|
Periodic solutions of singular systems without the strong force condition
Author(s):
Daniel
Franco;
Pedro
J.
Torres
Journal:
Proc. Amer. Math. Soc.
136
(2008),
1229-1236.
MSC (2000):
Primary 37J45, 34C25, 34B16.
Posted:
December 27, 2007
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
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.
References:
-
- 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)
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(2000):
37J45, 34C25, 34B16.
Retrieve articles in all Journals with MSC
(2000):
37J45, 34C25, 34B16.
Additional Information:
Daniel
Franco
Affiliation:
Departamento de Matemática Aplicada, UNED, ETSI Industriales, c/ Juan del Rosal 12, 28040 Madrid, Spain
Email:
dfranco@ind.uned.es
Pedro
J.
Torres
Affiliation:
Universidad de Granada, Departamento de Matemática Aplicada, 18071 Granada, Spain
Email:
ptorres@ugr.es
DOI:
10.1090/S0002-9939-07-09226-X
PII:
S 0002-9939(07)09226-X
Received by editor(s):
August 17, 2006
Posted:
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
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
Forward Citation(s): Information for authors on submitting citations The following works have cited this article Chu, Jifeng; Torres, Pedro J.; Zhang, Meirong, Periodic solutions of second order non-autonomous singular dynamical systems., J. Differential Equations (1) 239 (2007), 196-212. MR 2341553
Chu, Jifeng; Li, Ming, Positive periodic solutions of Hill's equations with singular nonlinear perturbations, Nonlinear Anal. (1) 69 (2008), 276-286. MR 2417870
|
Comments: webmaster@ams.org