## 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

### Keywords:

strong and weak singularity
Full Text:

### 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
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