Open Access

Hyperbolic monotonicity in the Hilbert ball

Fixed Point Theory and Applications20062006:78104

https://doi.org/10.1155/FPTA/2006/78104

Received: 17 August 2005

Accepted: 22 August 2005

Published: 26 February 2006

Abstract

We first characterize -monotone mappings on the Hilbert ball by using their resolvents and then study the asymptotic behavior of compositions and convex combinations of these resolvents.

[1234567891011121314151617181920]

Authors’ Affiliations

(1)
Institute of Mathematics, Czech Academy of Sciences
(2)
Institut für Analysis, Johannes Kepler Universität
(3)
Department of Mathematics, The Technion – Israel Institute of Technology

References

  1. Abate M: The infinitesimal generators of semigroups of holomorphic maps. Annali di Matematica Pura ed Applicata. Serie Quarta 1992,161(1):167–180. 10.1007/BF01759637MathSciNetView ArticleMATHGoogle Scholar
  2. Aharonov D, Elin M, Reich S, Shoikhet D: Parametric representations of semi-complete vector fields on the unit balls in and in Hilbert space. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni 1999,10(4):229–253.MathSciNetMATHGoogle Scholar
  3. Bauschke HH, Combettes PL, Reich S: The asymptotic behavior of the composition of two resolvents. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods 2005,60(2):283–301.MathSciNetView ArticleMATHGoogle Scholar
  4. Bauschke HH, Matoušková E, Reich S: Projection and proximal point methods: convergence results and counterexamples. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods 2004,56(5):715–738. 10.1016/j.na.2003.10.010MathSciNetView ArticleMATHGoogle Scholar
  5. Goebel K, Reich S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and Applied Mathematics. Volume 83. Marcel Dekker, New York; 1984:ix+170.Google Scholar
  6. Halpern BR, Bergman GM: A fixed-point theorem for inward and outward maps. Transactions of the American Mathematical Society 1968, 130: 353–358. 10.1090/S0002-9947-1968-0221345-0MathSciNetView ArticleMATHGoogle Scholar
  7. Kuczumow T: Nonexpansive retracts and fixed points of nonexpansive mappings in the Cartesian product of Hilbert balls. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods 1985,9(6):601–604. 10.1016/0362-546X(85)90043-4MathSciNetView ArticleMATHGoogle Scholar
  8. Kuczumow T, Stachura A: Extensions of nonexpansive mappings in the Hilbert ball with the hyperbolic metric. I. Commentationes Mathematicae Universitatis Carolinae 1988,29(3):399–402.MathSciNetMATHGoogle Scholar
  9. Kuczumow T, Stachura A: Extensions of nonexpansive mappings in the Hilbert ball with the hyperbolic metric. II. Commentationes Mathematicae Universitatis Carolinae 1988,29(3):403–410.MathSciNetMATHGoogle Scholar
  10. Martin RH Jr.: A global existence theorem for autonomous differential equations in a Banach space. Proceedings of the American Mathematical Society 1970, 26: 307–314. 10.1090/S0002-9939-1970-0264195-6MathSciNetView ArticleMATHGoogle Scholar
  11. Minty GJ: Monotone (nonlinear) operators in Hilbert space. Duke Mathematical Journal 1962,29(3):341–346. 10.1215/S0012-7094-62-02933-2MathSciNetView ArticleMATHGoogle Scholar
  12. Reich S: Minimal displacement of points under weakly inward pseudo-Lipschitzian mappings. Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali 1975,59(1–2):40–44 (1976).MathSciNetMATHGoogle Scholar
  13. Reich S: On fixed point theorems obtained from existence theorems for differential equations. Journal of Mathematical Analysis and Applications 1976,54(1):26–36. 10.1016/0022-247X(76)90232-8MathSciNetView ArticleMATHGoogle Scholar
  14. Reich S: Extension problems for accretive sets in Banach spaces. Journal of Functional Analysis 1977,26(4):378–395. 10.1016/0022-1236(77)90022-2View ArticleMathSciNetMATHGoogle Scholar
  15. Reich S: Averaged mappings in the Hilbert ball. Journal of Mathematical Analysis and Applications 1985,109(1):199–206. 10.1016/0022-247X(85)90187-8MathSciNetView ArticleMATHGoogle Scholar
  16. Reich S: The alternating algorithm of von Neumann in the Hilbert ball. Dynamic Systems and Applications 1993,2(1):21–25.MathSciNetMATHGoogle Scholar
  17. Reich S, Shafrir I: Nonexpansive iterations in hyperbolic spaces. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods 1990,15(6):537–558. 10.1016/0362-546X(90)90058-OMathSciNetView ArticleMATHGoogle Scholar
  18. Reich S, Shoikhet D: Generation theory for semigroups of holomorphic mappings in Banach spaces. Abstract and Applied Analysis 1996,1(1):1–44. 10.1155/S1085337596000012MathSciNetView ArticleMATHGoogle Scholar
  19. Reich S, Shoikhet D: Semigroups and generators on convex domains with the hyperbolic metric. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni 1997,8(4):231–250.MathSciNetMATHGoogle Scholar
  20. Shafrir I: Coaccretive operators and firmly nonexpansive mappings in the Hilbert ball. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods 1992,18(7):637–648. 10.1016/0362-546X(92)90003-WMathSciNetView ArticleMATHGoogle Scholar

Copyright

© Kopecká and Reich 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.