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Existence and multiplicity of periodic solutions for a class of second-order ordinary differential equations

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Abstract

In this paper, we study the existence of positive periodic solutions for a class of non-autonomous second-order ordinary differential equations

$$\begin{aligned} x''+\alpha x' +a(t)x^{n}-b(t)x^{n+1}+c(t)x^{n+2}=0, \end{aligned}$$

where \(\alpha \in {\mathbb {R}} \) is a constant, n is a finite positive integer, and a(t), b(t), c(t) are continuous periodic functions. By using Mawhin’s continuation theorem, we prove the existence and multiplicity of positive periodic solutions for these equations.

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References

  1. Cronin, J.: Biomathematical model of the circle of Willis: a quantitative analysis of the diffferential equation of Austin. Math. Biosci. 16, 209–225 (1973)

    Article  MathSciNet  Google Scholar 

  2. Austin, G.: Biomathematical model of the circle of Willis I: the Duffing equation and some approximate solutions. Math. Biosci. 11, 209–225 (1971)

    Article  Google Scholar 

  3. Faria, T., Oliveira, J.J.: A note on global attractivity of the periodic solution for a model of hematopoiesis. Appl. Math. Lett. 94, 1–7 (2019)

    Article  MathSciNet  Google Scholar 

  4. Chen, A., Tian, C., Huang, W.: Periodic solutions with equal period for the Friedmann-Robertson-Walker model. Appl. Math. Lett. 77, 101–107 (2018)

    Article  MathSciNet  Google Scholar 

  5. Liao, F.F.: Periodic solutions of Liebau-type differential equations. Appl. Math. Lett. 69, 8–14 (2017)

    Article  MathSciNet  Google Scholar 

  6. Araujo, A.L.A.: Periodic solutions for a nonautonomous ordinary differential equation. Nonlinear Anal. 75, 2897–2903 (2012)

    Article  MathSciNet  Google Scholar 

  7. Grossinho, M.R., Sanchez, L.: A note on periodic solutions of some nonautonomous differential equations. Bull. Austral. Math. Soc. 34, 253–265 (1986)

    Article  MathSciNet  Google Scholar 

  8. Grossinho, M.R., Sanchez, L.,St.A, Tersian.: Periodic solutions for a class of second order differential equations, in: Applications of Mathematics in Engineering, Sozopol, 1998, Heron Press, Sofia, pp. 67–70 (1999)

  9. de Araujo, A.L.A.: Existence of periodic solutions for anonautonomous differential equation. Bull. Belg. Math. Soc. Simon Stevin. 19, 305–310 (2012)

    Article  MathSciNet  Google Scholar 

  10. de Araujo, A.L.A.: Periodic solutions for a third-order differential equation without asymptotic behavior on the potential. Portugal. Math. 69, 85–94 (2012)

    Article  MathSciNet  Google Scholar 

  11. Feltrin, G., Zanolin, F.: Multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree. J. Differ. Equ. 262, 4255–4291 (2017)

    Article  MathSciNet  Google Scholar 

  12. Ma, R.: Bifurcation from infinity and multiple solutions for periodic boundary value problems. Nonlinear Anal. 42, 27–39 (2000)

    Article  MathSciNet  Google Scholar 

  13. Wang, H.: Periodic solutions to non-autonomous second-order systems. Nonlinear Anal. 71, 1271–1275 (2009)

    Article  MathSciNet  Google Scholar 

  14. Ma, R., Xu, J., Han, X.: Global structure of positive solutions for superlinear second-order periodic boundary value problems. Appl. Math. Comput. 218, 5982–5988 (2012)

    MathSciNet  MATH  Google Scholar 

  15. Yang, H., Han, X.: Existence and multiplicity of positive periodic solutions for fourth-order nonlinear differential equations. Electron. J. Differ. Equ. 119, 1–14 (2019)

    MathSciNet  Google Scholar 

  16. Li, Y.: Positive periodic solutions for fully third-order ordinary differential equations. Comput. Math. Appl. 59, 3464–3471 (2010)

    Article  MathSciNet  Google Scholar 

  17. Li, Y.: Existence and uniqueness of periodic solution for a class of semilinear evolution equations. J. Math. Anal. Appl. 349, 226–234 (2009)

    Article  MathSciNet  Google Scholar 

  18. Gaines, R.E., Mawhin, J.L.: Coincidence Degree and Nonlinear Differential Equations. Lecture Notes in Math, vol. 586. Springer, Berlin (1997)

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Authors and Affiliations

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

    Xiaoling Han & Hujun Yang

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

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Communicated by Adrian Constantin.

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Foundation term: This work is sponsored by the NNSF of China (No. 11561063)

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Han, X., Yang, H. Existence and multiplicity of periodic solutions for a class of second-order ordinary differential equations. Monatsh Math 193, 829–843 (2020). https://doi.org/10.1007/s00605-020-01465-w

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  • DOI: https://doi.org/10.1007/s00605-020-01465-w

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