On fixed point Theory and its applications to equilibrium models Текст научной статьи по специальности «Математика»

MSC 47H10, 54C10, 54E45, 91B50

DOI: 10.14529/ mmp 160102

ON FIXED POINT THEORY AND ITS APPLICATIONS TO EQUILIBRIUM MODELS

D.A. Serkov, Krasovskii Institute of Mathematics and Mechanics, Ural Federal University named after the first President of Russia B.N. Yeltsin, Yekaterinburg, Russian Federation, serkov@imm.uran.ru

For a given set and a given (generally speaking, multivalued) mapping of this set into itself, we study the problem on the existence of fixed points of this mapping, i.e., of points contained in their images. We assume that the given set is nonempty and the given mapping is defined on the entire set. In these conditions, we give the description (redefinition) of the set of fixed points in the set-theoretic terms. This general idea is concretized for cases where the set is endowed with a topological structure and the mapping has additional properties associated with this structure. In particular, we provide necessary and sufficient conditions for the existence of fixed points of mappings with closed graph in Hausdorif topological spaces as well as in metric spaces. An example illustrating the possibilities and advantages of the proposed approach is given. The immediate applications of these results to the search of equilibrium states in game problems are also given: we describe the sets of saddle points in the minimax problem (an analogue of the Fan theorem) and of Nash equilibrium points in the game with many participants in cases where the sets of strategies of players are Hausdorif spaces or metrizable topological spaces.

Keywords: multivalued mapping; fixed point; saddle point; Nash equilibrium.

Introduction

If one consider an automorphism F of a set X as the dynamics of a system with discrete time and the state space X, then many sufficient conditions for the existence of fixed points can be treated as conditions that prevent the formation of cycles (consisting of several points):

— "the contraction principle": the distance between the images of two arbitrary points is less than the distance between these points (a cycle has to be contracted to a point);

XF isotonic with respect to this order (only single-point cycles are possible).

In this case, either the linear structure (see the Banach principle of contractive mappings, the Kakutani theorem [1], and their generalization [2]), or the order structure (see theorems by Tarski [3], Kantorovich [4], Kleene [5, Theorem 1.2.17], and their

X

At the same time, in view of the known connection between fixed points and equilibrium points (see Lemmas 3 and 4), one would expect a fixed point result under conditions similar to the conditions of the Fan theorem [7]. We recall that this theorem is based only on topological properties of the domain of the quality function.

From this point of view, the idea of the conditions proposed in the paper consists in restricting the sizes of cycles due to the representation of the initial mapping by the set of its restrictions to a covering of its domain: a single-point cycle is present for any choice of such a covering; all other cycles, in dependence on the properties of the mapping (we require the closedness of the graph), are "cut off" under an appropriate choice of a family of coverings.

1. Definitions

1. For an arbitrary set X denote by 2X the set of all subsets of X. Consider a nonempty set X and a multivalued mapping X 3 x M F(x) E 2X. Denote by Fix(F) the set of all fixed points of the mapping F: Fix(F) :={x E X | x E F(x)}. For the given mapping F, define the mapping 2X 3 Y M F(Y) E 2X as follows:

F(Y ):=( U F (y)) f| Y = |J F (y) П Y.

\y€Y ' V&

Observe that F(Y) is the set of "candidates" from a set Y for the inclusion in the set of fixed points: by contradiction, it is easy to verify that the elements of Y not tying in F(Y)

Fix(F) {x} E 2X x E X

the set F({x}) is nonempty if and only if x E Fix(F).

Immediately from the definition, for any Y,Y' E 2X, we obtain

F(Y) C Y, (1)

Y' c Y ^ F(Y') C F(Y), (2)

Fix(F) n Y = Fix(F) n F(Y), (3)

(Y C Fix(F)) ^ (Y E Fix(F)).

