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Fixed points and stability of a nonconvolution equation

Author: T. A. Burton
Journal: Proc. Amer. Math. Soc. 132 (2004), 3679-3687
MSC (2000): Primary 34K20, 47H10
Published electronically: May 12, 2004
MathSciNet review: 2084091
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Abstract | References | Similar Articles | Additional Information

Abstract: In this note we consider an equation of the form

\begin{displaymath}x'(t)=-\int ^{t}_{t-r} a(t,s)g(x(s))ds\end{displaymath}

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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Additional Information

T. A. Burton
Affiliation: Northwest Research Institute, 732 Caroline St., Port Angeles, Washington 98362

Keywords: Delay equations, fixed points, stability
Received by editor(s): July 8, 2003
Received by editor(s) in revised form: September 3, 2003
Published electronically: May 12, 2004
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2004 American Mathematical Society