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A sufficient and necessary condition for Halpern-type strong convergence to fixed points of nonexpansive mappings

Author: Tomonari Suzuki
Journal: Proc. Amer. Math. Soc. 135 (2007), 99-106
MSC (2000): Primary 47H09; Secondary 47H10
Published electronically: June 13, 2006
MathSciNet review: 2280199
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Abstract: In this paper, we prove a Halpern-type strong convergence theorem for nonexpansive mappings in a Banach space whose norm is uniformly Gâteaux differentiable. Also, we discuss the sufficient and necessary condition about this theorem. This is a partial answer of the problem raised by Reich in 1983.

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Tomonari Suzuki
Affiliation: Department of Mathematics, Kyushu Institute of Technology, Sensuicho, Tobata, Kitakyushu 804-8550, Japan

Keywords: Nonexpansive mapping, fixed point, Halpern-type strong convergence theorem
Received by editor(s): March 1, 2005
Received by editor(s) in revised form: July 18, 2005
Published electronically: June 13, 2006
Additional Notes: The author was supported in part by Grants-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science and Technology.
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.