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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The set of common fixed points of a one-parameter continuous semigroup of mappings is $F \big ( T(1) \big ) \cap F \big ( T(\sqrt 2) \big )$


Author: Tomonari Suzuki
Journal: Proc. Amer. Math. Soc. 134 (2006), 673-681
MSC (2000): Primary 47H20, 47H10
DOI: https://doi.org/10.1090/S0002-9939-05-08361-9
Published electronically: September 28, 2005
MathSciNet review: 2180883
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Abstract: In this paper we prove the following theorem: Let $\{ T(t) : t \geq 0 \}$ be a one-parameter continuous semigroup of mappings on a subset $C$ of a Banach space $E$. The set of all fixed points of $T(t)$ is denoted by $F \big ( T(t) \big )$ for each $t \geq 0$. Then \[ \bigcap _{t \geq 0} F \big ( T(t) \big ) = F \big ( T(1) \big ) \cap F \big ( T(\sqrt 2) \big ) \] holds. Using this theorem, we discuss convergence theorems to a common fixed point of $\{ T(t) : t \geq 0 \}$.


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Additional Information

Tomonari Suzuki
Affiliation: Department of Mathematics, Kyushu Institute of Technology, Sensuicho, Tobata, Kitakyushu 804-8550, Japan
Email: suzuki-t@mns.kyutech.ac.jp

Keywords: Nonexpansive semigroup, common fixed point, irrational number
Received by editor(s): December 17, 2003
Published electronically: September 28, 2005
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: Jonathan M. Borwein
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.