Fixed Point Theory and Applications
Volume 2008 (2008), Article ID 167535, 14 pages
Research Article

Convergence on Composite Iterative Schemes for Nonexpansive Mappings in Banach Spaces

Jong Soo Jung

Department of Mathematics, Dong-A University, Busan 604-714, South Korea

Received 13 January 2008; Revised 5 April 2008; Accepted 3 May 2008

Academic Editor: Mohammed Khamsi

Copyright © 2008 Jong Soo Jung. 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 E be a reflexive Banach space with a uniformly Gâteaux differentiable norm. Suppose that every weakly compact convex subset of E has the fixed point property for nonexpansive mappings. Let C be a nonempty closed convex subset of E, f:CC a contractive mapping (or a weakly contractive mapping), and T:CC nonexpansive mapping with the fixed point set F(T). Let {xn} be generated by a new composite iterative scheme: yn=λnf(xn)+(1λn)Txn, xn+1=(1βn)yn+βnTyn, (n0). It is proved that {xn} converges strongly to a point in F(T), which is a solution of certain variational inequality provided that the sequence {λn}(0,1) satisfies limnλn=0 and n=1λn=, {βn}[0,a) for some 0<a<1 and the sequence {xn} is asymptotically regular.