Multiple periodic solutions of delay differential systems with 2k-1 lags via variational approach

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September 2016, 36(9): 4925-4943. doi: 10.3934/dcds.2016013

Multiple periodic solutions of delay differential systems with $2k-1$ lags via variational approach

1.	School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
2.	Department of Foundation Courses, Beijing Union University, Beijing 100101, China

Received August 2015 Revised February 2016 Published May 2016

Abstract
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By variational methods, this paper considers 4k-periodic solutions of a kind of differential delay systems with $2k-1$ lags. Our results reveal the fact that the number of $4k-$periodic orbits depends only upon the eigenvalues of both matrices $A_{\infty}$ and $A_0$. The conditions are more definite and easier to be examined. Moreover, two examples are given to illustrate the applications of the results.

Keywords: Periodic solution, critical point., variational approach, differential delay system.

Mathematics Subject Classification: Primary: 34K13; Secondary: 58E5.

Citation: Weigao Ge, Li Zhang. Multiple periodic solutions of delay differential systems with $2k-1$ lags via variational approach. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4925-4943. doi: 10.3934/dcds.2016013

References:

[1]	L. Fannio, Multiple periodic solutions of Hamiltonian systems with strong resonance at infinity,, Discrete and Cont. Dynamicals Sys., 3 (1997), 251. doi: 10.3934/dcds.1997.3.251.
[2]	G. Fei, Multiple periodic solutins of differential delay equations via Hamiltonian systems(I),, Nonlinear Anal., 65 (2006), 25. doi: 10.1016/j.na.2005.06.011.
[3]	G. Fei, Multiple periodic solutins of differential delay equations via Hamiltonian systems(II),, Nonlinear Anal., 65 (2006), 40. doi: 10.1016/j.na.2005.06.012.
[4]	W. Ge, On the existence of periodic solutions of the differential delay equations with multiple lags,, Acta Appl. Math. Sinica(in Chinese), 17 (1994), 173.
[5]	W. Ge, Oscillatory periodic solutions of differential delay equations with multiple lags,, Chinese Sci. Bull., 42 (1997), 444. doi: 10.1007/BF02882587.
[6]	W. Ge, Periodic solutions of the differential delay equation $x'(t) = -f(x(t-1))$,, Acta Math. Sinica(New Series), 12 (1996), 113. doi: 10.1007/BF02108151.
[7]	W. Ge, Two existence theorems of periodic solutions for differential delay equations,, Chinese Ann. Math., 15 (1994), 217.
[8]	Z. Guo and J. Yu, Multiple results for periodic solutions to delay differential equations via critical point theory,, J. Differential Equations, 218 (2005), 15. doi: 10.1016/j.jde.2005.08.007.
[9]	Z. Guo and J. Yu, Multiple results on periodic solutions to higher dimensional differential equations with multiple delays,, J. Dynam. Differential Equations, 23 (2011), 1029. doi: 10.1007/s10884-011-9228-z.
[10]	J. Kaplan and J. Yorke, Ordinary differential equations which yield periodic solution of delay equations,, J. Math. Anal. Appl., 48 (1974), 317. doi: 10.1016/0022-247X(74)90162-0.
[11]	J. Li and X. He, Multiple periodic solutions of differential delay equations created by asymptotically linear Hailtonian systems,, Nonlinear Anal., 31 (1998), 45. doi: 10.1016/S0362-546X(96)00058-2.
[12]	J. Li and X. He, Proof and gengeralization of Kaplan-Yorke' conjecture under the condition $f'(0)>0$ on periodic solution of differential delay equations,, Sci. China Ser. A, 42 (1999), 957. doi: 10.1007/BF02880387.
[13]	J. Li, X. He and Z. Liu, Hamiltonian symmetric group and multiple periodic solutions of differential delay equations,, Nonlinear Analysis, 35 (1999), 457. doi: 10.1016/S0362-546X(97)00623-8.
[14]	S. Li and J. Liu, Morse theory and asymptotically linear Hamiltonian systems,, J. Differential Equations, 78 (1989), 53. doi: 10.1016/0022-0396(89)90075-2.
[15]	J. Mawhen and M. Willem, Critical Point Theory and Hamiltonian System,, Springer-Verlag, (1989). doi: 10.1007/978-1-4757-2061-7.
[16]	R. D. Nussbaum, Periodic solutions of special differential delay equations: An example in nonlinear functional analysis,, Proc. Loyal Soc. Edingburgh, 81 (1978), 131. doi: 10.1017/S0308210500010490.

