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

with the linear operations on X in the following way: 1) the addition operation is continuous, i.e., the map

      is continuous; 2) the number multiplication operation is continuous, i.e., the map

      is continuous.

      If X is a linear topological space and A ⊂ X then the closure of the set coA is denoted by

. It is called the convex closure A and it is a least convex closed set containing A.

      The following Brouwer fixed point theorem holds true. If M is a convex closed subset of a finite-dimensional linear topological space then every continuous map f : M → M such that its range f(M) is bounded has at least one fixed point x ∈ M, x = f(x).

      We will suppose that the reader is familiar with the concepts of normed and Banach spaces as well as with a main information concerning their properties (see, e.g., [120], [124], [247], [371], [384] and others).

      Nevertheless, let us indicate the following facts that we will use in the sequel.

      Let A be a closed subset of a metric space X and Y a normed space. Then each continuous map f : A → Y has a continuous extension : X → Y and, moreover

(the Tietze-Dugundji theorem). The following assertion is an immediate consequence of this result. If Y is a normed space and A is its nonempty closed convex subset then there exists a continuous map (the retraction) r : Y → A such that r(y) = y for all y ∈ A.

      If X is a Banach space and A ⊂ X is a compact set then its convex closure

is also compact (the Mazur theorem).

      If A is a bounded subset of a normed space X then by the norm of the set A we mean the value

      We will assume also that the reader is familiar with the notions of the Lebesgue measure, a measurable and a Bochner integrable function with the values in a Banach space as well as with main properties of the space of integrable functions L¹(see, e.g., [124], [316], [371], [384], [408]).

      The sign := will denote the equality by definition.

      The end of the proof will be marked with the symbol ■.

Chapter 1

      Multivalued maps

      And I claim that it is sufficient to launch any fulcrum into space and to place a ladder to it. The road to heaven is open!

      —Stanislaw Jerzy Lec

1.1Some examples

      Mathematics is the part of physics in which the experiments are very cheap.

      — Vladimir Arnold

      Let X and Y be arbitrary sets; a multivalued map (multimap) F of a set X into a set Y is the correspondence which associates to every x ∈ X a nonempty subset F(x) ⊂ Y, called the value (or the image) of x. Denoting by P(Y) the collection of all nonempty subsets of Y we can write this correspondence as

      It is clear that the class of multivalued maps includes into itself usual single-valued maps: for them each value consists of a single point.

      In the sequel we will denote multimaps by capital letters.

      Definition 1.1.1. For any set A ⊂ X the set

is called the image of the set A under the multimap F.

Definition 1.1.2. Let F : X → P(Y) be a multimap. The set Γ_F in the Cartesian product X × Y,

      is called the graph of the multimap F.

      It is worth noting that the concept of a multimap is not something too unusual: after all, we encounter with maps of this kind already in elementary mathematics when trying to invert, for example, such functions as y = x² or y = sin x and others. However, here the “non-singlevaluedness” of the inverse function is perceived, rather, as a negative circumstance: the introduction of such notions as arithmetic value of the square root, or functions of type arcsin, arccos etc. is related precisely with the “liquidation” of this ambiguity.

      Consider a few examples of multimaps.

      Example 1.1.3. Denote pr₁, pr₂ the projections from X × Y onto X and Y respectively. Each subset Γ ⊂ X × Y such that pr₁ (Γ) = X defines the multimap F : X → P(Y) by the formula

. It is clear that the graph Γ_F of the multimap F coincides with Γ.

Example 1.1.4. Define the multimaps of the interval [0, 1] into itself assuming

      The graphs of these multimaps are presented in Fig. 1–3.

      Fig. 1: Graph F₁

      Fig. 2: Graph F₂

      Fig. 3: Graph F₃

      Denote

.

Example 1.1.5. Define the multimap

, assuming F(x) = [tan x, +∞) (Fig. 4).