On the existence and stability of periodic solutions for pendulum-like equations with friction and \phi-Laplacian

Previous Article
Some bifurcation results for rapidly growing nonlinearities
DCDS Home
This Issue
Next Article
Existence and qualitative properties of solutions for nonlinear Dirichlet problems

January 2013, 33(1): 141-152. doi: 10.3934/dcds.2013.33.141

On the existence and stability of periodic solutions for pendulum-like equations with friction and $\phi$-Laplacian

J. Ángel Cid ^1, and Pedro J. Torres ^2,

1.	Departamento de Matemáticas, Universidade de Vigo, Higher Technical School of Computer Engineering, 32004, Ourense, Spain
2.	Departamento de Matemática Aplicada, Universidad de Granada, Facultad de Ciencias, Granada, Spain

Received April 2011 Revised January 2012 Published September 2012

Abstract
Related Papers

In this paper we study the existence, multiplicity and stability of T-periodic solutions for the equation $\left(\phi(x')\right)'+c\, x'+g(x)=e(t)+s.$

Keywords: pendulum equation, Periodic solution, stability., $\phi$-Laplacian, degree theory.

Mathematics Subject Classification: Primary: 34C25.

Citation: J. Ángel Cid, Pedro J. Torres. On the existence and stability of periodic solutions for pendulum-like equations with friction and $\phi$-Laplacian. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 141-152. doi: 10.3934/dcds.2013.33.141

References:

[1]	H. Amann, "Ordinary Differential Equations. An Introduction to Nonlinear Analysis," Walter de Gruyter, 1990.
[2]	C. Bereanu, P. Jebelean and J. Mawhin, Periodic solutions of pendulum-like perturbations of singular and bounded $\Phi$-Laplacians, Journal of Dynamics and Differential Equations, 22 (2010), 463-471. doi: 10.1007/s10884-010-9172-3.
[3]	C. Bereanu and J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular $\phi$-Laplacian, J. Differential Equations, 243 (2007), 536-557. doi: 10.1016/j.jde.2007.05.014.
[4]	C. Bereanu and J. Mawhin, Multiple periodic solutions of ordinary differential equations with bounded nonlinearities and $\phi$-Laplacian, NoDEA, 15 (2008), 159-168. doi: 10.1007/s00030-007-7004-x.
[5]	C. Bereanu and J. Mawhin, Boundary value problems for some nonlinear systems with singular $\phi$-Laplacian, J. Fixed Point Theory Appl., 4 (2008), 57-75.
[6]	C. Bereanu and P. J. Torres, Existence of at least two periodic solutions of the forced relativistic pendulum, Proc. Am. Math. Soc. 140 (2012), 2713-2719.
[7]	H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum, Differential Integral Equations, 23, (2010), 801-810.
[8]	J. Čepička, P. Drábek and J. Jenšíková, On the stability of periodic solutions of the damped pendulum equation, J. Math. Anal. Appl., 209 (1997), 712-723. doi: 10.1006/jmaa.1997.5380.
[9]	H. Chen and Y. Li, Rate of decay of stable periodic solutions of Duffing equations, J. Differential Equations, 236 (2007), 493-503. doi: 10.1016/j.jde.2007.01.023.
[10]	J. Chu, J. Lei and M. Zhang, The stability of the equilibrium of a nonlinear planar system and application to the relativistic oscillator, J. Differential Equations, 247 (2009), 530-542.
[11]	J. A. Cid and P. J. Torres, Solvability for some boundary value problems with $\phi$-Laplacian operators, Discrete Contin. Dyn. Syst., 23 (2009), 727-732. doi: 10.3934/dcds.2009.23.727.
[12]	G. Hamel, Ueber erzwungene Schingungen bei endlischen Amplituden, Math. Ann., 86 (1922), 1-13. doi: 10.1007/BF01458566.
[13]	J. Mawhin, Global results for the forced pendulum equation, in "Handbook of Differential Equations" (eds. A. Ca\ nada, P. Drabek and A. Fonda), Elsevier, 1 (2004), 533-589.
[14]	J. Mawhin, Periodic solutions of the forced pendulum: classical vs relativistic, Le Matematiche, LXV (2010), 97-107.
[15]	J. Mawhin and M. Willem, Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations, J. Differential Equations, 52 (1984), 264-287. doi: 10.1016/0022-0396(84)90180-3.
[16]	D. S. Mitrinović, J. E. Pečarić and A. M. Fink, "Inequalities Involving Functions and Their Integrals and Derivatives," Kluwer Academic Publishers, 1991.
[17]	R. Ortega, Stability and index of periodic solutions of an equation of Duffing type, Boll. Un. Mat. Italiana, 3-B (1989), 533-546.
[18]	R. Ortega, Some applications of the topological degree to stability theory, in "Topological Methods in Differential Equations and Inclusions'' (eds. A. Granas and M. Frigon), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., 472, KluwerAcademic, (1995), 377-409.
[19]	R. Ortega, Stability of a periodic problem of Ambrosetti-Prodi type, Diff. Int. Equ., 3 (1990), 275-284.
[20]	P. J. Torres, Periodic oscillations of the relativistic pendulum with friction, Physics Letters A, 372 (2008), 6386-6387. doi: 10.1016/j.physleta.2008.08.060.
[21]	P. J. Torres, Nondegeneracy of the periodically forced Liénard differential equation with $\phi$-Laplacian, Communications in Contemporary Mathematics, 13 (2011), 283-292. doi: 10.1142/S0219199711004208.