From the definition of F and the above relations, for an arbitrary family (ZT)T C 2X, we get

U F(Zt) c U Zt, (4)

t&t t&T

Fix(F^ (J Zr = Fix(F^ (J F(ZT). (5)

t&T t&T

Here, inclusion (4) follows from inclusion (1) and equality (5) follows from equality

(3)-

2. For families in 2X define the relation to be inscribed which will be denoted by the symbol for arbitrary (ZT)T, (ZT,)T> C 2X, we say that the family (Z^)T> is inscribed into the family (ZT)T and denote this as (Z^)T> Q (ZT)T if, for arbitrary t' E T', there exists t E T such that ZT, C ZT. In the sequel, this order relation will be useful in view of the following property of the mapping F: it follows from implication (2) that if (ZT,)T', (Zt)t c 2X and (ZT, )t' Q (Zt)t, then

U F(Z't<) C (J F(Zt). (6)

T'eT' t&t

3. Denote by O(X) ^te set of all coverings of X, i.e., the set of all families (0) C 2X such that 0L = ^^et t(X) be a topology in X (the family of all open sets). Denote by 0fo(X) (0fc(X)) the set of all finite open (closed) coverings of X.

2. Criterion for the Existence of Fixed Points

1. The set of fixed points of the mapping F can formally be described as follows. Theorem 1. For any nonempty set X and for any mapping F : X m 2X, we have

Fix(F)= H U FO). (7)

(Oi)ieO(X) ei

Remark 1. Since among the coverings there exists a smallest element with respect to the inscribing relation, the covering ({x})X consisting of all singleton sets of the set X, by (6), for this coveriog and for any (Ol)I E O(X), we have U^X F({x}) C (JF(Ot) and, hence, Fix(F) = (J X F({x}). However, exactly relation (7) is important for us, since it

XF

2. The corollaries of equality (7) given below are based on the intuitively obvious fact that if a closed subset G of a compact set X x X has points arbitrarily close to the diagonal {(x,x) | x E X}, then G contains an element (x,x) of the diagonal. At the same time, if the set G is the graph of a multivalued mapping F, then the element x is a fixed point of

F

XF

graph. Then

Fix(F )= H UF(Oi)= 0 UF(Oi). (8)

(Oi)ieOfo(x) iei (Oi)ieoic(X) ei

Fix(F)

(V(Oi)i E Ofc(X))(3i E I) F(O,) = 0. (9)

Corollary 1. Let X be a compaet Hausdorff space, and let a function f : X m X

be continuous. Then Fix(f) = f|(Oi)ieOfo(x)Uei f (Oi) = fl(ot)ieOfc(x^ei f (O^)- In particular, the function f has a fixed point if and only if (V(Ol)i E Ofc(X))(3l E I)

f (O) = 03. Let the space X be endowed with a metric p : X2 m [0, to). Denote by d(x, A) ^te from a point x E X to a set A C X defined by the metric

p : d(x, A) := infae^ p(a,x). For any Y E 2X, define the diameter of the set Y: diam(Y) :=supy y,eY p(y,y')- Assume th&t the topology t(X) generated % the metric p is defined in X. For any 8> 0, denote by O£(X) Oo(X)) the subset of Ofc(X) (Ofo(X)) of closed (open) finite coverings (OJ.r of diameter not greater than 5: maxiei diam(Ot) ^ 8. Recall that if X is a compact metric space, then for any 8 > 0 the sets Ofc(X) and Ofo(X) are nonempty.

XF Then, for an arbitrary family of coverings

(Oki)ik E O'c(X) U O^ofc(X), 8k > 0, k E N, lim 8k = 0, (10)

the equality Fix(F) = P|keN (Jieik F(Okl) holds. In particular, the set Fix(F) is nonempty if and only if

(V8 > 0)(3xs E X) d(xs, F(xs)) ^ 8. (11)

Corollary 2. Let X be a compact metric space, and let a function f be continuous. Then, for an arbitrary family of coverings of the form (10), the equality Fix(f) = PlfceN U¿,&ik f (Oki) holds. In particular, the function f has a fixed point if and only if

(V5 > 0)(3x5 £ X) p(xs, f (xs)) ^ 5. (12)

Remark 2. Theorems 2 and 3 as well as Corollaries 1 and 2 remain true if we replace the condition of compactness of the space X by the condition of precompactness of the image of the space X under the mapping F (f ):

U F(x) c K (U f (x) C K )

xex \xex J

xÇ.X \x&X

where K £ 2X is a compact set in the space (X, т(X)).

