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Periodic solutions of singular systems without the strong force condition

Authors: Daniel Franco and Pedro J. Torres
Journal: Proc. Amer. Math. Soc. 136 (2008), 1229-1236
MSC (2000): Primary 37J45, 34C25, 34B16.
Published electronically: December 27, 2007
MathSciNet review: 2367097
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Abstract: We present sufficient conditions for the existence of at least a non-collision periodic solution for singular systems under weak force conditions. We deal with two different types of systems. First, we assume that the system is generated by a potential, and then we consider systems without such hypothesis. In both cases we use the same technique based on Schauder fixed point theorem. Recent results in the literature are significantly improved.

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  • 1. Adachi S., Non-collision periodic solutions of prescribed energy problem for a class of singular Hamiltonian systems, Topol. Methods Nonlinear Anal. 25 (2005) 275-296. MR 2154429 (2006e:37113)
  • 2. Adachi S., Tanaka K., Terui M., A remark on periodic solutions of singular Hamiltonian systems, NoDEA Nonlinear Differential Equations Appl. 12 (2005) 265-274. MR 2186332 (2006j:37071)
  • 3. Ambrosetti A., Coti Zelati V. Periodic solutions of singular Lagrangian systems, Birkhäuser Boston, Boston, MA, 1993. MR 1267225 (95b:58054)
  • 4. Brézis, H., Analyse fonctionnelle, Masson, Paris, 1983. MR 697382 (85a:46001)
  • 5. Franco D., Webb J.R.L., Collisionless orbits of singular and non singular dynamical systems, Discrete Contin. Dyn. Syst. 15 (2006) 747-757. MR 2220746 (2006m:34110)
  • 6. Gordon W.B. Conservative dynamical systems involving strong forces, Trans. Amer. Math. Soc. 204 (1975) 113-135. MR 0377983 (51:14152)
  • 7. Jiang D., Chu J., Zhang M., Multiplicity of positive periodic solutions to superlinear repulsive singular equations, J. Differential Equations 211 (2005) no. 2, 282-302. MR 2125544 (2005k:34087)
  • 8. Lazer A.C., Solimini S., On periodic solutions of nonlinear differential equations with singularities, Proc. Amer. Math. Soc. 99 (1987), 109-114. MR 866438 (87k:34064)
  • 9. 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 (2005) no. 2, 263-273. MR 2154428 (2006d:34050)
  • 10. Martin R.H., Nonlinear operators and differential equations in Banach spaces, John Wiley and Sons, New York, 1976. MR 0492671 (58:11753)
  • 11. Poincaré H., Sur les solutions périodiques et le priciple de moindre action, C.R. Math. Acad. Sci. Paris 22 (1896) 915-918.
  • 12. Qian D., Torres P.J., Periodic motions of linear impact oscillators via the successor map, SIAM J. Math. Anal. 36 no. 6 (2005), 1707-1725. MR 2178218 (2006g:34078)
  • 13. Ramos M., Terracini S., Noncollision periodic solutions to some singular dynamical systems with very weak forces, J. Differential Equations 118 (1995) no. 1, 121-152. MR 1329405 (96d:58115)
  • 14. Torres P.J., Weak singularities may help periodic solutions to exist, J. Differential Equations 232 (2007) 277-284. MR 2281196
  • 15. Torres P.J., Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem, J. Differential Equations 190 (2003) no. 2, 643-662. MR 1970045 (2003k:34082)
  • 16. Torres P.J., Zhang M., A monotone iterative scheme for a nonlinear second order equation based on a generalized anti-maximum principle. Math. Nachr. 251 (2003) 101-107. MR 1960807 (2004a:34032)
  • 17. Zhang M., Periodic solutions of damped differential systems with repulsive singular forces, Proc. Amer. Math. Soc. 127 (1999) 401-407. MR 1637460 (99k:34092)

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Additional Information

Daniel Franco
Affiliation: Departamento de Matemática Aplicada, UNED, ETSI Industriales, c/ Juan del Rosal 12, 28040 Madrid, Spain

Pedro J. Torres
Affiliation: Universidad de Granada, Departamento de Matemática Aplicada, 18071 Granada, Spain

Received by editor(s): August 17, 2006
Published electronically: December 27, 2007
Additional Notes: The first author was supported by D.G.I. MTM2004-06652-C03-03, Ministerio de Educación y Ciencia, Spain.
The second author was supported by D.G.I. MTM2005-03483, Ministerio de Educación y Ciencia, Spain.
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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