Nonlinear Semigroups, Fixed Points, and Geometry of Domains in Banach Spaces

NONLINEAR SEMIGROUPS, FIXED POINTS, AND GEOMETRY OF DOMAINS IN BANACH SPACES......Page 1
Half-title......Page 2
Title Page......Page 4
Copyright Page......Page 5
Preface......Page 6
Contents......Page 12
1.1.2 Neighborhoods......Page 18
1.1.3 Examples of topologies......Page 19
1.1.4 Interiors and closures. Limit points......Page 20
1.1.6 Induced topology. Subspaces......Page 21
1.1.7 Continuous mappings......Page 22
1.1.8 Compactness......Page 23
1.1.10 Topological vector spaces......Page 24
1.2.1 Metrics and pseudometrics (semimetrics)......Page 25
1.2.2 Examples......Page 28
1.3.1 Norms on a vector space......Page 29
1.3.2 Examples......Page 30
1.4.1 Scalar product......Page 31
1.4.2 Examples......Page 32
1.5.1 Convex sets and convex hulls......Page 34
1.5.2 Extreme points......Page 35
1.5.4 The topology induced by seminorms......Page 36
1.5.5 The Minkowski functional......Page 37
1.5.6 Locally convex spaces and seminorms......Page 38
1.6.1 Linear operators......Page 39
1.6.2 Examples......Page 41
1.6.4 Multilinear mappings and polynomials......Page 42
1.6.5 Banach algebra of linear operators......Page 44
1.6.6 Spectra and resolvents of linear operators......Page 46
1.6.7 Examples......Page 49
1.7.3 Weak topology and reflexivity......Page 50
1.7.4 The weak and weak* topologies......Page 52
1.8.1 The extension theorem......Page 53
1.8.3 Geometric Hahn–Banach separation theorems......Page 55
1.9.1 Mean ergodic theorem......Page 57
1.9.2 Uniform ergodic theorems in Banach spaces......Page 59
1.10.1 Lipschitzian and contraction mappings......Page 61
1.10.3 Uniformly Lipschitzian mappings......Page 62
1.10.4 Firmly nonexpansive mappings......Page 63
1.10.5 Monotone and accretive mappings......Page 64
2.1 Differentiable Mappings. Fréchet Derivatives......Page 68
2.1.1 Examples......Page 70
2.2 Holomorphic Mappings......Page 71
2.2.1 The Cauchy integral formula......Page 72
2.2.2 Power series representation......Page 73
2.2.3 The maximum modulus theorem......Page 74
2.3.1 T-topology and compact open topology on Hol(D, Y)......Page 77
2.3.2 Montel's theorem......Page 78
2.3.3 Vitali's theorem......Page 79
2.4.1 Symbolic calculus on Banach algebras......Page 80
2.4.2 The spectral mapping theorem,......Page 82
2.4.3 Some *-algebras......Page 83
2.4.4 l-analytic functions on unital J*-algebras......Page 85
2.5.1 The classical Schwarz Lemma and Carton's uniqueness theorem......Page 86
2.6.1 The unit disk......Page 89
2.6.3 The Euclidean ball in C^n and the Hilbert ball......Page 91
2.6.5 The Schwarz–Pick lemma......Page 93
3.1 The Poincaré Metric on the Unit Disk......Page 98
3.2 The Infinitesimal Poincaré Metric and Geodesies......Page 103
3.3 The Poincaré Metric on the Hilbert Ball and its Powers......Page 105
3.4.1 The Carathéodory pseudometric......Page 106
3.4.2 The Kobayashi pseudometric......Page 108
3.5 Infinitesimal Finsler Pseudometrics......Page 110
3.5.1 Examples......Page 112
3.6 Schwarz–Pick Systems of Pseudometrics......Page 114
3.7 Bounded Convex Domains and Metric Domains in Banach Spaces......Page 118
4.1 The Banach Principle......Page 124
4.2 The Theorems of Brouwer and Schauder......Page 127
4.3 Holomorphic Fixed Point Theorems......Page 128
4.4 Fixed Points in the Hilbert Ball......Page 132
4.5 Fixed Points in Finite Powers of the Hilbert Ball......Page 133
5.1.1 Iterates of holomorphic self-mappings of Δ with an interior fixed point......Page 136
5.1.2 Iterates of holomorphic self-mappings of Δ with no interior fixed point......Page 138
5.2 The Unit Hilbert Ball......Page 144
5.3 Convex Domains in C^n......Page 152
5.4 Domains in Banach Space......Page 155
5.5 Holomorphic Retracts and the Structure of the Fixed Point Sets......Page 161
6.1.1 Discrete and continuous flows on a domain......Page 174
6.1.2 Examples......Page 176
6.2 Linear semigroups......Page 184
6.3 Generated Semigroups of Nonexpansive and Holomorphic Mappings......Page 192
6.4 The Cauchy Problem and the Product Formula......Page 200
6.5 Nonlinear Resolvents, the Range Condition and Exponential Formulas......Page 207
7.1 Boundary Flow Invariance Conditions......Page 216
7.2 Numerical Range of Holomorphic Mappings......Page 219
7.3 Interior Flow Invariance Conditions......Page 224
7.4 Semi-Complete and Complete Vector Fields......Page 230
8.1 Generalities......Page 236
8.2 Generated Semigroups......Page 243
8.3 The Resolvent Method......Page 245
8.4 Null Point Free Generators......Page 249
8.5 The Structure of Null Point Sets of Holomorphic Generators. Retractions......Page 254
8.6 A Stabilization Phenomenon......Page 261
8.7.1 Cartan's uniqueness theorem......Page 265
8.7.2 Harris' spectrum of a semi-complete vector field......Page 266
9.1 Strongly Semi-Complete Vector Fields in Banach Spaces......Page 270
9.2 Asymptotic Behavior of Flows of p-Nonexpansive Mappings on the Hilbert Ball......Page 279
9.3 Flows of Holomorphic Mappings on the Hilbert Ball......Page 289
9.3.1 Interior stationary point......Page 290
9.3.2 Boundary sink point. Continuous version of the Julia–Wolff–Carathéodory theorem......Page 297
9.4 Admissible Lower and Upper Bounds and Rates of Convergence......Page 303
10.1 Biholomorphic Mappings in Banach Spaces and Generators on Biholomorphically Equivalent Domains......Page 314
10.2 Starlike, Convex, and Spirallike Mappings......Page 317
10.2.1 Starlike functions on the unit disk......Page 318
10.2.2 Convex and close-to-convex functions on the unit disk......Page 320
10.3 Higher-Dimensional Extensions and the Dynamical Approach......Page 321
10.4 Distortion Theorems for Starlike Mappings on the Unit Ball......Page 333
10.5 Differential Equations for Starlike and Spirallike Mappings in H = C^n......Page 341
Bibliography......Page 356
Index......Page 368