• Cai, Hua (College of Mathematics Jilin University) ;
  • Chang, Xiaojun (School of Mathematics and Statistics Northeast Normal University, College of Mathematics Jilin University) ;
  • Zhao, Xin (College of Information Technology Jilin Agricultural University)
  • Received : 2013.06.04
  • Published : 2014.09.30


In this paper we study the existence of multiple periodic solutions of second-order ordinary differential equations. New results of multiplicity of periodic solutions are obtained when the nonlinearity may cross multiple consecutive eigenvalues. The arguments are proceeded by a combination of variational and degree theoretic methods.


  1. H. Amann, A note on degree theory for gradient mappings, Proc. Amer. Math. Soc. 85 (1982), no. 4, 591-595.
  2. G. Barletta and N. S. Papageorgiou, Periodic problems with double resonance, NoDEA Nonlinear Differential Equations Appl. 19 (2012), no. 3, 303-328.
  3. A. Castro and J. Cossio, Multiple solutions for a nonlinear Dirichlet problem, SIAM J. Math. Anal. 25 (1994), no. 6, 1554-1561.
  4. X. J. Chang and Q. D. Huang, Two-point boundary value problems for Duffing equations across resonance, J. Optim. Theory Appl. 140 (2009), no. 3, 419-430.
  5. X. J. Chang and Y. Li, Existence and multiplicity of nontrivial solutions for semilinear elliptic Dirichlet problems across resonance, Topol. Methods Nonlinear Anal. 36 (2010), no. 2, 285-310.
  6. X. J. Chang, Y. Li, and S. G. Ji, Nonresonance conditions on the potential with respect to the Fucik spectrum for semilinear Dirichlet problems, Z. Angew. Math. Phys. 61 (2010), no. 5, 823-833.
  7. X. J. Chang and Y. Qiao, Existence of periodic solutions for a class of p-Laplacian equations, Bound. Value Probl. 2013 (2013), 11 pp.
  8. D. G. Costa and A. S. Oliveira, Existence of solution for a class of semilinear elliptic problems at double resonance, Bol. Soc. Brasil. Mat. 19 (1988), no. 1, 21-37.
  9. D. Del Santo and P. Omari, Nonresonance conditions on the potential for a semilinear elliptic problem, J. Differential Equations 108 (1994), no. 1, 120-138.
  10. T. Ding, R. Iannacci, and F. Zanolin, Existence and multiplicity results for periodic solutions of semilinear Duffing equations, J. Differential Equations 105 (1993), no. 2, 364-409.
  11. C. L. Dolph, Nonlinear integral equations of the Hammerstein type, Trans. Amer. Math. Soc. 66 (1949), 289-307.
  12. P. Drabek, Landesman-Lazer conditions for nonlinear problems with jumping nonlin-earities, J. Differential Equations 85 (1990), no. 1, 186-199.
  13. P. Habets and G. Metzen, Existence of periodic solutions of Duffing equations, J. Differential Equations 78 (1989), no. 1, 1-32.
  14. P. Habets, P. Omari, and F. Zanolin, Nonresonance conditions on the potential with respect to the Fuik spectrum for the periodic boundary value problem, Rocky Mountain J. Math. 25 (1995), no. 4, 1305-1340.
  15. C. Fabry and A. Fonda, Periodic solutions of nonlinear differential equations with double resonance, Ann. Mat. Pura Appl. (4) 157 (1990), 99-116.
  16. A. Fonda, On the existence of periodic solutions for scalar second order differential equations when only the asymptotic behaviour of the potential is known, Proc. Amer. Math. Soc. 119 (1993), no. 2, 439-445.
  17. A. Fonda and P. Habets, Periodic solutions of asymptotically positively homogeneous differential equations, J. Differential Equations 81 (1989), no. 1, 68-97.
  18. D. Y. Hao and S. W. Ma, Semilinear Duffing equations crossing resonance points, J. Differential Equations 133 (1997), no. 1, 98-116.
  19. H. Hofer, The topological degree at a critical point of mountain pass type, Nonlinear functional analysis and its applications, Part 1 (Berkeley, Calif., 1983), 501-509, Proc. Sympos. Pure Math., 45, Part 1, Amer. Math. Soc., Providence, RI, 1986.
  20. R. Iannacci and M. N. Nkashama, Unbounded perturbations of forced second order or-dinary differential equations at resonance, J. Differential Equations 69 (1987), no. 3, 289-301.
  21. E. M. Landesman and A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech. 19 (1969/1970), 609-623.
  22. A. C. Lazer and S. Solimini, Nontrivial solutions of operator equations and morse indices of critical points of min-max type, Nonlinear Anal. 12 (1988), no. 8, 761-775.
  23. W. B. Liu and Y. Li, Existence of 2-periodic solutions for the non-dissipative Duffing equation under asymptotic behaviors of potential function, Z. Angew. Math. Phys. 57 (2006), no. 1, 1-11.
  24. J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Spring-Verlag, Berlin, 1989.
  25. L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute of Mathemat-ical Sciences, New York University, New York, 1974.
  26. P. Omari and F. Zanolin, Nonresonance conditions on the potential for a second-order periodic boundary value problem, Proc. Amer. Math. Soc. 117 (1993), no. 1, 125-135.
  27. N. S. Papageorgiou and V. Staicu, Multiple nontrivial solutions for doubly resonant periodic problems, Canad. Math. Bull. 53 (2010), no. 2, 347-359.
  28. J. Su and L. Zhao, Multiple periodic solutions of ordinary differential equations with double resonance, Nonlinear Anal. 70 (2009), no. 4, 1520-1527.
  29. M. R. Zhang, Nonresonance conditions for asymptotically positively homogeneous differential systems: the Fucik spectrum and its generalization, J. Differential Equations 145 (1998), no. 2, 332-366.
  30. X. Zhao and X. J. Chang, Existence of anti-periodic solutions for second-order ordinary differential equations involving the Fucik spectrum, Bound. Value Probl. 2012 (2012), 12 pp.