, Volume 92, Issue 6, pp 602-613
Date: 22 May 2009

Reich’s problem concerning Halpern’s convergence

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Abstract.

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.

The author is supported in part by Grants-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science and Technology.
Received: 15 July 2008