Advertisement

Existence of positive periodic solutions for a neutral Liénard equation with a singularity of repulsive type

  • Shiping LuEmail author
  • Xingchen Yu
Article

Abstract

The periodic problem is studied in this paper for the neutral Liénard equation with a singularity of repulsive type
$$\begin{aligned} (x(t)-cx(t-\sigma ))''+f(x(t))x'(t)+\varphi (t)x(t-\tau )-\frac{r(t)}{x^{\mu }(t)}=h(t), \end{aligned}$$
where \(f:[0,+\infty )\rightarrow R\) is continuous, \(r: R\rightarrow (0,+\infty )\) and \(\varphi :R \rightarrow R\) are continuous with T-periodicity in the t variable, \(c,\mu ,\sigma ,\tau \) are constants with \(|c|>1,\mu >1,0<\sigma ,\tau <T\). Many authors obtained the existence of periodic solutions under the condition \(|c|<1\) , and we extend their results to the case of \(|c|>1\). The proof of the main result relies on a continuation theorem of coincidence degree theory established by Mawhin.

Keywords

Neutral functional differential equation periodic solution singularity continuation theorem 

Mathematics Subject Classification

34K13 34B16 34C25 

Notes

Acknowledgements

The authors are grateful to the referee for the careful reading of the paper and for useful suggestions. The authors gratefully acknowledge support from NSF of China (no. 11271197).

References

  1. 1.
    Torres, P.J.: Mathematical Models with Singularities—A Zoo of Singular Creatures. Atlantis Press, Amsterdam (2015)CrossRefGoogle Scholar
  2. 2.
    Bevc, V., Palmer, J.L., Süsskind, C.: On the design of the transition region of axi-symmetric magnetically focused beam valves. J. Br. Inst. Radio Eng. 18, 696–708 (1958)Google Scholar
  3. 3.
    Ye, Y., Wang, X.: Nonlinear differential equations in electron beam focusing theory. Acta Math. Appl. Sin. 1, 13–41 (1978). (in Chinese)Google Scholar
  4. 4.
    Huang, J., Ruan, S., Song, J.: Bifurcations in a predator–prey system of Leslie type with generalized Holling type III functional response. J. Differ. Equ. 257(6), 1721–1752 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Plesset, M.S., Prosperetti, A.: Bubble dynamic and cavitation. Annu. Rev. Fluid Mech. 9, 145–185 (1977)CrossRefGoogle Scholar
  6. 6.
    Jebelean, P., Mawhin, J.: Periodic solutions of singular nonlinear differential perturbations of the ordinary \(p\)-Laplacian. Adv. Nonlinear Stud. 2(3), 299–312 (2002)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Lazer, A.C., Solimini, S.: On periodic solutions of nonlinear dfferential equations with singularities. Proc. Am. Math. Soc. 99, 109–114 (1987)CrossRefGoogle Scholar
  8. 8.
    Li, X., Zhang, Z.: Periodic solutions for second order differential equations with a singular nonlinearity. Nonlinear Anal. 69, 3866–3876 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Lu, S., Guo, Y., Chen, L.: Periodic solutions for Liénard equation with an indefinite singularity. Nonlinear Anal. Real World Appl. 45, 542–556 (2019)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Jiang, D., Chu, J., Zhang, M.: Multiplicity of positive periodic solutions to superlinear repulsive singular equations. J. Differ. Equ. 211, 282–302 (2005)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Martins, R.: Existence of periodic solutions for second-order differential equations with singularities and the strong force condition. J. Math. Anal. Appl. 317, 1–13 (2006)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Zhang, M.: Periodic solutions of Liénard equations with singular forces of repulsive type. J. Math. Anal. Appl. 203(1), 254–269 (1996)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Yu, X., Lu, S.: A multiplicity result for periodic solutions of Liénard equations with an attractive singularity. Appl. Math. Comput. 346, 183–192 (2019)MathSciNetGoogle Scholar
  14. 14.
    Hakl, R., Zamora, M.: On the open problems connected to the results of Lazer and Solimini. Proc. R.Soc. Edinb. Sect. A. Math. 144, 109–118 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Torres, P.J.: Weak singularities may help periodic solutions to exist. J. Differ. Equ. 232, 277–284 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Chu, J., Torres, P.J., Zhang, M.: Periodic solutions of second order non-autonomous singular dynamical systems. J. Differ. Equ. 239, 196–212 (2007)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Peng, S.: Periodic solutions for p-Laplacian neutral Rayleigh equation with a deviating argument. Nonlinear Anal. 69, 1675–1685 (2008)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lu, S., Xu, Y., Xia, D.: New properties of the D-operator and its applications on the problem of periodic solutions to neutral functional differential system. Nonlinear Anal. 74, 3011–3021 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lu, S., Chen, L.: The problem of existence of periodic solutions for neutral functional differential system with nonlinear difference operator. J. Math. Anal. Appl. 387, 1127–1136 (2012)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Wang, Z.: Periodic solutions of Liénard equation with a singularity and a deviating argument. Nonlinear Anal. Real World Appl. 16(16), 227–234 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kong, F., Luo, Z., Lu, S.: Positive periodic solutions for singular high-order neutral functional differential equations. Math. Slovaca 68, 379–396 (2018)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kong, F., Lu, S., Liang, Z.: Existence of positive periodic solutions for neutral Liénard differential equations with a singularity. Electron. J. Differ. Equ. 242, 1–12 (2015)zbMATHGoogle Scholar
  23. 23.
    Kyrychko, Y.N., Blyuss, K.B., Gonzalez-Buelga, A., Hogan, S.J., Wagg, D.J.: Real-time dynamic sub structuring in a coupled oscillator–pendulum system. Proc. R. Soc. A 462, 1271–1294 (2006)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Hale, J., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer, New York (1993)CrossRefGoogle Scholar
  25. 25.
    Gaines, R., Mawhin, J.: Coincidence Degree and Nonlinear Differential Equation. Springer, Berlin (1977)CrossRefGoogle Scholar
  26. 26.
    Lu, S., Ge, W.: On the existence of periodic solutions for a kind of second order neutral functional differential equation. Appl. Math. Comput. 157, 433–448 (2004)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Hakl, R., Torres, P.J., Zamora, M.: Periodic solutions of singular second order differential equations: upper and lower functions. Nonlinear Anal. 74, 7078–7093 (2011)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Lu, S., Gui, Z.: On the existence of periodic solutions to \(p\)-Laplacian Rayleigh differential equation with a delay. J. Math. Anal. Appl. 325(1), 685–702 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsNanjing University of Information Science and TechnologyNanjingPeople’s Republic of China

Personalised recommendations