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1997 Annulus arguments in the stability theory for functional differential equations
László Hatvani
Differential Integral Equations 10(5): 975-1002 (1997). DOI: 10.57262/die/1367438629


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.


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László Hatvani. "Annulus arguments in the stability theory for functional differential equations." Differential Integral Equations 10 (5) 975 - 1002, 1997.


Published: 1997
First available in Project Euclid: 1 May 2013

zbMATH: 0897.34060
MathSciNet: MR1741762
Digital Object Identifier: 10.57262/die/1367438629

Primary: 34K20

Rights: Copyright © 1997 Khayyam Publishing, Inc.


Vol.10 • No. 5 • 1997
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