Continuous Selections of Multivalued Mappings - Springer Link

Continuous Selections of Multivalued Mappings

Mathematics and Its Applications

Managing Editor:

M. HAZEWINKEL

Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

Volume 455

Continuous Selections of Multivalued Mappings

by

Dusan Repovs

Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia

and

Pavel Vladimirovic Semenov

Department of Mathematics, Moscow City Pedagogical University, Moscow, Russia

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

A C.I.P. Catalogue record for this book is available from the Library of Congress ..

ISBN 978-90-481-5111-0 ISBN 978-94-017-1162-3 (eBook) DOi 10.1007/978-94-017-1162-3

Printed on acid-free paper

All Rights Reserved © 1998 Springer Science+ Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1998 Softcover reprint of the hardcover 1st edition 1998 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner

Contents

Preface............................................................... 1

A. T~~y 5

§0. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

0.1. Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

0.2. Topological vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

0.3. Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

0.4. Extensions of continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

0.5. Multivalued mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

§1. Convex-valued selection theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.1. Paracompactness of the domain as a necessary condition . . . . . . . . . . 33

1.2. The method of outside approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.3. The method of inside approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

1.4. Properties of paracompact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

1.5. Nerves of locally finite coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

1.6. Some properties of paracompact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

§2. Zero-dimensional selection theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.1. Zero-dimensionality of the domain as a necessary condition . . . . . . . 54

2.2. Proof of Zero-dimensional selection theorem . . . . . . . . . . . . . . . . . . . . . . 56

§3. Relations between Zero-dimensional and Convex-valued selection theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.1. Preliminaries. Probabilistic measure and integration . . . . . . . . . . . . . . 60

3.2. Milyutin mappings. Convex-valued selection theorem via Zero-dimen-sional theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.3. Existence of Milyutin mappings on the class of paracompact spaces 69

3.4. Zero-dimensionality of Xo and continuity off . . . . . . . . . . . . . . . . . . . . 71

§4. Compact-valued selection theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.1. Approach via Zero-dimensional theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2. Proof via inside approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.3. Method of coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

vi

§5. Finite-dimensional selection theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.1. en and LC" subsets of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.2. Shift selection theorem. Sketch of the proof . . . . . . . . . . . . . . . . . . . . . . . 91

5.3. Proofs of main lemmas and Controlled contractibility theorem . . . . . 95

5.4. From Nerve-weak shift selection theorem to Shift selection theorem 99

5.5. From Controlled extension theorem to Nerve-weak shift selection theorem ...................................................... 101

5.6. ,Controlled extension theorem ..................................... 104

5.7. Finite-dimensional selection theorem. Uniform relative version .... 108

5.8. From UELCn restrictions to ELCn restrictions. . ................. 112

§6. Examples and counterexamples ....................................... 115

§7. Addendum: New proof of Finite-dimensional selection theorem ....... 126

7.1. Filtered multivalued mappings. Statements of the results .......... 126

7.2. Singlevalued approximations of upper semicontinuous mappings ... 132

7.3. Separations of multivalued mappings. Proof of Theorem (17.6) .... 136

7.4. Enlargements of compact-valued mappings. Proof of Theorem (17. 7) .............................................................. 140

B. Results 146

§1. Characterization of normality-type properties ......................... 146

1.1. Some other convex-valued selection theorems ...................... 146

1.2. Characterizations via compact-valued selection theorems .......... 148

1.3. Dense families of selections. Characterization of perfect normality 150

1.4. Selections of nonclosed-valued equi-LCn mappings ................ 152

§2. Unified selection theorems ............................................ 155

2.1. Union of Finite-dimensional and Convex-valued theorems. Approxi­mative selection properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

2.2. "Countable" type selection theorems and their unions with other selection theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

§3. Selection theorems for non-lower semicontinuous mappings ............ 160

3.1. Lower semicontinuous selections and derived mappings ........... 160

3.2. Almost lower semicontinuity ..................................... 163

3.3. Quasi lower semicontinuity ....................................... 164

3.4. Further generalizations of lower semicontinuity . . . . . . . . . . . . . . . . . . . 169

3.5. Examples ........................................................ 170

vii

§4. Selection theorems for nonconvex-valued maps ................. ,. ...... 173

4.1. Paraconvexity. Function of non-convexity of closed subsets of normed spaces ....................................................... 173

