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

Abstract

Using Szulkin’s critical point theory, we prove that the relativistic forced pendulum with periodic boundary value conditions ( u ′ 1 − u ′ 2 ) ′ + μ sin ⁡ u = h ( t ) , u ( 0 ) − u ( T ) = 0 = u ′ ( 0 ) − u ′ ( T ) , \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 π 2\pi for any continuous function h : [ 0 , T ] → R h:[0,T]\to \mathbb {R} with ∫ 0 T h ( t ) d t = 0 \int _0^Th(t)dt=0 and any μ ≠ 0. \mu eq 0. The existence of at least one solution has been recently proved by Brezis and Mawhin.