Hybrid Methods for Solving Simultaneously an Equilibrium Problem and Countably Many Fixed Point Problems in a Hilbert Space

This paper presents a framework of iterative methods for finding a common solution to an equilibrium problem and a countable number of fixed point problems defined in a Hilbert space. A general strong convergence theorem is established under mild conditions. Two hybrid methods are derived from the proposed framework in coupling the fixed point iterations with the iterations of the proximal point method or the extragradient method, which are well-known methods for solving equilibrium problems. The strategy is to obtain the strong convergence from the weak convergence of the iterates without additional assumptions on the problem data. To achieve this aim, the solution set of the problem is outer approximated by a sequence of polyhedral subsets.