Let K be a closed convex nonempty subset of a Hilbert space H and let the set of the common fixed points of a finite family of Lipschitzian hemicontractive maps on K be nonempty. Sufficient conditions for the strong convergence of the sequence of successive approximations generated by an M-step Picard-like process to a common fixed point of the family are proved.

Hilbert space, common fixed point, finite family, hemicontraction, Picard process, strong convergence.