Extragradient method for solving variational inequality and equilibrium problem in Banach spaces

The extragradient method is well known because of its efficiency in numerical tests. In this paper, we propose a new extragradient algorithm for finding the solution of a variational inequality problem whose given operator is an $\alpha$-invers strongly monotone and constraint set is the element of the set of solutions of an equilibrium problem in Banach spaces. To obtain strong convergence for the sequences which are generated by our algorithm, we assume that the equilibrium function satisfies in $\phi$-Lipschitz-type .condition