Convex and Set-Valued Analysis: Selected Topics

✍ Scribed by Aram V. Arutyunov; Valeri Obukhovskii

• Publisher: De Gruyter
• Year: 2016
• Tongue: English
• Leaves: 210
• Category: Library

✦ Synopsis

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

• Contains many illustrative examples.
• An introduction for mathematicians, but also useful in mathematical economics and engineering.

✦ Table of Contents

Preface
Contents
Part I: Convex analysis
1 Convex sets and their properties
2 The convex hull of a set. The interior of convex sets
3 The affine hull of sets. The relative interior of convex sets
4 Separation theorems for convex sets
5 Convex functions
6 Closedness, boundedness, continuity, and Lipschitz property of convex functions
7 Conjugate functions
8 Support functions
9 Differentiability of convex functions and the subdifferential
10 Convex cones
11 A little more about convex cones in infinite-dimensional spaces
12 A problem of linear programming
13 More about convex sets and convex hulls
Part II: Set-valued analysis
14 Introduction to the theory of topological and metric spaces
15 The Hausdorff metric and the distance between sets
16 Some fine properties of the Hausdorff metric
16.1 Hausdorff distance between sets satisfying the Bolzano–Weierstrass condition
16.2 Hausdorff distance between subsets of normed spaces
17 Set-valued maps. Upper semicontinuous and lower semicontinuous set-valued maps
18 A base of topology of the spaceHc(X)
19 Measurable set-valued maps. Measurable selections and measurable choice theorems
20 The superposition set-valued operator
21 The Michael theorem and continuous selections. Lipschitz selections. Single-valued approximations
22 Special selections of set-valued maps
23 Differential inclusions
24 Fixed points and coincidences of maps in metric spaces
24.1 The case of single-valued maps
24.2 The case of set-valued maps
25 Stability of coincidence points and properties of covering maps
26 Topological degree and fixed points of set-valued maps in Banach spaces
26.1 Topological degree of single-valued maps
26.2 Topological degree of set-valued maps
27 Existence results for differential inclusions via the fixed point method
Notation
Bibliography
Index