�Xi ,�Xi ). Later on, if there exist KKM-structures of sets Xi ,

we define the product KKM-structure of X as above and say that there exists a product KKM-structure of X .



The following lemma can be checked easily.

Lemma 3.1 If Yi is a �Xi -convex subset of Xi for each i ∈ I , then Y = ∏i∈I Yi is a �X -convex subset of X.

Definition 3.2 Let I be a finite index set, Xi nonempty sets, X = ∏i∈I Xi , and Pi , Qi : X ⇒ Xi . Let

Fi = (�Xi ,�Xi ) be a KKM-structure of Xi and F = (�X ,�X ) = ∏i∈I (�Xi ,�Xi ).

(a) Qi is said to be �X -convex w.r.t. Pi if, for all x ∈ X , ϕN ∈ �X , and M ⊂ N with Mi ⊂ Pi (x), ϕNi (�Mi ) ⊂Qi (x).

(b) Qi is called �X -weak-convex w.r.t. Pi if, whenever ϕN ∈ �X , M ⊂ N , and x ∈ X satisfying x ∈ ϕN (�M)and Mi ⊂ Pi (x), one has ϕNi (�Mi ) ⊂ Qi (x).

(Here Ni (Mi , resp) is the i th component of N (M , resp) defined as above.

)Let I , Xi , X and Qi be as in Definition 3.2. A point x = (x1, x2, . . . , xn) ∈ X is called a fixed-component

point of the family {Qi }i∈I if xi ∈ Qi (x) for all i ∈ I .

Theorem 3.3 Let I be a finite index set, Xi nonempty sets, X = ∏i∈I Xi , and Qi : X ⇒ Xi . Then, the family

{Qi }i∈I has a fixed-component point if and only if there exist a product KKM-structure F = (�X ,�X ) of X andPi : X ⇒ Xi such that (X,�X ) is compact, and for each i ∈ I , Pi has nonempty values and �X -open fibers, andQi is �X -weak-convex w.r.t. Pi .

P r o o f. Necessity. Suppose that {Qi }i∈I has a fixed-component point x = (xi )i∈I ∈ X . For each i ∈ I ,let Pi : X ⇒ Xi be defined by Pi (x) = {xi } for all x ∈ X , and a KKM-structure Fi = (�Xi ,�Xi ) of Xi

be defined as follows: �Xi = {U ⊂ Xi | xi /∈ U } ∪ {Xi } and �Xi = {ϕNi : �|Ni | → Xi | ϕNi (e) = xi for all e ∈

�|Ni |, Ni ∈ 〈Xi 〉}. LetF = (�X ,�X ) = ∏

i∈I Fi = ∏i∈I (�Xi ,�Xi ). Then, it is not hard to check the conditions of

Theorem 3.3.Sufficiency. We define P, Q : X ⇒ X by, for x ∈ X ,

P(x) =∏i∈I

Pi (x) and Q(x) =∏i∈I

Qi (x).

For x = (xi )i∈I ∈ X , P−1(x) = {x ′ ∈ X | x ′ ∈ P−1

i (xi ), i ∈ I} = ⋂

i∈I P−1i (xi ) is �X -open, since the fibers of

Pi are �X -open. Moreover, for any a ∈ X and i ∈ I , since the values of Pi are nonempty, there exists xai ∈ Xi

such that a ∈ P−1(xai ). Putting xa = ⊗

i∈I xai , we have a ∈ ⋂

i∈I P−1(xai ) = P−1(xa). This means that X =⋃

x∈X P−1(x). Now, suppose ϕN ∈ �X , M = ⊗i∈I Mi ⊂ N = ⊗

i∈I Ni and a = ⊗i∈I ai ∈ X satisfying a ∈

ϕN (�M) and M ⊂ P(a). As M ⊂ P(a) we also have Mi ⊂ Pi (a) for each i ∈ I . Because Qi is �X -weak-convexw.r.t. Pi , one has ϕNi (�Mi ) ⊂ Qi (a). Note that �M = �Mi for all i ∈ I . Hence, ϕN (�M) = {⊗i∈I ϕNi (e) | e ∈�M} ⊂ ∏

i∈I ϕNi (�Mi ) ⊂ ∏i∈I Qi (a) = Q(a). Thus, Q is �X -weak-convex w.r.t. P . Applying Theorem 2.5, we

obtain a = (a1, a2, . . . , an) ∈ A such that a ∈ Q(a), i.e., ai ∈ Qi (a) for all i ∈ I . �

