A new three-step fixed point iteration scheme for asymptotically nonexpansive mappings

In the present paper, we define and study a new three-step iterative scheme inspired by Suantai [J. Math. Anal. Appl. 311 (2005) 506-517]. This scheme includes many well-known iterations, for examples, modified Mann-type, modified Ishikawa-type iterative schemes, and the three-step iterative scheme of Xu and Noor. Several convergence theorems of this scheme are established for asymptotically nonexpansive mappings. Our results extend and improve the recent ones announced by Schu [J. Math. Anal. Appl. 158 (1991) 407-413; Bull. Aust. Math. Soc. 43 (1991) 153-159], Xu and Noor [J. Math. Anal. Appl. 267 (2002) 444-453], Suantai [J. Math. Anal. Appl. 311 (2005) 506-517], and many others. A misleading conclusion of Theorem 2.6 in Suantai's paper is also corrected.