Stability theorems for nonlinear functional differential equations

A new approach in stability theory for a class of retarded nonlinear functional differential equations is discussed. The problem of stability of the zero solution is considered under the assumption that the system of interest has a trivial linearization, i.e., it is essentially nonlinear. Sufficient conditions for uniform asymptotic stability and instability are given by auxiliary functionals of Lyapunov-Krasovskii type. The method is also applicable to linear systems with a small parameter in the standard form. Some examples are given.