On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces
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- by Tomonari Suzuki PDF
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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 $u$.References
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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
- 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
- © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1963759