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

Bereanu, Cristian; Torres, Pedro

doi:10.1090/S0002-9939-2011-11101-8

Existence of at least two periodic solutions of the forced relativistic pendulum
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by Cristian Bereanu and Pedro J. Torres PDF

Proc. Amer. Math. Soc. 140 (2012), 2713-2719 Request permission

Abstract:

Using Szulkin’s critical point theory, we prove that the relativistic forced pendulum with periodic boundary value conditions \begin{equation*} \left (\frac {u’}{\sqrt {1-u’^2}}\right )’ +\mu \sin u=h(t), \quad u(0)-u(T)=0=u’(0)-u’(T), \end{equation*} 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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