A strong convergence theorem for asymptotically nonexpansive mappings in Banach spaces

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 $n=0,1,2,\dots \,.$