Solution Sets for Differential Equations and Inclusions
1st Edition
Published on 6. December 2012
XIX, 453 pages
978-3-11-029356-2 (ISBN)
Description
This monograph gives a systematic presentation of classical and recent results obtained in the last couple of years. It comprehensively describes the methods concerning the topological structure of fixed point sets and solution sets for differential equations and inclusions. Many of the basic techniques and results recently developed about this theory are presented, as well as the literature that is disseminated and scattered in several papers of pioneering researchers who developed the functional analytic framework of this field over the past few decades. Several examples of applications relating to initial and boundary value problems are discussed in detail.
The book is intended to advanced graduate researchers and instructors active in research areas with interests in topological properties of fixed point mappings and applications; it also aims to provide students with the necessary understanding of the subject with no deep background material needed. This monograph fills the vacuum in the literature regarding the topological structure of fixed point sets and its applications.
Reviews / Votes
"This interesting and self-contained book offers both classical and recent results on the existence of solutions for differential equations and inclusions, and on the topological structure of solution sets." Mathematical Reviews
"In this excellent book, a comprehensive description of methods concerning the topological structure of fixed point sets and solution sets for differential equations and inclusions is presented." Zentralblatt für Mathematik
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Smaïl Djebali | Lech Górniewicz | Abdelghani Ouahab
Solution Sets for Differential Equations and Inclusions
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Smaïl Djebali | Lech Górniewicz | Abdelghani Ouahab
Solution Sets for Differential Equations and Inclusions
Book
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1st Edition
De Gruyter
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Persons
Smäil Djebali, Ecole Normale Supérieure, Algiers, Algeria; Lech Górniewicz, Nicolaus Copernicus University, Torun, Poland; Abdelghani Ouahab, Sidi-Bel-Abbès University, Algeria.
Content
1 TOPOLOGICAL STRUCTURE OF FIXED POINT SETS 11
1.1 Case of single-valued mappings . . . . . . . . . . . . . . . . . . . . . . 11
1.1.1 Fundamental ?xed point theorems . . . . . . . . . . . . . . . . . 11
1.1.2 Approximation theorems . . . . . . . . . . . . . . . . . . . . . . 14
1.1.3 Browder{Gupta Theorems . . . . . . . . . . . . . . . . . . . . . 16
1.1.4 Acyclicity of the solution sets of operator equation . . . . . . . 21
1.1.5 Solution sets for nonexpansive maps . . . . . . . . . . . . . . . . 24
1.2 Case of multi-valued mappings . . . . . . . . . . . . . . . . . . . . . . . 25
1.2.1 Fixed point theorems . . . . . . . . . . . . . . . . . . . . . . . . 25
1.2.2 Multivalued contractions . . . . . . . . . . . . . . . . . . . . . . 27
1.2.3 Fixed point sets of multi-valued contractions . . . . . . . . . . . 29
1.2.4 Fixed point sets of multivalued condensing maps . . . . . . . . . 32
1.2.5 Approximation of multi-valued maps . . . . . . . . . . . . . . . 37
1.3 Admissible maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.3.2 Fixed point theorems for admissible multivalued maps . . . . . 48
1.3.3 Browder{Gupta type results for admissible mappings . . . . . . 54
1.4 Topological structure of ?xed point sets of inverse limit maps . . . . . . 58
1.4.1 De?nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
1.4.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
1.4.3 Multi-maps of inverse systems . . . . . . . . . . . . . . . . . . . 60
