Archiv der Mathematik

, Volume 92, Issue 6, pp 602-613

First online:

Reich’s problem concerning Halpern’s convergence

  • Tomonari SuzukiAffiliated withDepartment of Basic Sciences, Kyushu Institute of Technology Email author 

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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 47H09 Secondary 47H10, 47J25


Nonexpansive mapping fixed point convergence theorem