Some applications of multivalued mappings

6.

Some Applications of Multivalued Mappings

6. i. Applications in Game Theory and Mathematical Economics. Historically the first applications of fixed-point theorems of multivalued mappings were closely related to problems of the existence of optimal strategies in game theory. We illustrate this relation with the following example. n

Let XI, .... X n be compact topological spaces; l e t X - ~ X ~ .

Let {fih=1 ~ be continuous gain

i=l

functions defined on X. The game is that each player (i = 1 .... ,n) independently chooses a strategy xzGXz. Thins game leads player (k) to the gain fk(xl .... ,Xn). 2.6.1.

Definition.

A point x 0 = (x ~ ..... x~) is called an equilibrium point of the game

(X, {L},=~) ~ if

f,

(x~ . . . . .

x ~ ) -= m a x f ~ (x~ . . . .

y, . . . . .

x ~)

b'i~X t

for all i = 1 ..... n. To prove theorems of the existence of equilibrium points of the game we consider the mmapping Fi:X + K(Xi),

F , ( x l . . . . . x , ~ ) = { y i E X z I / i ( x l . . . . . y~. . . . . x,,)----max / z (xl . . . . . . y . . . . . Y~Xi

the

Using

maximum theorem

1.2.25

it

is

not

hard

to

x~)}.

tz

show that

--

F~HFz:X~K(X)

is

an

l=l

upper

(X,

semicontinuous m-mapping. the set Fix F.

It

is

clear

that

the

set

of

equilibrium

points

of

the

game

{fi}nl=, is

Let X be a subset in a topological vector space. 2.6.2.

Definition.

a r e convex for any a%R. 1, 2,...,n.

A game (X, {f~}~--1) is called quasiconcave if the sets

{ y , e x , [f~ (xl . . . . . y~. . . . . x~) > a} (xl . . . . . , Xi-1, Xi+I'' , xn)~xix.~=. ~ i l i ~ X i + l X - . . .

XXnand

for

any

i =

2.6.3. THEOREM (yon Neumann-Nash). Suppose X I ..... X n are convex compact subsets ofl a locally convex space and the game(X, {[i}}~i) is quasiconcave. Then there exists an equilibium point of the game. This theorem follows immediately from the Glicksberg-Ky

Fan fixed-point theorem.

Fixed-point theorems of m-mappings are applied in the theory of equilibrium of mathematical economics in a similar manner (see, for example, [64, 76]). 6.2. Dispersive Dynamical Systems. Dispersive dynamical systems describe the development of systems whose initial state does not uniquely determine their evolution in time. The definition of such systems is naturally formulated in terms of multivalued mappings. Various axiomatics hav$ been proposed to describe generalized dynamical systems. One of the most popular is the definition proposed by Barbashin [i].

P: XXR-+K(X)

Let X be a topological space; an m-mapping system if the following axioms are satisfied: i.

P(', 0):X + X is idx;

2.

for any x6X and tI,~ER, 11.t2~0,

defines a dispersive dynamical

P ( P ( x , t l ) , t2) = P ( x , t i q - t j ;

xEX, t~R, y~P(x, t)

implies that

x~P(y,--t);

3.

for any

4.

P is an upper semicontinuous m-mapping;

5.

P(x,.) :rR-->-K(X) is

a continuous mapping.

As an exa~nple of a dispersive dynamical system it is possible to consider the system generat%d by the autonomous differential inclusion x E F (x),

2804

where F" Rn-+Kv(R~)is upper semicontinuous. Suppose solutions of this inclusion are not locally continuable to the entire numerical axis. Then with it it is possible to associate a dispersive dynamical system (a shift operator along trajectories) P: R n x R - + K ( R n) defined according to the rule:P(xo,to)={y]N6R~,~ a solution x(t) of the inclusion such that x(0) = x0, x(t0) = y}. 2.6.4. Definition. A trajectory of a dispersive dynamical system P on a segment [a,b]6R n is a mapping ~:[a,b]->-X,which for any tl, t 2 satisfies the relation ~(t2)EP(.~(I1),t2--tl). It follows from axioms i-5 that the trajectory ~ is a continuous mapping. 2.6.5. THEOREM. For any ~a,b,x1@P(xO,b'a) on the segment [a, b] there exists a trajectory of the dynamical system P such that ~(a)=x0, ~(b)=xb Let A c X; we denote by S(P, A, [a, b]) the set of trajectories {~/}i~j of the system P on the segment [a, b] such that{~y(a)}j~jcA, i.e., S(P, A, [a, b]) is an integral cone of the dynamical system issuing from the set A. 2.6.6. THEOREM. Let X be a metric space; then for any [a, b]) is compact in the topology of uniform convergence.

