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Abstract and Applied Analysis
Volume 2012, Article ID 206345, 29 pages
Research Article

A New Modified Hybrid Steepest-Descent by Using a Viscosity Approximation Method with a Weakly Contractive Mapping for a System of Equilibrium Problems and Fixed Point Problems with Minimization Problems

1Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi KMUTT, Bangkok 10140, Thailand
2Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Chiang Rai, Chiang Rai 57120, Thailand

Received 8 July 2012; Revised 29 August 2012; Accepted 29 August 2012

Academic Editor: Yongfu Su

Copyright © 2012 Uamporn Witthayarat 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.


The purpose of this paper is to consider a modified hybrid steepest-descent method by using a viscosity approximation method with a weakly contractive mapping for finding the common element of the set of a common fixed point for an infinite family of nonexpansive mappings and the set of solutions of a system of an equilibrium problem. The sequence is generated from an arbitrary initial point which converges in norm to the unique solution of the variational inequality under some suitable conditions in a real Hilbert space. The results presented in this paper generalize and improve the results of Moudafi (2000), Marino and Xu (2006), Tian (2010), Saeidi (2010), and some others. Finally, we give an application to minimization problems and a numerical example which support our main theorem in the last part.