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

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

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)lim⁡n→∞αn=0;(C2)∑n=0∞αn=∞;(C6)0<lim⁡inf⁡n→∞βn≤lim⁡sup⁡n→∞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.