A new general algorithm for set-valued mappings and equilibrium problem

We consider a multistep algorithm to approximate a common element of the set of solutions of monotone and Lipschitztype continuous equilibrium problems, and the set of common fixed points of a finite family of setvalued mappings satisfying condition (E). We prove strong convergence theorems of such an iterative scheme in real Hilbert spaces. This common solution is the unique solution of a variational inequality problem and it satisfies the optimality condition for a minimization problem. The main result extends various results exiting in the literature.