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Infinitely Many Fast Homoclinic Solutions for Some Second-Order Nonautonomous Systems

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Ukrainian Mathematical Journal Aims and scope

We investigate the existence of infinitely many fast homoclinic solutions for a class of second-order nonautonomous systems. Our main tools are based on the variant fountain theorem. A criterion guaranteeing that the second-order system has infinitely many fast homoclinic solutions is obtained. Recent results from the literature are generalized and significantly improved.

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References

  1. P. H. Rabinowitz, “Homoclinic orbits for a class of Hamiltonian systems,” Proc. Roy. Soc. Edinburgh A, 114, No. 1-2, 33–38 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  2. V. Coti Zelati and P. H. Rabinowitz, “Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials,” J. Amer. Math. Soc., 4, 693–727 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  3. Y. Ding and M. Girardi, “Periodic and homoclinic solutions to a class of Hamiltonian systems with the potentials changing sign,” Dynam. Syst. Appl., 2, 131–145 (1993).

    MATH  MathSciNet  Google Scholar 

  4. G. Fei, “The existence of homoclinic orbits for Hamiltonian systems with the potential changing sign,” Chinese Ann. Math. Ser. B, 17, 403–410 (1996).

    MATH  MathSciNet  Google Scholar 

  5. M. Izydorek and J. Janczewska, “Homoclinic solutions for a class of second order Hamiltonian systems,” J. Different. Equat., 219, No. 2, 375–389 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  6. Y. Ding, “Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems,” Nonlin. Anal., 25, 1095–1113 (1995).

    Article  MATH  Google Scholar 

  7. P. Korman and A. C. Lazer, “Homoclinic orbits for a class of symmetric Hamiltonian systems,” Electron. J. Different. Equat., 1994, 1–10 (1994).

    MathSciNet  Google Scholar 

  8. Z. Ou and C. Tang, “Existence of homoclinic solutions for the second order Hamiltonian systems,” J. Math. Anal. Appl., 291, 203–213 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  9. W. Zou, “Infinitely many homoclinic orbits for the second order Hamiltonian systems,” Appl. Math. Lett., 16, 1283–1287 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  10. W. Omana and M. Willem, “Homoclinic orbits for a class of Hamiltonian systems,” Different. Integral Equat., 5, No. 5, 1115–1120 (1992).

    MATH  MathSciNet  Google Scholar 

  11. R. Yuan and Z. Zhang, “Infinitely many homoclinic orbits for the second order Hamiltonian systems with super-quadratic potentials,” Nonlinear Anal.: Real World Appl., 10, 1417–1423 (2009).

    Article  MathSciNet  Google Scholar 

  12. L. Yang, H. Chen, and J. Sun, “Infinitely many homoclinic solutions for some second order Hamiltonian systems,” Nonlinear Anal.: Real World Appl., 74, 6459–6468 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  13. Z. Zhang and R. Yuan, “Homoclinic solutions for a class of nonautonomous subquadratic second order Hamiltonian systems,” Nonlin. Anal., 71, 4125–4130 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  14. Z. Zhang and R. Yuan, “Homoclinic solutions of some second order nonautonomous systems,” Nonlin. Anal., 71, 5790–5798 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  15. Z. Zhang and R. Yuan, “Fast homoclinic solutions for some second order nonautonomous systems,” J. Math. Anal. Appl., 376, 51–63 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  16. W. Zou, “Variant fountain theorems and their applications,” Manuscr. Math., 104, 343–358 (2001).

    Article  MATH  Google Scholar 

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 3, pp. 404–414, March, 2014.

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Yang, L., Luo, L. & Luo, Z. Infinitely Many Fast Homoclinic Solutions for Some Second-Order Nonautonomous Systems. Ukr Math J 66, 454–466 (2014). https://doi.org/10.1007/s11253-014-0943-6

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  • DOI: https://doi.org/10.1007/s11253-014-0943-6

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