  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.

Functional Analysis