We introduce a finite family of nonspreading multivalued mappings and a finite family of nonexpansive multivalued mappings in Hilbert space. We establish some weak and strong convergence theorems of the sequences generated by our iterative process. Some new iterative sequences for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points it was introduced. The results improve and extend the corresponding results of Mohammad Eslamian (Optim. Lett., 7(3):547-557, 2012).