3.1 Applications in optimization-related problems

Now we apply Theorem 3.3 to prove existence conditions for solutions of an inequality and a non-cooperativegame.

Corollary 3.4 For i ∈ I = {1, 2, . . . , n}, let Xi be a nonempty set, X = ∏i∈I Xi , X j = ∏

i = j Xi , x j be theprojection of x on X j , fi : X → R, and αi ∈ R. Assume that there exists a product KKM-structure F = (�X ,�X )of X such that (X,�X ) is compact, and

(i) {y j ∈ X j | f j (y j , x j ) > α j } is nonempty and �X j -convex for each x ∈ X;(ii) for each y j ∈ X j , f j (y j , ·) is �X j -lower semicontinuous on X j , where �X j is the Tikhonov product of

topologies �Xi , i = j .

Then, there exists x = (x1, x2, . . . , xn) ∈ X such that f j (x) > α j for all j ∈ I .



P r o o f. Define Q j : X ⇒ X j by Q j (x) = {y j ∈ X j | f j (y j , x j ) > α j }. By assumption (i), Q j has nonemptyand �X j -convex values. For each x j ∈ X j , we have Q−1

j (y j ) = X j × {x j ∈ X j | f j (y j , x j ) > α j }. Since f j (y j , ·)is �X j -lower semicontinuous on X j , the set {x j ∈ X j | f j (y j , x j ) > α j } is �X j -open in X j . Hence, Q−1

j (y j ) is�X -open in X . Applying Theorem 3.3 for Q j = Pj for all j ∈ I , one has x = (x1, x2, . . . , xn) ∈ X such thatx j ∈ Q j (x) = {x j ∈ X j | f j (x j , x j ) > α j } for all j ∈ I . Then, f j (x) > α j for all j ∈ I . �

Assume that the i th player in a n-player game � has a strategy set Xi and a payoff function fi : X → R,where X = ∏

i∈I Xi . Denote X j = ∏i = j Xi . A point x = (x1, x2, . . . , xn) ∈ X is called a Nash equilibrium of �

(see [12]–[13]) if f j (x) = maxx j ∈X j f j (x j , x j ) for all j ∈ I .

Corollary 3.5 (Nash equilibrium.) Assume for the non-cooperative game � that there exist compactKKM-structures Fi = (�Xi ,�Xi ) of Xi with their product KKM-structure F = (�X ,�X ) such that eachfi is �X -continuous, and for any ε > 0 sufficiently close to 0 and x ∈ X, the set

{y j ∈ X j | f j (y j , x j ) >

maxy j ∈X j f j (y j , x j ) − ε}

is �X j -convex. Then, there exists a Nash equilibrium of �.

P r o o f. Define h j : X → R by h j (x) = f j (x) − maxy j ∈X j f j (y j , x j ). For any ε > 0, we easily check theassumptions of Corollary 3.4 for the functions h j and α j := −ε. Applying Corollary 3.4, we have x ∈ X suchthat h j (x) = f j (x) − maxy j ∈X j f j (y j , x j ) > −ε. Since ε is arbitrary and sufficiently close to 0, we have f j (x) −maxy j ∈X j f j (y j , x j ) ≥ 0 and hence f j (x) = maxy j ∈X j f j (y j , x j ). �

Acknowledgements This work was supported by the grant 101.01-2011-10 of the National Foundation for Science andTechnology Development (NAFOSTED). The authors are indebted to an anonymous referee for his valuable remarks.

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a topological characterization of the existence of fixed points and applications

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