Nonmonotone functionals in the Lyapunov direct method for equations of the neutral type

We present new tests for the stability and asymptotic stability of trivial solutions of equations with deviating argument of the neutral type. Unlike well-known results, here we use nonmonotone indefinite Lyapunov functionals. Our class of functionals contains both Lyapunov-Krasovskii functionals and Lyapunov-Razumikhin functions as natural special cases. This class of functionals is broad enough that, in a number of stability tests, we have been able to omit the a priori requirement of stability of the corresponding difference operator. In addition, we present tests for the asymptotic stability of solutions of equations of the neutral type with unbounded right-hand side and new estimates for the magnitude of perturbations that do not violate the asymptotic stability if it holds for the unperturbed equation. The obtained estimates single out domains of the phase space in which perturbations should be small and domains in which essentially no constraints are imposed on the perturbation magnitude.