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

The theory of multivalued maps and the theory of differential inclusions are closely connected and intensively developing branches of contemporary mathematics. They have effective and interesting applications in control theory, optimization, calculus of variations, non-smooth and convex analysis, game theory, mathematical economics and in other fields. This book presents a user-friendly and self-contained introduction to both subjects. It is aimed at 'beginners', starting with students of senior courses. The book will be useful both for readers whose interests lie in the sphere of pure mathematics, as well as for those who are involved in applicable aspects of the theory. In Chapter 0, basic definitions and fundamental results in topology are collected. Chapter 1 begins with examples showing how naturally the idea of a multivalued map arises in diverse areas of mathematics, continues with the description of a variety of properties of multivalued maps and finishes with measurable multivalued functions. Chapter 2 is devoted to the theory of fixed points of multivalued maps. The whole of Chapter 3 focuses on the study of differential inclusions and their applications in control theory. The subject of last Chapter 4 is the applications in dynamical systems, game theory, and mathematical economics. The book is completed with the bibliographic commentaries and additions containing the exposition related both to the sections described in the book and to those which left outside its framework. The extensive bibliography (including more than 400 items) leads from basic works to recent studies. Contents: PrefacePreliminariesMultivalued MapsFixed Points and Topological DegreeDifferential Inclusions and Control SystemsOn Some ApplicationsBibliographical Comments and AdditionsBibliographyIndex Readership: Researchers and practitioners in functional analysis, operator theory, topology, differential equations, mathematical control theory, optimization, game theory, mathematical economics and related fields. Graduate and undergraduate students in pure and applied mathematics and in engineering sciences.Multivalued Map;Differential Inclusion;Control System;Optimal Solution;Continuous Selection;Michael Theorem;Single-Valued Approximation;Measurable Multivalued Function;Measurable Selection;Multivalued Integral;Filippov Implicit Function Lemma;Fixed Point;Multivalued Contraction;Nadler Fixed Point Theorem;Topological Degree;Kakutani Fixed Point Theorem;Bohnenblust–Karlin Fixed Point Theorem;Measure of Noncompactness;Condensing Multivalued Map;Variational Inequality;Periodic Solution;Guiding Function;Generalized Dynamical System;Rest Point;Zero-Sum Game;Equilibrium Strategies;Matrix Game;Von Neumann Theorem;Equilibrium in a Competitive Economy0 Key Features: It is a short, clear introduction to various directions of an intensively developing area of contemporary mathematics and accessible for «beginners»It is for experts in «pure» mathematics as well as to scientists interested in applicable aspects of the theory, starting with students of senior coursesIt is useful for researchers in functional analysis, differential equations, optimal control, game theory, mathematical economics and other applied sciences as a self-contained source