Journal of Applied Mathematics
Volume 2012 (2012), Article ID 787419, 13 pages
http://dx.doi.org/10.1155/2012/787419
Research Article

Iterative Algorithm for Common Fixed Points of Infinite Family of Nonexpansive Mappings in Banach Spaces

Songnian He and Jun Guo

College of Science, Civil Aviation University of China, Tianjin 300300, China

Received 9 January 2012; Accepted 18 January 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Songnian He and Jun Guo. 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

Let 𝐶 be a nonempty closed convex subset of a real uniformly smooth Banach space 𝑋 , { 𝑇 𝑘 } ∞ 𝑘 = 1 ∶ 𝐶 → 𝐶 an infinite family of nonexpansive mappings with the nonempty set of common fixed points ⋂ ∞ 𝑘 = 1 F i x ( 𝑇 𝑘 ) , and 𝑓 ∶ 𝐶 → 𝐶 a contraction. We introduce an explicit iterative algorithm 𝑥 𝑛 + 1 = 𝛼 𝑛 𝑓 ( 𝑥 𝑛 ) + ( 1 − 𝛼 𝑛 ) 𝐿 𝑛 𝑥 𝑛 , where 𝐿 𝑛 = ∑ 𝑛 𝑘 = 1 𝜔 𝑘 / s 𝑛 𝑇 𝑘 , 𝑆 𝑛 = ∑ 𝑛 𝑘 = 1 𝜔 𝑘 , and 𝑤 𝑘 > 0 with ∑ ∞ 𝑘 = 1 𝜔 𝑘 = 1 . Under certain appropriate conditions on { 𝛼 𝑛 } , we prove that { 𝑥 𝑛 } converges strongly to a common fixed point 𝑥 ∗ of { 𝑇 𝑘 } ∞ 𝑘 = 1 , which solves the following variational inequality: ⟨ 𝑥 ∗ − 𝑓 ( 𝑥 ∗ ) , 𝐽 ( 𝑥 ∗ ⋂ − 𝑝 ) ⟩ ≤ 0 , 𝑝 ∈ ∞ 𝑘 = 1 F i x ( 𝑇 𝑘 ) , where 𝐽 is the (normalized) duality mapping of 𝑋 . This algorithm is brief and needs less computational work, since it does not involve 𝑊 -mapping.