1.5 Further results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
1.5.1 Semi-compactness in L1 . . . . . . . . . . . . . . . . . . . . . . 63
1.5.2 Decomposability in L1(T;E) . . . . . . . . . . . . . . . . . . . . 64
1.5.3 Michael family of subsets . . . . . . . . . . . . . . . . . . . . . . 66
2 EXISTENCE THEORY FOR DIFFERENTIAL EQUATIONS AND
INCLUSIONS 71
2.1 Case of di (R)erential equations . . . . . . . . . . . . . . . . . . . . . . . . 71
2.1.1 Existence and uniqueness results . . . . . . . . . . . . . . . . . 71
2.1.2 Picard-LindelAEof Theorem . . . . . . . . . . . . . . . . . . . . . . 72
2.1.3 Peano and Carath?eodory theorems . . . . . . . . . . . . . . . . 77
2.1.4 Global existence theorems . . . . . . . . . . . . . . . . . . . . . 79
2.1.5 Existence results on non-compact intervals . . . . . . . . . . . . 82
2.1.6 A boundary value problem on the half-line . . . . . . . . . . . . 89
2.2 Case of di (R)erential inclusions . . . . . . . . . . . . . . . . . . . . . . . 94
2.2.1 Initial value problem . . . . . . . . . . . . . . . . . . . . . . . . 94
2.2.2 A boundary value problem . . . . . . . . . . . . . . . . . . . . . 99
3 SOLUTIONS SETS FOR DIFFERENTIAL EQUATIONS AND IN-
CLUSIONS 105
3.1 Solutions sets for di (R)erential equations . . . . . . . . . . . . . . . . . . 105
3.1.1 Problems on bounded intervals . . . . . . . . . . . . . . . . . . 105
3.1.2 Problems on unbounded intervals . . . . . . . . . . . . . . . . . 107
3.1.3 Kneser-Hukuhara Theorem . . . . . . . . . . . . . . . . . . . . . 109
3.2 Aronszajn-type results for di (R)erential inclusions . . . . . . . . . . . . . 111
3.3 Application to neutral di (R)erential inclusions . . . . . . . . . . . . . . . 118
3.3.1 The convex case . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
3.3.2 The nonconvex case . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.3.3 Solutions sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
3.4 Application to second order di (R)erential inclusions . . . . . . . . . . . . 136
3.4.1 The convex case . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
3.4.2 The nonconvex case . . . . . . . . . . . . . . . . . . . . . . . . . 141
3.4.3 Solution sets to second-order di (R)erential equations . . . . . . . . 144
3.4.4 Solution sets to second-order di (R)erential inclusions . . . . . . . 146
3.5 Application to a nonlocal problem . . . . . . . . . . . . . . . . . . . . . 150
3.5.1 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . 150
3.5.2 Solutions set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
3.6 Application to a nonlocal viability problem . . . . . . . . . . . . . . . . 152
3.6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
3.6.2 Viable solutions on proximate retracts . . . . . . . . . . . . . . 154
3.7 Application to hyperbolic di (R)erential inclusions . . . . . . . . . . . . . 158
3.7.1 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . 158
3.7.2 Solution sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
3.8 Application to abstract Volterra operators . . . . . . . . . . . . . . . . 166
4 IMPULSIVE DIFFERENTIAL INCLUSIONS: EXISTENCE AND
SOLUTION SETS 169
4.1 Impulsive di (R)erential inclusions . . . . . . . . . . . . . . . . . . . . . . 169
4.1.1 C0!Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
4.1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
4.1.3 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . 174
4.1.4 Structure of solution sets . . . . . . . . . . . . . . . . . . . . . . 190
4.2 A periodic problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
4.2.1 Existence results: 1 2 1/2(T(b)) . . . . . . . . . . . . . . . . . . . 203
4.2.2 The convex case: direct approach . . . . . . . . . . . . . . . . . 204
4.2.3 The convex case: MNC approach . . . . . . . . . . . . . . . . . 211
4.2.4 The nonconvex case . . . . . . . . . . . . . . . . . . . . . . . . . 216
