• Previous Article
    Partial regularity result for non-autonomous elliptic systems with general growth
  • CPAA Home
  • This Issue
  • Next Article
    Asymptotic expansion of the ground state energy for nonlinear Schrödinger system with three wave interaction
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
$ \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 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.

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.

[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.

[1]

Qiong Meng, X. H. Tang. Multiple solutions of second-order ordinary differential equation via Morse theory. Communications on Pure and Applied Analysis, 2012, 11 (3) : 945-958. doi: 10.3934/cpaa.2012.11.945

[2]

Tomás Caraballo, Renato Colucci, Luca Guerrini. Bifurcation scenarios in an ordinary differential equation with constant and distributed delay: A case study. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2639-2655. doi: 10.3934/dcdsb.2018268

[3]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392

[4]

P. Dormayer, A. F. Ivanov. Symmetric periodic solutions of a delay differential equation. Conference Publications, 1998, 1998 (Special) : 220-230. doi: 10.3934/proc.1998.1998.220

[5]

Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369

[6]

Fathalla A. Rihan, Yang Kuang, Gennady Bocharov. From the guest editors: "Delay Differential Equations: Theory, Applications and New Trends". Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : i-iv. doi: 10.3934/dcdss.2020404

[7]

Nguyen Thi Hoai. Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 495-512. doi: 10.3934/naco.2020040

[8]

David Blázquez-Sanz, Juan J. Morales-Ruiz. Lie's reduction method and differential Galois theory in the complex analytic context. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 353-379. doi: 10.3934/dcds.2012.32.353

[9]

Emmanuel Frénod, Mathieu Lutz. On the Geometrical Gyro-Kinetic theory. Kinetic and Related Models, 2014, 7 (4) : 621-659. doi: 10.3934/krm.2014.7.621

[10]

Suqi Ma, Zhaosheng Feng, Qishao Lu. A two-parameter geometrical criteria for delay differential equations. Discrete and Continuous Dynamical Systems - B, 2008, 9 (2) : 397-413. doi: 10.3934/dcdsb.2008.9.397

[11]

Daniel G. Alfaro Vigo, Amaury C. Álvarez, Grigori Chapiro, Galina C. García, Carlos G. Moreira. Solving the inverse problem for an ordinary differential equation using conjugation. Journal of Computational Dynamics, 2020, 7 (2) : 183-208. doi: 10.3934/jcd.2020008

[12]

Richard A. Norton, G. R. W. Quispel. Discrete gradient methods for preserving a first integral of an ordinary differential equation. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1147-1170. doi: 10.3934/dcds.2014.34.1147

[13]

Lukáš Adam, Jiří Outrata. On optimal control of a sweeping process coupled with an ordinary differential equation. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2709-2738. doi: 10.3934/dcdsb.2014.19.2709

[14]

Jingxian Sun, Shouchuan Hu. Flow-invariant sets and critical point theory. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 483-496. doi: 10.3934/dcds.2003.9.483

[15]

Antonio Ambrosetti, Massimiliano Berti. Applications of critical point theory to homoclinics and complex dynamics. Conference Publications, 1998, 1998 (Special) : 72-78. doi: 10.3934/proc.1998.1998.72

[16]

Xiao Wang, Zhaohui Yang, Xiongwei Liu. Periodic and almost periodic oscillations in a delay differential equation system with time-varying coefficients. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6123-6138. doi: 10.3934/dcds.2017263

[17]

Ulrike Kant, Werner M. Seiler. Singularities in the geometric theory of differential equations. Conference Publications, 2011, 2011 (Special) : 784-793. doi: 10.3934/proc.2011.2011.784

[18]

Min Liu, Zhongwei Tang. Multiplicity and concentration of solutions for Choquard equation via Nehari method and pseudo-index theory. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3365-3398. doi: 10.3934/dcds.2019139

[19]

Editorial Office. Retraction: Xiao-Qian Jiang and Lun-Chuan Zhang, A pricing option approach based on backward stochastic differential equation theory. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 969-969. doi: 10.3934/dcdss.2019065

[20]

Anatoli F. Ivanov, Sergei Trofimchuk. Periodic solutions and their stability of a differential-difference equation. Conference Publications, 2009, 2009 (Special) : 385-393. doi: 10.3934/proc.2009.2009.385

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (165)
  • HTML views (158)
  • Cited by (0)

Other articles
by authors

[Back to Top]