Let be a nonempty closed convex subset of a Banach space . A mapping is said to be asymptotically nonexpansive if there exists a sequence of reals with such that for all . A continuous strictly increasing function defined on such that and is called a gauge. A duality mapping associated with gauge is defined as
is said to have a weakly continuous duality mapping if a gauge exists such that the duality mapping is single-valued and continuous from with the weak topology to with the weak topology.
In this paper, the authors design an iterative algorithm that converges strongly to common fixed points of a sequence of asymptotically nonexpansive mappings in a reflexive Banach space with weakly continuous duality mapping. The results proved in the paper show that the uniform smoothness requirement imposed on the space in the main results of J. G. O’Hara, P. Pillay and H.–K. Xu [Nonlinear Anal., Theory Methods Appl. 54A, No. 8, 1417–1426 (2003; Zbl 1052.47049)], J. S. Jung [J. Math. Anal. Appl. 302, No. 2, 509–520 (2005; Zbl 1062.47069)] and R. Wittmann [Arch. Math. 58, No. 5, 486–491 (1992; Zbl 0797.47036)] is not required.