Archiv der Mathematik

, Volume 72, Issue 5, pp 354–359 | Cite as

A strong convergence theorem for asymptotically nonexpansive mappings in Banach spaces

  • Naoki Shioji
  • Wataru Takahashi

Abstract.

Let C be a closed, convex subset of a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable and let T be an asymptotically nonexpansive mapping from C into itself such that the set F (T) of fixed points of T is nonempty. Let {a n } be a sequence of real numbers with \(0 \leq a_n \leq 1\), and let x and x 0 be elements of C. In this paper, we study the convergence of the sequence {x n } defined by¶¶\(x_{n+1}=a_n x + (1-a_n) {1\over n+1} \sum\limits_{j=0}^n T^j x_n\quad \) for \( n=0,1,2,\dots \,.\)

Keywords

Banach Space Real Number Convex Subset Convergence Theorem Nonexpansive Mapping 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag, Basel 1999

Authors and Affiliations

  • Naoki Shioji
    • 1
  • Wataru Takahashi
    • 2
  1. 1.Faculty of Engineering, Tamagawa University, Tamagawa-Gakuen, Machida, Tokyo 194, JapanJP
  2. 2.Department of Mathematical and Computer Sciences, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152, JapanJP

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