# American Institute of Mathematical Sciences

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

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 $\left(\phi(x')\right)'+c\, x'+g(x)=e(t)+s.$
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:
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##### 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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