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In this paper, using sunny generalized nonexpansive retraction, we proposenew extragradient and linesearch algorithms for finding a common element of theset of solutions of an equilibrium problem and the set of fixed points of arelatively nonexpansive mapping in Banach spaces. To prove strong convergenceof iterates in the extragradient method, we introduce a $\phi$-Lipschitz-typecondition and assume that the equilibrium bifunction satisfies in thiscondition. This condition is unnecessary when the linesearch method is usedinstead of the extragradient method. A numerical example is given to illustratethe usability of our results. Our results generalize, extend and enrich someexisting results in the literature.

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