American Institute of Mathematical Sciences

Advanced
August  2022, 21(8): 2723-2737. doi: 10.3934/cpaa.2022070

Stability, existence and non-existence of $ T $-periodic solutions of nonlinear delayed differential equations with $ \varphi $-Laplacian

Pablo Amster 1,, , Mariel Paula Kuna 1, and  Dionicio Santos 2,

1. 

Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires & IMAS-CONICET, Ciudad Universitaria. Pabellón I (1428), Buenos Aires, Argentina
2. 

Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional del Centro de la Provincia de Buenos Aires, Del Pinto 399 (7000), Tandil, Buenos Aires, Argentina

*Corresponding author

Received  November 2021 Revised  March 2022 Published  August 2022 Early access  April 2022

Fund Project: This work was supported by the projects CONICET PIP 11220130100006CO, UBACyT 20020160100002BA and TOMENADE, MATH- AmSud, 21-MATH-08

Using a Lyapunov-Krasovskii functional, new results concerning the global stability, boundedness of solutions, existence and non-existence of $ T $-periodic solutions for a kind of delayed equation for a $ \varphi $-Laplacian operator are obtained. An application is given for the well known sunflower equation.

Keywords: Functional equations, Lyapunov-Krasovskii functional, global stability, boundedness, existence of periodic solutions, non-existence of periodic solutions.
Mathematics Subject Classification: Primary: 34K20; 34K13.
Citation: Pablo Amster, Mariel Paula Kuna, Dionicio Santos. Stability, existence and non-existence of $ T $-periodic solutions of nonlinear delayed differential equations with $ \varphi $-Laplacian. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2723-2737. doi: 10.3934/cpaa.2022070
References:
[1]

P. Amster, Topological Methods in the Study of Boundary Value Problems, Springer-Verlag, New York, 2014. doi: 10.1007/978-1-4614-8893-4.

[2]

P. Amster, M. P. Kuna and D. P. Santos, Multiplicity of periodic solutions for dynamic liénard equations with delay and singular $\varphi$-laplacian of relativistic type, arXiv: 2005.12850.

[3] T. A. Burton, Stability and Periodic Solution of Ordinary and Functional Differential Equations, Academic Press, Orland, FL, 1985. 
[4]

T. A. Burton and L. Hatvani, Stability theorems for non autonomous functional differential equations by Lyapunov functionals, Tohoku Math. J., 41 (1989), 65-104.  doi: 10.2748/tmj/1178227868.

[5]

J. A. Cid, On the existence of periodic oscillations for pendulum-type equations, Adv. Nonlinear Anal., 10 (2021), 121-130.  doi: 10.1515/anona-2020-0222.

[6]

X. Huang and Z. Xiang, On existence of $2\pi$-periodic solutions for delay Duffing equation $x''+g(t, x(t-\tau(t)))=p(t)$, Chin. Sci. Bull., 39 (1994), 201-203. 

[7]

X. Liu, M. Tang and R. Martin, Periodic solutions for a kind of Liénard equation, J. Comput. Appl. Math., 219 (2008), 263-275.  doi: 10.1016/j.cam.2007.07.024.

[8]

S. Lu and W. Ge, Periodic solutions for a kind of second order differential equation with multiple deviating arguments, Appl. Math. Comput., 146 (2003), 195-209.  doi: 10.1016/S0096-3003(02)00536-2.

[9]

S. Lu and W. Ge, Sufficient conditions for the existence of periodic solutions to some second order differential equations with a deviating argument, Appl. Math. Comput., 308 (2005), 393-419.  doi: 10.1016/j.jmaa.2004.09.010.

[10] N. N. Krasovskii, Stability of Motion, Stanford University Press, 1963. 
[11]

J. Mawhin, Degré topologique et solutions périodiques des systèmes différentiels non linéaires, Bull. Sot. Roy. Sci. Liège, 38 (1969), 308-398. 

