Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2012, Article ID 474031, 15 pages doi:10.1155/2012/474031 Research Article Convergence of Implicit and Explicit Schemes for an Asymptotically Nonexpansive Mapping in q-Uniformly Smooth and Strictly Convex Banach Spaces Meng Wen, Changsong Hu, and Zhiyu Wu Department of Mathematics, Hubei Normal University, Hubei, Huangshi 435002, China Correspondence should be addressed to Changsong Hu, huchang1004@yahoo.com.cn Received 24 April 2012; Accepted 23 June 2012 Academic Editor: Hong-Kun Xu Copyright q 2012 Meng Wen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce a new iterative scheme with Meir-Keeler contractions for an asymptotically nonexpansive mapping in q-uniformly smooth and strictly convex Banach spaces. We also proved the strong convergence theorems of implicit and explicit schemes. The results obtained in this paper extend and improve many recent ones announced by many others. 1. Introduction Let E be a real Banach space. With J : E → 2 E ∗ , we denote the normalized duality mapping given by J x  f ∈ E ∗ : 〈x, f 〉  ‖x‖ 2 ,   f    ‖x‖  , 1.1 where 〈·, ·〉 denotes the generalized duality pairing and E ∗ the dual space of E. In the sequel we will donate single-valued duality mappings by j . Given q> 1, by J q we will denote the generalized duality mapping given by J q x  f ∈ E ∗ : 〈x, f 〉  ‖x‖ q ,   f    ‖x‖ q-1  . 1.2