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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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Additional Information

Cristian Bereanu
Affiliation: Institute of Mathematics “Simion Stoilow”, Romanian Academy 21, Calea Griviţei, RO-010702 Bucharest, Sector 1, România

Pedro J. Torres
Affiliation: Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain

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.