Solution Sets for Differential Equations and Inclusions

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

• 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