Ebook: Continuous Selections of Multivalued Mappings

This book is dedicated to the theory of continuous selections of multi­ valued mappings, a classical area of mathematics (as far as the formulation of its fundamental problems and methods of solutions are concerned) as well as !'J-n area which has been intensively developing in recent decades and has found various applications in general topology, theory of absolute retracts and infinite-dimensional manifolds, geometric topology, fixed-point theory, functional and convex analysis, game theory, mathematical economics, and other branches of modern mathematics. The fundamental results in this the­ ory were laid down in the mid 1950's by E. Michael. The book consists of (relatively independent) three parts - Part A: Theory, Part B: Results, and Part C: Applications. (We shall refer to these parts simply by their names). The target audience for the first part are students of mathematics (in their senior year or in their first year of graduate school) who wish to get familiar with the foundations of this theory. The goal of the second part is to give a comprehensive survey of the existing results on continuous selections of multivalued mappings. It is intended for specialists in this area as well as for those who have mastered the material of the first part of the book. In the third part we present important examples of applications of continuous selections. We have chosen examples which are sufficiently interesting and have played in some sense key role in the corresponding areas of mathematics.

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This book is the first systematic and comprehensive study of the theory of continuous selections of multivalued mappings. This interesting branch of modern topology was introduced by E.A. Michael in the 1950s and has since witnessed an intensive development with various applications outside topology, e.g. in geometry of Banach spaces, manifolds theory, convex sets, fixed points theory, differential inclusions, optimal control, approximation theory, and mathematical economics.
The work can be used in different ways: the first part is an exposition of the basic theory, with details. The second part is a comprehensive survey of the main results. Lastly, the third part collects various kinds of applications of the theory.
Audience: This volume will be of interest to graduate students and research mathematicians whose work involves general topology, convex sets and related geometric topics, functional analysis, global analysis, analysis on manifolds, manifolds and cell complexes, and mathematical economics.

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This book is the first systematic and comprehensive study of the theory of continuous selections of multivalued mappings. This interesting branch of modern topology was introduced by E.A. Michael in the 1950s and has since witnessed an intensive development with various applications outside topology, e.g. in geometry of Banach spaces, manifolds theory, convex sets, fixed points theory, differential inclusions, optimal control, approximation theory, and mathematical economics.
The work can be used in different ways: the first part is an exposition of the basic theory, with details. The second part is a comprehensive survey of the main results. Lastly, the third part collects various kinds of applications of the theory.
Audience: This volume will be of interest to graduate students and research mathematicians whose work involves general topology, convex sets and related geometric topics, functional analysis, global analysis, analysis on manifolds, manifolds and cell complexes, and mathematical economics.
Content:
Front Matter....Pages i-3
Preliminaries....Pages 5-32
Convex-Valued Selection Theorem....Pages 33-53
Zero-Dimensional Selection Theorem....Pages 54-59
Relations Between Zero-Dimensional and Convex-Valued Selection Theorems....Pages 60-74
Compact-Valued Selection Theorem....Pages 75-83
Finite-Dimensional Selection Theorem....Pages 84-114
Examples and Counterexamples....Pages 115-125
Addendum: New Proof of Finite-Dimensional Selection Theorem....Pages 126-145
Characterization of Normality-Type Properties....Pages 146-154
Unified Selection Theorems....Pages 155-159
Selection Theorems for Non-Lower Semicontinuous Mappings....Pages 160-172
Selection Theorems for Nonconvex-Valued Maps....Pages 173-185
Miscellaneous Results....Pages 186-214
Measurable Selections....Pages 215-231
First Applications....Pages 232-246
Regular Mappings and Locally Trivial Fibrations....Pages 247-260
Fixed-Point Theorems....Pages 261-273
Homeomorphism Group Problem....Pages 274-280
Soft Mappings....Pages 281-299
Metric Projections....Pages 300-310
Back Matter....Pages 330-359
Differential Inclusions....Pages 311-329

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This book is the first systematic and comprehensive study of the theory of continuous selections of multivalued mappings. This interesting branch of modern topology was introduced by E.A. Michael in the 1950s and has since witnessed an intensive development with various applications outside topology, e.g. in geometry of Banach spaces, manifolds theory, convex sets, fixed points theory, differential inclusions, optimal control, approximation theory, and mathematical economics.
The work can be used in different ways: the first part is an exposition of the basic theory, with details. The second part is a comprehensive survey of the main results. Lastly, the third part collects various kinds of applications of the theory.
Audience: This volume will be of interest to graduate students and research mathematicians whose work involves general topology, convex sets and related geometric topics, functional analysis, global analysis, analysis on manifolds, manifolds and cell complexes, and mathematical economics.
Content:
Front Matter....Pages i-3
Preliminaries....Pages 5-32
Convex-Valued Selection Theorem....Pages 33-53
Zero-Dimensional Selection Theorem....Pages 54-59
Relations Between Zero-Dimensional and Convex-Valued Selection Theorems....Pages 60-74
Compact-Valued Selection Theorem....Pages 75-83
Finite-Dimensional Selection Theorem....Pages 84-114
Examples and Counterexamples....Pages 115-125
Addendum: New Proof of Finite-Dimensional Selection Theorem....Pages 126-145
Characterization of Normality-Type Properties....Pages 146-154
Unified Selection Theorems....Pages 155-159
Selection Theorems for Non-Lower Semicontinuous Mappings....Pages 160-172
Selection Theorems for Nonconvex-Valued Maps....Pages 173-185
Miscellaneous Results....Pages 186-214
Measurable Selections....Pages 215-231
First Applications....Pages 232-246
Regular Mappings and Locally Trivial Fibrations....Pages 247-260
Fixed-Point Theorems....Pages 261-273
Homeomorphism Group Problem....Pages 274-280
Soft Mappings....Pages 281-299
Metric Projections....Pages 300-310
Back Matter....Pages 330-359
Differential Inclusions....Pages 311-329
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