A6K(X),

[a;b]cRthe

set S(P, A,

Using the theory of m-mappings, it is possible to study the stability and structure of certain classes of dispersive dynamical systems (see, for example, [83]). Other axiomatics of a dynamical system without uniqueness were proposed in the work [31]. 2.6.7. Definition. namical system if

An m-mapping

P XXR+-->-K(X} is

1.

P(', O):X + X is idx;

2.

P(P(x, h);t2)cP(x, h+t2) for any x~Xi tl, t2fiR+;

3.

P is upper semicontinuous.

called a one-sided generalized dy-

It is obvious that the restriction of a dispersive dynamical system to X X R + i s sided generalized dynamical system. 2.6.8. point

xo6X

a one-

Definition. A point of rest of a one-sided generalized dynamical system P is a such that xoEP(xo,t) for any t~R+.

Theorems on the existence of rest points of generalized dynamical systems are deduced from fixed-point theorems for m-mappings with the help of the following assertion. 2.6.9. LEMM_~. Let X be a regular topological space, and suppose the sequence of positive numbers {tn} , t n § 0 is such that each m-mapping Pn = P(', tn) has a fixed point x n. If x 0 is a limit point of the sequence {Xn} , then x0 iS a rest point of P . 2.6.10. THEOREM. Let X be a one-dimensional finite polyhedron whose Euler characteristic is nonzero, and suppose for some sequence {tn} , t n § 0 all the sets P(x, tn) (x~X; n=1, 2,...) are linearly connected. Then the dynamical system P has a rest point. If P: X X R + - + K ( X )

is acyclic, then the following assertions hold.

2.6.11. THEOREM. Suppose X is a finite polyhedron with nonzero Euler characteristic; then the dynamical system P has a rest point. 2.6.12. THEOREM. Suppose X is a compact ANR-space (an absolute neighborhood retract). if for some t o > 0 the set P(X, t o ) is acyclic in X, then the dynamical system P has a rest point. ping.

6.3 Periodic Problems for Differential Inclusions. The differential inclusion

16F(t, x)

Let F : R M R n ~ K v ( R

n) be an m-map-

(6.1)

generalizes the concept of an ordinary differential equation. We call an absolutely continuous function x(t) a solution of this inclusion on the interval [to, t !] if almost everywhere on this interval we have x(t)6P(t,x(t)). Suppose that solutions of the inclusion (6.1) are nonlocally continuable to the entire numerical axis. Suppose the m-mapping F is m-periodic in the first argument, F(t + m, x) = F(t, x), and is upper semicontinuous.

2805

We are interested in conditions under which the inclusion (6.1) has an m-periodic solu - " tion. Here, as in the theory of ordinary differential equations, two approaches are possible: by means of an integral operator and by means of a shift operator. We shall describe the first of these. We consider the m-mapping ~:C([o.ol.Rn)-+Kv(C([o,~l;~,9) , defined according to the following rule: t

(x) = x (~) + ~ ~ (x) (s) ds, 0

where

~ r (x) (s)---- {V (s) IV (~)GF (x, x (T)) f o r a l m o s t a l l *E[0, o]}. 2.6.13.

LEMMA. The m-mapping r i s c o m p l e t e l y c o n t i n u o u s .

It is obvious that the fixed points of this operator and only they are m-periodic solutions of the inclusion (6.1). 2o6.14. Definition. A point xo6R n is called a point of m-nonreturn of trajectory if for any solution x of the inclusion (6.1) with the initial condition x(0) = x 0 we have x(t) x0, t~ (0, o ] . 2.6.15. THEOREM. Let G ~ R ~ be a bounded set such that all points of ~G are points of m-nonreturn of trajectory. Suppose the m-mapping -F(0, ") has no fixed points on 8G; then

~(--F(O, .), OO)=?(~--O, Oa),

where e----{xfiC(io,ol,R.)[llxHsup{rlr=llx[l, x i s a s o l u t i o n , x(O)fiG}. We note that the finiteness of N follows from the boundedness of the integral cone S(G,

[0, m ] ) . A continuously differentiable f u n c t i o n ~ : Rn-+R is called nondegenerate ifigrad ~(x)-~=0 for ![xl[>~N0 for some N O > 0. The rotation of the vector field grad~ on the spheres llxll = R > N O is called the index of ~ and is denoted by ind(~, ~). 2.6.16.

