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

Abstract

LetCbe a closed, convex subset of a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable and letTbe an asymptotically nonexpansive mapping fromCinto itself such that the setF(T) of fixed points ofTis nonempty. In this paper, we show thatF(T) is a sunny, nonexpansive retract ofC. Using this result, we discuss the strong convergence of the sequence {x_n} defined byx_n=a_nx+(1−a_n) 1/(n+1) ∑ⁿ_j=0 T^jx_nforn=0, 1, 2, …, wherex∈Cand {a_n} is a real sequence in (0, 1].