On the asymptotic stability with respect to a part of variables of solutions of nonlinear systems with delay

A class of nonlinear time-delay systems is studied. With the aid of the Lyapunov direct method and the Razumikhin approach, conditions are found under which one can guarantee that the zero solutions of the considered systems are stable with respect to all variables and asymptotically stable with respect to a part of variables. An example is presented to demonstrate the effectiveness of the obtained results.