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Solution Sets for Differential Equations and Inclusions

E-BookPDFDRM AdobeE-Book
472 Seiten
Englisch
De Gruytererschienen am06.12.20121. Auflage

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.



Smäil Djebali, Ecole Normale Supérieure, Algiers, Algeria;Lech Górniewicz, Nicolaus Copernicus University, Torun, Poland; Abdelghani Ouahab, Sidi-Bel-Abbès University,Algeria.mehr
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Produkt

Klappentext
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.



Smäil Djebali, Ecole Normale Supérieure, Algiers, Algeria;Lech Górniewicz, Nicolaus Copernicus University, Torun, Poland; Abdelghani Ouahab, Sidi-Bel-Abbès University,Algeria.
Details
Weitere ISBN/GTIN9783110293562
ProduktartE-Book
EinbandartE-Book
FormatPDF
Format HinweisDRM Adobe
FormatE107
Erscheinungsjahr2012
Erscheinungsdatum06.12.2012
Auflage1. Auflage
ReiheDe Gruyter Series in Nonlinear Analysis and Applications
Reihen-Nr.18
Seiten472 Seiten
SpracheEnglisch
Illustrationen3 b/w ill.
Artikel-Nr.1196967
Rubriken
Genre9200

Inhalt/Kritik

Inhaltsverzeichnis
1;Preface;7
2;Notations;11
3;1 Topological structure of fixed point sets;21
3.1;1.1 Case of single-valued mappings;21
3.1.1;1.1.1 Fundamental fixed point theorems;21
3.1.1.1;1.1.1.1 Banach´s fixed point theorem;21
3.1.1.2;1.1.1.2 Brouwer´s fixed point theorem;25
3.1.1.3;1.1.1.3 Schauder´s fixed point theorem;27
3.1.2;1.1.2 Approximation theorems;31
3.1.3;1.1.3 Browder-Gupta theorems;33
3.1.4;1.1.4 Acyclicity of the solution sets of operator equations;40
3.1.5;1.1.5 Nonexpansive maps;43
3.1.5.1;1.1.5.1 Existence theory;43
3.1.5.2;1.1.5.2 Solution sets;46
3.2;1.2 The case of multi-valued mappings;47
3.2.1;1.2.1 Approximation of multi-valued maps;47
3.2.2;1.2.2 Fixed point theorems;50
3.2.3;1.2.3 Multi-valued contractions;53
3.2.4;1.2.4 Fixed point sets of multi-valued contractions;55
3.2.5;1.2.5 Fixed point sets of multi-valued nonexpansive maps;58
3.2.6;1.2.6 Fixed point sets of multi-valued condensing maps;59
3.2.6.1;1.2.6.1 Measure of noncompactness;59
3.2.6.2;1.2.6.2 Condensing maps;63
3.3;1.3 Admissible maps;64
3.3.1;1.3.1 Generalities;64
3.3.2;1.3.2 Fixed point theorems for admissible multi-valued maps;73
3.3.3;1.3.3 The general Brouwer fixed point theorem;78
3.3.4;1.3.4 Browder-Gupta type results for admissible mappings;80
3.3.5;1.3.5 Topological dimensions of solution sets;82
3.4;1.4 Topological structure of fixed point sets of inverse limit maps;85
3.4.1;1.4.1 Definition;85
3.4.2;1.4.2 Basic properties;86
3.4.3;1.4.3 Multi-maps of inverse systems;87
4;2 Existence theory for differential equations and inclusions;92
4.1;2.1 Fundamental theorems;92
4.1.1;2.1.1 Existence and uniqueness results;92
4.1.2;2.1.2 Picard-Lindelöf theorem;93
4.1.2.1;2.1.2.1 Maximal solutions;95
4.1.3;2.1.3 Peano and Carathéodory theorems;97
4.1.3.1;2.1.3.1 Peano theorem;97
4.2;2.2 The extendability problem;99
4.2.1;2.2.1 Global existence theorems;99
4.2.2;2.2.2 Existence results on noncompact intervals;102
4.2.2.1;2.2.2.1 The Lipschitz case;102
4.2.2.2;2.2.2.2 The Lipschitz-Nagumo case;103
4.2.2.3;2.2.2.3 The Nagumo case;106
4.2.3;2.2.3 A boundary value problem on the half-line;108
4.3;2.3 The case of differential inclusions;114
4.3.1;2.3.1 Initial value problems;114
4.3.1.1;2.3.1.1 A Nagumo type nonlinearity;114
4.3.1.2;2.3.1.2 A Lipschitz nonconvex nonlinearity;117
4.3.2;2.3.2 Boundary value problems;119
4.3.2.1;2.3.2.1 The convex case;120
4.3.2.2;2.3.2.2 The nonconvex case;123
5;3 Solution sets for differential equations and inclusions;125
5.1;3.1 General results;125
5.1.1;3.1.1 Kneser-Hukuhara theorem;125
5.1.2;3.1.2 Problems on bounded intervals;128
5.1.3;3.1.3 Problems on unbounded intervals;129
5.1.4;3.1.4 Second-order differential equations;131
5.1.5;3.1.5 Abstract Volterra equations;133
5.1.6;3.1.6 Aronszajn type results for differential inclusions;134
5.2;3.2 Second-order differential inclusions;142
5.2.1;3.2.1 The convex case;142
5.2.2;3.2.2 The nonconvex case;147
5.2.3;3.2.3 Solution sets;150
5.3;3.3 Higher-order differential inclusions;154
