*(English)*Zbl 1163.47055

Let $D$ be a nonempty closed convex subset of a Banach space $X$. A mapping $T:D\to D$ is said to be asymptotically nonexpansive if there exists a sequence $\langle {k}_{n}\rangle $ of reals with ${k}_{n}\to 1$ such that $\parallel {T}^{h}x-{T}^{hy}\parallel \le {k}_{n}\parallel x-y\parallel $ for all $x,y\in D$. A continuous strictly increasing function $\phi $ defined on ${\mathbb{R}}^{+}:=[0,\infty )$ such that $\phi \left(0\right)=0$ and ${lim}_{r\to \infty}\phi \left(r\right)=\infty $ is called a gauge. A duality mapping ${J}_{\phi}:x\to {x}^{*}$ associated with gauge $\phi $ is defined as

$X$ is said to have a weakly continuous duality mapping if a gauge $\phi $ exists such that the duality mapping ${J}_{\phi}$ is single-valued and continuous from $X$ with the weak topology to ${X}^{*}$ 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.

##### MSC:

47J25 | Iterative procedures (nonlinear operator equations) |

47H06 | Accretive operators, dissipative operators, etc. (nonlinear) |

47H09 | Mappings defined by “shrinking” properties |

47H10 | Fixed point theorems for nonlinear operators on topological linear spaces |