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Positive Periodic Solutions of Singular Third-Order Functional Differential Equations with p-Laplacian-Like Operators

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Abstract

The aim of this paper is to consider singular third-order functional differential equations with p-Laplacian-like operator of the form

$$\left( {{\varphi _p}\left( {x^{\prime \prime}\left( t \right)} \right)} \right)^{\prime} + f\left( {t,x\left( {t - \tau } \right)} \right) = e\left( t \right).$$

Unlike in previous works, f has a strong singularity at x = 0 and satisfies a small force condition at x = ∞. Based on a continuation theorem due to Mawhin, new results on the existence of positive periodic solutions are obtained, which makes it possible to refine and extend some related results in the literature. A typical example demonstrating the effectiveness and flexibility of the main results is given.

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References

  1. H. Amann, “Existence theorems for equations of Hammerstein type,” Appl. Anal. 2, 385–397 (1973).

    Article  MathSciNet  Google Scholar 

  2. D. Anderson, “Green’s function for a third-order generalized right focal problem,” J. Math. Anal. Appl. 288, 1–14 (2003).

    Article  MathSciNet  Google Scholar 

  3. J. Cronin, “Some mathematics of biological oscillations,” SIAM Rev. 19, 100–137 (1977).

    Article  MathSciNet  Google Scholar 

  4. J. Chu and Z. Zhou, “Positive solutions for singular non-linear third-order periodic boundary value problems,” Nonlinear Anal. 64 1528–1542 (2006).

    Article  MathSciNet  Google Scholar 

  5. Z. Du, W. Ge, and M. Zhou, “Singular perturbations for third-order nonlinear multi-point boundary value problem,” J. Differential Equations. 218, 69–90 (2005).

    Article  MathSciNet  Google Scholar 

  6. J. Ezeilo, “On the existence of periodic solutions of a certain third-order differential equation,” Proc. Cambridge Philos. Soc. 56, 381–389 (1960).

    Article  MathSciNet  Google Scholar 

  7. K. Friedrichs, “On nonlinear vibrations of the third-order,” in: Studies in Nonlinear Vibrations Theory, Inst. Math. Mech., New York University, 2, 65–103 (1946).

    MathSciNet  Google Scholar 

  8. Y. Feng, “On the existence and multiplicity of positive periodic solutions of a nonlinear third-order equation,” Appl. Math. Lett. 22, 1220–1224 (2009).

    Article  MathSciNet  Google Scholar 

  9. Y. Feng, “Existence and uniqueness results for a third-order implicit differential equation,” Comput. Math. Appl. 56, 2507–2514 (2008).

    Article  MathSciNet  Google Scholar 

  10. L. Guo, J. Sun, and Y. Zhao, “Existence of positive solutions for nonlinear third-order three-point boundary value problems,” Nonlinear Anal. 68, 3151–3158 (2008).

    Article  MathSciNet  Google Scholar 

  11. R. Hakl and P. J. Torres, “On periodic solutions of second-order differential equations with attractive-repulsive singularities,” J. Differential Equations. 248, 111–126 (2010).

    Article  MathSciNet  Google Scholar 

  12. L. Kong and S. Wang, “Positive solution of a singular nonlinear third-order periodic boundary value problem,” J. Comput. Appl. Math. 132, 247–253 (2001).

    Article  MathSciNet  Google Scholar 

  13. F. C. Kong, S. P. Lu, and Z. G. Luo, “Positive periodic solutions for singular higher order delay differential equations,” Results Math. 72, 71–86 (2017).

    Article  MathSciNet  Google Scholar 

  14. F. C. Kong, Z. G. Luo, and S. P. Lu, “Positive periodic solutions for singular high-order neutral functional differential equations,” Math. Slovaca. 68, 379–396 (2018).

    Article  MathSciNet  Google Scholar 

  15. B. Li, “Uniqueness and stability of a limit cycle for a third-order dynamical system arising in neuron modelling,” Nonlinear Anal. 5, 13–19 (1981).

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. S. P. Lu and X. W. Jia, “Homoclinic solutions for a second-order singular differential equation,” J. Fixed Point Theory Appl. 20 (3), 101 (2018).

