International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 29728, 10 pages
doi:10.1155/IJMMS/2006/29728
Strong convergence and control condition of modified
Halpern iterations in Banach spaces
1Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
2Department of Mathematics, Shijiazhuang
Mechanical Engineering College, Shijiazhuang 050003, China
Received 27 August 2005; Revised 14 February 2006; Accepted 28 February 2006
Copyright © 2006 Yonghong Yao et al. 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 C
be a nonempty closed convex subset of a real Banach space
X
which has a uniformly Gâteaux differentiable norm. Let
T∈ΓC
and f∈ΠC. Assume that {xt}
converges
strongly to a fixed point z
of T
as t→0, where
xt
is the unique element of C
which satisfies
xt=tf(xt)+(1−t)Txt. Let {αn}
and {βn} be two real sequences in (0,1) which satisfy the following conditions: (C1)limn→∞αn=0;(C2)∑n=0∞αn=∞;(C6)0<liminfn→∞βn≤limsupn→∞βn<1. For arbitrary x0∈C, let the sequence
{xn}
be defined iteratively by
yn=αnf(xn)+(1−αn)Txn, n≥0,
xn+1=βnxn+(1−βn)yn, n≥0. Then {xn}
converges strongly to a fixed point of T.