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Reich S., Shoikhet D. — Nonlinear Semigroups, Fixed Points, and Geometry of Domains in Banach Spaces
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Название: Nonlinear Semigroups, Fixed Points, and Geometry of Domains in Banach SpacesАвторы: Reich S., Shoikhet D. Аннотация: Nonlinear semigroup theory is not only of intrinsic interest, but is also important in the study of evolution problems. In the last forty years, the generation theory of flows of holomorphic mappings has been of great interest in the theory of Markov stochastic branching processes, the theory of composition operators, control theory, and optimization. It transpires that the asymptotic behavior of solutions to evolution equations is applicable to the study of the geometry of certain domains in complex spaces. Readers are provided with a systematic overview of many results concerning both nonlinear semigroups in metric and Banach spaces and the fixed point theory of mappings, which are nonexpansive with respect to hyperbolic metrics (in particular, holomorphic self-mappings of domains in Banach spaces). The exposition is organized in a readable and intuitive manner, presenting basic functional and complex analysis as well as very recent developments.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 2005Количество страниц: 354Добавлена в каталог: 17.06.2008Операции: Положить на полку |
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Предметный указатель
(continuously) invertible operator 28 (Nonlinear) resolvent 190 Accumulation point 3 Adjoint operator 66 Admissible curve 83 86 93 95 Algebra with involution 66 Angular derivative 124 132 281 Angular limit 124 Approximate semigroup property 177 189 191 Approximating curve 130 229 231 Attractive point 266 Attractive sink point 263 Banach algebra 28 Banach Contraction Principle 108 Banach space 13. Boundary 4 Boundary flow invariance condition 206 Boundary point 3 Bounded homogeneous polynomial 27 Brouwer's theorem 110 C*-algebra 66 Caratheodory pseudometric 89—91 97 Cartan's uniqueness theorem 71 Cauchy 12 Closure 3 Closure point 3 Cluster point 3 Common fixed point 220 Commutative algebra 28 Commuting family 220 Compact commuting property (CCP) 222 Compact open topology 61 complete 213 Complex extreme point 59 Composition operation 28 44 Continuity of scalar product 15 Continuous flows 157 Continuous semigroup 175 Continuous spectrum 31 Contraction 45 Convex closure 18 Convex hull 17 Curve, asymptotically normal 132 281 Denjoy — Wolff iteration property (DWIP) 224 Denjoy — Wolff point 137 Denjoy — Wolff theorem 119 126 Distance 9 Domain, homogeneous 100 Domain, p-hyperbolic 100 Domain, spiral-shaped 301 Dtationary point 219 223 270 Duality mapping 48 306 Earle — Hamilton theorem 206 Edelstein 109 Eigensubspace 31 Eigenvalue 31 Eigenvector 31 Elliptic automorphism 119 162 Equicontinuous family 41 Exponential type 270 272 Extreme point 18 58 Extreme subset 18 Fixed point 107 Flow invariance condition 199 201 202 213 Fractional linear transformation 162 166 Frechet derivative 51 Fundamental sequence 12 Gauge 20 38 Geodesic segment 87 Hausdorff 3 Hausdorff's maximality theorem 7 Hilbert space 15 Hille — Yosida theorem 173 196 206 Holomorphic, retract 146 Holomorphic, retraction 146 Homogeneous polynomial of order m 27 Hyperaccretive 49 Hyperbolic automorphisms 162 Hyperbolic domain 100 Hyperbolic length 87 Hyperbolic metric 101 257 Hyperbolic type 125 135 Implicit approximation property 231 Indefinite metric 163 164 Infinitesimal Caratheodory pseudometric 95 Infinitesimal Finsler metric 93 Infinitesimal Finsler pseudometric 93 Infinitesimal generator 160 175 215 Infinitesimal Poincare metric 86 95 164 Interior flow invariance condition 207 Interior point 3 Intersection 17 Involution property 73 Isometry 23 81 158 J*-algebra 67 Julia number 132 Julia — Caratheodory theorem 124 131 282 283 Julia — Wolff — Caratheodory theorem 125 131 135 280 285 286 Julia's lemma 122 123 282 284 285 Kobayashi pseudometric 91 Krein space 163 164 l-analytic function 68 Length 94 Limit point 3 Linear functional 