Solution Sets for Differential Equations and Inclusions

Inbunden, Engelska, 2012

Av Smaïl Djebali, Lech Górniewicz, Abdelghani Ouahab, Lech Djebali, Smaïl

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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.

Produktinformation

Utgivningsdatum2012-11-15
Mått170 x 240 x 31 mm
Vikt926 g
FormatInbunden
SpråkEngelska
SerieDe Gruyter Series in Nonlinear Analysis & Applications
Antal sidor472
Upplaga12001
FörlagDe Gruyter
ISBN9783110293449

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1 TOPOLOGICAL STRUCTURE OF FIXED POINT SETS 111.1 Case of single-valued mappings . . . . . . . . . . . . . . . . . . . . . . 111.1.1 Fundamental ¯xed point theorems . . . . . . . . . . . . . . . . . 111.1.2 Approximation theorems . . . . . . . . . . . . . . . . . . . . . . 141.1.3 Browder{Gupta Theorems . . . . . . . . . . . . . . . . . . . . . 161.1.4 Acyclicity of the solution sets of operator equation . . . . . . . 211.1.5 Solution sets for nonexpansive maps . . . . . . . . . . . . . . . . 241.2 Case of multi-valued mappings . . . . . . . . . . . . . . . . . . . . . . . 251.2.1 Fixed point theorems . . . . . . . . . . . . . . . . . . . . . . . . 251.2.2 Multivalued contractions . . . . . . . . . . . . . . . . . . . . . . 271.2.3 Fixed point sets of multi-valued contractions . . . . . . . . . . . 291.2.4 Fixed point sets of multivalued condensing maps . . . . . . . . . 321.2.5 Approximation of multi-valued maps . . . . . . . . . . . . . . . 371.3 Admissible maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401.3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401.3.2 Fixed point theorems for admissible multivalued maps . . . . . 481.3.3 Browder{Gupta type results for admissible mappings . . . . . . 541.4 Topological structure of ¯xed point sets of inverse limit maps . . . . . . 581.4.1 De¯nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581.4.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 581.4.3 Multi-maps of inverse systems . . . . . . . . . . . . . . . . . . . 601.5 Further results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631.5.1 Semi-compactness in L1 . . . . . . . . . . . . . . . . . . . . . . 631.5.2 Decomposability in L1(T;E) . . . . . . . . . . . . . . . . . . . . 641.5.3 Michael family of subsets . . . . . . . . . . . . . . . . . . . . . . 662 EXISTENCE THEORY FOR DIFFERENTIAL EQUATIONS ANDINCLUSIONS 712.1 Case of di®erential equations . . . . . . . . . . . . . . . . . . . . . . . . 712.1.1 Existence and uniqueness results . . . . . . . . . . . . . . . . . 712.1.2 Picard-LindelÄof Theorem . . . . . . . . . . . . . . . . . . . . . . 722.1.3 Peano and Carath¶eodory theorems . . . . . . . . . . . . . . . . 772.1.4 Global existence theorems . . . . . . . . . . . . . . . . . . . . . 792.1.5 Existence results on non-compact intervals . . . . . . . . . . . . 822.1.6 A boundary value problem on the half-line . . . . . . . . . . . . 892.2 Case of di®erential inclusions . . . . . . . . . . . . . . . . . . . . . . . 942.2.1 Initial value problem . . . . . . . . . . . . . . . . . . . . . . . . 942.2.2 A boundary value problem . . . . . . . . . . . . . . . . . . . . . 993 SOLUTIONS SETS FOR DIFFERENTIAL EQUATIONS AND IN-CLUSIONS 1053.1 Solutions sets for di®erential equations . . . . . . . . . . . . . . . . . . 1053.1.1 Problems on bounded intervals . . . . . . . . . . . . . . . . . . 1053.1.2 Problems on unbounded intervals . . . . . . . . . . . . . . . . . 1073.1.3 Kneser-Hukuhara Theorem . . . . . . . . . . . . . . . . . . . . . 1093.2 Aronszajn-type results for di®erential inclusions . . . . . . . . . . . . . 1113.3 Application to neutral di®erential inclusions . . . . . . . . . . . . . . . 1183.3.1 The convex case . . . . . . . . . . . . . . . . . . . . . . . . . . . 1193.3.2 The nonconvex case . . . . . . . . . . . . . . . . . . . . . . . . . 1253.3.3 Solutions sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1293.4 Application to second order di®erential inclusions . . . . . . . . . . . . 1363.4.1 The convex case . . . . . . . . . . . . . . . . . . . . . . . . . . . 1373.4.2 The nonconvex case . . . . . . . . . . . . . . . . . . . . . . . . . 1413.4.3 Solution sets to second-order di®erential equations . . . . . . . . 1443.4.4 Solution sets to second-order di®erential inclusions . . . . . . . 1463.5 Application to a nonlocal problem . . . . . . . . . . . . . . . . . . . . . 1503.5.1 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . 