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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A note on periodic solutions of the delay differential equation $ x'(t)=-f(x(t-1))$


Author: Jianshe Yu
Journal: Proc. Amer. Math. Soc. 141 (2013), 1281-1288
MSC (2010): Primary 34K13, 58E50
Published electronically: August 10, 2012
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Abstract: Consider the delay differential equation $ x'(t)=-f(x(t-1))$, where $ f\in C(\mathbb{R}, \mathbb{R})$ is odd and satisfies $ xf(x)>0$ for $ x\ne 0$. When $ \alpha =\lim _{x\to 0}\frac {f(x)}{x}$ and $ \beta =\lim _{x\to \infty }\frac {f(x)}{x}$ exist, there is at least one Kaplan-Yorke periodic solution with period $ 4$ if $ \min \{\alpha ,\beta \}<\frac {\pi }{2}<\max \{\alpha ,\beta \}$. When this condition is not satisfied, we present several sufficient conditions on the existence/nonexistence of such periodic solutions. It is worthy of mention that some results are on the existence of at least two Kaplan-Yorke periodic solutions with period $ 4$ and in some cases we do not require the limits $ \alpha $ and/or $ \beta $ to exist. Hence our results not only greatly improve but also complement existing ones. Moreover, some of the theoretical results are illustrated with examples.


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Additional Information

Jianshe Yu
Affiliation: College of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, People’s Republic of China
Email: jsyu@gzhu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11386-3
PII: S 0002-9939(2012)11386-3
Received by editor(s): March 21, 2011
Received by editor(s) in revised form: August 5, 2011
Published electronically: August 10, 2012
Additional Notes: This project was supported by the National Natural Science Foundation of China (11031002) and the grant DPFC (20104410110001).
Communicated by: Yingfei Yi
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.