Abstract and Applied Analysis
Volume 2010 (2010), Article ID 978137, 14 pages
doi:10.1155/2010/978137
Research Article
Variational Approaches for the Existence of Multiple Periodic Solutions of Differential Delay Equations
1College of Mathematics and Physics, Nanjing University of Information Science and Technology, Nanjing 210044, China
2College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
Received 30 March 2009; Revised 15 December 2009; Accepted 11 January 2010
Academic Editor: John Mallet-Paret
Copyright © 2010 Rong Cheng and Jianhua Hu. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Linked References
- T. Furumochi, “Existence of periodic solutions of one-dimensional differential-delay equations,” The Tôhoku Mathematical Journal. Second Series, vol. 30, no. 1, pp. 13–35, 1978. View at Zentralblatt MATH · View at MathSciNet
- W. J. Cunningham, “A nonlinear differential-difference equation of growth,” The Proceedings of the National Academy of Sciences, vol. 40, no. 4, pp. 708–713, 1954. View at Zentralblatt MATH · View at MathSciNet
- E. M. Wright, “A non-linear difference-differential equation,” Journal für die Reine und Angewandte Mathematik, vol. 194, no. 1, pp. 66–87, 1955. View at Zentralblatt MATH · View at MathSciNet
- R. May, Stablity and Complexity in Model Ecosystems, vol. 9, Princeton University Press, Princeton, NJ, USA, 1973.
- S. N. Chow and H. O. Walther, “Characteristic multipliers and stability of symmetric periodic solutions of ,” Transactions of the American Mathematical Society, vol. 307, no. 1, pp. 127–142, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- A. V. M. Herz, “Solutions of approach the Kaplan-Yorke orbits for odd sigmoid ,” Journal of Differential Equations, vol. 118, no. 1, pp. 36–53, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- J. L. Kaplan and J. A. Yorke, “Ordinary differential equations which yield periodic solutions of differential delay equations,” Journal of Mathematical Analysis and Applications, vol. 48, no. 2, pp. 317–324, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- J. L. Kaplan and J. A. Yorke, “On the stability of a periodic solution of a differential delay equation,” SIAM Journal on Mathematical Analysis, vol. 6, no. 2, pp. 268–282, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- J. A. Yorke, “Asymptotic stability for one dimensional differential-delay equations,” Journal of Differential Equations, vol. 7, no. 1, pp. 189–202, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- S. Chapin, “Periodic solutions of differential-delay equations with more than one delay,” The Rocky Mountain Journal of Mathematics, vol. 17, no. 3, pp. 555–572, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- H. O. Walther, “Homoclinic solution and chaos in ,” Nonlinear Analysis: Theory, Methods & Applications, vol. 5, no. 7, pp. 775–788, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- H. O. Walther, “Density of slowly oscillating solutions of ,” Journal of Mathematical Analysis and Applications, vol. 79, no. 1, pp. 127–140, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- G. S. Jones, “The existence of periodic solutions of ,” Journal of Mathematical Analysis and Applications, vol. 5, no. 3, pp. 435–450, 1962. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- R. D. Nussbaum, “Periodic solutions of special differential equations: an example in nonlinear functional analysis,” Proceedings of the Royal Society of Edinburgh. Section A, vol. 81, no. 1-2, pp. 131–151, 1978. View at Zentralblatt MATH · View at MathSciNet
- R. D. Nussbaum, “A Hopf global bifurcation theorem for retarded functional differential equations,” Transactions of the American Mathematical Society, vol. 238, no. 1, pp. 139–164, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- R. D. Nussbaum, “Uniqueness and nonuniqueness for periodic solutions of ,” Journal of Differential Equations, vol. 34, no. 1, pp. 25–54, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- J. Li, X.-Z. He, and Z. Liu, “Hamiltonian symmetric groups and multiple periodic solutions of differential delay equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 35, no. 4, pp. 457–474, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- J. Li and X.-Z. He, “Multiple periodic solutions of differential delay equations created by asymptotically linear Hamiltonian systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 31, no. 1-2, pp. 45–54, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- Z. Guo and J. Yu, “Multiplicity results for periodic solutions to delay differential equations via critical point theory,” Journal of Differential Equations, vol. 218, no. 1, pp. 15–35, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- J. Llibre and A.-A. Tarţa, “Periodic solutions of delay equations with three delays via bi-Hamiltonian systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 11, pp. 2433–2441, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- S. Jekel and C. Johnston, “A Hamiltonian with periodic orbits having several delays,” Journal of Differential Equations, vol. 222, no. 2, pp. 425–438, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- M. A. Han, “Bifurcations of periodic solutions of delay differential equations,” Journal of Differential Equations, vol. 189, no. 2, pp. 396–411, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- P. Dormayer, “The stability of special symmetric solutions of with small amplitudes,” Nonlinear Analysis: Theory, Methods & Applications, vol. 14, no. 8, pp. 701–715, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- F. E. Browder, “A further generalization of the Schauder fixed point theorem,” Duke Mathematical Journal, vol. 32, pp. 575–578, 1965. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- S. N. Chow and J. Mallet-Paret, “The Fuller index and global Hopf bifurcation,” Journal of Differential Equations, vol. 29, no. 1, pp. 66–85, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- J. Mawhin, “Periodic solutions of nonlinear functional differential equations,” Journal of Differential Equations, vol. 10, no. 2, pp. 240–261, 1971. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- J. Mawhin, “Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces,” Journal of Differential Equations, vol. 12, no. 3, pp. 610–636, 1972. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Washington, DC, USA, 1986. View at MathSciNet
- J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989. View at MathSciNet
- V. Benci, “On critical point theory for indefinite functionals in the presence of symmetries,” Transactions of the American Mathematical Society, vol. 274, no. 2, pp. 533–572, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- L. O. Fannio, “Multiple periodic solutions of Hamiltonian systems with strong resonance at infinity,” Discrete and Continuous Dynamical Systems, vol. 3, no. 2, pp. 251–264, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet