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Existence of multiple periodic solutions for a class of second-order delay differential equations. (English) Zbl 1190.34083

The paper considers the second order multi-dimensional differential delay equation
\[ x^{\prime\prime}(t)=-f(x(t-\tau)),\qquad x\in\mathbb R^n,\quad\tau>0 \]
with a particular symmetric behaviour of the vector-function \(f(x)\) at \(x=0+\) and \(x=+\infty\). A lower estimate for the number of periodic solutions of period \(2\tau\) in the system is given. The paper generalizes similar results derived for a like first order differential delay equation considered in [Z. M. Guo and J. S. Yu, J. Differ. Equations 218, No. 1, 15–35 (2005; Zbl 1095.34043)].

MSC:

34K13 Periodic solutions to functional-differential equations

Citations:

Zbl 1095.34043
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References:

[1] Benci, V., On critical point theory for indefinite functionals in the presence of symmetries, Trans. amer. math. soc, 274, 533-572, (1982) · Zbl 0504.58014
[2] Benci, V.; Rabinowitz, P.H., Critical point theorem for indefinite functionals, Invent. math, 53, 241-273, (1979) · Zbl 0465.49006
[3] Claeyssen, J.R., The integral-averaging bifurcation methods and the general one-delay equation, J. math. anal. appl, 78, 428-439, (1980)
[4] Chang, K.C., Infinite dimensional Morse theory and multiple solution problems, (1993), Birkhäuser Boston
[5] Chen, Y.S., The existence of periodic solutions of the equation \(x^\prime(t) = - f(x(t), x(t - r))\), J. math. anal. appl., 163, 227-237, (1992) · Zbl 0755.34063
[6] Chen, Y.S., The existence of periodic solutions for a class of neutral differential difference equations, Bull. austral. math. soc, 33, 508-516, (1992) · Zbl 0755.34062
[7] Fei, G.H., Multiple periodic solutions of differential delay equations via Hamiltonian systems (I), Nonlinear anal., 65, 25-39, (2006) · Zbl 1136.34329
[8] Fei, G.H., Multiple periodic solutions of differential delay equations via Hamiltonian systems (II), Nonlinear anal., 65, 40-58, (2006) · Zbl 1136.34330
[9] Fannio, L.O., Multiple periodic solution of Hamiltonian systems with strong resonance at infinity, Discrete and cont. dynamical syst., 3, 251-264, (1997) · Zbl 0989.37060
[10] Grafton, R., A periodicity theorem for autonomous functional differential equations, J. differential equations, 6, 87-109, (1969) · Zbl 0175.38503
[11] Gaines, R.; Mawhin, J., Coincide degree and nonlinear differential equation, (1977), Springer-Verlag Berlin · Zbl 0326.34021
[12] Ge, W.G., Number of simple periodic solutions of differential difference equation on \(x^\prime(t) = - f(x(t - 1))\), Chinese ann. math., 14A, 480-491, (1993)
[13] Guo, Z.M.; Yu, J.S., Multiplicity results for periodic solutions to delay differential difference equations via critical point theory, J. differential equations, 218, 15-35, (2005) · Zbl 1095.34043
[14] Hale, J.K., Theory of functional differential equations, (1977), Springer-Verlag
[15] Hale, J.K.; Lunel, S.M.V., Introduction to functional differential equations, (1993), Springer-Verlag
[16] Kaplan, J.L.; Yorke, J.A., Ordinary differential equations which yield periodic solution of delay equations, J. math. anal. appl., 48, 317-324, (1974) · Zbl 0293.34102
[17] Kaplan, J.L.; Yorke, J.A., On the stability of a periodic solution of a differential delay equation, SIAM J. math. anal., 6, 268-282, (1975) · Zbl 0241.34080
[18] Kaplan, J.L.; Yorke, J.A., On the nonlinear differential delay equation \(x^\prime(t) = - f(x(t), x(t - 1))\), J. differential equations, 23, 293-314, (1977) · Zbl 0307.34070
[19] Li, J.B.; He, X.Z., Multiple periodic solutions of differential delay equations created by asymptotically linear Hamiltonian systems, Nonlinear anal. TMA, 31, 45-54, (1998) · Zbl 0918.34066
[20] Li, J.B.; He, X.Z., Proof and generalization of kaplan – yorke’s conjecture on periodic solution of differential delay equations, Sci. China (ser.A), 42, 9, 957-964, (1999) · Zbl 0983.34061
[21] Li, J.B.; He, X.Z., Periodic solutions of some differential delay equations created by Hamiltonian systems, Bull. austral. math. soc., 60, 377-390, (1999) · Zbl 0946.34063
[22] Li, S.J.; Liu, J.Q., Morse theory and asymptotically linear Hamiltonian systems, J. differential equations, 78, 53-73, (1989) · Zbl 0672.34037
[23] Lu, S.P.; Ge, W.G., Periodic solutions of the second order differential equation with deviating arguments, Acta. mathematic sinica, 45, 811-818, (2002) · Zbl 1027.34079
[24] Long, Y.M.; Zehnder, E., Morse theory for forced oscillations of asymptotically linear Hamiltonian systems, (), 528-563
[25] Nussbaum, R.D., Periodic solutions of some nonlinear autonomous functional differential equations, Ann. math. pura. appl., 10, 263-306, (1974) · Zbl 0323.34061
[26] Schechter, M., Spectra of partial differential equation, (1971), North-Holland Amsterdam
[27] Shu, X.B.; Xu, Y.T., Multiple periodic solutions to a class of second-order functional differential equations of mixed type, Acta. math. appl. sin., 29, 5, 821-831, (2006)
[28] Wen, L.Z., The existence of periodic solutions of a class differential difference equations, Chinese bull. sci., 32, 934-935, (1987)
[29] Wang, G.Q.; Yan, J.R., Existence of periodic solutions for second order nonlinear neutral delay equations, Acta math. sinica, 47, 2, 379-384, (2004) · Zbl 1387.34100
[30] Wang, G.Q.; Cheng, S.S., Even periodic solutions of higher order Duffing differential equations, Czechoslovak math. J., 57, 132, 331-343, (2007) · Zbl 1174.34037
[31] Xu, Y.T.; Guo, Z.M., Applications of a \(Z_p\) index theory to periodic solutions for a class of functional differential equations, J. math. anal. appl, 257, 1, 189-205, (2001) · Zbl 0992.34051
[32] Xu, Y.T.; Guo, Z.M., Applications of a geometrical index theory to functional differential equations, Acta. math. sinica, 44, 6, 1027-1036, (2001) · Zbl 1027.34078
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