×
Xiao, Huafeng

A note on the minimal period problem for second order Hamiltonian systems. (English) Zbl 1470.34123

Abstr. Appl. Anal. 2014, Article ID 385381, 7 p. (2014).
Summary: We study periodic solutions of second order Hamiltonian systems with even potential. By making use of generalized Nehari manifold, some sufficient conditions are obtained to guarantee the multiplicity and minimality of periodic solutions for second order Hamiltonian systems. Our results generalize the outcome in the literature.

Cited in 2 Documents

MSC:

34C25 Periodic solutions to ordinary differential equations
PDF BibTeX XML Cite
\textit{H. Xiao}, Abstr. Appl. Anal. 2014, Article ID 385381, 7 p. (2014; Zbl 1470.34123)
Full Text: DOI

References:

[1] Rabinowitz, P. H., Periodic solutions of Hamiltonian systems, Communications on Pure and Applied Mathematics, 31, 2, 157-184 (1978) · Zbl 0358.70014
[2] Long, Y. M., The minimal period problem of classical Hamiltonian systems with even potentials, Annales de l’Institut Henri Poincaré. Analyse Non Linéaire, 10, 605-626 (1993) · Zbl 0804.58018
[3] Long, Y. M., The minimal period problem of periodic solutions for autonomous superquadratic second order Hamiltonian systems, Journal of Differential Equations, 111, 1, 147-174 (1994) · Zbl 0804.34043
[4] Long, Y. M., On the minimal period for periodic solutions of nonlinear Hamiltonian systems, Chinese Annals of Mathematics B, 18, 4, 481-484 (1997) · Zbl 0890.58069
[5] Fei, G.; Kim, S.; Wang, T., Solutions of minimal period for even classical Hamiltonian systems, Nonlinear Analysis. Theory, Methods & Applications, 43, 363-375 (2001) · Zbl 0964.37040
[6] Fei, G.; Wang, T., The minimal period problem for nonconvex even second order Hamiltonian systems, Journal of Mathematical Analysis and Applications, 215, 2, 543-559 (1997) · Zbl 0898.34042
[7] Fei, G.; Wang, T., Some results on the minimal period problem of nonconvex second order Hamiltonian systems, Chinese Annals of Mathematics B, 20, 1, 83-92 (1999) · Zbl 0931.37020
[8] Nehari, Z., On a class of nonlinear second-order differential equations, Transactions of the American Mathematical Society, 95, 101-123 (1960) · Zbl 0097.29501
[9] Nehari, Z., Characteristic values associated with a class of non-linear second-order differential equations, Acta Mathematica, 105, 141-175 (1961) · Zbl 0099.29104
[10] Pankov, A., Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan Journal of Mathematics, 73, 259-287 (2005) · Zbl 1225.35222
[11] Pankov, A., On decay of solutions to nonlinear Schrödinger equations, Proceedings of the American Mathematical Society, 136, 7, 2565-2570 (2008) · Zbl 1143.35093
[12] Szulkin, A.; Weth, T., Ground state solutions for some indefinite variational problems, Journal of Functional Analysis, 257, 12, 3802-3822 (2009) · Zbl 1178.35352
[13] Szulkin, A.; Weth, T.; Gao, D. Y.; Motreanu, D., The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications, 597-632 (2010), Boston, Mass, USA: International Press, Boston, Mass, USA · Zbl 1218.58010
[14] Willem, M., Minimax Theorems. Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, 24 (1996), Boston, Mass, USA: Birkhäuser, Boston, Mass, USA · Zbl 0856.49001
[15] Chen, G.; Ma, S., Periodic solutions for Hamiltonian systems without Ambrosetti-Rabinowitz condition and spectrum 0, Journal of Mathematical Analysis and Applications, 379, 2, 842-851 (2011) · Zbl 1218.37079
[16] Ye, Y. W.; Tang, C. L., Periodic solutions for second order Hamiltonian systems with general superquadratic potential, Bulletin of the Belgian Mathematical Society. Simon Stevin, 21, 1, 1-18 (2014) · Zbl 1406.37048
[17] Ambrosetti, A.; Mancini, G., Solutions of minimal period for a class of convex Hamiltonian systems, Mathematische Annalen, 255, 3, 405-421 (1981) · Zbl 0466.70022
[18] Xiao, Y. M., Periodic solutions with prescribed minimal period for the second order Hamiltonian systems with even potentials, Acta Mathematica Sinica, 26, 5, 825-830 (2010) · Zbl 1211.34051
[19] Deng, S. T., Minimal periodic solutions of a class of Hamiltonian systems, Acta Mathematica Sinica, 27, 5, 664-675 (1984) · Zbl 0577.58015
[20] Magrone, P., Critical point methods for indefinite nonlinear Elliptic equations and Hamiltonian systems [Ph.D. thesis] (2001), Universit a Degli Studi Di Roma “Tor Vergata”
[21] Wang, Q.; Wang, Z.; Shi, J., Subharmonic oscillations with prescribed minimal period for a class of Hamiltonian systems, Nonlinear Analysis. Theory, Methods & Applications, 28, 7, 1273-1282 (1997) · Zbl 0872.34022
[22] Mawhin, J.; Willem, M., Critical Point Theory and Hamiltonian Systems (1989), New York, NY, USA: Springer, New York, NY, USA · Zbl 0676.58017
[23] Fei, G., On periodic solutions of superquadratic Hamiltonian systems, Electronic Journal of Differential Equations, 2002, 1-12 (2002) · Zbl 0999.37039
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.