International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 19, Pages 3103-3110

Implicit iteration process of nonexpansive non-self-mappings

Somyot Plubtieng and Rattanaporn Punpaeng

Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

Received 14 February 2005; Revised 14 June 2005

Copyright © 2005 Somyot Plubtieng and Rattanaporn Punpaeng. 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.


Suppose C is a nonempty closed convex subset of real Hilbert space H. Let T:CH be a nonexpansive non-self-mapping and P is the nearest point projection of H onto C. In this paper, we study the convergence of the sequences {xn}, {yn}, {zn} satisfying xn=(1αn)u+αnT[(1βn)xn+βnTxn], yn=(1αn)u+αnPT[(1βn)yn+βnPTyn], and zn=P[(1αn)u+αnTP[(1βn)zn+βnTzn]], where {αn}(0,1), 0βnβ<1 and αn1 as n. Our results extend and improve the recent ones announced by Xu and Yin, and many others.