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Existence of at least two periodic solutions of the forced relativistic pendulum


Authors: Cristian Bereanu and Pedro J. Torres
Journal: Proc. Amer. Math. Soc. 140 (2012), 2713-2719
MSC (2010): Primary 34B15, 49J52, 49J35
Published electronically: November 23, 2011
MathSciNet review: 2910759
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Abstract: Using Szulkin's critical point theory, we prove that the relativistic forced pendulum with periodic boundary value conditions

$\displaystyle \left (\frac {u'}{\sqrt {1-u'^2}}\right )' +\mu \sin u=h(t), \quad u(0)-u(T)=0=u'(0)-u'(T),$    

has at least two solutions not differing by a multiple of $ 2\pi $ for any continuous function $ h:[0,T]\to \mathbb{R}$ with $ \int _0^Th(t)dt=0$ and any $ \mu \neq 0.$ The existence of at least one solution has been recently proved by Brezis and Mawhin.

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  • 1. Cristian Bereanu, Petru Jebelean, and Jean Mawhin, Radial solutions for Neumann problems involving mean curvature operators in Euclidean and Minkowski spaces, Math. Nachr. 283 (2010), no. 3, 379–391. MR 2643019, 10.1002/mana.200910083
  • 2. C. Bereanu, P. Jebelean, and J. Mawhin, Variational methods for nonlinear perturbations of singular $ \phi $-Laplacians, Rend. Lincei Mat. Appl. 22 (2011), 89-111.
  • 3. C. Bereanu, P. Jebelean, and J. Mawhin, Radial solutions of Neumann problems involving mean extrinsic curvature and periodic nonlinearities, preprint.
  • 4. C. Bereanu and J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular 𝜑-Laplacian, J. Differential Equations 243 (2007), no. 2, 536–557. MR 2371799, 10.1016/j.jde.2007.05.014
  • 5. Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011. MR 2759829
  • 6. Haïm Brezis and Jean Mawhin, Periodic solutions of the forced relativistic pendulum, Differential Integral Equations 23 (2010), no. 9-10, 801–810. MR 2675583
  • 7. E. N. Dancer, On the use of asymptotics in nonlinear boundary value problems, Ann. Mat. Pura Appl. (4) 131 (1982), 167–185. MR 681562, 10.1007/BF01765151
  • 8. N. Ghoussoub and D. Preiss, A general mountain pass principle for locating and classifying critical points, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989), no. 5, 321–330 (English, with French summary). MR 1030853
  • 9. Georg Hamel, Über erzwungene Schwingungen bei endlichen Amplituden, Math. Ann. 86 (1922), no. 1-2, 1–13 (German). MR 1512073, 10.1007/BF01458566
  • 10. Salvatore A. Marano and Dumitru Motreanu, A deformation theorem and some critical point results for non-differentiable functions, Topol. Methods Nonlinear Anal. 22 (2003), no. 1, 139–158. MR 2037271
  • 11. J. Mawhin and M. Willem, Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations, J. Differential Equations 52 (1984), no. 2, 264–287. MR 741271, 10.1016/0022-0396(84)90180-3
  • 12. F. Obersnel and P. Omari, Multiple bounded variation solutions of a periodicaly perturbed sine-curvature equation, preprint.
  • 13. Patrizia Pucci and James Serrin, Extensions of the mountain pass theorem, J. Funct. Anal. 59 (1984), no. 2, 185–210. MR 766489, 10.1016/0022-1236(84)90072-7
  • 14. Andrzej Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), no. 2, 77–109 (English, with French summary). MR 837231
  • 15. M. Willem, Oscillations forcées de l'équation du pendule, Pub. IRMA Lille, 3 (1981), V-1-V-3.

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Additional Information

Cristian Bereanu
Affiliation: Institute of Mathematics “Simion Stoilow”, Romanian Academy 21, Calea Griviţei, RO-010702 Bucharest, Sector 1, România
Email: cristian.bereanu@imar.ro

Pedro J. Torres
Affiliation: Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain
Email: ptorres@ugr.es

DOI: https://doi.org/10.1090/S0002-9939-2011-11101-8
Received by editor(s): December 15, 2010
Received by editor(s) in revised form: March 2, 2011
Published electronically: November 23, 2011
Additional Notes: Support of the first author from the Romanian Ministry of Education, Research, and Innovation (PN II Program, CNCSIS code RP 3/2008) is gratefully acknowledged.
Communicated by: Yingfei Yi
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.