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


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 \( n=0,1,2,\dots \,.\)


Banach Space Real Number Convex Subset Convergence Theorem Nonexpansive Mapping 
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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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