Multivalued Maps And Differential Inclusions: Elements Of Theory And Applications. Valeri Obukhovskii

Multivalued Maps And Differential Inclusions: Elements Of Theory And Applications

We will study differential inclusions, control systems and generalized dynamical systems in Chapters 3 and 4 in more detail.

Example 1.1.13. Metric projection. The following notion arises naturally in the theory of best approximations. Let (X, ϱ) be a metric space; C ⊂ X a nonempty closed subset. For x ∈ X, the set

(x) of points y ⊂ C such that ϱ(x, y) = ϱ(x, C) is called the metric projection of x onto C. Notice that the set

(x) may be empty. If

: X → P(C) which is also called the metric projection. As examples of proximinal sets may be considered compact sets as well as closed convex subsets of reflexive Banach spaces. Metric projections play an important role in various problems of the approximation theory, geometry of Banach spaces, fixed point theory, variational methods. Extensive literature is devoted to the study of their properties (see, e.g., [175], [380]).

Example 1.1.14. Approximate calculations. Suppose that at each point x of a certain set X some number characteristics y(x) = (y₁(x), . . . , y_n(x)) ∈

ⁿ are measured. By the nonhomogeneity of the set X, absolute errors of measurements δ_i (1 ≤ i ≤ n) depend on x : δ_i = δ_i(x). The multimap F : X → P(

ⁿ),

is called the field of values of characteristics.

Example 1.1.15. Theory of games.

(a) Zero-sum games.

Notice that first examples and applications of the notion of a multivalued map were connected with the new science, arising in the thirties-forties of the XX century, the theory of games. This branch of mathematics studies the mathematical models of conflict situations, i.e., such collisions in which the interests of participants do not coincide or are directly opposite. Situations of such kind emerge repeatedly in economics, military or political conflicts and in other spheres of human activity. Their simple and visual models are provided by chess, card games etc., from where the name of the discipline comes from. The base of the theory of games was laid by such prominent scientists as J. von Neumann, J.Nash, O.Morgenstern and others.

From the mathematical point of view the behavior of the participants of a conflict situation (let us call them players) is determined by the choice of a strategies, points from a certain sets of admissible strategies. The selection of a strategy completely defines the behavior of a player at each position which can arise in the process of a game. It is easy to see that even in very simple games there is an enormous number of possible strategies and so their analysis is not a very simple matter. What can be the main principles of such analysis?

For simplicity, let us consider the case of a game with two players, or the two-person game. Let all admissible strategies of the first player form a set X and Y be a set of all admissible strategies of the second player. By a game rule of the first player we mean the assignment to each strategy y ∈ Y of the second player the set of best strategies A(y) ⊂ X from which the first player chooses his strategy. Similarly, the game rule for the second player is defined by the sets of his strongest responses B(x) ⊂ Y to the strategies x ∈ X of the first player. This means that the game rule of the first player may be interpreted as the multimap A : Y → P(X), whereas the game rule of the second player is the multimap B : X → P(Y).

For a simple example of constructing of game rules we can consider a zero-sum or antagonistic game. The game of this type is determined by the payoff function f : X × Y →

defined on the Cartesian product of the spaces of strategies. It is supposed that after the choice by the first player of his strategy x ∈ X and by the second player of the strategy y ∈ Y, the payoff of the first player is equal to f(x, y) whereas the payoff of the second player is directly opposite and equals −f(x, y). This means that the first player is trying to maximize the value f(x, y) whereas the second one is making efforts to minimize it. In this case the game rules can be given explicitly:

of course, under condition that pointed out maximums and minimums exist.

Therefore, while the elaboration of suitable strategies, each of the players should analyze the multimaps A : X → P(Y) and B : Y → P(X). The consideration of the question how these multimaps may be used for the searching of optimal strategies for each player is postponed till the fourth chapter.

(b) Games with a complete information.

The language of multivalued maps allows also to simulate some game situations in the following way. Let X be a set of game positions partitioned into n subsets X₁, . . . , X_n in accordance with the number of players. For each player, a certain preference relation is given on X which allows him to compare positions from the point of view of their utility. Let {a} be any singleton, a ∉ X and F : X → P(X ∪ a) a multimap such that a ∈ F(x) implies F(x) = {a}. Let an initial position x₀ ∈ X_i be given, then the i-th player makes his move choosing a position x₁ in the set F(x₀). If x₁ ∈ Xj then the j-th player chooses a position in the set F(x₁) and so on. The game is over if any player chooses a position x such that F(x) = {a}. The goal of the game for an individual player may be formulated as, for example, the obtaining of a position being as profitable as possible at least once during the game.

Example 1.1.16. Mathematical economics.

(a) Multifunctions of productivity and demand.

Let an economic system include n categories of goods whose prices p = (p₁, . . . , p_n) can vary in frameworks of a set Δ ⊂

ⁿ. Let an enterprise–producer has a certain compact set Y ⊂

ⁿ of possible production plans for the output of goods (a technological set). The component y_j of the vector y ∈ Y corresponds to the amount of the j-th commodity produced in accordance with this plan. The profit of the producer after the realization of the plan y equals

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