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RETRACTED ARTICLE: Minimal Period Symmetric Solutions for Some Hamiltonian Systems Via the Nehari Manifold Method

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This article was retracted on 01 September 2020

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Abstract

For a given T > 0, we prove, under the global ARS-condition and using the Nehari manifold method, the existence of a T-periodic solution having the W-symmetry introduced in [21], for the hamiltonian system

$$\ddot{z}+V^\prime(z)=0,\;\;\;z\in\mathbb{R}^N\;\;\;N\in\mathbb{N}*.$$

Moreover, such a solution is shown to have T as a minimal period without relaying to any index theory. A multiplicity result is also proved under the same condition.

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References

  1. Ambrosetti A, Coti Zelati V. Solutions with Minimal Period for Hamiltonian Systems in a Potential Well. Ann Inst H Poincar Anal Non linéeaire, 1987, 4: 275–296

    Article  MathSciNet  Google Scholar 

  2. Ambrosetti A, Manchini G. Solutions of minimal period for a class of convex Hamiltonian systems. Math Ann, 1981, 255: 405–421

    Article  MathSciNet  Google Scholar 

  3. Bahri A, Lions P L. Morse Index of Min-Max Critical Points I. Applications to Multiplicity Results. Comm Pure Appl Math, 1988, 41(8): 1027–1037

    Article  MathSciNet  Google Scholar 

  4. Benci V, Rabinowitz P H. Critical point theorems for indefinite functionals. Invent Math, 1979, 52(3): 241–273

    Article  MathSciNet  Google Scholar 

  5. Calkovi´c L, Shu J L, Willem M. A note on Palais-Smale condition and convexity. Differential Integral Equations, 1990, 3: 799–800

    MathSciNet  Google Scholar 

  6. Dong Y, Long Y. Closed characteristics on partially symmetric compact convex hyper surfaces in R2n. J Differential Equations, 2004, 196: 226–248

    Article  MathSciNet  Google Scholar 

  7. Ekeland I. On the variational principle. J Math Anal Appl, 1974, 47: 324–354

    Article  MathSciNet  Google Scholar 

  8. Ekeland I, Hoffer H. Periodic Solutions with Prescribed Period for Autonomous Hamiltonian Systems. Invent Math, 1985, 81: 155–188

    Article  MathSciNet  Google Scholar 

  9. Girardi M, Matzeu M. Some results on solutions of minimal period to superquadratic hamiltonian equations. Nonlinear Anal TMA, 1983, 7: 475–482

    Article  Google Scholar 

  10. Girardi M, Matzeu M. Solutions of Minimal Period for a Class of Nonconvex Hamiltonian Systems and Applications to the Fixed Energy Problem. Nonlinear Anal TMA, 1986, 10: 371–383

    Article  MathSciNet  Google Scholar 

  11. Liu C, Zhang D. Multiple brake orbits on compact convex symmetric reversible hypersurfaces in ℝ2n. arXiv:1111.0722v1 [math.DS]_3 November (2011)

    Google Scholar 

  12. Long Y. The minimal period problem of periodic solutions for autonomous second order Hamiltonian systems. J Differential Equations, 1994, 111: 147–171

    Article  MathSciNet  Google Scholar 

  13. Long Y, Zhang D, Zhu C. Multiple brake orbits in bounded convex symmetric domains. Adv Math, 2006, 203: 568–635

    Article  MathSciNet  Google Scholar 

  14. Mawhin J, Willem M. Critical Point Theory and Hamiltonian Systems//Applied Mathematical Sciences. New York: Springer-Verlag, 1989, 74

    Book  Google Scholar 

  15. Rabinowitz P H. Minimax Methods in Critical Point Theory with Applications to Differential Equations. Providence, RI: CBMS Reg Conf Ser in Math Amer Math Soc, 1986, 65

    Google Scholar 

  16. Rabinowitz P H. Periodic solutions of Hamiltonian systems. Comm Pure Appl Math, 1978, 31: 157–184

    Article  MathSciNet  Google Scholar 

  17. Struwe M. Variational Methods. Berlin: Springer-Verlag, 1990, 65

    Book  Google Scholar 

  18. Szulkin A, Weth T. The method of Nehari manifold//Gao D Y, Motreanu D. Handbook of Nonconvex Analysis and Applications. Boston, Mass, USA: International Press, 2010: 597–632

    MATH  Google Scholar 

  19. Szulkin A. The method of Nehari manifold revisited. RIMS Kokyuroku, 2011, 1740: 89–102

    Google Scholar 

  20. Souissi C. Orbits with Minimal Period for a class of Autonomous Second Order One-dimensional Hamilto-nian System. Georgian Math J. DOI: https://doi.org/10.1515/gmj-2016-0054

  21. Souissi C. Generalized Ambrosetti-Rabinowitz condition for minimal period solutions of autonomous hamil-tonian system. Arch Math, 2017, 109: 73–82

    Article  MathSciNet  Google Scholar 

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Souissi, C. RETRACTED ARTICLE: Minimal Period Symmetric Solutions for Some Hamiltonian Systems Via the Nehari Manifold Method. Acta Math Sci 40, 614–624 (2020). https://doi.org/10.1007/s10473-020-0302-7

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  • DOI: https://doi.org/10.1007/s10473-020-0302-7

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