A general three-step fixed-point iteration for asymptotically nonexpansive mappings in a Banach space is studied. The method includes the (modified) Noor and Ishikawa iteration schemes [cf. B. Xu
and M. Aslam Noor
, J. Math. Anal. Appl. 267, No. 2, 444–453 (2002; Zbl 1011.47039
)]. The first main result shows strong convergence if the mapping is a nonexpansive, completely continuous mapping of a nonempty, closed, convex subset of a uniformly convex Banach space into itself. The second main result asserts weak convergence in case the mapping is not necessarily completely continuous but the Banach space possesses Opial’s property.