5.4;3.4 Neutral differential inclusions;155
5.4.1;3.4.1 The convex case;156
5.4.2;3.4.2 The nonconvex case;162
5.4.3;3.4.3 Solutions sets;166
5.5;3.5 Nonlocal problems;166
5.5.1;3.5.1 Main results;167
5.5.2;3.5.2 A viability problem;169
5.6;3.6 Hyperbolic differential inclusions;174
5.6.1;3.6.1 Existence results;175
5.6.1.1;3.6.1.1 The convex case;175
5.6.1.2;3.6.1.2 The nonconvex case;179
5.6.2;3.6.2 Solution sets;180
6;4 Impulsive differential inclusions: existence and solution sets;183
6.1;4.1 Motivation;183
6.1.1;4.1.1 Ecological model with impulsive control strategy;183
6.1.2;4.1.2 Leslie predator-prey system;184
6.1.3;4.1.3 Pulse vaccination model;185
6.2;4.2 Semi-linear impulsive differential inclusions;186
6.2.1;4.2.1 Existence results;186
6.2.1.1;4.2.1.1 The convex case;187
6.2.1.2;4.2.1.2 The nonconvex case;201
6.2.2;4.2.2 Structure of solution sets;206
6.3;4.3 A periodic problem;217
6.3.1;4.3.1 Existence results: 1 . .(T(b));218
6.3.2;4.3.2 The convex case: a direct approach;219
6.3.3;4.3.3 The convex case: an MNC approach;227
6.3.4;4.3.4 The nonconvex case;232
6.3.5;4.3.5 The parameter-dependant case;235
6.3.5.1;4.3.5.1 The convex case;235
6.3.5.2;4.3.5.2 The nonconvex case;237
6.3.6;4.3.6 Filippov´s Theorem;240
6.3.7;4.3.7 Existence of solutions: 1 . .(T(b));250
6.3.7.1;4.3.7.1 A nonlinear alternative;250
6.3.7.2;4.3.7.2 A Poincare translation operator;253
6.3.7.3;4.3.7.3 The MNC approach;253
6.4;4.4 Impulsive functional differential inclusions;256
6.4.1;4.4.1 Introduction;256
6.4.2;4.4.2 Existence results;257
6.4.3;4.4.3 Structure of the solution set;266
6.5;4.5 Impulsive differential inclusions on the half-line;270
6.5.1;4.5.1 Existence results and compactness of solution sets;270
6.5.1.1;4.5.1.1 The convex u.s.c. case;271
6.5.1.2;4.5.1.2 The nonconvex Lipschitz case;278
6.5.1.3;4.5.1.3 The nonconvex l.s.c. case;282
6.5.2;4.5.2 Topological structure via the projective limit;285
6.5.2.1;4.5.2.1 The nonconvex case;286
6.5.2.2;4.5.2.2 The convex case;291
6.5.2.3;4.5.2.3 The terminal problem;294
6.5.3;4.5.3 Using solution sets to prove existence results;303
7;5 Preliminary notions of topology and homology;308
7.1;5.1 Retracts, extension and embedding properties;308
7.2;5.2 Absolute retracts;314
7.3;5.3 Homotopical properties of spaces;316
7.4;5.4 Cech homology (cohomology) functor;324
7.5;5.5 Maps of spaces of finite type;326
7.6;5.6 Cech homology functor with compact carriers;333
7.7;5.7 Acyclic sets and Vietoris maps;335
7.8;5.8 Homology of open subsets of Euclidean spaces;339
7.9;5.9 Lefschetz number;343
7.10;5.10 The coincidence problem;350
8;6 Background in multi-valued analysis;357
8.1;6.1 Continuity of multi-valued mappings;359
8.1.1;6.1.1 Basic notions;359
8.1.2;6.1.2 Upper semi-continuity;361
8.1.2.1;6.1.2.1 Generalities;361
8.1.2.2;6.1.2.2 . - d u.s.c. mappings;364
8.1.2.3;6.1.2.3 U.s.c. maps and closed graphs;365
8.1.3;6.1.3 Lower semi-continuity;366
8.1.3.1;6.1.3.1 Generalities;366
8.1.3.2;6.1.3.2 . - d l.s.c. mappings;369
8.1.4;6.1.4 Hausdorff continuity;370
8.2;6.2 The selection problem;374
8.2.1;6.2.1 Michael´s selection theorem;375
8.2.2;6.2.2 Michael´s family of subsets;378
8.2.3;6.2.3 s-selectionable mappings;382
8.2.4;6.2.4 The Kuratowski-Ryll-Nardzewski selection theorem;386
8.2.5;6.2.5 Aumann and Filippov theorems;398
8.2.6;6.2.6 Hausdorff measurable multi-valued maps;402
8.2.7;6.2.7 Product-measurability and the Scorza-Dragoni property;403
8.3;6.3 Decomposable sets;410
8.3.1;6.3.1 The Bressan-Colombo-Fryszkowski selection theorem;410
8.3.2;6.3.2 Decomposability in L1(T,E);410
8.3.3;6.3.3 Integration of multi-valued maps;412
8.3.4;6.3.4 Nemytski operators;413
9;Appendix;419
9.1;A.1 Axioms of the Cech homology theory;419
9.2;A.2 The Bochner integral;420
9.3;A.3 Absolutely continuous functions;423
9.4;A.4 Compactness criteria in C([a,b], E), Cb([0,8), E), and PC ([a, b], E);425
9.5;A.5 Weak-compactness in L1;428
9.6;A.6 Proper maps and vector fields;430
9.7;A.7 Fundamental theorems in functional analysis;431
9.8;A.8 C0-Semigroups;432
10;References;435
11;Index;471mehr

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