Archiv der Mathematik

, Volume 72, Issue 5, pp 354–359

A strong convergence theorem for asymptotically nonexpansive mappings in Banach spaces

  • Naoki Shioji
  • Wataru Takahashi

DOI: 10.1007/s000130050343

Cite this article as:
Shioji, N. & Takahashi, W. Arch. Math. (1999) 72: 354. doi:10.1007/s000130050343


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 {an} be a sequence of real numbers with \(0 \leq a_n \leq 1\), and let x and x0 be elements of C. In this paper, we study the convergence of the sequence {xn} 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 \( n=0,1,2,\dots \,.\)

Copyright information

© Birkhäuser Verlag, Basel 1999

Authors and Affiliations

  • Naoki Shioji
    • 1
  • Wataru Takahashi
    • 3
  1. 1.Faculty of Engineering, Tamagawa University, Tamagawa-Gakuen, Machida, Tokyo 194, JapanJP
  2. 2.Current address: Department of Mathematics, Faculty of Engineering, Yokohama National University, Tokiwadai, Hodogaya-ku, Yokohama 240-8501, JapanJP
  3. 3.Department of Mathematical and Computer Sciences, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152, JapanJP