Convex and Set-Valued Analysis ：Selected Topics ( De Gruyter Textbook )

Publication subTitle ：Selected Topics

Publication series ：De Gruyter Textbook

Author： Arutyunov Aram V.;Obukhovskii Valeri  

Publisher： De Gruyter‎

Publication year： 2016

E-ISBN: 9783110460308

P-ISBN(Paperback): 9783110460285

Subject： O174.13 semilocally convex model theory

Keyword： 最优化的数学理论,数值分析

Language： ENG

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Description

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

Chapter

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

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