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In this paper, the weak convergence of an iterative twostage proximal method for the approximate solution of the equilibrium problem in a Hilbert space is investigated. This method was recently been developed by Vedel and Semenov and can be used to solve mathematical programming problems, variational inequalities and game theory problems. The analysis of the convergence of the method was carried out under the assumption of the existence of a solution of the equilibrium problem and under conditions weaker than the previously considered ones.

In this paper, we consider bilevel problem: variational inequality problem over the set of solutions the equilibrium problems. To solve this problem, an iterative algorithm is proposed that combines the ideas of a two-stage proximal method and iterative regularization. For monotone bifunctions of Lipschitz type and strongly monotone Lipschitz continuous operators, the theorem on strong convergence of sequences generated by the algorithm is proved.