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On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces


Author: Tomonari Suzuki
Journal: Proc. Amer. Math. Soc. 131 (2003), 2133-2136
MSC (2000): Primary 47H20; Secondary 47H10
DOI: https://doi.org/10.1090/S0002-9939-02-06844-2
Published electronically: December 30, 2002
MathSciNet review: 1963759
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Abstract: In this paper, we prove the following strong convergence theorem: Let $C$ be a closed convex subset of a Hilbert space $H$. Let $\{ T(t) : t \geq 0 \}$ be a strongly continuous semigroup of nonexpansive mappings on $C$ such that $\bigcap_{t \geq 0} F\big(T(t)\big) \neq \emptyset$. Let $\{ \alpha_n \}$ and $\{ t_n \}$ be sequences of real numbers satisfying $0 < \alpha_n < 1$, $t_n > 0$ and $\lim_n t_n = \lim_n \alpha_n / t_n = 0$. Fix $u \in C$ and define a sequence $\{ u_n \}$ in $C$ by $ u_n = (1 - \alpha_n) T(t_n) u_n + \alpha_n u $ for $n \in \mathbb{N} $. Then $\{ u_n \}$ converges strongly to the element of $\bigcap_{t \geq 0} F\big(T(t)\big)$ nearest to $u$.


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Additional Information

Tomonari Suzuki
Affiliation: Department of Mathematics and Information Science, Graduate School of Science and Technology, Niigata University, Niigata 950-2181, Japan
Email: tomonari@math.sc.niigata-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-02-06844-2
Keywords: Fixed point, nonexpansive semigroup
Received by editor(s): April 14, 2000
Received by editor(s) in revised form: February 12, 2002
Published electronically: December 30, 2002
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2002 American Mathematical Society

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