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Periodic solutions of second-order Liénard equations with-Laplacian-like operators

Journal of Inequalities and Applications volume 2006, Article number: 98685 (2006) Cite this article

Abstract

The existence of periodic solutions for second-order Liénard equations with-Laplacian-like operator is studied by applying new generalization of polar coordinates.

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References

  1. Burton TA, Townsend CG: On the generalized Liénard equation with forcing function. Journal of Differential Equations 1968,4(4):620–633. 10.1016/0022-0396(68)90012-0

    Article  MathSciNet  MATH  Google Scholar 

  2. del Pino MA, Manásevich RF, Murúa AE: Existence and multiplicity of solutions with prescribed period for a second order quasilinear ODE. Nonlinear Analysis. Theory, Methods & Applications 1992,18(1):79–92. 10.1016/0362-546X(92)90048-J

    Article  MathSciNet  MATH  Google Scholar 

  3. del Pino MA, Manásevich RF, Murúa A: On the number ofperiodic solutions forusing the Poincaré-Birkhoff theorem. Journal of Differential Equations 1992,95(2):240–258. 10.1016/0022-0396(92)90031-H

    Article  MathSciNet  MATH  Google Scholar 

  4. Ding TR, Iannacci R, Zanolin F: Existence and multiplicity results for periodic solutions of semilinear Duffing equations. Journal of Differential Equations 1993,105(2):364–409. 10.1006/jdeq.1993.1093

    Article  MathSciNet  MATH  Google Scholar 

  5. Fabry C, Fayyad D: Periodic solutions of second order differential equations with a -Laplacian and asymmetric nonlinearities. Rendiconti dell'Istituto di Matematica dell'Università di Trieste 1992,24(1–2):207–227 (1994).

    MathSciNet  MATH  Google Scholar 

  6. Fabry C, Mawhin J, Nkashama MN: A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations. Bulletin of the London Mathematical Society 1986,18(2):173–180. 10.1112/blms/18.2.173

    Article  MathSciNet  MATH  Google Scholar 

  7. Gossez J-P, Omari P: Periodic solutions of a second order ordinary differential equation: a necessary and sufficient condition for nonresonance. Journal of Differential Equations 1991,94(1):67–82. 10.1016/0022-0396(91)90103-G

    Article  MathSciNet  MATH  Google Scholar 

  8. Manásevich RF, Mawhin J: Periodic solutions for nonlinear systems with-Laplacian-like operators. Journal of Differential Equations 1998,145(2):367–393. 10.1006/jdeq.1998.3425

    Article  MathSciNet  MATH  Google Scholar 

  9. Mawhin J, Ward JR Jr.: Periodic solutions of some forced Liénard differential equations at resonance. Archiv der Mathematik 1983,41(4):337–351. 10.1007/BF01371406

    Article  MathSciNet  MATH  Google Scholar 

  10. Omari P, Villari G, Zanolin F: Periodic solutions of the Liénard equation with one-sided growth restrictions. Journal of Differential Equations 1987,67(2):278–293. 10.1016/0022-0396(87)90151-3

    Article  MathSciNet  MATH  Google Scholar 

  11. Sandqvist A, Andersen KM: A necessary and sufficient condition for the existence of a unique nontrivial periodic solution to a class of equations of Liénard's type. Journal of Differential Equations 1982,46(3):356–378. 10.1016/0022-0396(82)90100-0

    Article  MathSciNet  MATH  Google Scholar 

  12. Savel'ev PN: Dissipativity of the generalized Liénard equation. Differential Equations 1992,28(6):794–800.

    MathSciNet  MATH  Google Scholar 

  13. Sun W, Ge W: The existence of solutions to Sturm-Liouville boundary value problems with Laplacian-like operator. Acta Mathematicae Applicatae Sinica 2002,18(2):341–348. 10.1007/s102550200034

    Article  MathSciNet  MATH  Google Scholar 

  14. Villari G: On the existence of periodic solutions for Liénard's equation. Nonlinear Analysis 1983,7(1):71–78. 10.1016/0362-546X(83)90104-9

    Article  MathSciNet  MATH  Google Scholar 

  15. Wang Z: Periodic solutions of the second-order forced Liénard equation via time maps. Nonlinear Analysis. Theory, Methods & Applications. Ser. A: Theory Methods 2002,48(3):445–460.

    Article  MATH  Google Scholar 

  16. Wang Y, Ge W: Existence of periodic solutions for nonlinear differential equations with a-Laplacian-like operator. to appear in Applied Mathematics Letters to appear in Applied Mathematics Letters

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Authors and Affiliations

  1. The School of Mathematics, Beijing Institute of Technology, Beijing, 100081, China

    Youyu Wang & Weigao Ge

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  1. Youyu Wang
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  2. Weigao Ge
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Correspondence to Youyu Wang.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Wang, Y., Ge, W. Periodic solutions of second-order Liénard equations with-Laplacian-like operators. J Inequal Appl 2006, 98685 (2006). https://doi.org/10.1155/JIA/2006/98685

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  • DOI: https://doi.org/10.1155/JIA/2006/98685

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