A note on periodic solutions of the delay differential equation $xβ(t)=-f(x(t-1))$
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Abstract:
Consider the delay differential equation $xβ(t)=-f(x(t-1))$, 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.References
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Additional Information
- Jianshe Yu
- Affiliation: College of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, Peopleβs Republic of China
- MR Author ID: 259924
- Email: jsyu@gzhu.edu.cn
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 1281-1288
- MSC (2010): Primary 34K13, 58E50
- DOI: https://doi.org/10.1090/S0002-9939-2012-11386-3
- MathSciNet review: 3008875