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December  2021, 20(12): 4253-4269. doi: 10.3934/cpaa.2021159

## Periodic solutions for a class of second-order differential delay equations

 School of Mathematics and Information Science, Guangzhou University, Guangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou, Guangdong 510006, China

* Corresponding author

Received  June 2021 Revised  August 2021 Published  December 2021 Early access  September 2021

Fund Project: This project is supported by National Natural Science Foundation of China (No. 11871171)

In this paper, we study the existence of periodic solutions of the following differential delay equations
 $$$z^{\prime\prime}(t) = \sum\limits_{k = 1}^{M-1}(-1)^kf(z(t-k)), \notag$$$
where
 $f\in C(\mathbf{R}^N, \mathbf{R}^N)$
,
 $M,N\in \mathbf{N}$
and
 $M$
is odd. By making use of
 $S^1$
-geometrical index theory, we obtain an estimation about the number of periodic solutions in term of the difference between eigenvalues of asymptotically linear matrices at the origin and at infinity.
Citation: Xuan Wu, Huafeng Xiao. Periodic solutions for a class of second-order differential delay equations. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4253-4269. doi: 10.3934/cpaa.2021159
##### References:
 [1] M. Degiovanni and L. O. Fannio, Multiple periodic solutions of asymptotically linear Hamiltonian systems, Nonlinear Anal., 26 (1996), 1437-1446.  doi: 10.1016/0362-546X(94)00274-L. [2] G. Fei, Multiple periodic solutions 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 solutions of differential delay equations via Hamiltonian systems(II), Nonlinear Anal., 65 (2006), 40-58.  doi: 10.1016/j.na.2005.06.012. [4] C. Guo and Z. Guo, Existence of multiple periodic solutions for a class of second-order delay differential equations, Nonlinear Anal. Real World Appl., 10 (2009), 3285-3297.  doi: 10.1016/j.nonrwa.2008.10.023. [5] Z. Guo and J. Yu, Multiplicity results for periodic solutions to delay differential difference equation via critical point theory, J. Differ. Equ., 218 (2005), 15-35.  doi: 10.1016/j.jde.2005.08.007. [6] Z. Guo and J. Yu, Multiplicity results on period solutions to higher dimensional differential equations with multiple delays, J. Dynam. Differ. Equ., 23 (2011), 1029-1052.  doi: 10.1007/s10884-011-9228-z. [7] Z. Guo and X. Zhang, Multiplicity results for periodic solutions to delay differential difference equation via critical point theory, Commun. Pure Appl. Anal., 9 (2010), 1529-1542.  doi: 10.3934/cpaa.2010.9.1529. [8] U. Heiden, Periodic solutions of a nonlinear second order differential equation with delay, J. Math. Anal. Appl., 70 (1970), 599-609.  doi: 10.1016/0022-247X(79)90068-4. [9] J. Kaplan and J. Yorke, Ordinary differential equations which yield periodic solutions of differential delay equations, J. Math. Anal. Appl., 48 (1974), 317-324.  doi: 10.1016/0022-247X(74)90162-0. [10] J. Li and X. He, Periodic solutions of some differential delay equations created by Hamiltonian systems, Bull. Austral. Math. Soc., 60 (1999), 377-390.  doi: 10.1017/S000497270003656X. [11] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, New York, 1989. doi: 10.1007/978-1-4757-2061-7. [12] R. Nussbaum, Periodic solutions of special differential delay equations: an example in nonlinear functional analysis, BProc. Roy. Soc. Edinburgh Sect. A, 81 (1978), 131-151.  doi: 10.1017/S0308210500010490. [13] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics 65, American Mathematical Society, Providence, RI, 1986. doi: 10.1090/cbms/065. [14] S. Ruan and J. Wei, Periodic solutions of planar systems with two delays, BProc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 1017-1032.  doi: 10.1017/S0308210500031061. [15] Z. Sun, W. Ge and L. Li, Multiple periodic orbits of high-dimensional differential delay systems, Adv. Differ. Equ., 2019 (2019), 15 pp. doi: 10.1186/s13662-019-2427-3. [16] A. Szulkin, Cohomology and Morse theory for strongly indefinite functionals, Math. Z., 209 (1992), 375-418.  doi: 10.1007/BF02570842. [17] H. Xiao and Z. Guo, Multiplicity and minimality of periodic solutions to delay differential systems, Electron. J. Differ. Equ., 2014 (2014), 356-364. [18] J. Yu and Z. Guo, A survey on the periodic solutions to Kaplan-Yorke type delay differential equation-I, Ann. Differ. Equ., 30 (2014), 97-114. [19] B. Zheng and Z. Guo, Multiplicity results on periodic solutions to higher-dimensional differential equations with multiple delays, Rocky Mountain J. Math., 44 (2014), 1715-1744.  doi: 10.1216/RMJ-2014-44-5-1715.

