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Journal of Function Spaces
Volume 2017, Article ID 4247365, 5 pages
https://doi.org/10.1155/2017/4247365
Research Article

Odd Periodic Solutions of Fully Second-Order Ordinary Differential Equations with Superlinear Nonlinearities

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Correspondence should be addressed to Yongxiang Li; nc.ude.unwn@xyil

Received 17 January 2017; Revised 2 June 2017; Accepted 26 July 2017; Published 23 August 2017

Academic Editor: Lishan Liu

Copyright © 2017 Yongxiang Li and Lanjun Guo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. S. Leela, “Monotone method for second order periodic boundary value problems,” Nonlinear Analysis, vol. 7, no. 4, pp. 349–355, 1983. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. J. J. Nieto, “Nonlinear second-order periodic boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 130, no. 1, pp. 22–29, 1988. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. A. Cabada and J. J. Nieto, “A generalization of the monotone iterative technique for nonlinear second order periodic boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 151, no. 1, pp. 181–189, 1990. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. A. Cabada, “The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 185, no. 2, pp. 302–320, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. J.-P. Gossez and P. Omari, “Periodic solutions of a second order ordinary differential equation: a necessary and sufficient condition for nonresonance,” Journal of Differential Equations, vol. 94, no. 1, pp. 67–82, 1991. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. P. Omari, G. Villari, and F. Zanolin, “Periodic solutions of the Li\'enard equation with one-sided growth restrictions,” Journal of Differential Equations, vol. 67, no. 2, pp. 278–293, 1987. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. W. G. Ge, “On the existence of harmonic solutions of Lienard systems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 16, no. 2, pp. 183–190, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  8. A. Fonda and R. Toader, “Periodic orbits of radially symmetric Keplerian-like systems: a topological degree approach,” Journal of Differential Equations, vol. 244, no. 12, pp. 3235–3264, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. D. Tian, “Multiple positive periodic solutions for second-order differential equations with a singularity,” Acta Applicandae Mathematicae, vol. 144, pp. 1–10, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. J. Mawhin and M. Willem, “Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations,” Journal of Differential Equations, vol. 52, no. 2, pp. 264–287, 1984. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. V. Coti Zelati, “Periodic solutions of dynamical systems with bounded potential,” Journal of Differential Equations, vol. 67, no. 3, pp. 400–413, 1987. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. L. Lassoued, “Periodic solutions of a second order superquadratic system with a change of sign in the potential,” Journal of Differential Equations, vol. 93, no. 1, pp. 1–18, 1991. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. J. Su and L. Zhao, “Multiple periodic solutions of ordinary differential equations with double resonance,” Nonlinear Analysis, vol. 70, no. 4, pp. 1520–1527, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  14. C. Li, R. P. Agarwal, and D. sca, “Infinitely many periodic solutions for a class of new superquadratic second-order Hamiltonian systems,” Applied Mathematics Letters. An International Journal of Rapid Publication, vol. 64, pp. 113–118, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  15. D. Tongren, “Nonlinear oscillations at resonance,” Science in China, vol. 12A, pp. 1–12, 1984. View at Google Scholar
  16. L. Weigu, “A necessary and sufficient condition on the existence and uniqueness of 2π-periodic solution of Duffing equation,” Chinese Annals of Mathematics, vol. 11, no. 3, pp. 342–345, 1990. View at Google Scholar · View at MathSciNet
  17. A. Fonda and L. Ghirardelli, “Multiple periodic solutions of scalar second order differential equations,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, vol. 72, no. 11, pp. 4005–4015, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  18. A. Fonda and A. Sfecci, “Periodic bouncing solutions for nonlinear impact oscillators,” Advanced Nonlinear Studies, vol. 13, no. 1, pp. 179–189, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. A. Calamai and A. Sfecci, “Multiplicity of periodic solutions for systems of weakly coupled parametrized second order differential equations,” NoDEA. Nonlinear Differential Equations and Applications, vol. 24, no. 1, Art. 4, 17 pages, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  20. F. Merdivenci Atici and G. S. Guseinov, “On the existence of positive solutions for nonlinear differential equations with periodic boundary conditions,” Journal of Computational and Applied Mathematics, vol. 132, no. 2, pp. 341–356, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. Y. X. Li, “Positive periodic solutions of nonlinear second order ordinary differential equations,” Acta Mathematica Sinica. Chinese Series, vol. 45, no. 3, pp. 481–488, 2002. View at Google Scholar · View at MathSciNet
  22. P. J. Torres, “Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem,” Journal of Differential Equations, vol. 190, no. 2, pp. 643–662, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. H. Zhu and S. Li, “Multiplicity of positive periodic solutions to nonlinear boundary value problems with a parameter,” Journal of Applied Mathematics and Computing, vol. 51, no. 1-2, pp. 245–256, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. Y. Li, “Positive periodic solutions of first and second order ordinary differential equations,” Chinese Annals of Mathematics. Series B, vol. 25, no. 3, pp. 413–420, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. F. Li and Z. Liang, “Existence of positive periodic solutions to nonlinear second order differential equations,” Applied Mathematics Letters. An International Journal of Rapid Publication, vol. 18, no. 11, pp. 1256–1264, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. J. R. Graef, L. Kong, and H. Wang, “Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem,” Journal of Differential Equations, vol. 245, no. 5, pp. 1185–1197, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. Z. Wang, “Periodic solutions of the second order differential equations with asymmetric nonlinearities depending on the derivatives,” Discrete and Continuous Dynamical Systems. Series A, vol. 9, no. 3, pp. 751–770, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  28. M. Zamora, “New asymptotic stability and uniqueness results on periodic solutions of second order differential equations using degree theory,” Advanced Nonlinear Studies, vol. 15, no. 2, pp. 433–446, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. R. Hakl, P. J. Torres, and M. Zamora, “Periodic solutions of singular second order differential equations: upper and lower functions,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, vol. 74, no. 18, pp. 7078–7093, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  30. Y. Li and X. Jiang, “Positive periodic solutions for second-order ordinary differential equations with derivative terms and singularity in nonlinearities,” Journal of Function Spaces and Applications, Article ID 945467, 2012. View at Publisher · View at Google Scholar · View at Scopus
  31. Y. X. Li, “Oscillatory periodic solutions of nonlinear second order ordinary differential equations,” Acta Mathematica Sinica (English Series), vol. 21, no. 3, pp. 491–496, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  32. K. Deimling, Nonlinear Functional Analysis, Springer, Berlin , Germany, 1985. View at Publisher · View at Google Scholar · View at MathSciNet