Development of the direct Lyapunov method for functional-differential equations with infinite delay

We propose new sufficient conditions for the uniform asymptotic stability of the zero solution of a retarded functional-differential equation with unbounded (infinite) delay. The equation can be nonlinear and nonautonomous. The conditions are formulated in terms of Razumikhin-type functions, and in this case, a function is coupled with a functional related to this function by a certain dependence. In the results presented here, because of additional restrictions imposed on the right-hand side of the equation and the use of the limiting equation techniques, the classical requirements stating that the function and its derivative must be of fixed sign along the solution are weakened to the requirements that the function and its derivative must be of constant signs.