Multiplicity of positive periodic solutions of Rayleigh equations with singularities

Zaitao Liang; Xuemeng Shan; Hui Wei; Zaitao Liang; Xuemeng Shan; Hui Wei

doi:10.3934/math.2021377

AIMS Mathematics

2021, Volume 6, Issue 6: 6422-6438. doi: 10.3934/math.2021377

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Research article

Multiplicity of positive periodic solutions of Rayleigh equations with singularities

Department of Mathematics, Anhui University of Science and Technology, Huainan, Anhui 232001, China

Received: 25 November 2020 Accepted: 09 April 2021 Published: 13 April 2021
MSC : 34K13, 34B16

The existence of positive periodic solutions of the Rayleigh equations $ x''+f(x')+g(x) = e(t) $ with singularities is investigated in this paper. Based on the continuation theorem of coincidence degree theory and the method of upper and lower solutions, the multiple periodic solutions of the singular Rayleigh equations can be determined under the weak conditions of the term $ g $. We discuss both the repulsive singular case and the attractive singular case. Some results in the literature are generalized and improved. Moreover, some examples and numerical simulations are given to illustrate our theoretical analysis.
- Rayleigh equations,
- singularities,
- periodic solutions,
- coincidence degree theory
Citation: Zaitao Liang, Xuemeng Shan, Hui Wei. Multiplicity of positive periodic solutions of Rayleigh equations with singularities[J]. AIMS Mathematics, 2021, 6(6): 6422-6438. doi: 10.3934/math.2021377

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Abstract

The existence of positive periodic solutions of the Rayleigh equations $ x''+f(x')+g(x) = e(t) $ with singularities is investigated in this paper. Based on the continuation theorem of coincidence degree theory and the method of upper and lower solutions, the multiple periodic solutions of the singular Rayleigh equations can be determined under the weak conditions of the term $ g $. We discuss both the repulsive singular case and the attractive singular case. Some results in the literature are generalized and improved. Moreover, some examples and numerical simulations are given to illustrate our theoretical analysis.

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