Journal of Applied Mathematics
Volume 2012 (2012), Article ID 750732, 19 pages
http://dx.doi.org/10.1155/2012/750732
Research Article

A Modified Halpern-TypeIterative Method of a System of Equilibrium Problems and a Fixed Point for a Totally Quasi- 𝝓 -Asymptotically Nonexpansive Mapping in a Banach Space

Preedaporn Kanjanasamranwong,1 Poom Kumam,2 and Siwaporn Saewan1

1Department of Mathematics and Statistics, Faculty of Science, Thaksin University (TSU), Pa Phayom, Phatthalung 93110, Thailand
2Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand

Received 17 February 2012; Accepted 28 April 2012

Academic Editor: Yeol Je Cho

Copyright © 2012 Preedaporn Kanjanasamranwong 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.

Abstract

The purpose of this paper is to introduce the modified Halpern-type iterative method by the generalized f-projection operator for finding a common solution of fixed-point problem of a totally quasi- 𝜙 -asymptotically nonexpansive mapping and a system of equilibrium problems in a uniform smooth and strictly convex Banach space with the Kadec-Klee property. Consequently, we prove the strong convergence for a common solution of above two sets. Our result presented in this paper generalize and improve the result of Chang et al., (2012), and some others.