2.1. Example

We give as much as possible simple example of the use of Theorem 3. Consider the closed interval [0,1] as a compact metric space X with the natural metric p(x,y) := \x — y\. Choose and fix an infinite disjoint sequence ([a^ h^N of closed intervals in X of nonzero length: ai < hi, [ai,bi] fl [aj,bj] = 0, i,j £ N i = j- Define the mapping G : X m- 2x, setting

G(x) := U GK Al(x). GM(x) ly — x\• * £ ^ hj-

i&N {0, x £ [a, h]-

Define the mapping F : X m 2X ^ follows: the graph of the mapping F is the closure in R2 of the graph of the mapping G. ^^^e that the values of mapping F are nonconvex or

x £ X

F

disjointness of intervals [aj., hi], their lengths tend to zero as i m to and, hence, for any 5 > 0, the inequality d(ai,F(ai)) ^ 5 holds for a sufficiently large i. Consequently, by

F

sequence (ai)N are fixed points. Certainly, such limits exist in view of the compactness X

In connection with the considered example, we also note that

— the mapping F is not fc-contractive (it suffices to consider the passage of the argument through the center of any interval [ai, hi]), and, therefore, the Nadler theorem [8] is not applicable;

— the mapping F is not a-covering, since is not surjective, and, hence, the theorem on coincidence points [9] is not applicable;

F0 Kakutani theorem [1] is not applicable.

3. Applications to Equilibrium Theorems

Using the known method connecting the equilibrium positions with fixed points of multivalued mappings (see Lemmas 3 and 4), we can obtain new versions of equilibrium

theorems from the above theorems on fixed points. The very conditions describing the set of equilibriums are rather formal. However, their particular case, a property of Cournot approximations, may be of interest for applications.

1. From Theorem 2, we can obtain one more theorem on a criterion of the equality of the minimax and maximin values in conditions similar to the conditions of the Fan theorem [7].

Let U and V be (topological) spaces of strategies of two players, and let the standard topology of the product and an outcome function p : U x V m R with scalar values be given on the product X := U x V. The player choosing strategies u E U tends to minimize the outcome of the game, and the player choosing strategies v E V tends to increase the outcome. Define S(p) E 2U x 2V as the set of saddle points of function p, i.e., as the set of pairs (u*,v*) E U x V satisfying the conditions p(u*,v) ^ p(u,v*) V(u,v) E U x V. Consider the multivalued mapping X 3 (u,v) m F^(u,v) E 2X of the form

Fv(u, v) := argminp(u',v) x argmaxp(u,v'), (u,v) E X. (13)

u eu v'ev

Theorem 4. Let U and V be compact Hausdorff spaces. Let, for any v E V, the function p(-,v) : U m R be lower semicontinu,ous on U, and let, for any u E U, the function

p(u, ■) : V m R be upper semicontinu ous on V. Then, for the set S (p) of saddle points of p

S(p) = Fix(F^) = H U F<p(OL) = H U F^(Oi).

(Oi)ieOfo(x) iei (Oi)ieo^(x) iei

In particular, the equality

maxmin p(u, v) = minmax p(u,v) (14)

vev ueu ueu vev

holds if and only if for an arbitrary covering (Ol)i E Ofc(X), there exists a set Ol} i E I, containing two successive Cournot approximations x,x' E Oz, x' E Fv(x).

2. Passing to a more general case of the game, we obtain one more version of the theorem on the existence of Nash equilibrium. Let (X, J) be a game with n players in the normal form: X := Xi x ... x Xn and J :=( J\,... Jn) ■ Here, Ji is the payoff function of the zth player: Ji : X m R i E 1..n, and in the case where Xi are

X

(y,x-i) :=(xi,... ,x—i,y,xi+i,... ,xn), y E Xi, x = (xi,... ,x,n) E X. Denote by N (J) the set of elements x* E X for which the Nash equilibrium is attained, i.e., for which the following conditions are satisfied:

Ji(yi,x-i) ^ Ji(x*), Vyi E Xi, Vi E 1..n. (15)

