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

Suzuki, Tomonari

doi:10.1090/S0002-9939-05-08361-9

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

Proc. Amer. Math. Soc. 134 (2006), 673-681

DOI: https://doi.org/10.1090/S0002-9939-05-08361-9

Published electronically: September 28, 2005

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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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