4.2.5 The parameter-dependant case . . . . . . . . . . . . . . . . . . 219
4.2.6 Filippov's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 224
4.2.7 Existence of solutions: 1 62 1/2(T(b)) . . . . . . . . . . . . . . . . 232
4.3 Impulsive Functional Di (R)erential Inclusions . . . . . . . . . . . . . . . . 238
4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
4.3.2 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . 239
4.3.3 Structure of the solution set . . . . . . . . . . . . . . . . . . . . 247
4.4 Impulsive di (R)erential inclusions on the half-line . . . . . . . . . . . . . 251
4.4.1 Existence results and compactness of solution sets . . . . . . . . 252
4.4.2 Topological structure via the projective limit . . . . . . . . . . . 266
4.4.3 Using solution sets to prove existence results . . . . . . . . . . . 282
I SUPPLEMENTS 287
5 PRELIMINARY NOTIONS OF TOPOLOGY 289
5.1 Extension and embedding properties . . . . . . . . . . . . . . . . . . . 289
5.2 Homotopical properties of spaces . . . . . . . . . . . . . . . . . . . . . 296
5.3 ?Cech homology (cohomology) functor . . . . . . . . . . . . . . . . . . . 303
5.4 Maps of spaces of ?nite type . . . . . . . . . . . . . . . . . . . . . . . . 304
5.5 ?Cech homology functor with compact carriers . . . . . . . . . . . . . . 311
5.6 Acyclic sets and Vietoris maps . . . . . . . . . . . . . . . . . . . . . . . 313
5.7 Homology of open subsets of Euclidean spaces . . . . . . . . . . . . . . 317
5.8 Lefschetz number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
5.9 Coincidence problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
6 BACKGROUND IN MULTI-VALUED ANALYSIS 335
6.1 Continuity of multivalued mappings . . . . . . . . . . . . . . . . . . . . 337
6.1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
6.1.2 Upper semi-continuity . . . . . . . . . . . . . . . . . . . . . . . 339
6.1.3 Lower semi-continuity . . . . . . . . . . . . . . . . . . . . . . . 344
6.1.4 Hausdor (R) continuity . . . . . . . . . . . . . . . . . . . . . . . . 347
6.2 Selection theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
6.2.1 Partitions of unity . . . . . . . . . . . . . . . . . . . . . . . . . 349
6.2.2 Michael's selection theorem . . . . . . . . . . . . . . . . . . . . 350
6.2.3 3/4!selectionable mappings . . . . . . . . . . . . . . . . . . . . . 353
6.2.4 The Kuratowski-Ryll-Nardzewski selection theorem . . . . . . . 356
6.2.5 Hausdor (R)-measurable multivalued maps . . . . . . . . . . . . . 371
6.2.6 The Scorza-Dragoni property . . . . . . . . . . . . . . . . . . . 373
6.2.7 The Bressan-Colombo-Fryszkowski selection theorem . . . . . . 379
6.3 The Bochner integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
6.3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
6.3.2 Nemytski?i operators . . . . . . . . . . . . . . . . . . . . . . . . 383
6.3.3 Integration of multivalued maps . . . . . . . . . . . . . . . . . . 386
6.4 Compactness in C([a; b];E) and PC([a; b];E) . . . . . . . . . . . . . . . 388
6.5 Further auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . 391
1.1 Case of single-valued mappings . . . . . . . . . . . . . . . . . . . . . . 11
1.1.1 Fundamental ?xed point theorems . . . . . . . . . . . . . . . . . 11
1.1.2 Approximation theorems . . . . . . . . . . . . . . . . . . . . . . 14
1.1.3 Browder{Gupta Theorems . . . . . . . . . . . . . . . . . . . . . 16
1.1.4 Acyclicity of the solution sets of operator equation . . . . . . . 21
1.1.5 Solution sets for nonexpansive maps . . . . . . . . . . . . . . . . 24
1.2 Case of multi-valued mappings . . . . . . . . . . . . . . . . . . . . . . . 25
1.2.1 Fixed point theorems . . . . . . . . . . . . . . . . . . . . . . . . 25
1.2.2 Multivalued contractions . . . . . . . . . . . . . . . . . . . . . . 27
