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Proceedings of the American Mathematical Society

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



Asymptotic behavior of nonexpansive sequences and mean points

Authors: Jong Soo Jung and Jong Seo Park
Journal: Proc. Amer. Math. Soc. 124 (1996), 475-480
MSC (1991): Primary 47H09
MathSciNet review: 1291776
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Abstract: Let $E$ be a real Banach space with norm $\Vert \cdot \Vert$ and let $\{x_n\}_{n \ge 0}$ be a nonexpansive sequence in $E$ (i.e., $\Vert x_{i + 1} - x_{j + 1}\Vert \le \Vert x_i - x_j\Vert$ for all $i, j \ge 0$). Let $K = \bigcap _{n = 1}^{\infty }\overline {co}\{\{x_i - x_{i - 1}\}_{i \ge n}\}$. We deal with the mean point of $\{\frac {x_n}{n}\}$ concerning a Banach limit. We show that if $E$ is reflexive and $d = d(0,K)$, then $d = d(0,\overline {co}\{\frac {x_n - x_0}{n}\})$ and there exists a unique point $z_0$ with $\Vert z_0\Vert = d$ such that $z_0 \in \overline {co}\{\frac {x_n - x_0}{n}\}$. This result is applied to obtain the weak and strong convergence of $\{\frac {x_n}{n}\}$.

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Additional Information

Jong Soo Jung
Affiliation: Department of Mathematics, Dong–A University, Pusan 604–714, Korea

Jong Seo Park
Affiliation: Department of Mathematics, Graduate School, Dong-A University, Pusan 604–714, Korea

Keywords: Asymptotic behavior, Banach limit, mean point, nonexpansive sequence
Received by editor(s): March 24, 1994
Received by editor(s) in revised form: August 22, 1994
Additional Notes: This research was supported by the Korea Science and Engineering Foundation, project number 941-0100-035-2.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society