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

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. © 2011 Saewan and Kumam; licensee Springer.