Fixed Point Theory and Applications
Volume 2009 (2009), Article ID 314581, 19 pages
Research Article

Stability and Convergence Results Based on Fixed Point Theory for a Generalized Viscosity Iterative Scheme

IIDP. Faculty of Science and Technology, University of the Basque Country, Campus of Leioa (Bizkaia), P.O. Box 644, 48080 Bilbao, Spain

Received 18 February 2009; Accepted 27 April 2009

Academic Editor: Tomas Domínguez Benavides

Copyright © 2009 M. De la Sen. 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.


A generalization of Halpern's iteration is investigated on a compact convex subset of a smooth Banach space. The modified iteration process consists of a combination of a viscosity term, an external sequence, and a continuous nondecreasing function of a distance of points of an external sequence, which is not necessarily related to the solution of Halpern's iteration, a contractive mapping, and a nonexpansive one. The sum of the real coefficient sequences of four of the above terms is not required to be unity at each sample but it is assumed to converge asymptotically to unity. Halpern's iteration solution is proven to converge strongly to a unique fixed point of the asymptotically nonexpansive mapping.