Convergence of three-step iterations for asymptotically nonexpansive mappings

In this paper, weak and strong convergence theorems of three-step iterations are established for asymptotically nonexpansive mappings in Banach spaces. The results obtained in this paper extend and improve the recent ones announced by Xu and Noor [B.L. Xu, M.A. Noor, Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 267 (2002) 444-453], Suthep Suantai [S. Suantai, Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings, J. Math. Anal. Appl. 311 (2005) 506-517], Weerayuth Nilsrakoo and Satit Saejung [W. Nilsrakoo, S. Saejung, A new three-step fixed point iteration scheme for asymptotically nonexpansive mappings, Appl. Math. Comput., in press, doi:10.1016/j.amc.2006.01.063] and others.