Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
1998 | 130 | 3 | 231-244
Tytuł artykułu

Averages of holomorphic mappings and holomorphic retractions on convex hyperbolic domains

Treść / Zawartość
Warianty tytułu
Języki publikacji
Let D be a hyperbolic convex domain in a complex Banach space. Let the mapping F ∈ Hol(D,D) be bounded on each subset strictly inside D, and have a nonempty fixed point set ℱ in D. We consider several methods for constructing retractions onto ℱ under local assumptions of ergodic type. Furthermore, we study the asymptotic behavior of the Cesàro averages of one-parameter semigroups generated by holomorphic mappings.
Słowa kluczowe
Opis fizyczny
  • Department of Mathematics, The Technion-Israel Institute of Technology, 32000 Haifa, Israel
  • Department of Applied Mathematics, International College of Technology, P.O. Box 78, 20101 Karmiel, Israel
  • [1] M. Abate and J.-P. Vigué, Common fixed points in hyperbolic Riemann surfaces and convex domains, Proc. Amer. Math. Soc. 112 (1991), 503-512.
  • [2] M. Abd-Alla, L'ensemble des points fixes d'une application holomorphe dans un produit fini de boules-unités d'espaces de Hilbert et une sous-variété banachique complexe, Ann. Mat. Pura Appl. (4) 153 (1988), 63-75.
  • [3] T. Y. Azizov, V. Khatskevich, and D. Shoikhet, On the number of fixed points of a holomorphism, Sibirsk. Mat. Zh. 31 (1990), no. 6, 192-195 (in Russian).
  • [4] H. Cartan, Sur les rétractions d'une variété, C. R. Acad. Sci. Paris 303 (1986), 715-716.
  • [5] G.-N. Chen, Iteration for holomorphic maps of the open unit ball and the generalized upper half-plane of $ℂ^n$, J. Math. Anal. Appl. 98 (1984), 305-313.
  • [6] Do Duc Thai, The fixed points of holomorphic maps on a convex domain, Ann. Polon. Math. 56 (1992), 143-148.
  • [7] C. J. Earle and R. S. Hamilton, A fixed-point theorem for holomorphic mappings, in: Proc. Sympos. Pure Math. 16, Amer. Math. Soc., Providence, R.I., 1970, 61-65.
  • [8] G. Fischer, Complex Analytic Geometry, Lecture Notes in Math. 538, Springer, Berlin, 1976.
  • [9] T. Franzoni and E. Vesentini, Holomorphic Maps and Invariant Distances, North-Holland, Amsterdam, 1980.
  • [10] K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings, Marcel Dekker, New York, 1984.
  • [11] I. Gohberg and A. Markus, Characteristic properties of a pole of the resolvent of a linear closed operator, Uchenye Zapiski Bel'tskogo Gosped. 5 (1960), 71-76 (in Russian).
  • [12] L. F. Heath and T. J. Suffridge, Holomorphic retracts in complex n-space, Illinois J. Math. 25 (1981), 125-135.
  • [13] M. Hervé, Analyticity in Infinite Dimensional Spaces, de Gruyter, Berlin, 1989.
  • [14] J. M. Isidro and L. L. Stacho, z Holomorphic Automorphism Groups in Banach Spaces: An Elementary Introduction, North-Holland, Amsterdam, 1984.
  • [15] V. Khatskevich, S. Reich, and D. Shoikhet, Ergodic type theorems for nonlinear semigroups with holomorphic generators, in: Recent Developments in Evolution Equations, Pitman Res. Notes Math. 324, Longman, 1995, 191-200.
  • [16] V. Khatskevich, S. Reich, and D. Shoikhet, Asymptotic behavior of solutions to evolution equations and the construction of holomorphic retractions, Math. Nachr. 189 (1998), 171-178.
  • [17] V. Khatskevich and D. Shoikhet, Fixed points of analytic operators in a Banach space and applications, Sibirsk. Mat. Zh. 25 (1984), no. 1, 189-200 (in Russian).
  • [18] V. Khatskevich and D. Shoikhet, Differentiable Operators and Nonlinear Equations, Birkhäuser, Basel, 1994.