[12]

R. Ortega, A counterexample for the damped pendulum equation, Acad. Roy. Belg. Bull. Cl. Sci., 73 (1987), 405-409. 

[13]

A. Somolinos and A. Casal, Forced Oscillations for the Sunflower Equation, Entrainment, Nonlinear Anal. Theory. Methods Appl., 4 (1982), 397-414.  doi: 10.1016/0362-546X(82)90025-6.

[14]

A. Somolinos, Periodic solutions of the sunflower equation: $\ddot x + (a/r) \dot x + (b/r)\sin x(t-r) =0$, Q. Appl. Math., 35 (1978), 465-478.  doi: 10.1090/qam/465265.

[15]

J. M. Zhao, K. L. Huang and Q. S. Lu, The existence of periodic solutions for a class of functional-differential equations and their application, Appl. Math. Mech., 15 (1994), 49-58.  doi: 10.1007/BF02451981.

show all references

References:
[1]

P. Amster, Topological Methods in the Study of Boundary Value Problems, Springer-Verlag, New York, 2014. doi: 10.1007/978-1-4614-8893-4.

[2]

P. Amster, M. P. Kuna and D. P. Santos, Multiplicity of periodic solutions for dynamic liénard equations with delay and singular $\varphi$-laplacian of relativistic type, arXiv: 2005.12850.

[3] T. A. Burton, Stability and Periodic Solution of Ordinary and Functional Differential Equations, Academic Press, Orland, FL, 1985. 
[4]

T. A. Burton and L. Hatvani, Stability theorems for non autonomous functional differential equations by Lyapunov functionals, Tohoku Math. J., 41 (1989), 65-104.  doi: 10.2748/tmj/1178227868.

[5]

J. A. Cid, On the existence of periodic oscillations for pendulum-type equations, Adv. Nonlinear Anal., 10 (2021), 121-130.  doi: 10.1515/anona-2020-0222.

[6]

X. Huang and Z. Xiang, On existence of $2\pi$-periodic solutions for delay Duffing equation $x''+g(t, x(t-\tau(t)))=p(t)$, Chin. Sci. Bull., 39 (1994), 201-203. 

[7]

X. Liu, M. Tang and R. Martin, Periodic solutions for a kind of Liénard equation, J. Comput. Appl. Math., 219 (2008), 263-275.  doi: 10.1016/j.cam.2007.07.024.

[8]

S. Lu and W. Ge, Periodic solutions for a kind of second order differential equation with multiple deviating arguments, Appl. Math. Comput., 146 (2003), 195-209.  doi: 10.1016/S0096-3003(02)00536-2.

[9]

S. Lu and W. Ge, Sufficient conditions for the existence of periodic solutions to some second order differential equations with a deviating argument, Appl. Math. Comput., 308 (2005), 393-419.  doi: 10.1016/j.jmaa.2004.09.010.

[10] N. N. Krasovskii, Stability of Motion, Stanford University Press, 1963. 
[11]

J. Mawhin, Degré topologique et solutions périodiques des systèmes différentiels non linéaires, Bull. Sot. Roy. Sci. Liège, 38 (1969), 308-398. 

[12]

R. Ortega, A counterexample for the damped pendulum equation, Acad. Roy. Belg. Bull. Cl. Sci., 73 (1987), 405-409. 

[13]

A. Somolinos and A. Casal, Forced Oscillations for the Sunflower Equation, Entrainment, Nonlinear Anal. Theory. Methods Appl., 4 (1982), 397-414.  doi: 10.1016/0362-546X(82)90025-6.

[14]

A. Somolinos, Periodic solutions of the sunflower equation: $\ddot x + (a/r) \dot x + (b/r)\sin x(t-r) =0$, Q. Appl. Math., 35 (1978), 465-478.  doi: 10.1090/qam/465265.

[15]

J. M. Zhao, K. L. Huang and Q. S. Lu, The existence of periodic solutions for a class of functional-differential equations and their application, Appl. Math. Mech., 15 (1994), 49-58.  doi: 10.1007/BF02451981.

