Approximation of the Nearest Common Fixed Point of Asymptotically Nonexpansive Mappings in Banach Spaces

Abstract

In this paper, the iteration $x_{n+1} = α_ny + (1 − α_n)T^{k(n)}_{i(n)} x_n$ for a family of asymptotically nonexpansive mappings $T_1, T_2, · · · , T_N$ is originally introduced in a uniformly convex Banach space. Motivated by recent papers, we prove that under suitable conditions the iteration scheme converges strongly to the nearest common fixed point of the family of asymptotically nonexpansive mappings. The results presented in this paper expand and improve corresponding ones from Hilbert spaces to uniformly convex Banach spaces, or from nonexpansive mappings to asymptotically nonexpansive mappings.