4.2. Axiomatic definition of convex structures in metric spaces. Geodesic structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

4.3. Topological convex structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

4.4. Decomposable subsets of spaces of measurable functions . . . . . . . . . . . 183

§5. Miscellaneous results ................................................. 186

5.1. Metrizability of the range of a multivalued mapping ............... 186

5.2. A weakening of the metrizability of the ranges .................... 190

5.3. Hyperspaces, selections and orderability ........................... 195

5.4. Densely defined selections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

5.5. Continuous multivalued approximations of semicontinuous multival-ued mappings .................. .' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

5.6. Various results on selections ..................................... 207 5. 7. Recent results .................................................... 212

§6. Measurable selections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

6.1. Uniformization problem .......................................... 215

6.2. Measurable multivalued mappings ................................. 218

6.3. Measurable selections of semicontinuous mappings ................ 222

6.4. Caratheodory conditions. Solutions of differential inclusions . . . . . . 225

C. Applications

§1. First applications

232

232

1.1. Extension theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 1.2. Bartle- Graves type theorems. Theory of liftings ................... 233

1.3. Homeomorphism problem for separable Banach spaces ............ 237

1.4. Applications of Zero-dimensional selection theorem ............... 238

1.5. Continuous choice in continuity type definitions .................. 240

1.6. Paracompactness of OW-complexes ............................... 242

1.7. Miscellaneous results ............................................. 243

§2. Regular mappings and locally trivial fibrations ....................... 247

2.1. Dyer-Hamstrom theorem ......................................... 247

2. 2. Regular mappings with fibers homeomorphic to the interval . . . . . . . 249

2.3. Strongly regular mappings ........................................ 252

2.4. Noncompact fibers. Exact Milyutin mappings ..................... 253

viii

§3. Fixed-point theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

3.1. Fixed-point theorems and fixed-point sets for convex-valued mappings ••••••••••••••••••••••••••••••••••• 0 ••• 0 •••••••••••••••••••••• 261

3.2. Fixed-point sets of nonconvex valued mappings ................... 265

3.3. Hilbert space case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

3.4. An application of selections in the finite-dimensional case ......... 270

3.5. Fixed-point theorem for decomposable-valued contractions ......... 271

§4. Ho~eomorphism Group Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4

4.1. Statement of the problem. Solution for n = 1 ..................... 274

4.2. The space of all self-homeomorphisms of the disk ................. 277

§5. Soft mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

5.1. Dugundji spaces and AE(O)-compacta ............................ 281

5.2. Dugundji mappings and 0-soft mappings .......................... 286

5.3. General concept of softness. Adequacy problem . . . . . . . . . . . . . . . . . . . 290

5.4. Parametric versions of Vietoris- Wazewski- Wojdyslawski theorem .. 292

5.5. Functor of probabilistic measures ................................. 295

§6. Metric projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

6.1. Proximinal and Cebysev subsets of normed spaces ................. 300

6.2. Continuity of metric projections and c:-projections . . . . . . . . . . . . . . . . 303

6.3. Continuous self.xtions of metric projections in spaces of continuous functions and Lp-spaces ...................................... 307

6.4. Rational c:-approximations in spaces of continuous functions and Lp-spaces .................................................... 308

§7. Differential inclusions ................................................ 311

7.1. Decomposable sets in functional spaces ........................... 311

7.2. Selection approach to differential inclusions. Preliminary results .. 317

7.3. Selection theorems for decomposable valued mappings ............. 321

7.4. Directionally continuous selections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330

Subject Index ......................................................... 353

1

Preface

This book is dedicated to the theory of continuous selections of multi­valued 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 the­ory 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. The necessary prerequisites can all be found in the first part. It is intended for researchers in general and geometric topology, functional and convex analysis, approximation theory and fixed­-point theory, differen~ial inclusions, and mathematical economics.

