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