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Existence of Periodic Solutions for the Forced Pendulum Equations of Variable Length

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Abstract

This article investigates the forced pendulum equations of variable length

$$\begin{aligned} x''+k x' + a(t) \sin x= e(t), \end{aligned}$$

where a(t), e(t) are continuous T-periodic functions, \(k \in {\mathbb {R}}\) is a constant. Under suitable assumptions on the a(t), e(t) and T, we prove the existence of T-periodic solutions to the forced pendulum equations using Mawhin’s continuation theorem. Finally, some specific examples and numerical simulations are given to illustrate the applicability of the conclusions of this paper.

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Acknowledgements

The authors warmly thank the anonymous referee for his/her careful reading of the article and many pertinent remarks that lead to various improvements to this article. The authors thank the help from the editor too. This work is supported by the National Natural Science Foundation of China(No. 12161079) and Natural Science Foundation of Gansu Province(No. 20JR10RA086).

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  1. Department of Mathematics, Northwest Normal University, Lanzhou, 730070, Gansu Province, People’s Republic of China

    Hujun Yang & Xiaoling Han

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  1. Hujun Yang
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  2. Xiaoling Han
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Correspondence to Xiaoling Han.

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Yang, H., Han, X. Existence of Periodic Solutions for the Forced Pendulum Equations of Variable Length. Qual. Theory Dyn. Syst. 22, 20 (2023). https://doi.org/10.1007/s12346-022-00723-6

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