    Article  MathSciNet  Google Scholar 

  18. S. P. Lu, Y. Y. Guo, and L. J. Chen, “Periodic solutions for Linard equation with an indefinite singularity,” Nonlinear Anal. Real World Appl. 45, 542–556 (2019).

    Article  MathSciNet  Google Scholar 

  19. S. P. Lu and R. Y. Xue, “Periodic solutions for a singular Liénard equation with indefinite weight,” Topol. Methods Nonlinear Anal. 54, 203–218 (2019).

    MathSciNet  MATH  Google Scholar 

  20. R. Mulholland and M. Keener, “Analysis of linear compartment models for ecosystems,” J. Theoret. Biol. 44, 106–116 (1974).

    Article  Google Scholar 

  21. J. Mawhin, Topological Degree and Boundary-Value Problems for Nonlinear Differential Equations, in Lecture Notes in Mathematics, Vol. 1537: Topologic Methods for Ordinary Differential Equations, M. Furi and P. Zecca (eds.) (Springer, New York, 1993).

    Google Scholar 

  22. H. N. Pishkenari, M. Behzad, and A. Meghdari, “Nonlinear dynamic analysis of atomic force microscopy under deterministic and random excitation,” Chaos Solitons Fractals. 37 (3), 748–762 (2008).

    Article  Google Scholar 

  23. L. Rauch, “Oscillations of a third order nonlinear autonomous system”, in: Contributions to the Theory of Nonlinear Oscillations, in: Ann. of Math. Stud. 20, 39–88 (1950).

    MathSciNet  Google Scholar 

  24. S. Rützel, S. I. Lee, and A. Raman, “Nonlinear dynamics of atomic-force-microscope probes driven in Lennard-Jones potentials,” Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 459, 1925–1948 (2003).

    Article  Google Scholar 

  25. J. Sun and Y. Liu, “Multiple positive solutions of singular third-order periodic boundary value problem,” Acta Math. Sci. 25, 81–88 (2005).

    Article  MathSciNet  Google Scholar 

  26. G. Villari, “Soluzioni periodiche di una classa di equazioni differenziali de terzòrdine quasi lineari,” Ann. Mat. Pura Appl. 73 (4), 103–110 (1966).

    Article  MathSciNet  Google Scholar 

  27. G. Yang, J. Lu, and A. C. J. Luo, “On the computation of Lyapunov exponents for forced vibration of a Lennard-Jones oscillator,” Chaos Solitons Fractals. 23, 833–841 (2005).

    Article  Google Scholar 

  28. H. Yu and M. Pei, “Solvability of a nonlinear third-order periodic boundary value problem,” Appl. Math. Lett. 23, 892–896 (2010).

    Article  MathSciNet  Google Scholar 

  29. X. C. Yu and S. P. Lu, “A multiplicity result for periodic solutions of Liénard equations with an attractive singularity,” Appl. Math. Comput. 346, 183–192 (2019).

    MathSciNet  MATH  Google Scholar 

  30. Z. Zhang and Z. Wang, “Periodic solutions of the third order functional differential equations,” J. Math. Anal. Appl. 292, 115–134 (2004).

    Article  MathSciNet  Google Scholar 

  31. M. Zhang, “Nonuniform nonresonance at the first eigenvalue of the p-Laplacian,” Nonlinear Anal. 29(1), 41–51 (1997).

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

We would like to thank the editors and the anonymous referees for carefully reading the original manuscript and for constructive comments and suggestions, which have helped to improve the presentation of this paper. This work was supported by the Doctoral Fund of Anhui Normal University (project No. 751965).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Anhui Normal University, Wuhu, Anhui, 241000, P. R. China

    Fanchao Kong

  2. College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, P. R. China

    Shiping Lu

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  1. Fanchao Kong
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  2. Shiping Lu
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Correspondence to Fanchao Kong.

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Kong, F., Lu, S. Positive Periodic Solutions of Singular Third-Order Functional Differential Equations with p-Laplacian-Like Operators. Math Notes 107, 759–769 (2020). https://doi.org/10.1134/S0001434620050053

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  • DOI: https://doi.org/10.1134/S0001434620050053

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