American Institute of Mathematical Sciences

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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

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 & Continuous Dynamical Systems - A, 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).   Google Scholar

[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.  doi: 10.1007/s10884-010-9172-3.  Google Scholar

[3]

C. Bereanu and J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular $\phi$-Laplacian,, J. Differential Equations, 243 (2007), 536.  doi: 10.1016/j.jde.2007.05.014.  Google Scholar

[4]

C. Bereanu and J. Mawhin, Multiple periodic solutions of ordinary differential equations with bounded nonlinearities and $\phi$-Laplacian,, NoDEA, 15 (2008), 159.  doi: 10.1007/s00030-007-7004-x.  Google Scholar

[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.   Google Scholar

[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), 140 (2012), 2713.   Google Scholar

[7]

H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum,, Differential Integral Equations, 23 (2010), 801.   Google Scholar

[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.  doi: 10.1006/jmaa.1997.5380.  Google Scholar

[9]

H. Chen and Y. Li, Rate of decay of stable periodic solutions of Duffing equations,, J. Differential Equations, 236 (2007), 493.  doi: 10.1016/j.jde.2007.01.023.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.3934/dcds.2009.23.727.  Google Scholar

[12]

G. Hamel, Ueber erzwungene Schingungen bei endlischen Amplituden,, Math. Ann., 86 (1922), 1.  doi: 10.1007/BF01458566.  Google Scholar

[13]

J. Mawhin, Global results for the forced pendulum equation,, in, 1 (2004), 533.   Google Scholar

[14]

J. Mawhin, Periodic solutions of the forced pendulum: classical vs relativistic,, Le Matematiche, LXV (2010), 97.   Google Scholar

[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.  doi: 10.1016/0022-0396(84)90180-3.  Google Scholar

[16]

D. S. Mitrinović, J. E. Pečarić and A. M. Fink, "Inequalities Involving Functions and Their Integrals and Derivatives,", Kluwer Academic Publishers, (1991).   Google Scholar

[17]

R. Ortega, Stability and index of periodic solutions of an equation of Duffing type,, Boll. Un. Mat. Italiana, 3-B (1989), 533.   Google Scholar

[18]

R. Ortega, Some applications of the topological degree to stability theory,, in, 472 (1995), 377.   Google Scholar

[19]

R. Ortega, Stability of a periodic problem of Ambrosetti-Prodi type,, Diff. Int. Equ., 3 (1990), 275.   Google Scholar

[20]

P. J. Torres, Periodic oscillations of the relativistic pendulum with friction,, Physics Letters A, 372 (2008), 6386.  doi: 10.1016/j.physleta.2008.08.060.  Google Scholar

[21]

P. J. Torres, Nondegeneracy of the periodically forced Liénard differential equation with $\phi$-Laplacian,, Communications in Contemporary Mathematics, 13 (2011), 283.  doi: 10.1142/S0219199711004208.  Google Scholar

show all references

References:
[1]

H. Amann, "Ordinary Differential Equations. An Introduction to Nonlinear Analysis,", Walter de Gruyter, (1990).   Google Scholar

[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.  doi: 10.1007/s10884-010-9172-3.  Google Scholar

[3]

C. Bereanu and J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular $\phi$-Laplacian,, J. Differential Equations, 243 (2007), 536.  doi: 10.1016/j.jde.2007.05.014.  Google Scholar

[4]

C. Bereanu and J. Mawhin, Multiple periodic solutions of ordinary differential equations with bounded nonlinearities and $\phi$-Laplacian,, NoDEA, 15 (2008), 159.  doi: 10.1007/s00030-007-7004-x.  Google Scholar

[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.   Google Scholar

[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), 140 (2012), 2713.   Google Scholar

[7]

H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum,, Differential Integral Equations, 23 (2010), 801.   Google Scholar

[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.  doi: 10.1006/jmaa.1997.5380.  Google Scholar

[9]

H. Chen and Y. Li, Rate of decay of stable periodic solutions of Duffing equations,, J. Differential Equations, 236 (2007), 493.  doi: 10.1016/j.jde.2007.01.023.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.3934/dcds.2009.23.727.  Google Scholar

[12]

G. Hamel, Ueber erzwungene Schingungen bei endlischen Amplituden,, Math. Ann., 86 (1922), 1.  doi: 10.1007/BF01458566.  Google Scholar

[13]

J. Mawhin, Global results for the forced pendulum equation,, in, 1 (2004), 533.   Google Scholar

[14]

J. Mawhin, Periodic solutions of the forced pendulum: classical vs relativistic,, Le Matematiche, LXV (2010), 97.   Google Scholar

[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.  doi: 10.1016/0022-0396(84)90180-3.  Google Scholar

[16]

D. S. Mitrinović, J. E. Pečarić and A. M. Fink, "Inequalities Involving Functions and Their Integrals and Derivatives,", Kluwer Academic Publishers, (1991).   Google Scholar

[17]

R. Ortega, Stability and index of periodic solutions of an equation of Duffing type,, Boll. Un. Mat. Italiana, 3-B (1989), 533.   Google Scholar

[18]

R. Ortega, Some applications of the topological degree to stability theory,, in, 472 (1995), 377.   Google Scholar

[19]

R. Ortega, Stability of a periodic problem of Ambrosetti-Prodi type,, Diff. Int. Equ., 3 (1990), 275.   Google Scholar

[20]

P. J. Torres, Periodic oscillations of the relativistic pendulum with friction,, Physics Letters A, 372 (2008), 6386.  doi: 10.1016/j.physleta.2008.08.060.  Google Scholar

[21]

P. J. Torres, Nondegeneracy of the periodically forced Liénard differential equation with $\phi$-Laplacian,, Communications in Contemporary Mathematics, 13 (2011), 283.  doi: 10.1142/S0219199711004208.  Google Scholar

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