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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

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.
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.

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##### 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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