International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 29728, 10 pages

Strong convergence and control condition of modified Halpern iterations in Banach spaces

Yonghong Yao,1 Rudong Chen,1 and Haiyun Zhou2

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.


Let C be a nonempty closed convex subset of a real Banach space X which has a uniformly Gâteaux differentiable norm. Let TΓC and fΠC. Assume that {xt} converges strongly to a fixed point z of T as t0, where xt is the unique element of C which satisfies xt=tf(xt)+(1t)Txt. Let {αn} and {βn} be two real sequences in (0,1) which satisfy the following conditions: (C1)limnαn=0;(C2)n=0αn=;(C6)0<liminfnβnlimsupnβn<1. For arbitrary x0C, let the sequence {xn} be defined iteratively by yn=αnf(xn)+(1αn)Txn, n0, xn+1=βnxn+(1βn)yn, n0. Then {xn} converges strongly to a fixed point of T.