show all references

References:

[1]	L. Fannio, Multiple periodic solutions of Hamiltonian systems with strong resonance at infinity,, Discrete and Cont. Dynamicals Sys., 3 (1997), 251. doi: 10.3934/dcds.1997.3.251.
[2]	G. Fei, Multiple periodic solutins of differential delay equations via Hamiltonian systems(I),, Nonlinear Anal., 65 (2006), 25. doi: 10.1016/j.na.2005.06.011.
[3]	G. Fei, Multiple periodic solutins of differential delay equations via Hamiltonian systems(II),, Nonlinear Anal., 65 (2006), 40. doi: 10.1016/j.na.2005.06.012.
[4]	W. Ge, On the existence of periodic solutions of the differential delay equations with multiple lags,, Acta Appl. Math. Sinica(in Chinese), 17 (1994), 173.
[5]	W. Ge, Oscillatory periodic solutions of differential delay equations with multiple lags,, Chinese Sci. Bull., 42 (1997), 444. doi: 10.1007/BF02882587.
[6]	W. Ge, Periodic solutions of the differential delay equation $x'(t) = -f(x(t-1))$,, Acta Math. Sinica(New Series), 12 (1996), 113. doi: 10.1007/BF02108151.
[7]	W. Ge, Two existence theorems of periodic solutions for differential delay equations,, Chinese Ann. Math., 15 (1994), 217.
[8]	Z. Guo and J. Yu, Multiple results for periodic solutions to delay differential equations via critical point theory,, J. Differential Equations, 218 (2005), 15. doi: 10.1016/j.jde.2005.08.007.
[9]	Z. Guo and J. Yu, Multiple results on periodic solutions to higher dimensional differential equations with multiple delays,, J. Dynam. Differential Equations, 23 (2011), 1029. doi: 10.1007/s10884-011-9228-z.
[10]	J. Kaplan and J. Yorke, Ordinary differential equations which yield periodic solution of delay equations,, J. Math. Anal. Appl., 48 (1974), 317. doi: 10.1016/0022-247X(74)90162-0.
[11]	J. Li and X. He, Multiple periodic solutions of differential delay equations created by asymptotically linear Hailtonian systems,, Nonlinear Anal., 31 (1998), 45. doi: 10.1016/S0362-546X(96)00058-2.
[12]	J. Li and X. He, Proof and gengeralization of Kaplan-Yorke' conjecture under the condition $f'(0)>0$ on periodic solution of differential delay equations,, Sci. China Ser. A, 42 (1999), 957. doi: 10.1007/BF02880387.
[13]	J. Li, X. He and Z. Liu, Hamiltonian symmetric group and multiple periodic solutions of differential delay equations,, Nonlinear Analysis, 35 (1999), 457. doi: 10.1016/S0362-546X(97)00623-8.
[14]	S. Li and J. Liu, Morse theory and asymptotically linear Hamiltonian systems,, J. Differential Equations, 78 (1989), 53. doi: 10.1016/0022-0396(89)90075-2.
[15]	J. Mawhen and M. Willem, Critical Point Theory and Hamiltonian System,, Springer-Verlag, (1989). doi: 10.1007/978-1-4757-2061-7.
[16]	R. D. Nussbaum, Periodic solutions of special differential delay equations: An example in nonlinear functional analysis,, Proc. Loyal Soc. Edingburgh, 81 (1978), 131. doi: 10.1017/S0308210500010490.

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