By analogy with (13), introduce the multivalued mapping X 3 x m F(X J)(x) E 2X:

F{x,j)(x) := argmax Ji(yi,x-i) x ... x argmax Jn(yn,x-n), x E X.

yieXi yneXn

Xi i E 1 .. n x E X

i E 1..n, the function Ji(^,x-i) : Xi m R be upper semicontinu ous on Xi. Then, for the

set N (J) of Nash equilibriums, we have

N(J) = Fix(F(x ,j))= H U PX,J)(Oi) = H U PX,J)(O)

(Oi)I&Oio(X) i&I (OI)I&Oic(X) i&I

In particular, Nash equilibrium (15) is attained if and only if for an arbitrary covering (Ol)I £ Ofc(X), there exists a set Oz £ (Ol)I containing two successive Cournot approximations:

(V(Oi)i £ Ofc(X))(3t £ I)(lx,x' £ Oz) x' £ F(x ,J)(x). (16)

X

(16) is naturally transformed into condition (11), taking the form (W5 > 0)(3xs £ X) d(xs, F(x,j) (xs)) ^ 5.

Remark 4. Note that Theorem 5, as well as Theorem 4 and the Fan theorem [7], has the character of a criterion.

4. Proofs

4.1. Proof of Theorem 1

The inclusion Fix(F) C P|(e0(X) UL&I ^(Ot) follows from (5). Indeed, by (5), for any (Oi)i £ O(X), we have the equalises Fix(F) = Fix(F) f (JL&I Ot = Fix(F) f UleI F(OL) and, consequently, Fix(F) C UieI F(OL).

Let us show the converse inclusion. Since {Fix(F), {x} \ x £ X,x £ Fix(F)} £ O(X), the belonging of x to the right-hand side of equality (7) yields, in particular, the inclusion x £ F(Fix(F)) U Ux&X\Fix(F) i^({x}) which, in view of (1) and the equalities F({x}) = 0

for x £ X \ Fix(F^ implies ^te inclusions x £ F(Fix(F)) C Fix(F).

4.2. Proof of Theorem 2 and Corollary 1

By virtue of Theorem 1 and the inclusions Ofo(X), Ofc(X) C O(X), for the proof of the assertion, it suffices to establish that

Fix(F) d H U PO), (17)

Fix(F) D H U P(Oi). (18)

(oi)I€0fc(X) i&I

1. Justify inclusion (17). Let x belong to the right-hand side of (17). For an arbitrary local base of the topology (Os(x))^ of the compact set X at the point x, we construct a family of coverings (Osl)Is, s £ S, from Ofo(X) such that, for any s £ S, we have

(lis £ Is : Os(x) = Oss)&(Vi £ Is\{is} x £ Osi). (19)

Fix an arbitrary s £ S and, using the Hausdorff property, for every element y £ X \ Os(x), choose disjoint open neighborhoods Osy and Osx of the elements ^d x. The family

{Os(x),Osy \ y £ X \ Os(x)} (20)

is an open covering of X. For the chosen s, we take a finite subcovering of covering (20) as the covering (OSi)Is. By construction, this covering has all the above properties.

Relations (4) and (19) along with the assumption that x belongs to the right-hand side of (8) imply that the inclusions

x e U ^ F O ) \ u ^ Osi c U F (Ose) \ U ^ F (OSL) c F (Os(x)) =

Vy^x) F(y^ n Os(x) (21)

hold for all s e S.

Let us show that x e Fix(F). Assume the contrary: x e F(x). The closedness of the graph of F yields the closedness of the set F (x) c X. This, in view of the fact that X is a compact Hausdorff space, implies the existence of open neighborhoods O'(x) and O'(F(x)) of the point x and of the closed set F(x) such that

O'(x)p| O'(F(x)) = 0. (22)

F X F

is upper semicontinuous (see [10, Sect. 43,1,Theorem 4] and [11, Theorem 1.2.32]), i.e., for the neighborhood O'(F(x)), there exists a neighborhood O''(x) such that, for all y e O''(x), we have

F (y) c O' (F (x)). (23)

It follows from the definition of the family (Os(x))ses that the inclusion

O5(x) C O' (x) n O''(x) (24)

holds for some s e S. Relations (22), (23), and (24) yield the inclusions

( U F (y) J H Os(x) c( U F (y) J H O (x) C O (F (x)) f O' (x) = 0

\y£Os(x) J \yeO"(x) J

which lead to the contradiction with (21) for s = s.

