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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

An example in holomorphic fixed point theory


Author: Monika Budzynska
Journal: Proc. Amer. Math. Soc. 131 (2003), 2771-2777
MSC (2000): Primary 32A10, 46G20, 47H09, 47H10
Published electronically: March 11, 2003
MathSciNet review: 1974334
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Abstract | References | Similar Articles | Additional Information

Abstract: If $B$ is the open unit ball in the Cartesian product $l^2 \times l^2$ furnished with the $l^p$-norm $\Vert\cdot\Vert$, where $1 <p < \infty$ and $ p \neq 2$, then a holomorphic self-mapping $f$ of $B$ has a fixed point if and only if $\sup_{n\in \mathbb{N}} \Vert f^n (x)\Vert <1$ for some $x\in B.$


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  • 1. M. Budzynska & T. Kuczumow, A strict convexity of the Kobayashi distance, Fixed Point Theory and Applications, Vol. 4 (Ed. Y. J. Cho), Nova Science Publishers, Inc., to appear.
  • 2. S. Dineen, The Schwarz Lemma, Oxford University Press, 1989. MR 91f:46064
  • 3. C. J. Earle & R. S. Hamilton, A fixed point theorem for holomorphic mappings, Proc. Symp. Pure Math., Vol. 16, Amer. Math. Soc., 61-65 (1968). MR 42:918
  • 4. M. Edelstein, The construction of an asymptotic center with a fixed point property, Bull. Amer. Math. Soc. 78, 206-208 (1972). MR 45:1005
  • 5. T. Franzoni & E. Vesentini, Holomorphic maps and invariant distances, North-Holland, 1980. MR 82a:32032
  • 6. K. Goebel & W. A. Kirk, Topics in metric fixed point theory, Cambridge University Press, 1990. MR 92c:47070
  • 7. K. Goebel & S. Reich, Uniform convexity, hyperbolic geometry and nonexpansive mappings, Marcel Dekker, 1984. MR 86d:58012
  • 8. K. Goebel, T. Sekowski & A. Stachura, Uniform convexity of the hyperbolic metric and fixed points of holomorphic mappings in the Hilbert ball, Nonlinear Analysis 4, 1011-1021 (1980). MR 82f:32039
  • 9. L. A. Harris, Schwarz-Pick systems of pseudometrics for domains in normed linear spaces, Advances in Holomorphy, North Holland, 345-406 (1979). MR 80j:32043
  • 10. M. Jarnicki & P. Pflug, Invariant distances and metrics in complex analysis, Walter de Gruyter, 1993. MR 94k:32039
  • 11. W. Kaup & H. Upmeier, Banach spaces with biholomorphically equivalent unit balls are isomorphic, Proc. Amer. Math. Soc. 58, 129-133 (1976). MR 54:10690
  • 12. T. Kuczumow, Fixed points of holomorphic mappings in the Hilbert ball, Colloq. Math. 55, 101-107 (1988). MR 90f:47088
  • 13. T. Kuczumow, S. Reich & D. Shoikhet, The existence and non-existence of common fixed points for commuting families of holomorphic mappings, Nonlinear Analysis 43, 45-49 (2001). MR 2001h:47094
  • 14. T. Kuczumow, S. Reich & D. Shoikhet, Fixed points of holomorphic mappings: a metric approach, Handbook of Metric Fixed Point Theory (Eds. W. A. Kirk and B. Sims), Kluwer Academic Publishers, 2001, 437-515.
  • 15. T. Kuczumow, S. Reich & A. Stachura, Holomorphic retracts of the open ball in the $l_{\infty }$-product of Hilbert spaces, Recent advances in metric fixed point theory (T. Domínguez Benavides, Ed.), Universidad de Sevilla, Serie: Ciencias, Núm. 48, 99-110 (1996). MR 98b:46061
  • 16. T. Kuczumow & A. Stachura, Iterates of holomorphic and $k_{D}$-nonexpansive mappings in convex domains in $\mathbb{C}^{n}$, Adv. in Math. 81, 90-98 (1990). MR 91d:32037
  • 17. S. Reich, Averaged mappings in the Hilbert ball, J. Math. Anal. Appl. 109, 199-206 (1985). MR 87b:47061
  • 18. W. Rudin, Function theory on the unit ball in $\mathbb{C}^{n}$, Springer, 1980. MR 82i:32002
  • 19. T. Sekowski & A. Stachura, Holomorphic non-equivalence of balls in Banach spaces $l_{p}$ and $L_{2}$ from the geometrical point of view, Ann. Univ. Mariae Curie-Sk\lodowska Sect. A 50, 213-218 (1996). MR 98f:58019
  • 20. J.-P. Vigué, La métrique infinitésimale de Kobayashi et la caractérisation des domaines convexes bornés, J. Math. Pures Appl. (9) 78, 867-876 (1999). MR 2001i:32022
  • 21. J.-P. Vigué, Stricte convexité des domaines bornés et unicité des géodésiques complexes, Bull. Sci. Math. 125, 297-310 (2001). MR 2002b:32024

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Additional Information

Monika Budzynska
Affiliation: Instytut Matematyki UMCS, 20-031 Lublin, Poland
Email: monikab@golem.umcs.lublin.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-03-06982-X
PII: S 0002-9939(03)06982-X
Keywords: Fixed points, holomorphic mappings, $k_D$-nonexpansive mappings, the Kobayashi distance, strict convexity, uniform convexity
Received by editor(s): March 28, 2001
Received by editor(s) in revised form: April 3, 2001, and March 29, 2002
Published electronically: March 11, 2003
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2003 American Mathematical Society