Open Access Research

The shrinking projection method for solving generalized equilibrium problems and common fixed points for asymptotically quasi-ϕ-nonexpansive mappings

Siwaporn Saewan and Poom Kumam*

Author Affiliations

Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand

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Fixed Point Theory and Applications 2011, 2011:9  doi:10.1186/1687-1812-2011-9

Published: 23 June 2011

Abstract

In this article, we introduce a new hybrid projection iterative scheme based on the shrinking projection method for finding a common element of the set of solutions of the generalized mixed equilibrium problems and the set of common fixed points for a pair of asymptotically quasi-ϕ-nonexpansive mappings in Banach spaces and set of variational inequalities for an α-inverse strongly monotone mapping. The results obtained in this article improve and extend the recent ones announced by Matsushita and Takahashi (Fixed Point Theory Appl. 2004(1):37-47, 2004), Qin et al. (Appl. Math. Comput. 215:3874-3883, 2010), Chang et al. (Nonlinear Anal. 73:2260-2270, 2010), Kamraksa and Wangkeeree (J. Nonlinear Anal. Optim.: Theory Appl. 1(1):55-69, 2010) and many others.

AMS Subject Classification: 47H05, 47H09, 47J25, 65J15.

Keywords:
Generalized mixed equilibrium problem; Asymptotically quasi-ϕ-nonexpansive mapping; Strong convergence theorem; Variational inequality; Banach spaces