2. To justify inclusion (18), we prove two auxiliary assertions.

Lemma 1. Let X be a compact Hausdorff space. Then, for any element x e X and for any neighborhood O e t(X), x e O, there exist a closed set B and a neighborhood O' e t(X) such that x G O' C B C O.

Proof. Let x E O E t(X). Since the set X\O is closed and does not contain x, by virtue of the fact that X is a compact Hausdorff space, there exist O', O" E t(X) such that x E O', X \ O c O", and O' n O'' = 0.

Then, for the closed set B := X \ O'' and the neighborhood O', we have the required relations: x E O' C B c O.

□

Lemma 2. Let X be a compact Hausdorff space. Let x E X, O E t(X), and a closed set B E X be such that x E O C B. Then there exists a covering (Ol)I E Ofc(X) such that, for some 1 E I, we have

B = Oz, x E X \ |J O. (25)

iei\{i}

Proof. Let x E X and O,B E 2X satisfy the conditions of the lemma. Using the Hausdorff property of X, for any point y E X \ {x}, construct neighborhoods Oxy, Oyx E т(X) such that x E Oxy С O, y E Oyx, and Oxy П Oyx = 0.

Using the compactness of X, we select a finite subcovering {Oxyi, Oyjx | i E I,j E J} from the open covering {Oxy, Oyx | y E X \ {x}} and consider the finite family of closed sets {B,Bj I j E J}, where Bj denotes the closure of the set Oyjx. Since Oxyi С O С B,

i E I, this family is a covering of X. By construction, (UjeJ OyjП (Пге/ Oxyi) = 0-Hence, taking into account that x E P|ш Oxyi, we obtain x E X\UBj. We have verified the last of the required properties of the covering {B,Bj | j E J}.

□

x

(18) and x E F(x). Since the graph of F is closed, the set F(x) is closed. Therefore, in

X

O'(x), O'(F(x)) E т(X) of the point x and of the set F(x): O'(x) П O'(F(x)) = 0. The

F X F

semicontinuous with respect to inclusion (see [10, Sect. 43,1,Theorem 4] and [11, Theorem 1.2.32]), i.e., there exists a neighborhood O''(x) E т(X) such that Uyeo '' ( тЛ

F(y) С

O (F (x)).

Let, by Lemma 1, an open neighborhood O and a closed set B such that x E O С B С O'(x) П O''(x) be found. Let, by Lemma 2, a covering (Ol)i E Ofc(X) satisfy conditions (25). Then these conditions, the assumption x E П(Oi)i&Ofc(X)Ul£i F(Oi), and property (4) imply that

x E (J FOU \ I (J Oj С (U FO)) \ I U P(Oi-) I С F(OZ) = F(B).

B

F(B) := B n(U F (y)\ С O (x) П I J F (y) ) С O (x) f| O (F (x)) = 0.

\y€B J \y&O" (x) J

x E 0

4. For the proof of the second part of the theorem, first we will establish that the sets

J F(Oi), (Oi)i E Oic(X), (26)

are closed and centered if condition (9) is satisfied. This property along with the

X

The closedness of these sets follows from the definitions of the family Ofc(X) and the mapping F and from the closedness of the mapping F (see [11, Theorem 1.2.33]): for any closed set Y С X, the set F(y) is closed.

Let us verify that sets (26) are centered. Let (Oki)ik E Ofc(X), k E l..n, n E N. We define (Oz)j :={Oit1 П ... П Onin | E Ik,k E 1..n}. In other words, (Oz)j is the greatest lower bound of the finite set of covering (Okl)ik with respect to the inscribing relation. By

construction, (Oi)i E Ofc(X) and is inscribed into all the families (Okl)Ik: (Oi) j Q (Oki)Ik, k E 1..n. Therefore (see (6)), for any k E 1..n, the inclusion Ule/ F(Ox) C HJeIk F(Okl) eolds. Hence, in view of the condition |Jle-F(Ot) = 0, we obtain 0 = |Jle-F(Of) C Plfcei n UteIk F(Okl). Thue, subsets (2Hi of the compact space X are closed and centered.

