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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Differential delay equations that have periodic solutions of long period


Author: Steven A. Chapin
Journal: Trans. Amer. Math. Soc. 310 (1988), 555-566
MSC: Primary 34K15; Secondary 34C25
DOI: https://doi.org/10.1090/S0002-9947-1988-0948188-7
MathSciNet review: 948188
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Abstract: If $ f:{\mathbf{R}} \to {\mathbf{R}}$ is a continuous odd function satisfying $ xf(x) > 0$, $ x \ne 0$, and $ f(x) = o({x^{ - 2}})$ as $ x \to \infty $, then so-called periodic solutions of long period seem to play a prominent role in the dynamics of $ ({\ast})$

$\displaystyle x'(t) = - \alpha f(x(t - 1)),\qquad \alpha > 0.$

In this paper we prove the existence of long-period periodic solutions of $ ({\ast})$ for a class of nonodd functions that decay "rapidly" to 0 at infinity and satisfy $ xf(x) \geqslant 0$. These solutions have quite different qualitative features than in the odd case.

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DOI: https://doi.org/10.1090/S0002-9947-1988-0948188-7
Article copyright: © Copyright 1988 American Mathematical Society

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