On the asymptotic stability of solutions of a class of systems of nonlinear differential equations with delay

We study systems of differential equations with delay whose right-hand sides are represented as sums of potential and gyroscopic components of vector fields. We assume that in the absence of a delay zero solutions of considered systems are asymptotically stable. By the Lyapunov direct method, using the Razumikhin approach, we prove that in the case of essentially nonlinear equations the asymptotic stability of zero solutions is preserved for any value of the delay.