An extragradient approximation method for system of equilibrium problems and variational inequality problems

Chaichana Jaiboon, Poom Kumam

Abstract

The purpose of this paper is to investigate the problem of finding the common element of the set of common fixed points of an infinite family of nonexpansive mappings, the set of solutions of a system of equilibrium problems and the set of solutions of the variational inequality problem for a monotone and $\zeta$-Lipschitz continuous mapping in Hilbert spaces. Then, we prove that the strong convergence of the proposed iterative algorithm to the unique solutions of variational inequality, which is the optimality condition for a minimization problem. Our results extend and improve the corresponding results of Colao, Marino and Xu [V. Colao, G. Marino and H.K. Xu b, An iterative method for finding common solutions of equilibrium and fixed point problems, J. Math. Anal. Appl. 344 (2008) 340-352] and Peng and Yao [J.W. Peng and J.C. Yao, A viscosity approximation scheme for system of equilibrium problems, nonexpansive mappings and monotone mappings, Nonlinear Analysis. Doi.org/10.1016/j.na.2009.05.028] and many others.