Article Contents
Article Contents

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

• 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.
Mathematics Subject Classification: Primary: 34K13; Secondary: 58E50.

 Citation:

•  [1] L. Fannio, Multiple periodic solutions of Hamiltonian systems with strong resonance at infinity, Discrete and Cont. Dynamicals Sys., 3 (1997), 251-264.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), 25-39.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), 40-58.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), 173-181. [5] W. Ge, Oscillatory periodic solutions of differential delay equations with multiple lags, Chinese Sci. Bull., 42 (1997), 444-447.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), 113-121.doi: 10.1007/BF02108151. [7] W. Ge, Two existence theorems of periodic solutions for differential delay equations, Chinese Ann. Math., 15 (1994), 217-224. [8] Z. Guo and J. Yu, Multiple results for periodic solutions to delay differential equations via critical point theory, J. Differential Equations, 218 (2005), 15-35.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), 1029-1052.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), 317-324.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), 45-54.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), 957-964.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, TMA, 35 (1999), 457-474.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), 53-73.doi: 10.1016/0022-0396(89)90075-2. [15] J. Mawhen and M. Willem, Critical Point Theory and Hamiltonian System, Springer-Verlag, New Yorke, 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), 131-151.doi: 10.1017/S0308210500010490.