Abstract and Applied Analysis
Volume 2006 (2006), Article ID 58684, 10 pages
doi:10.1155/AAA/2006/58684
Common fixed points of one-parameter nonexpansive semigroups in strictly convex Banach spaces
Department of Mathematics, Kyushu Institute of Technology, Sensuicho, Tobata, Kitakyushu 804-8550, Japan
Received 20 December 2003; Accepted 10 July 2005
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
One of our main results is the following convergence
theorem for one-parameter nonexpansive semigroups: let C be a bounded closed convex subset of a Hilbert space E, and let {T(t):t∈ℝ+} be a strongly continuous semigroup of nonexpansive mappings on C. Fix u∈C and t1,t2∈ℝ+ with t1<t2. Define a sequence {xn} in C by xn=(1−αn)/(t2−t1)∫t1t2T(s)xnds+αnu for n∈ℕ, where {αn} is a sequence in (0,1) converging to 0. Then {xn} converges strongly to a common fixed point of {T(t):t∈ℝ+}.