The style of exp~sition changes as we pass from one part of the book to another. Proofs in Theory are given in details. Here, our philosophy was to present "the minimum of facts with the maximum of proofs". In Results, however, proofs are, as a rule, omitted or are only sketched. In other words, as it is usual for advanced expositions, we give here "the maximum of facts with the minimum of proofs". Finally, in every paragraph of Applications the part concerning selections is studied in details whereas the rests of the argument is usually only sketched. So the style is of mixed type.

Next, we wish to explain the methodical approach in Theory. We have presented the proofs in some fixed strudurized form. More precisely, every theorem is proved in two steps; Part I: Construction and Part II: Verification. The first part lists all steps of the proof and in the sequel we formulate the necessary properties of the construction. The second part brings a detailed

2

verification of each of the statements of the first part. In this way, an experienced reader can only browse through the first part and then skip the second part altogether, whereas a beginner may well wish to pause after Construction and try to verify all steps by himself. In this way, Construction part can also be regarded as a set of exercises on selection theory.

Consequently, there are no special exercise sections in Theory after each paragraph: instead, we have organized each proof as a sequence of exercises. We have also provided Theory with a separate introduction, where we explain the ways in which multivalued mappings and their continuous selections arise in different areas of mathematics.

Some comments concerning terminology and notations: A multivalued mapping to a space Y can be defined as a singlevalued mapping into a suitable space of subsets of Y. Such approach forces us to introduce a special notation for specific classes of subsets of Y. In the following table we have collected various notations which one can find in the literature:

Classes of subsets of Y Notations

all subsets A(Y), 2y, P(Y), B(Y)

all non'empty subsets 2y, exp Y, N(Y)

closed exp Y, .F(Y), Cl(Y), C(Y)

compact Cp(Y), C(Y), K(Y), Comp(Y), C(Y)

finite F(Y), K(Y), exp~(Y), .1(Y)

convex Cv(Y), C(Y), K(Y), Conv(Y)

closed convex Fc(Y), CC(Y), Cc(Y), CK(Y)

compact convex Kv(Y), CK(Y), ComC(Y), Uk(Y), convO(Y)

complete CMP(Y), II(Y)

bounded B(Y), Bd(Y)

combination of above BdF(Y), IIK(Y), IICK(Y), ...

We have solved the problem of the choice of notations in a very simple way: we did not make any choice. More precisely, we prefer the language instead of abbreviations and we always (except in some places in Results) use phrases of the type "let F : X ---+ Y be a multivalued mapping with closed (compact, bounded, etc.) values ... ". The only general agreement is that all values of any multi valued mapping F : X ---+ Y are nonempty subsets of Y.

According to our decision, we systematically use the notation "F :X---+ Y" and associate with it the term "multivalued mapping", although from purely pedagogical point of view the last term should be related to notions of the type "F : X ---+ 2Y, F : X ---+ Fc(Y), etc." Finally, a word about cross-references in our book: when we are e.g. in Part B: Results and refer to say, Theorem (A.3.9) (resp. Definition (C.7.1)), we mean Theorem (3.9) of Part A: Theory (resp. Definition (7.1) of Part C: Applications).

3

We conclude by some comments concerning the existing literature. There already are some textbooks and monographs where some attention is also given to certain aspects of the theory of selections [16,17,25,30,131,280,298, 326,405]. However, none of them contains a systematic treatment of the

· theory and so to the best of our knowledge, the present monograph is the first one which is devoted exclusively to this subject.

Preliminary versions of the book were read by several of our col­leagues. In particular, we acknowledge remarks by S. M. Ageev, V. Gutev, S. V. Konyagin, V. I. Levin, and E. Michael. The manuscript was prepared using 'lEX by M. Zemljic and we are very grateful for his technical help and assistance through all these years. The first author acknowledges the support of the Ministry for Science and Technology of the Republic of Slovenia grants No. P1-0214-101-92 and No. J1-7039-0101-95, and the second author the sup­port of the International Science G. Soros Foundation grant No. NFUOOO and the Russian Basic Research Foundation grant No. 96-01-01166a.

D. Repovs and P. V. Semenov