  • [19] J. J. Koliha, Some convergence theorems in Banach algebras, Pacific J. Math. 52, (1974), 467-473.
  • [20] M. A. Krasnosel'skiĭ and P. P. Zabreĭko, Geometrical Methods of Nonlinear Analysis, Springer, Berlin, 1984.
  • [21] Y. Kubota, Iteration of holomorphic maps of the unit ball into itself, Proc. Amer. Math. Soc. 88 (1983) 476-480.
  • [22] T. Kuczumow and A. Stachura, Iterates of holomorphic and $K_D$-nonexpansive mappings in convex domains in $ℂ^n$, Adv. Math. 81 (1990), 90-98.
  • [23] K. B. Laursen and M. Mbekhta, Operators with finite chain length and the ergodic theorem, Proc. Amer. Math. Soc. 123 (1995), 3443-3448.
  • [24] L. Lempert, Holomorphic retracts and intrinsic metrics in convex domains, Anal. Math. 8 (1982), 257-261.
  • [25] Yu. Lyubich and J. Zemánek, Precompactness in the uniform ergodic theory, Studia Math. 112 (1994), 89-97.
  • [26] B. D. MacCluer, Iterates of holomorphic self-maps of the unit ball in $C^N$, Michigan Math. J. 30 (1983), 97-106.
  • [27] P. Mazet, Les points fixes d'une application holomorphe d'un domaine borné dans lui-même admettent une base de voisinages convexes stable, C. R. Acad. Sci. Paris 314 (1992), 197-199.
  • [28] P. Mazet et J.-P. Vigué, Points fixes d'une application holomorphe d'un domaine borné dans lui-même, Acta Math. 166 (1991), 1-26.
  • [29] P. Mazet et J.-P. Vigué, Convexité de la distance de Carathéodory et points fixes d'applications holomorphes, Bull. Sci. Math. 116 (1992), 285-305.
  • [30] P. R. Mercer, Complex geodesics and iterates of holomorphic maps on convex domains in $ℂ^n$, Trans. Amer. Math. Soc. 338 (1993), 201-211.
  • [31] S. Reich and D. Shoikhet, Metric domains, holomorphic mappings and nonlinear semigroups, Technion Preprint Series No. MT-1018, 1997.
  • [32] W. Rudin, The fixed-point set of some holomorphic maps, Bull. Malaysian Math. Soc. 1 (1978), 25-28.
  • [33] W. Rudin, Functional Analysis, McGraw-Hill, New York, 1973.
  • [34] D. Shoikhet, Some properties of Fredholm mappings in Banach analytic manifolds, Integral Equations Operator Theory 16 (1993), 430-451.
  • [35] T. J. Suffridge, Common fixed points of commuting holomorphic maps of the hyperball, Michigan Math. J. 21 (1974), 309-314.
  • [36] A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, Wiley, New York, 1980.
  • [37] E. Vesentini, Complex geodesics and holomorphic maps, Sympos. Math. 26 (1982), 211-230.
  • [38] E. Vesentini, Su un teorema di Wolff e Denjoy, Rend. Sem. Mat. Fis. Milano 53 (1983), 17-25.
  • [39] E. Vesentini, Iterates of holomorphic mappings, Uspekhi Mat. Nauk 40 (1985), no. 4, 13-16 (in Russian).
  • [40] J.-P. Vigué, Points fixes d'applications holomorphes dans un produit fini de boules-unités d'espaces de Hilbert, Ann. Mat. Pura Appl. 137 (1984), 245-256.
  • [41] J.-P. Vigué, Points fixes d'applications holomorphes dans un domaine borné convexe de $ℂ^n$, Trans. Amer. Math. Soc. 289 (1985), 345-353.
  • [42] J.-P. Vigué, Sur les points fixes d'applications holomorphes, C. R. Acad. Sci. Paris 303 (1986), 927-930.
  • [43] J.-P. Vigué, Fixed points of holomorphic mappings in a bounded convex domain in $ℂ^n$, in: Proc. Sympos. Pure Math.
Typ dokumentu
Identyfikator YADDA
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.