doi: 10.3934/cpaa.2021159
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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 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
$ \begin{equation} z^{\prime\prime}(t) = \sum\limits_{k = 1}^{M-1}(-1)^kf(z(t-k)), \notag \end{equation} $
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 & Applied Analysis, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[11]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, New York, 1989. doi: 10.1007/978-1-4757-2061-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

A. Szulkin, Cohomology and Morse theory for strongly indefinite functionals, Math. Z., 209 (1992), 375-418.  doi: 10.1007/BF02570842.  Google Scholar

[17]

H. Xiao and Z. Guo, Multiplicity and minimality of periodic solutions to delay differential systems, Electron. J. Differ. Equ., 2014 (2014), 356-364.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, New York, 1989. doi: 10.1007/978-1-4757-2061-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

A. Szulkin, Cohomology and Morse theory for strongly indefinite functionals, Math. Z., 209 (1992), 375-418.  doi: 10.1007/BF02570842.  Google Scholar

[17]

H. Xiao and Z. Guo, Multiplicity and minimality of periodic solutions to delay differential systems, Electron. J. Differ. Equ., 2014 (2014), 356-364.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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