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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Asymptotic behavior of nonexpansive sequences and mean points
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by Jong Soo Jung and Jong Seo Park PDF
Proc. Amer. Math. Soc. 124 (1996), 475-480 Request permission

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
  • Email: jungjs@seanghak.donga.ac.kr.
  • Jong Seo Park
  • Affiliation: Department of Mathematics, Graduate School, Dong-A University, Pusan 604–714, Korea
  • 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
  • © Copyright 1996 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 124 (1996), 475-480
  • MSC (1991): Primary 47H09
  • DOI: https://doi.org/10.1090/S0002-9939-96-03039-0
  • MathSciNet review: 1291776