Abstract

In this paper we propose and we study two algorithmic methods for finding a common solution of an equilibrium problem and a fixed point problem in a Hilbert space. The strategy is to replace the proximal point iteration used in most papers by an extragradient procedure with or without an Armijo-backtracking linesearch. The strong convergence of the iterates generated by each method is obtained thanks to a shrinking projection method and under the assumptions that the fixed point mapping is a $\xi$-quasi-strict pseudo-contraction and the equilibrium function is monotone and Lipschitz-continuous for the pure extragradient method and pseudomonotone and weakly continuous for the extragradient method with linesearches. The particular case when the equilibrium problem is a variational inequality problem is considered in the last section.