show all references

References:

[1]	H. Amann, "Ordinary Differential Equations. An Introduction to Nonlinear Analysis," Walter de Gruyter, 1990.
[2]	C. Bereanu, P. Jebelean and J. Mawhin, Periodic solutions of pendulum-like perturbations of singular and bounded $\Phi$-Laplacians, Journal of Dynamics and Differential Equations, 22 (2010), 463-471. doi: 10.1007/s10884-010-9172-3.
[3]	C. Bereanu and J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular $\phi$-Laplacian, J. Differential Equations, 243 (2007), 536-557. doi: 10.1016/j.jde.2007.05.014.
[4]	C. Bereanu and J. Mawhin, Multiple periodic solutions of ordinary differential equations with bounded nonlinearities and $\phi$-Laplacian, NoDEA, 15 (2008), 159-168. doi: 10.1007/s00030-007-7004-x.
[5]	C. Bereanu and J. Mawhin, Boundary value problems for some nonlinear systems with singular $\phi$-Laplacian, J. Fixed Point Theory Appl., 4 (2008), 57-75.
[6]	C. Bereanu and P. J. Torres, Existence of at least two periodic solutions of the forced relativistic pendulum, Proc. Am. Math. Soc. 140 (2012), 2713-2719.
[7]	H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum, Differential Integral Equations, 23, (2010), 801-810.
[8]	J. Čepička, P. Drábek and J. Jenšíková, On the stability of periodic solutions of the damped pendulum equation, J. Math. Anal. Appl., 209 (1997), 712-723. doi: 10.1006/jmaa.1997.5380.
[9]	H. Chen and Y. Li, Rate of decay of stable periodic solutions of Duffing equations, J. Differential Equations, 236 (2007), 493-503. doi: 10.1016/j.jde.2007.01.023.
[10]	J. Chu, J. Lei and M. Zhang, The stability of the equilibrium of a nonlinear planar system and application to the relativistic oscillator, J. Differential Equations, 247 (2009), 530-542.
[11]	J. A. Cid and P. J. Torres, Solvability for some boundary value problems with $\phi$-Laplacian operators, Discrete Contin. Dyn. Syst., 23 (2009), 727-732. doi: 10.3934/dcds.2009.23.727.
[12]	G. Hamel, Ueber erzwungene Schingungen bei endlischen Amplituden, Math. Ann., 86 (1922), 1-13. doi: 10.1007/BF01458566.
[13]	J. Mawhin, Global results for the forced pendulum equation, in "Handbook of Differential Equations" (eds. A. Ca\ nada, P. Drabek and A. Fonda), Elsevier, 1 (2004), 533-589.
[14]	J. Mawhin, Periodic solutions of the forced pendulum: classical vs relativistic, Le Matematiche, LXV (2010), 97-107.
[15]	J. Mawhin and M. Willem, Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations, J. Differential Equations, 52 (1984), 264-287. doi: 10.1016/0022-0396(84)90180-3.
[16]	D. S. Mitrinović, J. E. Pečarić and A. M. Fink, "Inequalities Involving Functions and Their Integrals and Derivatives," Kluwer Academic Publishers, 1991.
[17]	R. Ortega, Stability and index of periodic solutions of an equation of Duffing type, Boll. Un. Mat. Italiana, 3-B (1989), 533-546.
[18]	R. Ortega, Some applications of the topological degree to stability theory, in "Topological Methods in Differential Equations and Inclusions'' (eds. A. Granas and M. Frigon), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., 472, KluwerAcademic, (1995), 377-409.
[19]	R. Ortega, Stability of a periodic problem of Ambrosetti-Prodi type, Diff. Int. Equ., 3 (1990), 275-284.
[20]	P. J. Torres, Periodic oscillations of the relativistic pendulum with friction, Physics Letters A, 372 (2008), 6386-6387. doi: 10.1016/j.physleta.2008.08.060.
[21]	P. J. Torres, Nondegeneracy of the periodically forced Liénard differential equation with $\phi$-Laplacian, Communications in Contemporary Mathematics, 13 (2011), 283-292. doi: 10.1142/S0219199711004208.

