Weak and strong convergence theorems for fixed points of asymptotically nonexpensive mappings

Let E be a uniformly convex Banach space, K a nonempty closed convex subset of E and T : K → K an asymptotically nonexpansive mapping with a nonempty fixed-point set. Weak and strong convergence theorems for the iterative approximation of fixed points of T are proved. Our results show that the boundedness requirement imposed on the subset K in recent results of Huang [1], Rhoades [2], and Schu [3,4] can be dropped. Furthermore, our results extend these results to more satisfactory modified Mann and Ishikawa iteration methods with errors in the sense of Xu [5].