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Periodic Solutions to Non-Autonomous Second-Order Dynamical Systems
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Advances in Pure Mathematics·Volume 01 (2011)·Pages 90–94·Published 19 May 2011·DOI10.4236/apm.2011.13020
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Abstract
We study the multiple existence of periodic solutions for a second-order non-autonomous dynamical systems (1). Using the method of invariant sets of descending flow and chain of rings theorem, we obtain the existence of seven -periodic solutions.
KeywordsCritical PointInvariant Sets of Descending Flow(PS) Condition
- Z. Liu and J. Sun, “Invariant Sets of Descending Flow in Criti-cal Point Theory with Applications to Nonlinear Differential Equations,” Journal of Differential Equations, Vol. 172, No. 2, 2001, pp. 257-299. doi:10.1006/jdeq.2000.3867
- H. Y. Wang, “Periodic Solu-tions to Non-Autonomous Second-Order Systems,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 71, No. 3-4, 2009, pp. 1271-1275.
- M. Schechter, “Periodic Non-Autonomous Second-Order Dynamical Systems,” Journal of Differential Equations, Vol. 223, No. 2, 2006, pp. 290-302. doi:10.1016/j.jde.2005.02.022
- Z. Y. Wang, J. H. Zhang and Z. T Zhang, “Periodic Solutions of Second Order Non-Autonomous Hamiltonian Systems with Local Super-quadratic Potential,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 70, No. 10, 2009, pp. 3672-3681. doi:10.1006/jmaa.1995.1002
- S. J. Li and M. Willem, “Applications of Local Linking to Critical Point Theory,” Journal of Mathematical Analysis and Applications, Vol. 189, No. 1, 1995, pp. 6-32.
- A. Ambrosetti and P. H. Rabi-nowitz, “Dual Variational Methods in Critical Point Theory and Applications,” Journal of Functional Analysis, Vol. 14, No. 4, 1973, pp. 349-381. doi:10.1016/0022-1236(73)90051-7
- J. X. Sun and J. L. Sun, “Existence Theorems of Multiple Critical Points and Applications,” Nonlinear Analysis, Vol. 61, No. 8, 2005, pp. 1303-1317. doi:10.1016/j.na.2005.01.082
- J. R. Graef, L. J. Kong and H. Y. Wang, “Existence, Multiplicity and Dependence on a Parameter for a Periodic Boundary Value Problem,” Journal of Differential Equations, Vol. 245, No. 5, 2008, pp. 1185-1197. doi:10.1016/j.jde.2008.06.012
- F. Zhao and X. Wu, “Ex-istence and Multiplicity of Periodic Solutions for Non-Autonomous Second-Order Systems with Linear Nonlin-earity,” Nonlinear Analysis, Vol. 60, No. 2, 2005, pp. 325-335.
- J. Mawhin and M. Willem, “Critical Point The-ory and Hamiltonian Systems,” Springer, Berlin, New York, 1989.
- P. H. Rabinowitz, “Minimax Methods in Critical Point Theory with Applications to Differential Equations,” Expository Lectures from the CBMS Regional Conference, Series in Mathematics, American Mathematical Society, Vol. 65, 1986.
- C. L. Tang, “Periodic Solutions of Nonautono-mous Second-Order Systems with Sublinear Nonlinearity,” Proceedings of the American Mathematical Society, Vol. 126, No. 11, 1998, pp. 3263-3270. doi:10.1090/S0002-9939-98-04706-6