[1]	Michele V. Bartuccelli, G. Gentile, Kyriakos V. Georgiou. Kam theory, Lindstedt series and the stability of the upside-down pendulum. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 413-426. doi: 10.3934/dcds.2003.9.413
[2]	Peng Mei, Zhan Zhou, Genghong Lin. Periodic and subharmonic solutions for a 2$n$th-order $\phi_c$-Laplacian difference equation containing both advances and retardations. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2085-2095. doi: 10.3934/dcdss.2019134
[3]	Jean Mawhin. Periodic solutions of second order Lagrangian difference systems with bounded or singular $\phi$-Laplacian and periodic potential. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 1065-1076. doi: 10.3934/dcdss.2013.6.1065
[4]	Feng Wang, Jifeng Chu, Zaitao Liang. Prevalence of stable periodic solutions in the forced relativistic pendulum equation. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4579-4594. doi: 10.3934/dcdsb.2018177
[5]	Cristian Bereanu, Petru Jebelean, Jean Mawhin. Radial solutions for Neumann problems with $\phi$-Laplacians and pendulum-like nonlinearities. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 637-648. doi: 10.3934/dcds.2010.28.637
[6]	Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107
[7]	Roman Srzednicki. On periodic solutions in the Whitney's inverted pendulum problem. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2127-2141. doi: 10.3934/dcdss.2019137
[8]	Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991
[9]	Francesca Papalini. Strongly nonlinear multivalued systems involving singular $\Phi$-Laplacian operators. Communications on Pure and Applied Analysis, 2010, 9 (4) : 1025-1040. doi: 10.3934/cpaa.2010.9.1025
[10]	R. Kannan, S. Seikkala. Existence of solutions to some Phi-Laplacian boundary value problems. Conference Publications, 2001, 2001 (Special) : 211-217. doi: 10.3934/proc.2001.2001.211
[11]	Xiying Sun, Qihuai Liu, Dingbian Qian, Na Zhao. Infinitely many subharmonic solutions for nonlinear equations with singular $ \phi $-Laplacian. Communications on Pure and Applied Analysis, 2020, 19 (1) : 279-292. doi: 10.3934/cpaa.20200015
[12]	J. Ángel Cid, Pedro J. Torres. Solvability for some boundary value problems with $\phi$-Laplacian operators. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 727-732. doi: 10.3934/dcds.2009.23.727
[13]	Mi-Young Kim. Uniqueness and stability of positive periodic numerical solution of an epidemic model. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 365-375. doi: 10.3934/dcdsb.2007.7.365
[14]	Jean Mawhin. Multiplicity of solutions of variational systems involving $\phi$-Laplacians with singular $\phi$ and periodic nonlinearities. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 4015-4026. doi: 10.3934/dcds.2012.32.4015
[15]	Rafael Ortega. Stability and index of periodic solutions of a nonlinear telegraph equation. Communications on Pure and Applied Analysis, 2005, 4 (4) : 823-837. doi: 10.3934/cpaa.2005.4.823
[16]	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
[17]	Hui Huang, Jian-Guo Liu. Well-posedness for the Keller-Segel equation with fractional Laplacian and the theory of propagation of chaos. Kinetic and Related Models, 2016, 9 (4) : 715-748. doi: 10.3934/krm.2016013
[18]	Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3659-3683. doi: 10.3934/dcdss.2021023
[19]	Chengxin Du, Changchun Liu. Time periodic solution to a two-species chemotaxis-Stokes system with $ p $-Laplacian diffusion. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4321-4345. doi: 10.3934/cpaa.2021162
[20]	Alberto Cabada, J. Ángel Cid. Heteroclinic solutions for non-autonomous boundary value problems with singular $\Phi$-Laplacian operators. Conference Publications, 2009, 2009 (Special) : 118-122. doi: 10.3934/proc.2009.2009.118

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
J. Ángel Cid Pedro J. Torres
on Google Scholar
J. Ángel Cid Pedro J. Torres

[Back to Top]