Frontiers of Mathematics in China

, Volume 4, Issue 3, pp 585–598 | Cite as

From ODE to DDE

Research Article
First Online:
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Accepted:

Abstract

In this paper, by considering ordinary differential equation (ODE) as a special case and a starting point of delay differential equation (DDE), we will show that some typical topological methods such as continuation theorems can be applied to detect some dynamics of DDE like periodic solutions. Several problems will be presented.

Keywords

Ordinary differential equation (ODE) delay differential equation (DDE) periodic solution continuation theorem Sobolev constant non-degeneracy 

MSC

34K13 34L05 58J20 
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References

  1. 1.
    Adams R A, Fournier J J F. Sobolev Spaces. 2nd Ed. Pure Appl Math (Amsterdam), Vol 140. Amsterdam: Elsevier/Academic Press, 2003MATHGoogle Scholar
  2. 2.
    Arnold V I. Mathematical Methods of Classical Mechanics. Graduate Texts Math, Vol 60. New York-Heidelberg: Springer-Verlag, 1978MATHGoogle Scholar
  3. 3.
    Capietto A, Mawhin J, Zanolin F. The coincidence degree of some functional-differential operators in spaces of periodic functions and related continuation theorems. In: Delay Differential Equations and Dynamical Systems (Claremont, CA, 1990). Lecture Notes Math, Vol 1475. Berlin: Springer, 1991, 76–87CrossRefGoogle Scholar
  4. 4.
    Capietto A, Mawhin J, Zanolin F. Periodic solutions of some superlinear functional differential equations. In: Yoshizawa T, Kato J, eds. Proc Intern Symp Functional Differential Equations (Kyoto, Japan, 30 Aug–2 Sept, 1990). Singapore: World Scientific, 1991, 19–31Google Scholar
  5. 5.
    Gaines R E, Mawhin J L. Coincidence Degree and Nonlinear Differential Equations. Lecture Notes Math, Vol 568. Berlin-New York: Springer-Verlag, 1977MATHGoogle Scholar
  6. 6.
    Guo Z, Yu J. Multiplicity results for periodic solutions of delay differential equations via critical point theory. J Differential Equations, 2005, 218: 15–35MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Hale J K. Theory of Functional Differential Equations. New York: Springer-Verlag, 1977MATHGoogle Scholar
  8. 8.
    Jiang M -Y. A Landesman-Lazer type theorem for periodic solutions of the resonant asymmetric p-Laplacian equation. Acta Math Sinica, Engl Ser, 2005, 21: 1219–1228MATHCrossRefGoogle Scholar
  9. 9.
    Leach P G L, Andropoulos K. The Ermakov equation, a commentary. Appl Anal Discrete Math, 2008, 2: 146–157MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Lei J, Li X, Yan P, Zhang M. Twist character of the least amplitude periodic solution of the forced pendulum. SIAM J Math Anal, 2003, 35: 844–867MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Li J, He X. Proof and generalization of Kaplan-Yorke’s conjecture on periodic solutions of differential delay equations. Sci China, Ser A, 1999, 42: 957–964MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Llibre J, Ortega R. On the families of periodic orbits of the Sitnikov problem. SIAM J Appl Dynam Syst, 2008, 7: 561–576MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Mallet-Paret J, Sell G R. Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions. J Differential Equations, 1996, 125: 385–440MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Mawhin J. Continuation theorems and periodic solutions of ordinary differential equations. In: Topological Methods in Differential Equations and Inclusions. Dordrecht: Kluwer, 1995, 291–375Google Scholar
  15. 15.
    Meng G, Yan P, Lin X, Zhang M. Non-degeneracy and periodic solutions of semilinear differential equations with deviation. Adv Nonlinear Stud, 2006, 6: 563–590MATHMathSciNetGoogle Scholar
  16. 16.
    Morris G R. An infinite class of periodic solutions of x″ + 2x 3 = p(t). Proc Cambridge Phil Soc, 1965, 61: 157–164MATHCrossRefGoogle Scholar
  17. 17.
    Ortega R, Zhang M. Some optimal bounds for bifurcation values of a superlinear periodic problem. Proc Royal Soc Edinburgh, Sect A, 2005, 135: 119–132MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Ward J R. Asymptotic conditions for periodic solutions of ordinary differential equations. Proc Amer Math Soc, 1981, 81: 415–420MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Whyburn G T. Analytic Topology. Amer Math Soc Colloq Publ, Vol 28. New York: Amer Math Soc, 1942MATHGoogle Scholar
  20. 20.
    Yan P, Zhang M. Higher order non-resonance for differential equations with singularities. Math Meth Appl Sci, 2003, 26: 1067–1074MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Zhang M. Certain classes of potentials for p-Laplacian to be non-degenerate. Math Nachr, 2005, 278: 1823–1836MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Zhang M. Periodic solutions of equations of Emarkov-Pinney type. Adv Nonlinear Stud, 2006, 6: 57–67MathSciNetGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag GmbH 2009

Authors and Affiliations

  1. 1.Department of Mathematical Sciences and Zhou Pei-Yuan Center for Applied MathematicsTsinghua UniversityBeijingChina

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