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##### References:
 [1] M. Degiovanni and L. O. Fannio, Multiple periodic solutions of asymptotically linear Hamiltonian systems, Nonlinear Anal., 26 (1996), 1437-1446.  doi: 10.1016/0362-546X(94)00274-L. [2] G. Fei, Multiple periodic solutions 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 solutions of differential delay equations via Hamiltonian systems(II), Nonlinear Anal., 65 (2006), 40-58.  doi: 10.1016/j.na.2005.06.012. [4] C. Guo and Z. Guo, Existence of multiple periodic solutions for a class of second-order delay differential equations, Nonlinear Anal. Real World Appl., 10 (2009), 3285-3297.  doi: 10.1016/j.nonrwa.2008.10.023. [5] Z. Guo and J. Yu, Multiplicity results for periodic solutions to delay differential difference equation via critical point theory, J. Differ. Equ., 218 (2005), 15-35.  doi: 10.1016/j.jde.2005.08.007. [6] Z. Guo and J. Yu, Multiplicity results on period solutions to higher dimensional differential equations with multiple delays, J. Dynam. Differ. Equ., 23 (2011), 1029-1052.  doi: 10.1007/s10884-011-9228-z. [7] Z. Guo and X. Zhang, Multiplicity results for periodic solutions to delay differential difference equation via critical point theory, Commun. Pure Appl. Anal., 9 (2010), 1529-1542.  doi: 10.3934/cpaa.2010.9.1529. [8] U. Heiden, Periodic solutions of a nonlinear second order differential equation with delay, J. Math. Anal. Appl., 70 (1970), 599-609.  doi: 10.1016/0022-247X(79)90068-4. [9] J. Kaplan and J. Yorke, Ordinary differential equations which yield periodic solutions of differential delay equations, J. Math. Anal. Appl., 48 (1974), 317-324.  doi: 10.1016/0022-247X(74)90162-0. [10] J. Li and X. He, Periodic solutions of some differential delay equations created by Hamiltonian systems, Bull. Austral. Math. Soc., 60 (1999), 377-390.  doi: 10.1017/S000497270003656X. [11] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, New York, 1989. doi: 10.1007/978-1-4757-2061-7. [12] R. Nussbaum, Periodic solutions of special differential delay equations: an example in nonlinear functional analysis, BProc. Roy. Soc. Edinburgh Sect. A, 81 (1978), 131-151.  doi: 10.1017/S0308210500010490. [13] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics 65, American Mathematical Society, Providence, RI, 1986. doi: 10.1090/cbms/065. [14] S. Ruan and J. Wei, Periodic solutions of planar systems with two delays, BProc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 1017-1032.  doi: 10.1017/S0308210500031061. [15] Z. Sun, W. Ge and L. Li, Multiple periodic orbits of high-dimensional differential delay systems, Adv. Differ. Equ., 2019 (2019), 15 pp. doi: 10.1186/s13662-019-2427-3. [16] A. Szulkin, Cohomology and Morse theory for strongly indefinite functionals, Math. Z., 209 (1992), 375-418.  doi: 10.1007/BF02570842. [17] H. Xiao and Z. Guo, Multiplicity and minimality of periodic solutions to delay differential systems, Electron. J. Differ. Equ., 2014 (2014), 356-364. [18] J. Yu and Z. Guo, A survey on the periodic solutions to Kaplan-Yorke type delay differential equation-I, Ann. Differ. Equ., 30 (2014), 97-114. [19] B. Zheng and Z. Guo, Multiplicity results on periodic solutions to higher-dimensional differential equations with multiple delays, Rocky Mountain J. Math., 44 (2014), 1715-1744.  doi: 10.1216/RMJ-2014-44-5-1715.
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