On two classes of partial stability problems

For nonlinear nonautonomous systems of ordinary differential equations, we consider two basic classes of partial stability problems: (i) problems of stability with respect to part variables of the zero equilibrium position (Lyapunov - Rumyantsev partial stability problems); and (ii) problems of stability of partial equilibrium positions. The corresponding definitions of stability are unified so that the solvability conditions for these two classes of problems are identical in the context of the Lyapunov function method. We present a number of such common stability conditions. Also we consider problem of stability with respect to part variables of partial equilibrium positions. The results are illustrated by examples.