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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
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.
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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