A New Three-Step Mean Value Iterations with Errors for Asymptotically Nonexpansive Mappings in Banach Spaces

Hossein Dehghan, Jamal Rooin

Abstract

In this paper, we define and study a new three-step iterative scheme inspired by Nilsrakoo and Saejung [Appl. Math. Comput. 181 (2006) 1026–1034]. Many well-known iterations, for examples, Mann-type, Ishikawa-type and Noor-type iterations are covered by the new iterative scheme. Several convergence theorems of this scheme are established for asymptotically nonexpansive mappings in Banach spaces. 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], Nilsrakoo and Saejung [Appl. Math. Comput. 181 (2006) 1026–1034; Appl. Math. Comput. 190 (2007) 1472–1478] and many others.