Аннотация:The authors investigate the system (*) z ˙=f(z,ω), φ ˙=ω, where z=(x,y), dim x=n, dim y=m, dim φ =r, dim ω =r, and f(z,ω) is periodic in ω and x with period 2π. It is proved that a Lyapunov function, which is constructed in the form of a functional of average type considered along solutions of (*), has extreme values at the stable resonance solutions. The result is applied to the study of the motion of a satellite.