Convergence theorems for uniformly quasi-ϕ-asymptotically nonexpansive mappings, generalized equilibrium problems, and variational inequalities

In this article, we introduce an iterative algorithm for finding a common element of the set of common fixed points of an infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings, the set of the variational inequality for an α-inverse-strongly monotone operator, and the set of solutions of the generalized equilibrium problems. We obtain a strong convergence theorem for the sequences generated by this process in a 2-uniformly convex and uniformly smooth Banach space. The results presented in this article improve and extend the recent results of Zegeye [Nonlinear Anal. 72, 2136-2146 (2010)], Wattanawitoon and Kumam [Nonlinear Anal. Hybrid Syst. 3(1), 11-20 (2009)] and many others.