8 22 Lipschitz constant 44 Locally convex space 21 Locally uniformly attractive fixed point 119 163 253 Lumer — Phillips theorem 173 174 204—206 Mapping k-contraction 108 Mapping m-linear 25 Mapping, biholomorphic 69 Mapping, bounded 23 Mapping, continuous 5 Mapping, contractive 109 Mapping, differentiable 51 Mapping, fixed point free 127 Mapping, holomorphic 54 Mapping, holomorphically accretive 203 205 Mapping, holomorphically conservative 203 Mapping, holomorphically dissipative 203 205 Mapping, Lipschitzian 44 Mapping, locally Lipschitzian 44 Mapping, monotone 48 Mapping, multilinear 25 Mapping, nonexpansive 45 108 Mapping, power convergent 119 Mapping, strongly accretive 48 Mapping, strongly monotone 48 Maximum Modulus Principle 57 Mazur's theorem 18 Mean ergodic theorem 41 Metric 8 Metric domain 193 Metric space 8 Metric topology 9 Minkowski functional 20 Moebius transformation 88 120 Montel space 12 Multiplicative 69 Neighborhood 1 Neighborhood base 3 Nontangential approach region 124 Norm 20 Norm of a linear operator 23 Normed space 13 Null point 25 Numerical range 202 203 One-parameter group 158 159 One-parameter groups of automorphisms 162 One-parameter semigroup 158 One-parameter semigroup of composition operators 166 One-parameter, continuous semigroup 158 One-parameter, discrete group 158 One-parameter, discrete semigroup 158 Operator, accretive 48 49 159 173 Operator, bounded 23 Operator, conservative 174 215 Operator, continuously invertible 28 Operator, dissipative 173 Operator, hermitian 66 Operator, m-accretive 49 240 Operator, strongly accretive 49 257 Operator, unbounded 23 Orthogonality 15 Parabolic automorphisms 163 Parabolic type 125 Parallelogram identity 15 Periodic point 220 223 Poincare metric 82 85 88—90 92 95 97 292 Point spectrum 31 Poutryagin space 164 Producing function 69 Product formula 189 Pseudohyperbolic distance 81 Pseudometric 89 Quasi-regular, fixed point 144 Quasi-regular, null point 299 Quasi-regular, point 239 radiaL 18 Range condition 190 193 194 Reflexive 34 Regular fixed point 144 148 Regular null point 239 299 Relative topology 5 Residual spectrum 31 Resolvent 30 49 Resolvent identity 190 192 Resolvent set 29 Restricted limit 132 Riesz — Dunford integral 68 Scalar product 14 Schwarz inequality 15 Schwarz lemma 76 209 Schwarz — Pick Inequality 77 122 Schwarz — Pick Lemma 81 162 Schwarz — Pick system 97 257 Self-mapping 44 Semi-complete vector field 213 298 Semigroups of fractional linear transformations 164 Seminorm 12 Separated 3 Sequentially compact 6 Set, absorbing 18 Set, balanced 18 Set, circled 18 Set, closed 2 Set, compact 6 Set, complete 12 Set, convex 17 Set, cover of 6 Set, dense 4 Set, diameter of 9 Set, open 1 Set, partially ordered 7 Set, relatively compact 6 Set, starshaped 300 Set, symmetric 19 Sink point 125 233 270 281 Space, compact 6 Space, complex 59 Space, conjugate 33 Space, dual 33 34 Space, locally convex 41 Space, metrizable 8 Space, real 18 Space, separable 4 Spectral mapping theorem 65 Spectrum 29 Spirallike mapping 301 Spirallike mapping on 301 Spirallike mapping with respect to a boundary point 301 Star-shaped about zero 19 Starlike function 302 Starlike mapping 300 Starlike mapping with respect to a boundary point 300 Starlike mapping with respect to an interior point 300 Stolz angle 124 Strict contraction 45 108 Strictly inside 102 Strictly regular point 251 253 257 299 Strong maximum modulus principle 57 60 Strongly accretive 261 Strongly p-monotone 262 Strongly semi-complete 257 309 Strongly semi-complete vector field 262 290 Topological space 1 Topological vector space 8 topology 1 Topology of locally uniform convergence over V 60 Topology of uniform convergence on compact subsets 61 Topology, generated by the family of seminorms 21 Totally ordered 7 Uniform convergence 25 Uniform operator topology 25 Uniformly continuous semigroup 160 172 Uniformly Lipschitzian 45 unique 220 Unital 68 Vector space 7 Volterra integral operator 32 Weak maximum modulus theorem 57 Weak topology 34 35 Wolff — Schwarz Boundary Lemma 122 Wolff's point 125
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