Fixed Point Theory and Applications
Volume 2007 (2007), Article ID 28619, 8 pages
doi:10.1155/2007/28619
Research Article
An Iteration Method for Nonexpansive Mappings in Hilbert Spaces
Department of Mathematics, Kunming Teachers College, Kunming, Yunnan 650031, China
Received 22 August 2006; Revised 2 November 2006; Accepted 2 November 2006
Academic Editor: Nan-Jing Huang
Copyright © 2007 Lin Wang. 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
In real Hilbert space H, from an arbitrary initial point x0∈H, an explicit iteration scheme is defined as follows:
xn+1=αnxn+(1−αn)Tλn+1xn,n≥0, where Tλn+1xn=Txn−λn+1μF(Txn), T:H→H is a nonexpansive mapping such that F(T)={x∈K:Tx=x} is nonempty, F:H→H is a η-strongly
monotone and k-Lipschitzian mapping, {αn}⊂(0,1), and {λn}⊂[0,1). Under some suitable conditions, the sequence {xn} is shown to converge strongly to a fixed
point of T and the necessary and sufficient conditions that
{xn} converges strongly to a fixed point of T are
obtained.