1.2.3 Fixed point sets of multi-valued contractions . . . . . . . . . . . 29
1.2.4 Fixed point sets of multivalued condensing maps . . . . . . . . . 32
1.2.5 Approximation of multi-valued maps . . . . . . . . . . . . . . . 37
1.3 Admissible maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.3.2 Fixed point theorems for admissible multivalued maps . . . . . 48
1.3.3 Browder{Gupta type results for admissible mappings . . . . . . 54
1.4 Topological structure of ?xed point sets of inverse limit maps . . . . . . 58
1.4.1 De?nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
1.4.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
1.4.3 Multi-maps of inverse systems . . . . . . . . . . . . . . . . . . . 60
1.5 Further results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
1.5.1 Semi-compactness in L1 . . . . . . . . . . . . . . . . . . . . . . 63
1.5.2 Decomposability in L1(T;E) . . . . . . . . . . . . . . . . . . . . 64
1.5.3 Michael family of subsets . . . . . . . . . . . . . . . . . . . . . . 66
2 EXISTENCE THEORY FOR DIFFERENTIAL EQUATIONS AND
INCLUSIONS 71
2.1 Case of di (R)erential equations . . . . . . . . . . . . . . . . . . . . . . . . 71
2.1.1 Existence and uniqueness results . . . . . . . . . . . . . . . . . 71
2.1.2 Picard-LindelAEof Theorem . . . . . . . . . . . . . . . . . . . . . . 72
2.1.3 Peano and Carath?eodory theorems . . . . . . . . . . . . . . . . 77
2.1.4 Global existence theorems . . . . . . . . . . . . . . . . . . . . . 79
2.1.5 Existence results on non-compact intervals . . . . . . . . . . . . 82
2.1.6 A boundary value problem on the half-line . . . . . . . . . . . . 89
2.2 Case of di (R)erential inclusions . . . . . . . . . . . . . . . . . . . . . . . 94
2.2.1 Initial value problem . . . . . . . . . . . . . . . . . . . . . . . . 94
2.2.2 A boundary value problem . . . . . . . . . . . . . . . . . . . . . 99
3 SOLUTIONS SETS FOR DIFFERENTIAL EQUATIONS AND IN-
CLUSIONS 105
3.1 Solutions sets for di (R)erential equations . . . . . . . . . . . . . . . . . . 105
3.1.1 Problems on bounded intervals . . . . . . . . . . . . . . . . . . 105
3.1.2 Problems on unbounded intervals . . . . . . . . . . . . . . . . . 107
3.1.3 Kneser-Hukuhara Theorem . . . . . . . . . . . . . . . . . . . . . 109
3.2 Aronszajn-type results for di (R)erential inclusions . . . . . . . . . . . . . 111
3.3 Application to neutral di (R)erential inclusions . . . . . . . . . . . . . . . 118
3.3.1 The convex case . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
3.3.2 The nonconvex case . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.3.3 Solutions sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
3.4 Application to second order di (R)erential inclusions . . . . . . . . . . . . 136
3.4.1 The convex case . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
3.4.2 The nonconvex case . . . . . . . . . . . . . . . . . . . . . . . . . 141
3.4.3 Solution sets to second-order di (R)erential equations . . . . . . . . 144
3.4.4 Solution sets to second-order di (R)erential inclusions . . . . . . . 146
3.5 Application to a nonlocal problem . . . . . . . . . . . . . . . . . . . . . 150
3.5.1 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . 150
3.5.2 Solutions set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
3.6 Application to a nonlocal viability problem . . . . . . . . . . . . . . . . 152
3.6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
3.6.2 Viable solutions on proximate retracts . . . . . . . . . . . . . . 154
3.7 Application to hyperbolic di (R)erential inclusions . . . . . . . . . . . . . 158
3.7.1 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . 158
3.7.2 Solution sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
3.8 Application to abstract Volterra operators . . . . . . . . . . . . . . . . 166
4 IMPULSIVE DIFFERENTIAL INCLUSIONS: EXISTENCE AND
