Algorithm for equilibrium problems in Banach space

We consider the equilibrium programming problems in 2-uniformly convex and uniformly smooth Banach spaces. We obtained a theorem about weak convergence of the two-stage proximal algorithm for pseudo-monotone equilibrium programming problems in 2-uniformly convex and uniformly smooth Banach spaces. We proposed an adaptive two-stage proximal algorithm for equilibrium programming problems. The parameter update rule does not use the values of the Lipschitz constants of the bifunction. In contrast to the rules of the linear search type, it does not require calculations of the bifunction values at additional points. For pseudo-monotone bifunctions of the Lipschitz type, we prove the theorem on weak convergence of the sequences generated by the algorithm.