(grad ~ ( x ) , y ) > O

Definition. A function ~ is called directional for the inclusion (6.1) if for a l l y ~ F ( t ; x ) .

2.6.17. THEOREM.

Let ~ be a directing function of the inclusion (6.1); then for any

ball B0(0)~R n, p > N 0 , the following equality holds:

( " F (0, .), OB o (0)) -----(-- 1)" ind (% 0). From Theorems 2.6.15 and 2.6.17 we o b t a i n 2.6.18. THEOREM. Suppose there exists a directing function of nonzero index for the inclusion (6.1). Then the inclusion (6.1) has at least one m-periodic solution. 2.6.19.

Definition.

A directing function:~ is called increasing if [Ixll + ~ implies

l~(x) L-,-oo. If ~ is an increasing directing function, then

[ind(,~, oo)]=1. 2.6.20. THEOREM. If for the inclusion (6.1) it is possible to find an increasing directing function, then this inclusion has at least one m=periodic solution. Theorem 2.6o15 makes it possible to further develop the method of directing functions. A second approach to the study of m-periodic solutions is based on the study of fixed points of the shift operator along trajectories. In differential inclusions the study of the shift operator is impeded by the fact that it is an m-mapping which, generally speaking, has nonconvex (even nonacyclic) images. However, this operator is acyc!ic in the generalized sense, which makes it possible to study it by means of the degree theory developed in Sec. 2. 2.6.21. Definition. The shift operator along trajectories of the inclusion (6.1) is the m-mapping Uo: R~-+K(Rn),

u~(x)={vFv=~(~), ~GS(x, [o, where S(x,

~1)},

[0, m]) is the integral cone of the inclusion (6.1) issuing from the point x.

It is not hard to see that fixed points of the shift operator and only they are initial values of m-periodic solutions of the inclusion (6.1). 2806

From the continuous dependence of solutions of differential inclusions on the initial data it follows that the m-mapping U w is upper semicontinuous. We consider the m-mapping~S:Rn-+K(C(io.~i,Rn)),S ix)~=S(x, [0, ~]) , i.e., to the point x~R ~ we assign the integral cone issuing from this point. This m-mapping is acyclic (see [71]). Since!U~(x)~aoS(X), where :a:C(io.~l.Rn)-+R n,a(x) x(~), the operator U m is acyclic in the generalized sense. 2.6.22. THEOREM. If there exists a bali B6R~]isuch that ential inclusion (6ol) has an m-periodic solution.

gf~(aB)cff, then

the differ-

The connection between rotations of the fields F(0, .) and U m can be established in a manner similar to the way this is done in ordinary differential equations (see [29]); on the basis of this it is also possible to develop the method of directing functions. An___nptated Bib___!lior~ The space of closed subsets (see Sec. i, Chap. i) can be equipped with the structure of a topological space in various ways (see, for example, [41, 67, 68, 72]). The concepts of semicontinuous mappings (see Sec. 2, Chap. I) were introduced by Kuratowski and Buligan. Various types and criteria of continuity of m-mappings have been studied by many authors (see the bibliography in [i0]). The operations on multivalued mappings and their properties are expounded in the monographs of Kuratowski [67, 68] and Berge [49, 50]. We note however, that the assertion of Berge regarding lower semicontinuity of the intersection of lower semicontinuous m-mappings is erroneous. For some sufficient conditions for lower semicontinuity of the intersection of m-mappings see, for example, [i0, 74]. Regarding other operations on m-mappings and a bibliography see [i0]. The classical Theorem 1.3.3 (see Sec. 3, Chap. i) is due to Michael [73]. A proof of it is presented in [10]. Proofs of Theorems 1.3.5 and 1.3.6 are contained in the work [22]. The construction of steplike approximations for the case of upper semicontinuous m-mappings was proposed in [31] and was further developed in the works [ii, 23]. An obstruction to the construction of s-approximations of upper semicontinuous m-mappings is set forth in [18]. Theorem 1.3.22 was established in the work [70]; see also [9]. The concept of a measurable m-mapping (see Sec. 4, Chap. i) was apparently first introduced in the work of Pli~ [80]. A proof of Theorem 1.4.3 and bibliographies can be found in [25-27, 55, 62], while a proof of Theorem 1.4.5 can be found in the work [27]. Theorem 1.4.7 was proved by Kikuchi [66]. Theorem 1.4.7 generalizes a lemma of Filippov [45]. Theorem 1.4.10 was proved by Kukuchi [65], while Theorem 1.4.11 was proved by Castaing [54]. Theorem 1.4.13 goes back to Lasota and Opial [69]. More bibliography on measurable m-mappings and their applications can be found in [I0]. The proof of Theorem 1.5.7 (see Sec. 5, Chap. i) is contained, for example, in [77]. All theorems of Sec. i, Chap. 2 are proved in the work [3]. !Rp+1\0~was formulated in the work [4].

