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Periodic Solutions of Second Order Superlinear Singular Dynamical Systems

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Abstract

We study the existence of periodic solutions of second order superlinear dynamical systems with a singularity of repulsive type. The proof is based on a well-known fixed point theorem for completely continuous operators. We do not need to consider so-called strong force conditions. Recent results in the literature are generalized and significantly improved.

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References

  1. Adachi, S.: Non-collision periodic solutions of prescribed energy problem for a class of singular Hamiltonian systems. Topol. Methods Nonlinear Anal. 25, 275–296 (2005)

    MATH  MathSciNet  Google Scholar 

  2. Ambrosetti, A., Coti Zelati, V.: Critical points with lack of compactness and singular dynamical systems. Ann. Mat. Pura Appl. 149, 237–259 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ambrosetti, A., Coti Zelati, V.: Perturbation of Hamiltonian systems with Keplerian potentials. Math. Z. 201, 227–242 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ambrosetti, A., Coti Zelati, V.: Periodic Solutions of Singular Lagrangian Systems. Birkhäuser Boston, Boston (1993)

    MATH  Google Scholar 

  5. Bahri, A., Rabinowitz, P.H.: A minimax method for a class of Hamiltonian systems with singular potentials. J. Funct. Anal. 82, 412–428 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bevc, V., Palmer, J.L., Süsskind, C.: On the design of the transition region of axi-symmetric magnetically focusing beam valves. J. Br. Inst. Radio Eng. 18, 696–708 (1958)

    Google Scholar 

  7. Bonheure, D., De Coster, C.: Forced singular oscillators and the method of lower and upper solutions. Topol. Methods Nonlinear Anal. 22, 297–317 (2003)

    MATH  MathSciNet  Google Scholar 

  8. Chu, J., Franco, D.: Non-collision periodic solutions of second order singular dynamical systems. J. Math. Anal. Appl. 344, 898–905 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chu, J., Torres, P.J., Zhang, M.: Periodic solutions of second order non-autonomous singular dynamical systems. J. Differ. Equ. 239, 196–212 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Coti Zelati, V.: Periodic solutions for a class of planar, singular dynamical systems. J. Math. Pures Appl. 68, 109–119 (1989)

    MathSciNet  Google Scholar 

  11. del Pino, M., Manásevich, R.: Infinitely many T-periodic solutions for a problem arising in nonlinear elasticity. J. Differ. Equ. 103, 260–277 (1993)

    Article  MATH  Google Scholar 

  12. Ding, T.: A boundary value problem for the periodic Brillouin focusing system. Acta Scientarium Naturalium Universitatis Pekinensis 11, 31–38 (1965) [In Chinese]

    Google Scholar 

  13. Ferrario, D.L., Terracini, S.: On the existence of collisionless equivariant minimizers for the classical n-body problem. Invent. Math. 155, 305–362 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  14. Fonda, A., Toader, R.: Periodic orbits of radially symmetric Keplerian-like systems: A topological degree approach. J. Differ. Equ. 244, 3235–3264 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Franco, D., Webb, J.R.L.: Collisionless orbits of singular and nonsingular dynamical systems. Discrete Contin. Dyn. Syst. 15, 747–757 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Franco, D., Torres, P.J.: Periodic solutions of singular systems without the strong force condition. Proc. Am. Math. Soc. 136, 1229–1236 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. Gordon, W.B.: Conservative dynamical systems involving strong forces. Trans. Am. Math. Soc. 204, 113–135 (1975)

    Article  MATH  Google Scholar 

  18. Habets, P., Sanchez, L.: Periodic solution of some Liénard equations with singularities. Proc. Am. Math. Soc. 109, 1135–1144 (1990)

    Article  MathSciNet  Google Scholar 

  19. Jiang, D., Chu, J., Zhang, M.: Multiplicity of positive periodic solutions to superlinear repulsive singular equations. J. Differ. Equ. 211, 282–302 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Krasnosel’skii, M.A.: Positive Solutions of Operator Equations. Groningen, Noordhoff (1964)

    Google Scholar 

  21. Lazer, A.C., Solimini, S.: On periodic solutions of nonlinear differential equations with singularities. Proc. Am. Math. Soc. 99, 109–114 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  22. Lin, X., Jiang, D., O’Regan, D., Agarwal, R.P.: Twin positive periodic solutions of second order singular differential systems. Topol. Methods Nonlinear Anal. 25, 263–273 (2005)

    MATH  MathSciNet  Google Scholar 

  23. Majer, P.: Ljusternik-Schnirelman theory with local Palais-Smale condition and singular dynamical systems. Ann. Inst. Henri Poincaré Anal. Non Linéaire 8, 459–476 (1991)

    MATH  MathSciNet  Google Scholar 

  24. Rachůnková, I., Tvrdý, M., Vrkoc̆, I.: Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems. J. Differ. Equ. 176, 445–469 (2001)

    Article  MATH  Google Scholar 

  25. Ramos, M., Terracini, S.: Noncollision periodic solutions to some singular dynamical systems with very weak forces. J. Differ. Equ. 118, 121–152 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  26. Schechter, M.: Periodic non-autonomous second-order dynamical systems. J. Differ. Equ. 223, 290–302 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  27. Solimini, S.: On forced dynamical systems with a singularity of repulsive type. Nonlinear Anal. 14, 489–500 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  28. Tanaka, K.: A note on generalized solutions of singular Hamiltonian systems. Proc. Am. Math. Soc. 122, 275–284 (1994)

    Article  MATH  Google Scholar 

  29. Terracini, S.: Remarks on periodic orbits of dynamical systems with repulsive singularities. J. Funct. Anal. 111, 213–238 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  30. Torres, P.J.: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem. J. Differ. Equ. 190, 643–662 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  31. Torres, P.J.: Non-collision periodic solutions of forced dynamical systems with weak singularities. Discrete Contin. Dyn. Syst. 11, 693–698 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  32. Zhang, M.: Periodic solutions of damped differential systems with repulsive singular forces. Proc. Am. Math. Soc. 127, 401–407 (1999)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, College of Science, Hohai University, Nanjing, 210098, China

    Jifeng Chu

  2. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Ziheng Zhang

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  1. Jifeng Chu
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  2. Ziheng Zhang
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Correspondence to Jifeng Chu.

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Chu, J., Zhang, Z. Periodic Solutions of Second Order Superlinear Singular Dynamical Systems. Acta Appl Math 111, 179–187 (2010). https://doi.org/10.1007/s10440-009-9539-9

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