Differential and Integral Equations

Annulus arguments in the stability theory for functional differential equations

László Hatvani

Abstract

An annulus argument is a method of proof which can detect that a curve in $\mathbb{R}^n$ crosses an annulus around the origin infinitely many times. In this paper we give annulus arguments not requiring the boundedness of the derivatives of the functions involved. Using these results we establish Lyapunov type theorems for the attractivity, asymptotic stability, and partial stability properties of the zero solution of nonautonomous functional differential equations whose right hand sides are not bounded with respect to the time. We apply these results to the scalar equation $$x'(t) = -c(t)x(t) + b(t)x(t-h)\qquad(c(t) \ge 0),$$ the scalar equation with several delays $$x'(t) = -c(t)x(t) + \sum^n_{i=1} b_i(t)x(t - h_i)\qquad(c(t) \ge 0),$$ as well as to the system $$x'(t) = B(t)x(t-h)-C(t)x(t),$$ where $B (t)$ and $C(t)$ are continuous matrix functions.

Article information

Source
Differential Integral Equations, Volume 10, Number 5 (1997), 975-1002.

Dates
First available in Project Euclid: 1 May 2013

https://projecteuclid.org/euclid.die/1367438629

Mathematical Reviews number (MathSciNet)
MR1741762

Zentralblatt MATH identifier
0897.34060

Subjects
Primary: 34K20: Stability theory

Citation

Hatvani, László. Annulus arguments in the stability theory for functional differential equations. Differential Integral Equations 10 (1997), no. 5, 975--1002. https://projecteuclid.org/euclid.die/1367438629