Theorem 2.1.3 for the case

The concept of the rotation (the degree of a single-valued, finite-dimensional vector field) goes back to Kronecker and Poincar~ (see Sec. 2, Chap. 2). The basic properties of this concept are described by Brouwer. A detailed exposition of the theory of rotation of single-valued vector fields is given in the works of Krasnosel'skii and his students (see, for example, [28, 30]), Nirenberg [77], Browder [52], etc. The method of single-valued approximations in the theory of rotation of m.v.-fields with convex images is developed in the works [7, 8, 56, 9, i0] where detailed proofs can be found. Ilomotopy groups of the space of subsets and the degree of multivalued mappings were studied in the works [3, 4]. The topological characteristic of an m.v.-field was introduced and studied in the works [19~ 9]~ This concept generalizes to the case of completely continuous m.v.-fields in

2807

infinite-dimensional spaces (see [20]). found in the survey [9].

Detailed proofs and bibliographic references can be

The concept of a fundamental set of a mapping was introduced in the work [24] (see Sec. 3, Chap. 2). Its application to the construction of the rotation of impermeable vector fields was first indicated in the work of Sapronov [43]. In the work [32] this concept was used to define rotation of impermeable m.v.-fields in Fr~chet spaces. The theory of the rotation of fundamentally restrictable m.v.-fields is expounded in the works [32, 40, 13, 9]. We note that Theorems 2.3.8 and 2.3.9 generalize the bijection principle of Sapronov for single-valued impermeable vector fields [43]. The construction of the degree of fundamentally restrictable, generalized almost acyclic m.v.-fields was proposed by Obukhovskii [78, 38]. More details and proofs of the basic facts (see Sec. 4, Chap. 2) of the theory of linear Fredholm operators can be found, for example, in [79]. The basic properties of nonlinear Fredholm mappings and the construction of the degree can be found, for example, in [12, 4, 6,

77]. The construction of the degree for a mapping of the form of a "nonlinear Fredholm operator plus a completely continuous multivalued operator with convex images" was proposed by Borisovich [5] and generalized by Bennofadar and Gel'man [2] to the case of a "nonlinear Fredholm operator plus a completely continuous generalized acyclic operator." The exposition of Subsec. 2 of the construction of the topological characteristic follows the work [5] where further references are given. For single-valued continuous operators in 1959 Krasnosel'skii and A. I. Perov proved the connectivity principle (see, for example, [30]) which found wide application in the study of the structure of the set of solutions of various classes of equations (see Sec. 5, Chap. 2)~ This principle is an operator generalization of Knezer's theorem regarding the connectivity of an integral cone. In 1942 Arronszajn [47] proved that the integral cone of an ordinary differential equation is an acyclic set. These questions were developed in the study of various classes of integral and differential inclusions. For example, in the works [14, 15] the connectivity of the set of solutions of some classes of integral inclusions was studied, while in the works [71, 61] theorems on the acyclicity of the set of solutions of some classes of differential inclusions were proved. We note also the generalization of the connectivity principle of Krasnosel'skii-Perov for one class of operator inclusions obtained in the work [39]. A proof of Theorem 2.6.3 (see Sac. 6, Chap. 2) can be found in [49]. There and also, for example, in the monographs [64, 76] a detailed exposition is given of other applications of multivalued mappings in game theory and mathematical economics (see also the bibliography in the surveys [9, i0]). We note that in the work [36] the concept of the rotation of an m.v.-field is applied to prove the existence of positive equilibrium in a model of competing economics. For the proofs of Theorems 2.6.5 and 2.6.6 see [I]. Theorem 2.6.10 is proved in [31] and Theorem 2.6.12 in [35]. Theorem 2.6.15 is proved in [9]. The concept of a directing function goes back to the works of Krasnosel'skii (see, for example, [29]). Theorem 2.6.20 was proved by Gango and Povolotskii [16]; for Theorem 2.6.22 see [9]. The method of integral operators in the problem of periodic solutions of differential inclusions and control systems with retardation is developed in the works [33, 34]. LITERATURE CITED 1.

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2808

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