Multivalued Maps and Differential Inclusions
Elements of Theory and Applications
Multivalued Maps and Differential Inclusions
Elements of Theory and Applications
Valeri Obukhovskii
Voronezh State Pedagogical University, Russia
Boris Gel’man
Voronezh State University, Russia
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MULTIVALUED MAPS AND DIFFERENTIAL INCLUSIONS
Elements of Theory and Applications
Copyright © 2020 by World Scientific Publishing Co. Pte. Ltd.
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To Irina, Andrey, Alexey and Barbara, Maxim and Sergey with love...
Preface
The theory of multivalued maps and the theory of differential inclusions are closely related branches of contemporary mathematics which stood out as independent scientific disciplines around the middle of the last century. At the present time they continue to develop very rapidly, finding new supporters and new applications. A highly effective use of ideas and methods of the multivalued analysis and the theory of differential inclusions in such directions as theory of optimization and control, calculus of variations, nonsmooth and convex analysis, theory of differential equations, theory of games, mathematical economics and others, has become generally recognized and attracts the attention of many researchers all over the world.
In spite of several monographs and a huge number of other publications, in our opinion, the idea of a small book that provides a fairly basic introduction to the subject for “beginners”, starting with students of senior courses and graduate students, is sufficiently relevant. We hope that the given book will be of interest both for theorists and persons who are involved in applied aspects of science.
Basing on the stuff included in this book, the authors were repeatedly lecturing for students of the Voronezh State University and the Voronezh State Pedagogical University.
The main exposition is preceded with the zero chapter, where necessary definitions and preliminary information, mainly from general topology are presented.
The first chapter of the book begins with examples showing how naturally the idea of a multivalued map arises in various branches of mathematics. Further, the types of continuity of multivalued maps as well as various operations over them and their properties are described. Then the concept of a continuous single-valued selection of a multivalued map is introduced and the classical Michael continuous selection theorem is proved. The concept of a single-valued approximation is discussed and the corresponding existence theorem is presented. We give analogues of above statements for multivalued maps with decomposable values. The first chapter concludes with the description of the properties of measurable multivalued functions. We give here the proof of the Filippov implicit function lemma which is known by its applications in control theory and study in detail the properties of the multivalued superposition operator which will be used while the discussing of differential inclusions. A statement on properties of a multivalued integral and the description of a superposition multioperator generated by an almost lower semicontinuous multimap are given.
The second chapter is devoted to the fixed point theory of multivalued maps. We produce here the Nadler theorem, a multivalued analogue of the classical Banach contraction map principle. We consider the properties of contraction multivalued maps depending on a parameter and describe the topological structure of the set of fixed points for a contractive multimap. Further, fixed point results are applied to the study of some classes of equations with surjective linear operators. We give also versions of the Caristi and Nemytski fixed point theorems. Further, we expose the topological degree theory for compact multivalued vector fields with convex values in a Banach space and present its applications to a number of fixed point results including the known Kakutani–Bohnenblust–Karlin theorem. Then the generalization of this theory to the case of condensing multivalued vector fields is described. The chapter is concluded with the study of topological properties of fixed point sets and the Browder–Ky Fan fixed point theorem and its application to the solving of variational inequalities.
The entire third chapter is devoted to the study of differential inclusions and their applications in control theory. We start with a series of examples, illustrating the appearance of differential inclusions in the description of control systems, in differential equations with discontinuous right-hand part and in mathematical economics. Then, based on the topological methods developed in the previous chapter, we give theorems on the existence of a solution to the Cauchy problem for differential inclusions of various types and differential equations with discontinuous right-hand part and describe the properties of solution sets. The topological degree theory finds the systematic application in the study of a periodic problem for differential inclusions. In particular, we expose in sufficient detail the method of guiding functions which at the present time is one of the most effective tools for the solving of periodic problems. Next we consider the question on the equivalence of control systems and differential inclusions and consider on that base applications to some optimization problems.
The last chapter is devoted to applications in the theory of dynamical systems, theory of games and mathematical economics. We describe main properties of generalized dynamical systems and their trajectories. This section incudes also the question on rest points of dynamical systems, which is solved by using the technique of the fixed point theory. Further, by applying a fixed point theorem for multivalued maps we prove the general equilibrium theorem in a two-person game which yields, as a consequences, the classical results on equilibrium in a zero-sum game and in a matrix game. It is shown also how the same fixed point methods provide the existence of an equilibrium in the Arrow–Debreu–McKenzie model of a competitive economy.
The book is concluded with the chapter named “Bibliographical comments and additions”. We provide here fairly detailed comments that relate both to the sections described in the book and to those which left outside of its framework. For example, we give clues to the topological degree theory for multivalued maps with non-convex values, to the theory of differential