Extragradient and linesearch algorithms for solving equilibrium problems and fixed point problems in Banach spaces

In this paper, using sunny generalized nonexpansive retraction, we propose new extragradient and linesearch algorithms for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a relatively nonexpansive mapping in Banach spaces. To prove strong convergence of iterates in the extragradient method, we introduce a $\phi$-Lipschitz-type condition and assume that the equilibrium bifunction satisfies in this condition. This condition is unnecessary when the linesearch method is used instead of the extragradient method. A numerical example is given to illustrate the usability of our results. Our results generalize, extend and enrich some existing results in the literature.