Abstract and Applied Analysis

Variational Approaches for the Existence of Multiple Periodic Solutions of Differential Delay Equations

Rong Cheng and Jianhua Hu

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Abstract

The existence of multiple periodic solutions of the following differential delay equation x ( t ) = f ( x ( t r ) ) is established by applying variational approaches directly, where x , f C ( , ) and r > 0 is a given constant. This means that we do not need to use Kaplan and Yorke's reduction technique to reduce the existence problem of the above equation to an existence problem for a related coupled system. Such a reduction method introduced first by Kaplan and Yorke in (1974) is often employed in previous papers to study the existence of periodic solutions for the above equation and its similar ones by variational approaches.

Article information

Source
Abstr. Appl. Anal., Volume 2010 (2010), Article ID 978137, 14 pages.

Dates
First available in Project Euclid: 2 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1267538587

Digital Object Identifier
doi:10.1155/2010/978137

Mathematical Reviews number (MathSciNet)
MR2587611

Zentralblatt MATH identifier
1195.34104

Citation

Cheng, Rong; Hu, Jianhua. Variational Approaches for the Existence of Multiple Periodic Solutions of Differential Delay Equations. Abstr. Appl. Anal. 2010 (2010), Article ID 978137, 14 pages. doi:10.1155/2010/978137. https://projecteuclid.org/euclid.aaa/1267538587


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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.
  • 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.
  • E. M. Wright, “A non-linear difference-differential equation,” Journal für die Reine und Angewandte Mathematik, vol. 194, no. 1, pp. 66–87, 1955.
  • 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 $\dotx=g(x(t-1))$,” Transactions of the American Mathematical Society, vol. 307, no. 1, pp. 127–142, 1988.
  • A. V. M. Herz, “Solutions of $\dotx=-g(x(t-1))$ approach the Kaplan-Yorke orbits for odd sigmoid $g$,” Journal of Differential Equations, vol. 118, no. 1, pp. 36–53, 1995.
  • 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.
  • 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.
  • J. A. Yorke, “Asymptotic stability for one dimensional differential-delay equations,” Journal of Differential Equations, vol. 7, no. 1, pp. 189–202, 1970.
  • 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.
  • H. O. Walther, “Homoclinic solution and chaos in $\dotx(t)=f(x(t-1))$,” Nonlinear Analysis: Theory, Methods & Applications, vol. 5, no. 7, pp. 775–788, 1981.
  • H. O. Walther, “Density of slowly oscillating solutions of $\dotx(t)=-f(x(t-1))$,” Journal of Mathematical Analysis and Applications, vol. 79, no. 1, pp. 127–140, 1981.
  • G. S. Jones, “The existence of periodic solutions of $f^\prime (x)=-\alpha f(x(t-1))\1+f(x)\$,” Journal of Mathematical Analysis and Applications, vol. 5, no. 3, pp. 435–450, 1962.
  • 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.
  • 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.
  • R. D. Nussbaum, “Uniqueness and nonuniqueness for periodic solutions of $x^\prime (t)=-g(x(t-1))$,” Journal of Differential Equations, vol. 34, no. 1, pp. 25–54, 1979.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • M. A. Han, “Bifurcations of periodic solutions of delay differential equations,” Journal of Differential Equations, vol. 189, no. 2, pp. 396–411, 2003.
  • P. Dormayer, “The stability of special symmetric solutions of $\dotx(t)=\alpha f(x(t-1))$ with small amplitudes,” Nonlinear Analysis: Theory, Methods & Applications, vol. 14, no. 8, pp. 701–715, 1990.
  • F. E. Browder, “A further generalization of the Schauder fixed point theorem,” Duke Mathematical Journal, vol. 32, pp. 575–578, 1965.
  • 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.
  • J. Mawhin, “Periodic solutions of nonlinear functional differential equations,” Journal of Differential Equations, vol. 10, no. 2, pp. 240–261, 1971.
  • 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.
  • 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.
  • J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989.
  • 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.
  • 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.