Archiv der Mathematik

, Volume 92, Issue 6, pp 602–613

Reich’s problem concerning Halpern’s convergence


DOI: 10.1007/s00013-009-2945-4

Cite this article as:
Suzuki, T. Arch. Math. (2009) 92: 602. doi:10.1007/s00013-009-2945-4


We discuss Halpern’s convergence for nonexpansive mappings in Hilbert spaces. We prove that one of the conditions in [R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. (Basel), 58 (1992), 486–491] is the weakest sufficient condition among the conditions known to us. We also improve a necessary condition, which is close to Wittmann’s. This is one step to solve the problem raised by Reich in 1974 and 1983.

Mathematics Subject Classification (2000).

Primary 47H09Secondary 47H10, 47J25


Nonexpansive mappingfixed pointconvergence theorem

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Department of Basic SciencesKyushu Institute of TechnologyKitakyushuJapan