On the stability of invariant sets of functional differential equations

A system of functional differential equations with delay dz/dt=Z(t,z(t)), where z=(x, y),x epsilon R y epsilon R-m, and Z is the vector-valued functional, is considered. It is supposed that this system has a positive invariant set x=0. Definitions of its stability, asymptotic stability, and uniform asymptotic stability are given. Theorems on the uniform asymptotic stability are formulated and proved. (C) 2003 Published by Elsevier Ltd.