Journal of Applied Mathematics
Volume 2012 (2012), Article ID 327434, 17 pages
http://dx.doi.org/10.1155/2012/327434
Research Article

Common Fixed Points for Asymptotic Pointwise Nonexpansive Mappings in Metric and Banach Spaces

P. Pasom1 and B. Panyanak1,2

1Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 26 October 2011; Accepted 8 December 2011

Academic Editor: Rudong Chen

Copyright © 2012 P. Pasom and B. Panyanak. 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 bounded closed convex subset of a complete CAT(0) space 𝑋 . We prove that the common fixed point set of any commuting family of asymptotic pointwise nonexpansive mappings on 𝐶 is nonempty closed and convex. We also show that, under some suitable conditions, the sequence { 𝑥 𝑘 } ∞ 𝑘 = 1 defined by 𝑥 𝑘 + 1 = ( 1 − 𝑡 𝑚 𝑘 ) 𝑥 𝑘 ⊕ 𝑡 𝑚 𝑘 𝑇 𝑛 𝑘 𝑚 𝑦 ( 𝑚 − 1 ) 𝑘 , 𝑦 ( 𝑚 − 1 ) 𝑘 = ( 1 − 𝑡 ( 𝑚 − 1 ) 𝑘 ) 𝑥 𝑘 ⊕ 𝑡 ( 𝑚 − 1 ) 𝑘 𝑇 𝑛 𝑘 𝑚 − 1 𝑦 ( 𝑚 − 2 ) 𝑘 , 𝑦 ( 𝑚 − 2 ) 𝑘 = ( 1 − 𝑡 ( 𝑚 − 2 ) 𝑘 ) 𝑥 𝑘 ⊕ 𝑡 ( 𝑚 − 2 ) 𝑘 𝑇 𝑛 𝑘 𝑚 − 2 𝑦 ( 𝑚 − 3 ) 𝑘 , … , 𝑦 2 𝑘 = ( 1 − 𝑡 2 𝑘 ) 𝑥 𝑘 ⊕ 𝑡 2 𝑘 𝑇 𝑛 𝑘 2 𝑦 1 𝑘 , 𝑦 1 𝑘 = ( 1 − 𝑡 1 𝑘 ) 𝑥 𝑘 ⊕ 𝑡 1 𝑘 𝑇 𝑛 𝑘 1 𝑦 0 𝑘 , 𝑦 0 𝑘 = 𝑥 𝑘 , 𝑘 ∈ ℕ , converges to a common fixed point of 𝑇 1 , 𝑇 2 , … , 𝑇 𝑚 where they are asymptotic pointwise nonexpansive mappings on 𝐶 , { 𝑡 𝑖 𝑘 } ∞ 𝑘 = 1 are sequences in [ 0 , 1 ] for all 𝑖 = 1 , 2 , … , 𝑚 , and { 𝑛 𝑘 } is an increasing sequence of natural numbers. The related results for uniformly convex Banach spaces are also included.