Consequently, Fix(F) = f|mieofc(x) UeI F(Oi) = 0.

Conversely, let x E Fix(F). Then, for any covering (Ol)I E O(X), there exists a set Oi E (Ol)I such that x E O-L. Hence, by definition, x E F(Oj). Therefore, F(Oz) = 0.

For the justification of Corollary 1, we note that the condition of closedness of the mapping graph is equivalent to the condition of function continuity.

4.3. Proof of Theorem 3 and Corollary 2

In this subsection, for arbitrary 8 > 0 x E X, and Y E 2X, we define

O-(x) :={y E X | p(x, y) ^ 8}, Os(Y) :={y E X | d(y, Y) ^ 8}.

1. By virtue of Theorem 1 and the inclusions (Okl)Ik C O(X), k E N, for the proof of the first part, it suffices to establish th at Fix(F) D P| fce^U ¿,e Ik F (Oki).

Assume that x E P|fce^UeIk F(Okl). Then there is a sequence

Ok := OkLk, x E F(Ok) C Ok, diam(Ofc) ^ 8k, k E N. (27)

Let us show that x E Fix(F) Suppose the contrary: x E F(x). Define 1:= d(x,F(x))/3. Since the graph of F is closed, the set F(x) is closed. Therefore, 1 > 0 Choose 8 E (0,1] such that the relations F(x) C O-(F(x)) hold for all x E O-(x). We can make this, since

FF

e

time, due to the choice of the values ^nd 1, we have O- (x) P| O-(F(x)) = 0. From these constructions, we obtain

F(O-(x)) := O--(x) H i U F(x) I C O--(x) H O-(F(x)) = 0. (28)

\xeo-s(x) J

In view of (10), choose 1 E N such that the inclusion 0- C O- (x) holds. From these inclusions and relations (28) and (2), we obtain F(Ok) C F(O-(x)) = 0, which contradicts (27) for k = k. The first part of the theorem is proved.

2. Let us consider the second part of the assertion. The necessity of condition (12) follows immediately. Now, we verify the sufficiency of this condition. Using the compactness of X, construct a sequence (Okt)Ik, k E N, of inscribed open coverings of the form:

(Ok+ii)Ik+1 Q (Oki)Ik, (Oki)Ik E Ol/k(X), k E N.

Then the sequence of closed finite coverings (Oki)Ik, k E N, where Oki is the closure of the set Okl in X, satisfies conditions (10):

(Oh+u)Ik+1 Q (Oki)Ik, (Oki)Ik E Ol/k(X), k E N.

In this case, by (6), the sequence of the closed sets \JieIk F(Oki) is monotonically decreasing by inclusion. Using condition (12), we verify the nonemptiness of the terms of this sequence and, hence, the nonemptiness of their intersection (by a part of the first theorem, this is the set of fixed points of F).

For any k E N take 5k equal to the half of the Lebesgue number of the covering (Okl)Ik■ Then condition (12) implies the existence of a pair xk, yk E X such that p(xk, yk) ^ and yk E F(xk). Moreover, due to the choice of there exists an element Okz of the covering (Okl)Ik such that xk,yk E Okz C Okz. From the last two relations, we obtain F(Okz) = 0. Consequently, |JteIk F(Oki) = 0.

For the justification of Corollary 2, note that the condition of closedness of the mapping graph is equivalent to the condition of function continuity.

4.4. Proof of Theorem 4

Let us prove the following auxiliary assertion.

