Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 157278, 18 pages
Research Article

On a Hybrid Method for Generalized Mixed Equilibrium Problem and Fixed Point Problem of a Family of Quasi-ϕ-Asymptotically Nonexpansive Mappings in Banach Spaces

Department of Mathematics, Yibin University, Yibin, Sichuan 644007, China

Received 30 November 2009; Accepted 19 January 2010

Academic Editor: Nanjing Huang

Copyright © 2010 Min Liu 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.


We prove a strong convergence theorem by using a hybrid method for finding a common element of the set of solutions for generalized mixed equilibrium problems, the set of fixed points of a family of quasi-ϕ-asymptotically nonexpansive mappings in strictly convex reflexive Banach space with the Kadec-Klee property and, a Fréchet differentiable norm under weaker conditions. The method of the proof is different from, S. Takahashi and W. Takahashi that by (2008) and that by Takahashi and Zembayashi (2008) and see references. It also shows that the type of projection used in the iterative method is independent of the properties of the mappings. The results presented in the paper improve and extend some recent results.