Strong Convergence of Averaged Approximants for Asymptotically Nonexpansive Mappings in Banach Spaces

✍ Scribed by Naoki Shioji; Wataru Takahashi

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 129 KB
• Volume: 97
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1996.3251

✦ Synopsis

Let C be a closed, convex subset of a uniformly convex Banach space whose norm is uniformly Ga^teaux 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. In this paper, we show that F(T ) is a sunny, nonexpansive retract of C. Using this result, we discuss the strong convergence of the sequence [x n ] defined by x n =a n x+(1&a n ) 1Â(n+1) n j=0 T j x n for n=0, 1, 2, ..., where x # C and [a n ] is a real sequence in (0, 1].