Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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MULTIPLE PERIODIC SOLUTIONS OF SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS ACROSS RESONANCE

  • 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
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Abstract

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.

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References

  1. H. Amann, A note on degree theory for gradient mappings, Proc. Amer. Math. Soc. 85 (1982), no. 4, 591-595. https://doi.org/10.1090/S0002-9939-1982-0660610-2
  2. G. Barletta and N. S. Papageorgiou, Periodic problems with double resonance, NoDEA Nonlinear Differential Equations Appl. 19 (2012), no. 3, 303-328. https://doi.org/10.1007/s00030-011-0130-5
  3. A. Castro and J. Cossio, Multiple solutions for a nonlinear Dirichlet problem, SIAM J. Math. Anal. 25 (1994), no. 6, 1554-1561. https://doi.org/10.1137/S0036141092230106
  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. https://doi.org/10.1007/s10957-008-9461-8
  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. https://doi.org/10.1007/s00033-009-0051-x
  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. https://doi.org/10.1186/1687-2770-2013-11
  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. https://doi.org/10.1007/BF02584819
  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. https://doi.org/10.1006/jdeq.1994.1028
  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. https://doi.org/10.1006/jdeq.1993.1093
  11. C. L. Dolph, Nonlinear integral equations of the Hammerstein type, Trans. Amer. Math. Soc. 66 (1949), 289-307. https://doi.org/10.1090/S0002-9947-1949-0032923-4
  12. P. Drabek, Landesman-Lazer conditions for nonlinear problems with jumping nonlin-earities, J. Differential Equations 85 (1990), no. 1, 186-199. https://doi.org/10.1016/0022-0396(90)90095-7
  13. P. Habets and G. Metzen, Existence of periodic solutions of Duffing equations, J. Differential Equations 78 (1989), no. 1, 1-32. https://doi.org/10.1016/0022-0396(89)90073-9
  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. https://doi.org/10.1216/rmjm/1181072148
  15. C. Fabry and A. Fonda, Periodic solutions of nonlinear differential equations with double resonance, Ann. Mat. Pura Appl. (4) 157 (1990), 99-116. https://doi.org/10.1007/BF01765314
  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. https://doi.org/10.1090/S0002-9939-1993-1154246-4
  17. A. Fonda and P. Habets, Periodic solutions of asymptotically positively homogeneous differential equations, J. Differential Equations 81 (1989), no. 1, 68-97. https://doi.org/10.1016/0022-0396(89)90178-2
  18. D. Y. Hao and S. W. Ma, Semilinear Duffing equations crossing resonance points, J. Differential Equations 133 (1997), no. 1, 98-116. https://doi.org/10.1006/jdeq.1996.3193
  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. https://doi.org/10.1016/0022-0396(87)90121-5
  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. https://doi.org/10.1016/0362-546X(88)90037-5
  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. https://doi.org/10.4153/CMB-2010-030-4
  28. J. Su and L. Zhao, Multiple periodic solutions of ordinary differential equations with double resonance, Nonlinear Anal. 70 (2009), no. 4, 1520-1527. https://doi.org/10.1016/j.na.2008.02.031
  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. https://doi.org/10.1006/jdeq.1997.3403
  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. https://doi.org/10.1186/1687-2770-2012-12