Fixed Point Theory and Applications
Volume 2008 (2008), Article ID 672301, 13 pages
doi:10.1155/2008/672301

Weak convergence theorems of three iterative methods for strictly pseudocontractive mappings of Browder-Petryshyn type

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.