Fixed points and stability of a nonconvolution equation

Burton, T.

doi:10.1090/S0002-9939-04-07497-0

Fixed points and stability of a nonconvolution equation
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by T. A. Burton

Proc. Amer. Math. Soc. 132 (2004), 3679-3687

DOI: https://doi.org/10.1090/S0002-9939-04-07497-0

Published electronically: May 12, 2004

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Abstract:

In this note we consider an equation of the form \[ x’(t)=-\int ^{t}_{t-r} a(t,s)g(x(s))ds\] and give conditions on $a$ and $g$ to ensure that the zero solution is asymptotically stable. When applied to the classical case of $a(t,s)=a(t-s)$, these conditions do not require that $a(r)=0$, nor do they involve the sign of $a(t)$ or the sign of any derivative of $a(t)$.

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