Asymptotic behavior of control systems with periodic vector nonlinearities and denumerable sets of equibliria is examined. Both continuous systems, described by ordinary differential equations, and discrete systems, described by difference equations, are considered. Certain generalization of periodic Lyapunov-type functions and sequences is offered. By means of generalized Lyapunov-type functions and Yakubovich-Kalman theorem new frequency-domain criteria which guarantee that every solution of the system tends to an equilibrium are obtained.