SOLUTION SETS 169
4.1 Impulsive di (R)erential inclusions . . . . . . . . . . . . . . . . . . . . . . 169
4.1.1 C0!Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
4.1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
4.1.3 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . 174
4.1.4 Structure of solution sets . . . . . . . . . . . . . . . . . . . . . . 190
4.2 A periodic problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
4.2.1 Existence results: 1 2 1/2(T(b)) . . . . . . . . . . . . . . . . . . . 203
4.2.2 The convex case: direct approach . . . . . . . . . . . . . . . . . 204
4.2.3 The convex case: MNC approach . . . . . . . . . . . . . . . . . 211
4.2.4 The nonconvex case . . . . . . . . . . . . . . . . . . . . . . . . . 216
4.2.5 The parameter-dependant case . . . . . . . . . . . . . . . . . . 219
4.2.6 Filippov's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 224
4.2.7 Existence of solutions: 1 62 1/2(T(b)) . . . . . . . . . . . . . . . . 232
4.3 Impulsive Functional Di (R)erential Inclusions . . . . . . . . . . . . . . . . 238
4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
4.3.2 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . 239
4.3.3 Structure of the solution set . . . . . . . . . . . . . . . . . . . . 247
4.4 Impulsive di (R)erential inclusions on the half-line . . . . . . . . . . . . . 251
4.4.1 Existence results and compactness of solution sets . . . . . . . . 252
4.4.2 Topological structure via the projective limit . . . . . . . . . . . 266
4.4.3 Using solution sets to prove existence results . . . . . . . . . . . 282
I SUPPLEMENTS 287
5 PRELIMINARY NOTIONS OF TOPOLOGY 289
5.1 Extension and embedding properties . . . . . . . . . . . . . . . . . . . 289
5.2 Homotopical properties of spaces . . . . . . . . . . . . . . . . . . . . . 296
5.3 ?Cech homology (cohomology) functor . . . . . . . . . . . . . . . . . . . 303
5.4 Maps of spaces of ?nite type . . . . . . . . . . . . . . . . . . . . . . . . 304
5.5 ?Cech homology functor with compact carriers . . . . . . . . . . . . . . 311
5.6 Acyclic sets and Vietoris maps . . . . . . . . . . . . . . . . . . . . . . . 313
5.7 Homology of open subsets of Euclidean spaces . . . . . . . . . . . . . . 317
5.8 Lefschetz number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
5.9 Coincidence problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
6 BACKGROUND IN MULTI-VALUED ANALYSIS 335
6.1 Continuity of multivalued mappings . . . . . . . . . . . . . . . . . . . . 337
6.1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
6.1.2 Upper semi-continuity . . . . . . . . . . . . . . . . . . . . . . . 339
6.1.3 Lower semi-continuity . . . . . . . . . . . . . . . . . . . . . . . 344
6.1.4 Hausdor (R) continuity . . . . . . . . . . . . . . . . . . . . . . . . 347
6.2 Selection theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
6.2.1 Partitions of unity . . . . . . . . . . . . . . . . . . . . . . . . . 349
6.2.2 Michael's selection theorem . . . . . . . . . . . . . . . . . . . . 350
6.2.3 3/4!selectionable mappings . . . . . . . . . . . . . . . . . . . . . 353
6.2.4 The Kuratowski-Ryll-Nardzewski selection theorem . . . . . . . 356
6.2.5 Hausdor (R)-measurable multivalued maps . . . . . . . . . . . . . 371
6.2.6 The Scorza-Dragoni property . . . . . . . . . . . . . . . . . . . 373
6.2.7 The Bressan-Colombo-Fryszkowski selection theorem . . . . . . 379
6.3 The Bochner integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
6.3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
6.3.2 Nemytski?i operators . . . . . . . . . . . . . . . . . . . . . . . . 383
6.3.3 Integration of multivalued maps . . . . . . . . . . . . . . . . . . 386
6.4 Compactness in C([a; b];E) and PC([a; b];E) . . . . . . . . . . . . . . . 388
6.5 Further auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . 391
