Discrete Dynamics in Nature and Society
Volume 2011 (2011), Article ID 487864, 16 pages
doi:10.1155/2011/487864
Research Article

Iterative Approximation of Common Fixed Points of Two Nonself Asymptotically Nonexpansive Mappings

1Department of Mathematics, Faculty of Science, Ataturk University, 25240 Erzurum, Turkey
2Department of Mathematics, Statistics and Physics, Qatar University, Doha 2713, Qatar

Received 11 October 2010; Accepted 17 February 2011

Academic Editor: Xiaohui Liu

Copyright © 2011 Esref Turkmen 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.

Abstract

Suppose that 𝐾 is nonempty closed convex subset of a uniformly convex and smooth Banach space 𝐸 with 𝑃 as a sunny nonexpansive retraction and 𝐹 = 𝐹 ( 𝑇 1 ) 𝐹 ( 𝑇 2 ) = { 𝑥 𝐾 𝑇 1 𝑥 = 𝑇 2 𝑥 = 𝑥 } . Let 𝑇 1 , 𝑇 2 𝐾 𝐸 be two weakly inward nonself asymptotically nonexpansive mappings with respect to 𝑃 with two sequences { 𝑘 𝑛 ( 𝑖 ) } [ 1 , ) satisfying 𝑛 = 1 ( 𝑘 𝑛 ( 𝑖 ) 1 ) < ( 𝑖 = 1 , 2 ) , respectively. For any given 𝑥 1 𝐾 , suppose that { 𝑥 𝑛 } is a sequence generated iteratively by 𝑥 𝑛 + 1 = ( 1 𝛼 𝑛 ) ( 𝑃 𝑇 1 ) 𝑛 𝑦 𝑛 + 𝛼 𝑛 ( 𝑃 𝑇 2 ) 𝑛 𝑦 𝑛 , 𝑦 𝑛 = ( 1 𝛽 𝑛 ) 𝑥 𝑛 + 𝛽 𝑛 ( 𝑃 𝑇 1 ) 𝑛 𝑥 𝑛 , 𝑛 , where { 𝛼 𝑛 } and { 𝛽 𝑛 } are sequences in [ 𝑎 , 1 𝑎 ] for some 𝑎 ( 0 , 1 ) . Under some suitable conditions, the strong and weak convergence theorems of { 𝑥 𝑛 } to a common fixed point of 𝑇 1 and 𝑇 2 are obtained.