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

in Banach spaces and to other actual parts of the theory.

      We would like to express our deep gratitude to our colleagues and friends Aram Arutyunov, Anatoly Baskakov, Irene Benedetti, Yuri Gliklikh, Mikhail Kamenskii, Sergei Kornev, Yeong-Cheng Liou, Zhenhai Liu, Nguyen Van Loi, Luisa Malaguti, Paolo Nistri, Garik Petrosyan, Valentina Taddei, Ngai-Ching Wong, Jen-Chih Yao, Pietro Zecca for valuable and helpful discussions and support while working at this book. We are obliged to Alexei Gel’man and Irina Obukhovskaya for their help in preparation of the manuscript.

      At last, we have the pleasure to express our thanks to the editors of the World Scientific Publishing for their constructive cooperation.

	V. Obukhovskii, B. Gel’man
	Voronezh, February 2020.

      Contents

       Preface

       0.Preliminaries

       1.Multivalued maps

       1.1Some examples

       1.2Continuity of multivalued maps

       1.2.1Small and complete preimages of a set

       1.2.2Upper and lover semicontinuity, continuity, closedness of multimaps

       1.2.3Multivalued maps into a metric space

       1.3Operations on multivalued maps

       1.3.1Set-theoretic operations

       1.3.2Algebraic and other operations

       1.3.3Theorem of maximum

       1.4Continuous selections and approximations of multivalued maps

       1.5Measurable multivalued functions and the superposition multioperator

       1.5.1Measurable multifunctions and a multivalued integral

       1.5.2The Carathéodory conditions and the Filippov implicit function lemma

       1.5.3The superposition multioperator

       2.Fixed points and topological degree

       2.1Fixed points of contractive multimaps

       2.1.1The Nadler theorem

       2.1.2Contractive multimaps depending on a parameter

       2.1.3Equations with surjective linear operators

       2.1.4Inequalities of Caristi type and α-contractive multimaps

       2.1.5Fixed points of weakly α-contractive multimaps

       2.2Topological degree of compact multivalued vector fields

       2.3Topological degree of condensing multivalued vector fields

       2.4Some properties of the fixed point set

       2.5The Browder–Ky Fan fixed point theorem and variational inequalities

       3.Differential inclusions and control systems

       3.1Differential inclusions. Some examples

       3.2Existence theorems and properties of the solution sets

       3.3Periodic solutions of differential inclusions

       3.4Control systems

       4.On some applications

       4.1Generalized dynamical systems

       4.1.1General properties

       4.1.2Rest points of one-sided dynamical systems

       4.2On applications in theory of games and mathematical economics

       4.2.1Optimal strategies in zero-sum games

       4.2.2An equilibrium in a model of a competitive economics

       Bibliographical comments and additions

       Bibliography

       Index

Chapter 0

      Preliminaries

      In order to know something you first need to know something.

      —Stanislaw Lem

      This chapter contains the preliminary information mainly from the general topology which is necessary for the further reading (details may be found, for example, in [1], [131], [135], [241], [270]). A reader familiar with these topics can pass on directly to Chapter 1.

      We will use standard symbols x ∈ X (x ∉ X), X ⊂ Y,

X = Y \ X to denote the belonging (not belonging) of an element