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

Be guided by considerations to obtain the maximal gain under the prices p ∈ Δ, the producer will choose production plans from the set

      The multimap Ψ : Δ → P(

ⁿ) defined in such a way is called the productive multifunction of the enterprise.

      On the other hand, let at given prices p ∈ Δ for the enerprise-customer a compact set X(p) ⊂

ⁿ of consumption vectors be accessible. The component xj of the vector x ∈ X(p) corresponds to the consumption of the j-th product. The preference of one or other consumption vectors is characterized by a certain function u :

ⁿ →

which is called the utility index. Trying to purchase at the given prices the most useful collection of goods, the customer will make his choice in the set

      The multimap Φ : Δ → P(

ⁿ) is called the demand multifunction of the customer.

      A more detailed description of an economic model of that type and an application of the multimaps techniques to the finding of an equilibrium in it will be carried out in the fourth chapter.

      (b) Economic dynamics.

      Suppose that in an economic system a vector x_(t) ∈

ⁿ characterizes the collection of goods produced by the moment t during the preceding unit time interval (for example, a year). A part of this collection, y_(t) comes to consumption, whereas the remaining part z_(t) = x_(t) → y_(t) is spend to the accumulation, i.e., serves as the resource for the obtaining a new outcome vector x_(t+1). The pair (y_(t), z_(t)) is called the state of economics at the moment t. By investing the resource z_(t) into accumulation, it is possible to produce by the moment t + 1 one of collections of goods in the frameworks of a certain set B_t(z_(t)) ⊂

ⁿ. The multimap B_t :

ⁿ → P(

ⁿ) called productive characterizes the technology of the system at the moment t. So, starting from the state of economics y_(t), z_(t) it is possible to obtain by the next moment one of states filling the set

      

      Multimaps A_t :

ⁿ ×

ⁿ → P(

ⁿ ×

ⁿ) play an important role in the study of models of mathematical economics.

Example 1.1.17. Non-smooth optimization.

      In contemporary optimization theory it is necessary very often to find the maximums and minimums of functions which are not differentiable. Functions of that kind arise, for example, while the transfer to suprema and infima of families of smooth functions. (So “classical” non-differentiable at zero function y = | x | can be obtained as a supremum of functions y = x and y = −x). For the searching of extrema of such functions, the notion of a derivative must be extended.

      Let, for example, E be a finite-dimensional linear space; f : E →

a convex functional. For a given x ∈ E the set ∂f(x) ⊂ E of all points ξ ∈ E such that for all v ∈ E we have

      is called the subdifferential of a functional f at x.

      So, for a given functional instead of an ordinary derivative we have to deal with a modified derivative, expressed by the multimap x → ∂f(x). The classical Fermat rule in this situation takes the following form: if x₀ is a point of a local extremum of a functional f then 0 ∈ ∂f(x₀).

      It is easy to see that for the function y = | x | the subdifferential is evaluated by the formula:

      Concerning the problems of non-smooth analysis and methods for their solving see the monographs [24], [26], [27], [62], [104], [117], [118], [119], [120], [129], [216], [310], [311], [367] and others.

1.2Continuity of multivalued maps

      He who wants to get to the source must swim against the current.

      —Stanislaw Jerzy Lec

      The classical concept of continuity of a single-valued map splits into different notions when generalized to multimaps and each of these types of continuity has its own specific properties. This variety is based on the fact that the usual set-theoretic notion of the inverse image of a set can be interpreted differently when being applied to multimaps. We will start with the study of this notion.

1.2.1Small and complete preimages of a set

      Let X, Y be sets, F : X → P(Y) a multimap.

      Definition 1.2.1. The small preimage of a set D ⊂ Y is the set

      Definition 1.2.2. The complete preimage