[1]

Nguyen Minh Man, Nguyen Van Minh. On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations. Communications on Pure and Applied Analysis, 2004, 3 (2) : 291-300. doi: 10.3934/cpaa.2004.3.291
[2]

Jifeng Chu, Zaitao Liang, Fangfang Liao, Shiping Lu. Existence and stability of periodic solutions for relativistic singular equations. Communications on Pure and Applied Analysis, 2017, 16 (2) : 591-609. doi: 10.3934/cpaa.2017029
[3]

Keisuke Matsuya, Tetsuji Tokihiro. Existence and non-existence of global solutions for a discrete semilinear heat equation. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 209-220. doi: 10.3934/dcds.2011.31.209
[4]

Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167
[5]

Daoyi Xu, Yumei Huang, Zhiguo Yang. Existence theorems for periodic Markov process and stochastic functional differential equations. Discrete and Continuous Dynamical Systems, 2009, 24 (3) : 1005-1023. doi: 10.3934/dcds.2009.24.1005
[6]

Fuqin Sun, Mingxin Wang. Non-existence of global solutions for nonlinear strongly damped hyperbolic systems. Discrete and Continuous Dynamical Systems, 2005, 12 (5) : 949-958. doi: 10.3934/dcds.2005.12.949
[7]

Bilgesu A. Bilgin, Varga K. Kalantarov. Non-existence of global solutions to nonlinear wave equations with positive initial energy. Communications on Pure and Applied Analysis, 2018, 17 (3) : 987-999. doi: 10.3934/cpaa.2018048
[8]

Shu-Yu Hsu. Non-existence and behaviour at infinity of solutions of some elliptic equations. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 769-786. doi: 10.3934/dcds.2004.10.769
[9]

Qiang Li, Mei Wei. Existence and asymptotic stability of periodic solutions for neutral evolution equations with delay. Evolution Equations and Control Theory, 2020, 9 (3) : 753-772. doi: 10.3934/eect.2020032
[10]

Yoshihiro Hamaya. Stability properties and existence of almost periodic solutions of volterra difference equations. Conference Publications, 2009, 2009 (Special) : 315-321. doi: 10.3934/proc.2009.2009.315
[11]

Yingshu Lü. Symmetry and non-existence of solutions to an integral system. Communications on Pure and Applied Analysis, 2018, 17 (3) : 807-821. doi: 10.3934/cpaa.2018041
[12]

Denis Pennequin. Existence of almost periodic solutions of discrete time equations. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 51-60. doi: 10.3934/dcds.2001.7.51
[13]

Jifeng Chu, Meirong Zhang. Rotation numbers and Lyapunov stability of elliptic periodic solutions. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1071-1094. doi: 10.3934/dcds.2008.21.1071
[14]

Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125
[15]

Xuewei Cui, Mei Yu. Non-existence of positive solutions for a higher order fractional equation. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1379-1387. doi: 10.3934/dcds.2019059
[16]

Maoji Ri, Shuibo Huang, Canyun Huang. Non-existence of solutions to some degenerate coercivity elliptic equations involving measures data. Electronic Research Archive, 2020, 28 (1) : 165-182. doi: 10.3934/era.2020011
[17]

Luca Rossi. Non-existence of positive solutions of fully nonlinear elliptic equations in unbounded domains. Communications on Pure and Applied Analysis, 2008, 7 (1) : 125-141. doi: 10.3934/cpaa.2008.7.125
[18]

Meili Li, Maoan Han, Chunhai Kou. The existence of positive periodic solutions of a generalized. Mathematical Biosciences & Engineering, 2008, 5 (4) : 803-812. doi: 10.3934/mbe.2008.5.803
[19]

Hongbin Chen, Yi Li. Existence, uniqueness, and stability of periodic solutions of an equation of duffing type. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 793-807. doi: 10.3934/dcds.2007.18.793
[20]

M.I. Gil’. Existence and stability of periodic solutions of semilinear neutral type systems. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 809-820. doi: 10.3934/dcds.2001.7.809

2021 Impact Factor: 1.273

Tools

Metrics

  • PDF downloads (113)
  • HTML views (59)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy