Journal of Applied Mathematics
Volume 2012 (2012), Article ID 961560, 22 pages
http://dx.doi.org/10.1155/2012/961560
Research Article

Generalized Mixed Equilibrium Problems and Fixed Point Problem for a Countable Family of Total Quasi- 𝜙 -Asymptotically Nonexpansive Mappings in Banach Spaces

Jinhua Zhu,1 Shih-Sen Chang,2 and Min Liu1

1Department of Mathematics, Yibin University, Yibin, Sichuan 644007, China
2Department of Mathematics, College of Statistics and Mathematics, Yunnan University of Finance and Economics, Yunnan, Kunming 650221, China

Received 4 September 2011; Accepted 25 October 2011

Academic Editor: Giuseppe Marino

Copyright © 2012 Jinhua Zhu 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 first to introduce the concept of total quasi- 𝜙 -asymptotically nonexpansive mapping which contains many kinds of mappings as its special cases and then to use a hybrid algorithm to introduce a new iterative scheme for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems and the set of common fixed points for a countable family of total quasi- 𝜙 -asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in an uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in the paper improve and extend some recent results.