A stability theorem in functional-differential equations

Abstract

Consider a system of functional differential equations with finite delay $$ x'(t)=F(t,x_t),\quad x\in\Bbb R^n. \tag1 $$ We extend a result of Hering and show that if there exists a continuous functional $V:\Bbb R^+\times C_H\to\Bbb R^+$ and a constant $\gamma>0$ such that $$ \begin{align} &W_1(\phi(0)|)\le V(t,\phi)\le W_2(|\phi|_h)+W_3(\|\phi\|),\\ &V'_{(1)}(t,\phi)\le-W_4(|\phi|_h),\quad\text{and}\quad W_1(r)-W_3(r)>0\quad\text{for }r\in(0,\gamma), \end{align} $$ then the zero solution of (1) is uniformly asymptotically stable. Here $|\cdot|_h$ is a seminorm on the space $C([-h,0],\Bbb R^n)$. This result generalizes some very important theorems in the literature including Burton and Krasovskii's theorems. We also show that Krasovskii's second theorem is a corollary of his first theorem.