Fixed Point Theory and Applications
Volume 2006 (2006), Article ID 59692, 16 pages
doi:10.1155/FPTA/2006/59692

Browder's type strong convergence theorems for infinite families of nonexpansive mappings in Banach spaces

Tomonari Suzuki

Department of Mathematics, Kyushu Institute of Technology, Sensuicho, Kitakyushu 804-8550, Tobata, Japan

Received 19 August 2005; Revised 24 February 2006; Accepted 26 February 2006

Copyright © 2006 Tomonari Suzuki. 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 prove Browder's type strong convergence theorems for infinite families of nonexpansive mappings. One of our main results is the following: let C be a bounded closed convex subset of a uniformly smooth Banach space E. Let {Tn:n∈ℕ} be an infinite family of commuting nonexpansive mappings on C. Let {αn} and {tn} be sequences in (0,1/2) satisfying limntn=limnαn/tnℓ=0 for ℓ∈ℕ. Fix u∈C and define a sequence {un} in C by un=(1−αn)((1−∑k=1ntnk)T1un+∑k=1ntnkTk+1un)+αnu for n∈ℕ. Then {un} converges strongly to Pu, where P is the unique sunny nonexpansive retraction from C onto ∩n=1∞F(Tn).