×

Periodic solutions of second order non-autonomous singular dynamical systems. (English) Zbl 1127.34023

The authors use topological methods to prove the existence of positive solutions for some non-autonomous singular second order systems. These are solutions whose components all take only positive values. The singularity can be either of strong or of weak type.

MSC:

34C25 Periodic solutions to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
47H10 Fixed-point theorems
PDF BibTeX XML Cite
Full Text: DOI

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) · Zbl 1077.37040
[2] Ambrosetti, A.; Coti Zelati, V., Periodic solutions of singular Lagrangian systems, (1993), Birkhäuser Boston Boston, MA · Zbl 0785.34032
[3] Bonheure, D.; De Coster, C., Forced singular oscillators and the method of lower and upper solutions, Topol. methods nonlinear anal., 22, 297-317, (2003) · Zbl 1108.34033
[4] Chu, J.; Lin, X.; Jiang, D.; O’Regan, D.; Agarwal, P.R., Multiplicity of positive solutions to second order differential equations, Bull. austral. math. soc., 73, 175-182, (2006) · Zbl 1096.34518
[5] J. Chu, P.J. Torres, Applications of Schauder’s fixed point theorem to singular differential equations, Bull. London Math. Soc., in press · Zbl 1128.34027
[6] del Pino, M.; Manásevich, R.; Montero, A., T-periodic solutions for some second order differential equations with singularities, Proc. roy. soc. Edinburgh sect. A, 120, 231-243, (1992) · Zbl 0761.34031
[7] del Pino, M.; Manásevich, R., Infinitely many T-periodic solutions for a problem arising in nonlinear elasticity, J. differential equations, 103, 260-277, (1993) · Zbl 0781.34032
[8] Ferrario, D.L.; Terracini, S., On the existence of collisionless equivariant minimizers for the classical n-body problem, Invent. math., 155, 305-362, (2004) · Zbl 1068.70013
[9] Fonda, A.; Manásevich, R.; Zanolin, F., Subharmonic solutions for some second order differential equations with singularities, SIAM J. math. anal., 24, 1294-1311, (1993) · Zbl 0787.34035
[10] Franco, D.; Webb, J.R.L., Collisionless orbits of singular and nonsingular dynamical systems, Discrete contin. dyn. syst., 15, 747-757, (2006) · Zbl 1120.34029
[11] D. Franco, P.J. Torres, Periodic solutions of singular systems without the strong force condition, Proc. Amer. Math. Soc., in press · Zbl 1129.37033
[12] Gordon, W.B., Conservative dynamical systems involving strong forces, Trans. amer. math. soc., 204, 113-135, (1975) · Zbl 0276.58005
[13] Habets, P.; Sanchez, L., Periodic solution of some Liénard equations with singularities, Proc. amer. math. soc., 109, 1135-1144, (1990)
[14] Jiang, D.; Chu, J.; O’Regan, D.; Agarwal, R.P., Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces, J. math. anal. appl., 286, 563-576, (2003) · Zbl 1042.34047
[15] Jiang, D.; Chu, J.; Zhang, M., Multiplicity of positive periodic solutions to superlinear repulsive singular equations, J. differential equations, 211, 282-302, (2005) · Zbl 1074.34048
[16] Lazer, A.C.; Solimini, S., On periodic solutions of nonlinear differential equations with singularities, Proc. amer. math. soc., 99, 109-114, (1987) · Zbl 0616.34033
[17] 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) · Zbl 1098.34032
[18] O’Regan, D., Existence theory for nonlinear ordinary differential equations, (1997), Kluwer Academic Dordrecht · Zbl 1077.34505
[19] Poincaré, H., Sur LES solutions périodiques et le priciple de moindre action, C. R. math. acad. sci. Paris, 22, 915-918, (1896) · JFM 27.0608.02
[20] Rachunková, I.; Tvrdý, M.; Vrkoč, I., Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems, J. differential equations, 176, 445-469, (2001) · Zbl 1004.34008
[21] Ramos, M.; Terracini, S., Noncollision periodic solutions to some singular dynamical systems with very weak forces, J. differential equations, 118, 121-152, (1995) · Zbl 0826.34039
[22] Schechter, M., Periodic non-autonomous second-order dynamical systems, J. differential equations, 223, 290-302, (2006) · Zbl 1099.34042
[23] Solimini, S., On forced dynamical systems with a singularity of repulsive type, Nonlinear anal., 14, 489-500, (1990) · Zbl 0708.34041
[24] Tanaka, K., A note on generalized solutions of singular Hamiltonian systems, Proc. amer. math. soc., 122, 275-284, (1994) · Zbl 0812.58038
[25] Terracini, S., Remarks on periodic orbits of dynamical systems with repulsive singularities, J. funct. anal., 111, 213-238, (1993) · Zbl 0778.34031
[26] 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, 643-662, (2003) · Zbl 1032.34040
[27] Torres, P.J., Non-collision periodic solutions of forced dynamical systems with weak singularities, Discrete contin. dyn. syst., 11, 693-698, (2004) · Zbl 1063.34035
[28] Torres, P.J., Weak singularities may help periodic solutions to exist, J. differential equations, 232, 277-284, (2007) · Zbl 1116.34036
[29] Zhang, M., Periodic solutions of damped differential systems with repulsive singular forces, Proc. amer. math. soc., 127, 401-407, (1999) · Zbl 0908.34024
[30] Zhang, M.; Li, W., A Lyapunov-type stability criterion using \(L^\alpha\) norms, Proc. amer. math. soc., 130, 3325-3333, (2002) · Zbl 1007.34053
[31] Zhang, M., Periodic solutions of equations of emarkov – pinney type, Adv. nonlinear stud., 6, 57-67, (2006) · Zbl 1107.34037
[32] Zhang, S.; Zhou, Q., Nonplanar and noncollision periodic solutions for N-body problems, Discrete contin. dyn. syst., 10, 679-685, (2004) · Zbl 1238.70006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.