1503.5.2 Solutions set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1513.6 Application to a nonlocal viability problem . . . . . . . . . . . . . . . . 1523.6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1523.6.2 Viable solutions on proximate retracts . . . . . . . . . . . . . . 1543.7 Application to hyperbolic di®erential inclusions . . . . . . . . . . . . . 1583.7.1 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . 1583.7.2 Solution sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1643.8 Application to abstract Volterra operators . . . . . . . . . . . . . . . . 1664 IMPULSIVE DIFFERENTIAL INCLUSIONS: EXISTENCE ANDSOLUTION SETS 1694.1 Impulsive di®erential inclusions . . . . . . . . . . . . . . . . . . . . . . 1694.1.1 C0¡Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 1704.1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1724.1.3 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . 1744.1.4 Structure of solution sets . . . . . . . . . . . . . . . . . . . . . . 1904.2 A periodic problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2034.2.1 Existence results: 1 2 ½(T(b)) . . . . . . . . . . . . . . . . . . . 2034.2.2 The convex case: direct approach . . . . . . . . . . . . . . . . . 2044.2.3 The convex case: MNC approach . . . . . . . . . . . . . . . . . 2114.2.4 The nonconvex case . . . . . . . . . . . . . . . . . . . . . . . . . 2164.2.5 The parameter-dependant case . . . . . . . . . . . . . . . . . . 2194.2.6 Filippov's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 2244.2.7 Existence of solutions: 1 62 ½(T(b)) . . . . . . . . . . . . . . . . 2324.3 Impulsive Functional Di®erential Inclusions . . . . . . . . . . . . . . . . 2384.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2384.3.2 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . 2394.3.3 Structure of the solution set . . . . . . . . . . . . . . . . . . . . 2474.4 Impulsive di®erential inclusions on the half-line . . . . . . . . . . . . . 2514.4.1 Existence results and compactness of solution sets . . . . . . . . 2524.4.2 Topological structure via the projective limit . . . . . . . . . . . 2664.4.3 Using solution sets to prove existence results . . . . . . . . . . . 282I SUPPLEMENTS 2875 PRELIMINARY NOTIONS OF TOPOLOGY 2895.1 Extension and embedding properties . . . . . . . . . . . . . . . . . . . 2895.2 Homotopical properties of spaces . . . . . . . . . . . . . . . . . . . . . 2965.3 ·Cech homology (cohomology) functor . . . . . . . . . . . . . . . . . . . 3035.4 Maps of spaces of ¯nite type . . . . . . . . . . . . . . . . . . . . . . . . 3045.5 ·Cech homology functor with compact carriers . . . . . . . . . . . . . . 3115.6 Acyclic sets and Vietoris maps . . . . . . . . . . . . . . . . . . . . . . . 3135.7 Homology of open subsets of Euclidean spaces . . . . . . . . . . . . . . 3175.8 Lefschetz number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3215.9 Coincidence problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3276 BACKGROUND IN MULTI-VALUED ANALYSIS 3356.1 Continuity of multivalued mappings . . . . . . . . . . . . . . . . . . . . 3376.1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3376.1.2 Upper semi-continuity . . . . . . . . . . . . . . . . . . . . . . . 3396.1.3 Lower semi-continuity . . . . . . . . . . . . . . . . . . . . . . . 3446.1.4 Hausdor® continuity . . . . . . . . . . . . . . . . . . . . . . . . 3476.2 Selection theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3496.2.1 Partitions of unity . . . . . . . . . . . . . . . . . . . . . . . . . 3496.2.2 Michael's selection theorem . . . . . . . . . . . . . . . . . . . . 3506.2.3 ¾¡selectionable mappings . . . . . . . . . . . . . . . . . . . . . 3536.2.4 The Kuratowski-Ryll-Nardzewski selection theorem . . . . . . . 3566.2.5 Hausdor®-measurable multivalued maps . . . . . . . . . . . . . 3716.2.6 The Scorza-Dragoni property . . . . . . . . . . . . . . . . . . . 3736.2.7 The Bressan-Colombo-Fryszkowski selection theorem . . . . . . 3796.3 The Bochner integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3806.3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3806.3.2 Nemytski·i operators . . . . . . . . . . . . . . . . . . . . . . . . 3836.3.3 Integration of multivalued maps . . . . . . . . . . . . . . . . . . 3866.4 Compactness in C([a; b];E) and PC([a; b];E) . . . . . . . . . . . . . . . 3886.5 Further auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . 391