Let be a uniformly convex Banach space and a nonempty subset of . A mapping is said to be asymptotically nonexpansive mapping if there exists a sequence with and such that for all and for all . In this paper, if is a nonempty closed convex subset of and is a nonexpansive asymptotically mapping with a nonempty fixed point set, weak and strong convergence theorems for the iterative approximation of fixed points of are proved.
Furthermore, the results by this paper show that the boundedness requirement imposed on the subset in recent results by Z. Huang [Comput. Math. Appl. 37, No. 3, 1-7 (1999; Zbl 0942.47046)]; B. E. Rhoades [J. Mat. Anal. Appl. 183, No. 1, 118-120 (1994; Zbl 0807.47045)]; J. Schu [J. Math. Anal. Appl. 158, No. 2, 407-413 (1991; Zbl 0734.47036); Bull. Aust. Math. Soc. 43, No. 1, 153-159 (1991; Zbl 0709.47051)], can be dropped. The main results are the following:
Theorem 1: Let be a uniformly convex Banach space sastisfying Opial’s condition and let be a nonempty closed convex subset of . Let be an asymptotically nonexpansive mapping with and sequence such that and . Let and be bounded sequences in and let , , , , and be real sequence in satisfying the conditions:
(i) , ;
(ii) , ;
(iv) , .
Then the sequence generated from an arbitrary by , , , converges weakly to some fixed point of .
Theorem 2. Let be a uniformly convex Banach space and a nonempty closed subset of . Let be an asymptotically nonexpansive mapping with and sequence such that and . Suppose is compact for some . Let and be bounded sequence in and let , , , , and be as in Theorem 1. Then the sequence generate from an arbitrary as in Theorem 1 converges strongly to some fixed point of .