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

Xuan Wu; Huafeng Xiao

doi:10.3934/cpaa.2021159

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2021, Volume 20, Issue 12: 4253-4269. Doi: 10.3934/cpaa.2021159

This issue Previous Article Asymptotic expansion of the ground state energy for nonlinear Schrödinger system with three wave interaction Next Article Partial regularity result for non-autonomous elliptic systems with general growth

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

Xuan Wu and
Huafeng Xiao^,

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

^* Corresponding author
^* Corresponding author

Received: May 31, 2021

Revised: July 31, 2021

Early access: September 2021

Published: December 2021

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 34K13; Secondary: 58E05.

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[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.