## Differential and Integral Equations

- Differential Integral Equations
- Volume 9, Number 1 (1996), 199-208.

### 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.

#### Article information

**Source**

Differential Integral Equations, Volume 9, Number 1 (1996), 199-208.

**Dates**

First available in Project Euclid: 7 May 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1367969996

**Mathematical Reviews number (MathSciNet)**

MR1364042

**Zentralblatt MATH identifier**

0840.34086

**Subjects**

Primary: 34K20: Stability theory

Secondary: 34D20: Stability

#### Citation

Zhang, Bo. A stability theorem in functional-differential equations. Differential Integral Equations 9 (1996), no. 1, 199--208. https://projecteuclid.org/euclid.die/1367969996