Journal of Applied Mathematics
Volume 2011 (2011), Article ID 953903, 28 pages
http://dx.doi.org/10.1155/2011/953903
Research Article

Common Solutions of Generalized Mixed Equilibrium Problems, Variational Inclusions, and Common Fixed Points for Nonexpansive Semigroups and Strictly Pseudocontractive Mappings

Poom Kumam,1 Usa Hamphries,1 and Phayap Katchang2

1Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand
2Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand

Received 24 April 2011; Accepted 12 July 2011

Academic Editor: Yansheng Liu

Copyright © 2011 Poom Kumam 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

We introduce a new iterative scheme by shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of common solutions of variational inclusion problems with set-valued maximal monotone mappings and inverse-strongly monotone mappings, the set of solutions of fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above four sets under some mind conditions. Furthermore, by using the above result, an iterative algorithm for solution of an optimization problem was obtained. Our results improve and extend the corresponding results of Martinez-Yanes and Xu (2006), Shehu (2011), Zhang et al. (2008), and many authors.