Valeri Obukhovskii's Books

Multivalued Maps And Differential Inclusions: Elements Of Theory And Applications

By: Valeri Obukhovskii,Boris Gel'man

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Multivalued Maps And Differential Inclusions: Elements Of Theory And Applications

By: Valeri Obukhovskii,Boris Gel'man

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.

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Page Count

221

Publisher

World Scientific

Published Date

ISBN 10

9811220239

ISBN 13

9789811220234

Convex and Set-Valued Analysis

Selected Topics

By: Aram V. Arutyunov,Valeri Obukhovskii

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Convex and Set-Valued Analysis

Selected Topics

By: Aram V. Arutyunov,Valeri Obukhovskii

This textbook is devoted to a compressed and self-contained exposition of two important parts of contemporary mathematics: convex and set-valued analysis. In the first part, properties of convex sets, the theory of separation, convex functions and their differentiability, properties of convex cones in finite- and infinite-dimensional spaces are discussed. The second part covers some important parts of set-valued analysis. There the properties of the Hausdorff metric and various continuity concepts of set-valued maps are considered. The great attention is paid also to measurable set-valued functions, continuous, Lipschitz and some special types of selections, fixed point and coincidence theorems, covering set-valued maps, topological degree theory and differential inclusions. Contents: Preface Part I: Convex analysis Convex sets and their properties The convex hull of a set. The interior of convex sets The affine hull of sets. The relative interior of convex sets Separation theorems for convex sets Convex functions Closedness, boundedness, continuity, and Lipschitz property of convex functions Conjugate functions Support functions Differentiability of convex functions and the subdifferential Convex cones A little more about convex cones in infinite-dimensional spaces A problem of linear programming More about convex sets and convex hulls Part II: Set-valued analysis Introduction to the theory of topological and metric spaces The Hausdorff metric and the distance between sets Some fine properties of the Hausdorff metric Set-valued maps. Upper semicontinuous and lower semicontinuous set-valued maps A base of topology of the spaceHc(X) Measurable set-valued maps. Measurable selections and measurable choice theorems The superposition set-valued operator The Michael theorem and continuous selections. Lipschitz selections. Single-valued approximations Special selections of set-valued maps Differential inclusions Fixed points and coincidences of maps in metric spaces Stability of coincidence points and properties of covering maps Topological degree and fixed points of set-valued maps in Banach spaces Existence results for differential inclusions via the fixed point method Notation Bibliography Index

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Page Count

244

Publisher

Walter de Gruyter GmbH & Co KG

Published Date

ISBN 10

3110460416

ISBN 13

9783110460414

Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces

By: Mikhail I. Kamenskii,Valeri V. Obukhovskii,Pietro Zecca

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Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces

By: Mikhail I. Kamenskii,Valeri V. Obukhovskii,Pietro Zecca

The theory of set-valued maps and of differential inclusion is developed in recent years both as a field of his own and as an approach to control theory. The book deals with the theory of semilinear differential inclusions in infinite dimensional spaces. In this setting, problems of interest to applications do not suppose neither convexity of the map or compactness of the multi-operators. These assumption implies the development of the theory of measure of noncompactness and the construction of a degree theory for condensing mapping. Of particular interest is the approach to the case when the linear part is a generator of a condensing, strongly continuous semigroup. In this context, the existence of solutions for the Cauchy and periodic problems are proved as well as the topological properties of the solution sets. Examples of applications to the control of transmission line and to hybrid systems are presented.

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Page Count

245

Publisher

Walter de Gruyter

Published Date

ISBN 10

3110870894

ISBN 13

9783110870893

Method of Guiding Functions in Problems of Nonlinear Analysis

By: Valeri Obukhovskii,Pietro Zecca,Nguyen Van Loi,Sergei Kornev

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Method of Guiding Functions in Problems of Nonlinear Analysis

By: Valeri Obukhovskii,Pietro Zecca,Nguyen Van Loi,Sergei Kornev

This book offers a self-contained introduction to the theory of guiding functions methods, which can be used to study the existence of periodic solutions and their bifurcations in ordinary differential equations, differential inclusions and in control theory. It starts with the basic concepts of nonlinear and multivalued analysis, describes the classical aspects of the method of guiding functions, and then presents recent findings only available in the research literature. It describes essential applications in control theory, the theory of bifurcations, and physics, making it a valuable resource not only for “pure” mathematicians, but also for students and researchers working in applied mathematics, the engineering sciences and physics.

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