Fixed Point Theory and Applications
Volume 2008 (2008), Article ID 672301, 13 pages
doi:10.1155/2008/672301
Research Article
Weak Convergence Theorems of Three Iterative Methods for Strictly Pseudocontractive Mappings of Browder-Petryshyn Type
School of Mathematics and Physics, North China Electric Power University, Baoding, Hebei 071003, China
Received 25 September 2007; Revised 29 January 2008; Accepted 28 February 2008
Academic Editor: Huang Nan-Jing
Copyright © 2008 Ying Zhang and Yan 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 E be a real q-uniformly smooth Banach space which is also uniformly convex (e.g., Lp
or lp spaces (1<p<∞)), and K a nonempty closed convex subset of E. By constructing nonexpansive mappings, we elicit the weak convergence of Mann's algorithm for a κ-strictly pseudocontractive mapping of Browder-Petryshyn type on K in condition thet the control sequence {αn} is chosen so that (i) μ≤αn<1,n≥0; (ii) ∑n=0∞(1−αn)[qκ−Cq(1−αn)q−1]=∞, where μ∈[max{0,1−(qκ/Cq)1/(q−1)},1). Moreover, we consider to find a common fixed point of a finite family of strictly pseudocontractive mappings and consider the parallel and cyclic algorithms for solving this problem. We will prove the weak convergence of these algorithms.