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Oscillatory periodic solutions for two differential-difference equations arising in applications. (English) Zbl 1217.34113

Summary: We study the existence of oscillatory periodic solutions for two nonautonomous differential-difference equations (which arise in a variety of applications) of the following form:
\[ \dot x(t)=-f(t,x(t-r)) \]
and
\[ \dot x(t)=-f(t,x(t-s))-f(t,x(t-2s), \]
where \(f\in C(\mathbb R\times\mathbb R,\mathbb R)\) is odd with respect to \(x\), and \(r,s>0\) are two given constants. By using a symplectic transformation and a result for Hamiltonian systems, the existence of oscillatory periodic solutions of the above-mentioned equations is established.

MSC:

34K13 Periodic solutions to functional-differential equations
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References:

[1] T. Furumochi, “Existence of periodic solutions of one-dimensional differential-delay equations,” The Tôhoku Mathematical Journal, vol. 30, no. 1, pp. 13-35, 1978. · Zbl 0376.34057 · doi:10.2748/tmj/1178230094
[2] W. J. Cunningham, “A nonlinear differential-difference equation of growth,” Proceedings of the National Academy of Sciences of the United States of America, vol. 40, no. 8, pp. 708-713, 1954. · Zbl 0055.31601 · doi:10.1073/pnas.40.8.708
[3] E. M. Wright, “A non-linear difference-differential equation,” Journal für die Reine und Angewandte Mathematik, vol. 194, pp. 66-87, 1955. · Zbl 0064.34203 · doi:10.1515/crll.1955.194.66
[4] 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, pp. 435-450, 1962. · Zbl 0106.29504 · doi:10.1016/0022-247X(62)90017-3
[5] R. May, Stablity and Complexity in Model Ecosystems, Princeton University Press, Princeton, NJ, USA, 1973.
[6] D. Saupe, “Global bifurcation of periodic solutions of some autonomous differential delay equation,” in Forschungsschwerpunkt Dynamische Systems, Report Nr. 71, University of Bremen, 1982. · Zbl 0522.34067
[7] 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, pp. 317-324, 1974. · Zbl 0293.34102 · doi:10.1016/0022-247X(74)90162-0
[8] R. D. Nussbaum, “Periodic solutions of special differential equations: an example in nonlinear functional analysis,” Proceedings of the Royal Society of Edinburgh A, vol. 81, no. 1-2, pp. 131-151, 1978. · Zbl 0402.34061 · doi:10.1017/S0308210500010490
[9] 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. · Zbl 0404.34057 · doi:10.1016/0022-0396(79)90016-0
[10] 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. · Zbl 0644.34064 · doi:10.1216/RMJ-1987-17-3-555
[11] H.-O. Walther, “Density of slowly oscillating solutions of x\?(t)= - f(x(t - 1)),” Journal of Mathematical Analysis and Applications, vol. 79, no. 1, pp. 127-140, 1981. · Zbl 0451.34063 · doi:10.1016/0022-247X(81)90014-7
[12] 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, pp. 268-282, 1975. · Zbl 0241.34080 · doi:10.1137/0506028
[13] J. A. Yorke, “Asymptotic stability for one dimensional differential-delay equations,” Journal of Differential Equations, vol. 7, pp. 189-202, 1970. · Zbl 0184.12401 · doi:10.1016/0022-0396(70)90132-4
[14] P. Dormayer, “The stability of special symmetric solutions of x\?(t)=\alpha f(x(t - 1)) with small amplitudes,” Nonlinear Analysis: Theory, Methods & Applications, vol. 14, no. 8, pp. 701-715, 1990. · Zbl 0704.34086 · doi:10.1016/0362-546X(90)90045-I
[15] H.-O. Walther, “Homoclinic solution and chaos in x\?(t)=f(x(t - 1)),” Nonlinear Analysis: Theory, Methods & Applications, vol. 5, no. 7, pp. 775-788, 1981. · Zbl 0459.34040 · doi:10.1016/0362-546X(81)90052-3
[16] R. D. Nussbaum, “A Hopf global bifurcation theorem for retarded functional differential equations,” Transactions of the American Mathematical Society, vol. 238, pp. 139-164, 1978. · Zbl 0389.34050 · doi:10.2307/1997801
[17] M. Han, “Bifurcations of periodic solutions of delay differential equations,” Journal of Differential Equations, vol. 189, no. 2, pp. 396-411, 2003. · Zbl 1027.34081 · doi:10.1016/S0022-0396(02)00106-7
[18] 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. · Zbl 0920.34061 · doi:10.1016/S0362-546X(97)00623-8
[19] 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. · Zbl 0918.34066 · doi:10.1016/S0362-546X(96)00058-2
[20] G. Fei, “Multiple periodic solutions of differential delay equations via Hamiltonian systems (I),” Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 1, pp. 25-39, 2006. · Zbl 1136.34329 · doi:10.1016/j.na.2005.06.011
[21] G. Fei, “Multiple periodic solutions of differential delay equations via Hamiltonian systems (II),” Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 1, pp. 40-58, 2006. · Zbl 1136.34330 · doi:10.1016/j.na.2005.06.012
[22] J. Llibre and A.-A. Tar\cta, “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. · Zbl 1101.34055 · doi:10.1016/j.na.2005.08.023
[23] 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. · Zbl 1163.34386 · doi:10.1016/j.jde.2005.08.013
[24] J. Bélair and M. C. Mackey, “Consumer memory and price fluctuations in commodity markets: an integrodifferential model,” Journal of Dynamics and Differential Equations, vol. 1, no. 3, pp. 299-325, 1989. · Zbl 0682.34050 · doi:10.1007/BF01053930
[25] R. Cheng, “Symplectic transformations and a reduction method for asymptotically linear Hamiltonian systems,” Acta Applicandae Mathematicae, vol. 110, no. 1, pp. 209-214, 2010. · Zbl 1233.37031 · doi:10.1007/s10440-008-9400-6
[26] S. J. Li and J. Q. Liu, “Morse theory and asymptotic linear Hamiltonian system,” Journal of Differential Equations, vol. 78, no. 1, pp. 53-73, 1989. · Zbl 0672.34037 · doi:10.1016/0022-0396(89)90075-2
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