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

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.

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. LiuM. 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. ZhaoK. 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. LiuM. 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. ZhaoK. 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.

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