Lemma 3. The set of fixed points of mapping Fv, i.e., the set of pairs (u*,v*) E U x V satisfying the condition

(u*,v*) E Fv(u*,v*), (29)

coincides with the set of saddle points of the function i.e., with the set of pairs

(u*,v*) E U x V satisfying the inequalities

y(u*,v) ^ v(u,v*) V(u,v) E U x V. (30)

Proof. Assume that inclusion (29) holds. It means that the following inequalities are valid:

p(u*,v*) ^ inf p(u',v*) ^ p(u,v*) Vu E U, (31)

«'eu

p(u*,v) ^ sup p(u*,v') ^ p(u*,v*) Vv E V. (32)

v'ev

Since the left-hand side of the inequality in (31) equals to the right-hand side of the inequality in (32), we have relations (30) which mean that the pair (u*,v*) is a saddle point. To complete the proof, note that the above arguments are invertible.

□

The properties of the function ф(^) and of the spaces U and V required in the condition of Theorem 4 imply that the graph of the mapping Fv : U x V м- 2UxV is closed and U x V is a compact Hausdorff space with the topology of the product of the topological spaces U and V. Thus, conditions of Theorem 2 are satisfied for the mapping Fv.

Let the mapping F,ф satisfy condition (9). Then it follows from Theorem 2 and Lemma 3 that the set of saddle points for the function ф (30) is nonempty. Hence, equality (14) holds.

Conversely, let equality (14) hold. Then, by virtue of the conditions on the function ф(^) and on the spaces U and V, the set argminMeU max^eV ф(и, v) x argmax^eV minMeU ф(и, v) is nonempty and coincides with the set of saddle points of the mapping Fv. In this case, Lemma 3 and Theorem 2 imply that condition (9) holds.

4.5. Scheme of the Proof of Theorem 5

The lemma given below follows from the definition of mapping F.

Lemma 4. N (J) = Fix(F(XJ)).

The properties of functions J and spaces X^ i E l..n, required in the condition of Theorem 5 imply that the graph of mapping F(XjJ) : X м- 2X is closed and X is a

X

the mapping F(XjJу

After this remark, it is easy to see that the first part of Theorem 5 follows from Lemma 4 and Theorem 2. The equivalence of conditions (16) and (9) follows from the definition of function F(XjJ) % mapping F(X>J).

References / Литература

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Received June 30, 2015

УДК 517.952+517.977 DOI: 10.14529/mmpl60102

К ТЕОРИИ НЕПОДВИЖНЫХ ТОЧЕК И ЕЕ ПРИЛОЖЕНИЙ К МОДЕЛЯМ РАВНОВЕСИЙ

Д.А. Серков

Для заданных множества и (вообще говоря, многозначного) отображения этого множества в себя рассматривается вопрос о существовании неподвижных точек такого отображения, то есть точек, содержащихся в своем образе. Относительно заданных множества и отображения предполагается, что множество не пусто, а отображение определено на всем множестве. В этих условиях дается описание (переопределение) множества неподвижных точек в теоретико-множественных терминах. Это общее представление конкретизируется для случаев, когда множество наделено той или иной топологической структурой, а отображение имеет дополнительные свойства с ней свя-ЗБ.ННЫ6. В частности, предложены необходимые и достаточные условия существования неподвижных точек для случая отображений с замкнутым графиком как в хаусдор-фовых топологических пространствах, так и в метрических пространствах. Приведен пример, иллюстрирующий возможности и преимущества предлагаемого подхода. Также даны непосредственные приложения этих результатов к поиску равновесных состояний в игровых задачах: описаны множества седловых точек (аналог теоремы Фана) в задаче о минимаксе и точек равновесия по Нэшу в игре со многими участниками для случаев, когда множества стратегий игроков являются хаусдорфовыми или метризуе-мыми топологическими пространствами.

Ключевые слова: многозначное отображение; неподвижная точка; седловая точка; равновесие по Нэшу.

Дмитрий Александрович Серков, доктор физико-математических наук, ведущий научный сотрудник, Институт математики и механики им. H.H. Красовско-го УрО РАН; профессор, кафедра «Вычислительные методы и уравнения математической физики:», Институт радиоэлектроники и информационных технологий, Уральский федеральный университет им. Б.И. Е ЛЬЦИНсЦ (г. Екатеринбург, Россия), serkov@imm.uran.ru.

Поступила в редакцию 30 июня 2015 г.