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

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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
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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.

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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.

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