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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 929037, 18 pages
http://dx.doi.org/10.1155/2011/929037
Research Article

Strong Convergence Theorems for a Countable Family of Nonexpansive Mappings in Convex Metric Spaces

1Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
2Materials Science Research Center, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Received 3 February 2011; Accepted 20 June 2011

Academic Editor: Stefan Siegmund

Copyright © 2011 Withun Phuengrattana and Suthep Suantai. 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

We introduce a new modified Halpern iteration for a countable infinite family of nonexpansive mappings {𝑇𝑛} in convex metric spaces. We prove that the sequence {𝑥𝑛} generated by the proposed iteration is an approximating fixed point sequence of a nonexpansive mapping when {𝑇𝑛} satisfies the AKTT-condition, and strong convergence theorems of the proposed iteration to a common fixed point of a countable infinite family of nonexpansive mappings in CAT(0) spaces are established under AKTT-condition and the SZ-condition. We also generalize the concept of W-mapping for a countable infinite family of nonexpansive mappings from a Banach space setting to a convex metric space and give some properties concerning the common fixed point set of this family in convex metric spaces. Moreover, by using the concept of W-mappings, we give an example of a sequence of nonexpansive mappings defined on a convex metric space which satisfies the AKTT-condition. Our results generalize and refine many known results in the current literature.