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Existence of at least two periodic solutions of the forced relativistic pendulum. (English) Zbl 1275.34057
Summary: Using Szulkin’s critical point theory, we prove that the relativistic forced pendulum under periodic boundary value conditions $\left(\frac{u^\prime}{\sqrt{1-u\prime ^2}}\right)^\prime +\mu \sin u=h(t), \quad u(0)-u(T)=0=u'(0)-u'(T)$ 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 recently been proved by Brezis and Mawhin.

##### MSC:
 34C25 Periodic solutions to ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
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##### References:
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