On a result of Pazy concerning the asymptotic behaviour of nonexpansive mappings

  • Heinz H. Bauschke
  • Graeme R. Douglas
  • Walaa M. Moursi


In 1971, Pazy [Israel J. Math. 9 (1971), 235–240] presented a beautiful trichotomy result concerning the asymptotic behaviour of the iterates of a nonexpansive mapping. In this note, we analyze the fixedpoint- free case in more detail. Our results and examples give credence to the conjecture that the iterates always converge cosmically. The relationship to recent work by Lins [Proc. Amer. Math. Soc. 137 (2009), 2387–2392] is also discussed.


Cosmic convergence firmly nonexpansive mapping nonexpansive mapping Poincaré metric projection operator 

Mathematics Subject Classification

Primary 47H09 Secondary 90C25 


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© Springer International Publishing 2015

Authors and Affiliations

  • Heinz H. Bauschke
    • 1
  • Graeme R. Douglas
    • 2
  • Walaa M. Moursi
    • 3
    • 4
  1. 1.Mathematics DepartmentUniversity of British ColumbiaKelownaCanada
  2. 2.Computer Science DepartmentUniversity of British ColumbiaKelownaCanada
  3. 3.Mathematics DepartmentUniversity of British ColumbiaKelownaCanada
  4. 4.Mathematics Department, Faculty of ScienceMansoura UniversityMansouraEgypt

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